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Tertiary mathematics and content connections in the development of mathematical knowledge for teaching Radzimski, Vanessa E. 2020

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Tertiary Mathematics and Content Connections in theDevelopment of Mathematical Knowledge for TeachingbyVanessa E. RadzimskiB.Sc., Florida State University, 2012M.Sc., University of British Columbia, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Curriculum Studies)The University of British Columbia(Vancouver)April 2020c© Vanessa E. Radzimski, 2020The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Tertiary Mathematics and Content Connections in the Development ofMathematical Knowledge for Teachingsubmitted by Vanessa E. Radzimski in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Curriculum Studies.Examining Committee:Cynthia Nicol, Curriculum StudiesSupervisorAnn Anderson, Curriculum StudiesCo-SupervisorAlejandro Adem, MathematicsSupervisory Committee MemberLeah Edelstein-Keshet, MathematicsUniversity ExaminerMarina Milner-Bolotin, Curriculum StudiesUniversity ExamineriiAbstractEmphasis on the importance of subject matter expertise in teaching secondarymathematics is found in the research literature and in policy. In the United States,for instance, the No Child Left Behind Act, calls for secondary teachers to be certi-fied in a subject specialization. In Canada, admission to secondary teacher educa-tion programs requires extensive subject-specific university coursework. However,it is unclear if or how extensive subject matter expertise impacts the practices ofteachers in a secondary classroom.This study aims to explore how advanced coursework in mathematics, beyondthe scope of the high school curriculum, impacts the ways prospective teachers un-derstand and teach secondary content. Using a qualitative case study methodology,five prospective secondary mathematics teachers participated, with data obtainedthrough document analysis and semi-structured task-based interviews. Participantsengaged with classroom-relevant tasks and were explicitly asked how they coulddraw upon advanced mathematics to inform their teaching. Participants also de-tailed their perceptions of the role advanced mathematics plays in their develop-ment as teachers.Results from this study reveal that participants saw little value in the contentof advanced mathematics to their teaching, but expressed value towards the be-liefs and values gained through advanced mathematics, such as problem solvingand rigour. Some participants demonstrated misconceptions at the secondary level,which had direct connections to content from their post-secondary mathematicscoursework. For example, all participants made the false claim that a real-valuediiipolynomial can be factored if and only if it has a root.Results extend the literature through rich empirical data which illuminates howprospective secondary mathematics teachers perceive and use advanced mathemat-ics in understanding the secondary curriculum. While participants held contentknowledge beyond the secondary curriculum, this knowledge was not integratedin a way that impacted their understanding of secondary mathematics. An under-standing of post-secondary mathematics has the potential to be of value to sec-ondary teachers in the classroom, but this potential needs a space to be unlocked. Iargue that mathematicians and teacher educators need to work together to build op-portunities for prospective teachers to build connections between the mathematicsthey know and the mathematics they need to teach.ivLay SummaryI explore the role of advanced mathematics knowledge in the pedagogy of fu-ture secondary mathematics teachers. This study utilized a qualitative case studymethodology to understand what five teacher candidates perceived as the role oftheir advanced mathematics expertise, as well as connections they built betweensecondary and post-secondary content. Results revealed that participants did notview the content from their post-secondary degrees as being relevant to classroompractice. This was supported through participants’ engagement in task-based in-terviews, where they expressed limited connections between secondary and post-secondary mathematics, as well as content misconceptions at both levels. Thisstudy extends the literature in suggesting that advanced mathematical courseworkmay play a very limited role in impacting the practice of future teachers. Resultssuggest a need for further investigation into the ways mathematicians and teachereducators support the integration of post-secondary mathematics knowledge intothe mathematical knowledge for teaching of future teachers.vPrefaceThis dissertation is an original intellectual product of the author, V. Radzimski. Theresearch reported in Chapters 4-8 was covered by UBC Human Ethics CertificateID H17-01767.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . 71.6 Personal Statement . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11vii2 Literature Review and Theoretical Framework . . . . . . . . . . . . 132.1 Teachers’ Content Knowledge: Beginnings . . . . . . . . . . . . 152.2 Mathematical Knowledge for Teaching . . . . . . . . . . . . . . . 192.3 Advanced Mathematics Knowledge in Teaching . . . . . . . . . . 242.4 Studies on Teacher Knowledge . . . . . . . . . . . . . . . . . . . 272.5 Frameworks for Teacher Knowledge . . . . . . . . . . . . . . . . 322.5.1 Framework of Silverman and Thompson . . . . . . . . . . 362.5.2 Mathematical Knowledge for Teaching . . . . . . . . . . 412.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Research Methodology: Case Study and Interview . . . . . . . . . 513.1.1 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . 523.1.2 Interviews as Data . . . . . . . . . . . . . . . . . . . . . 543.1.3 Variations on the Interview . . . . . . . . . . . . . . . . . 563.1.4 Debates on the Interview as Data . . . . . . . . . . . . . . 573.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.1 Participants . . . . . . . . . . . . . . . . . . . . . . . . . 623.3 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.1 Experiences in Post-Secondary Mathematics . . . . . . . 653.3.2 Connections Between Secondary and Post-Secondary Math-ematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.3 Secondary Mathematics Instrument . . . . . . . . . . . . 713.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.5 Validity and Reliability . . . . . . . . . . . . . . . . . . . . . . . 793.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 Perceptions of the Role of Advanced Mathematics in PedagogicalDevelopment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.1 Perceptions of Mathematics . . . . . . . . . . . . . . . . . . . . . 834.2 Role of Advanced Knowledge for Teachers . . . . . . . . . . . . 884.3 Advanced Knowledge in Teacher Education . . . . . . . . . . . . 94viii4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 The Overextension of Familiar Mathematical Ideas: A Case of Poly-nomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . 1045.2 Participant Understandings . . . . . . . . . . . . . . . . . . . . . 1115.3 Post-secondary Connections . . . . . . . . . . . . . . . . . . . . 1275.4 An Experience of Abstract Algebra . . . . . . . . . . . . . . . . . 1335.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366 The Role of Limits, Infinity, and Formal Definitions in SecondaryMathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.2 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.2.1 Mathematical Background . . . . . . . . . . . . . . . . . 1406.2.2 Participant Understandings . . . . . . . . . . . . . . . . . 1436.2.3 Post-Secondary Connections . . . . . . . . . . . . . . . . 1536.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.3.1 Mathematical Background . . . . . . . . . . . . . . . . . 1576.3.2 Participant Understandings . . . . . . . . . . . . . . . . . 1616.3.3 Post-Secondary Connections . . . . . . . . . . . . . . . . 1646.4 Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.4.1 Mathematical Background . . . . . . . . . . . . . . . . . 1706.4.2 Participant Understandings . . . . . . . . . . . . . . . . . 1716.4.3 Post-Secondary Connections . . . . . . . . . . . . . . . . 1796.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867 The Tensions of Proof and Applications Observed Through Geomet-ric Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.1 The Square Root of Two . . . . . . . . . . . . . . . . . . . . . . 1897.1.1 Mathematical Background . . . . . . . . . . . . . . . . . 1907.1.2 Participant Understandings and Post-Secondary Connections1927.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1987.2.1 Mathematical Background . . . . . . . . . . . . . . . . . 199ix7.2.2 Participant Understandings . . . . . . . . . . . . . . . . . 2027.2.3 Post-Secondary Connections . . . . . . . . . . . . . . . . 2047.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2088 Conclusion and Implications . . . . . . . . . . . . . . . . . . . . . . 2108.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2108.1.1 The Role of University Mathematics in Teacher Development2118.1.2 Content Connections Between University and SecondaryMathematics . . . . . . . . . . . . . . . . . . . . . . . . 2148.1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 2218.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228.2.1 Implications in Research . . . . . . . . . . . . . . . . . . 2228.2.2 Implications in Practice . . . . . . . . . . . . . . . . . . . 2248.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 228References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231A Interview Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 243B Recruitment emails and documents . . . . . . . . . . . . . . . . . . 247xList of TablesTable 3.1 Summary of participant backgrounds . . . . . . . . . . . . . . 64Table 3.2 Tasks chosen by participants . . . . . . . . . . . . . . . . . . . 71Table 3.3 Interview transcript codes and themes . . . . . . . . . . . . . . 75Table 5.1 Partial fraction decomposition guidelines . . . . . . . . . . . . 107Table 7.1 Truth table for logical equivalence of P and ¬P =⇒ Q∧¬Q . 194xiList of FiguresFigure 2.1 Mathematical Knowledge for Teaching (MKT) . . . . . . . . 21Figure 2.2 Assessment item for fraction as quantity (Simon, 2006, p.368) 38Figure 5.1 Casey’s work on quadratics and air travel . . . . . . . . . . . 114Figure 5.2 Bailey’s factoring of x5−1 . . . . . . . . . . . . . . . . . . . 120Figure 5.3 Taylor’s factoring of x5−1 . . . . . . . . . . . . . . . . . . . 120Figure 5.4 Adrian’s factoring approach . . . . . . . . . . . . . . . . . . 122Figure 5.5 Casey’s factoring of x4+1 . . . . . . . . . . . . . . . . . . . 127Figure 5.6 Casey’s factoring of cubics . . . . . . . . . . . . . . . . . . . 131Figure 6.1 Casey’s work for the “inverse” of y = x2 . . . . . . . . . . . . 150Figure 6.2 Casey’s written work for the inverse of sin(x) . . . . . . . . . 151Figure 6.3 Casey’s work for circles and inverses . . . . . . . . . . . . . 153Figure 6.4 Adrian’s work for the value of e . . . . . . . . . . . . . . . . 155Figure 6.5 Taylor’s work for 0.999 . . .= 1 . . . . . . . . . . . . . . . . . 162Figure 6.6 Adrian’s work for 0.999 . . .= 1 . . . . . . . . . . . . . . . . 165Figure 6.7 Bailey’s work for the limiting behaviour of 0.999 . . . . . . . . 166Figure 6.8 Bailey’s work explaining rational exponents . . . . . . . . . . 173Figure 6.9 Adrian’s work finding a value of 2√3 . . . . . . . . . . . . . . 174Figure 6.10 Taylor’s work for work defining 2√3 . . . . . . . . . . . . . . 175Figure 6.11 Adrian’s work extending exponents to irrationals . . . . . . . 177Figure 6.12 Taylor’s work on exponents and logarithms . . . . . . . . . . 185Figure 7.1 Taylor’s proof that√2 is irrational . . . . . . . . . . . . . . . 193Figure 7.2 Bailey’s picture of “zooming in” to√2 . . . . . . . . . . . . 197xiiFigure 7.3 Casey’s work on symmetries of an equilateral triangle . . . . 206xiiiGlossaryAMK Advanced Mathematical Knowledge, a framework used specifi-cally for knowledge of mathematics beyond the secondary cur-riculum.HCK Horizon Content Knowledge, a component of Mathematical Knowl-edge for Teaching defined as the knowledge of the mathematicsbeyond the curriculum being taught.KDU Key Developmental Understandings, knowledge that allows a learnerto work with a particular concept in new ways and settings.MKT Mathematical Knowledge for Teaching, a framework for under-standing the knowledge used in the teaching of mathematics.PCK Pedagogical Content Knowledge, a framework which extends thenotion of content knowledge to content knowledge in teaching.xivAcknowledgmentsI would first and foremost like to thank my supervisors, Drs. Ann Anderson andCynthia Nicol. You believed in my passion for mathematics education and openedmy eyes to the possibilities of what a rich mathematics education can look like,from pre-school to the university level. Your support, sympathy, and wisdom haveallowed me the space to grow not only as a researcher, but as a person.I would be amiss to not thank my mom. You have been my biggest cheerleaderin whatever I’ve done in my life, from pointe shoes and tutus to textbooks and gradschool. Your creative spirit has always inspired me to look at the world differently.Thank you for the arts-based environment you worked so hard to foster at homeand know that it played a fundamental role in growing into the woman I am today.To my dearest Paul, you’ve dealt with me in times where I don’t know howyou have. Your love, kindness, and encouragement have kept me going in times ofextreme self-doubt. Through good times and bad, you continue to love and supportme in reaching my goals. I love you more than I can put into words.A final thanks to the Killam Trust and their generous fellowship that supportedmy studies. It was a great honour to receive their support for the work I was soexcited to embark on.xvDedicationTo my father, Michael Jerome Radzimski, January 21, 1951 - October 1, 2018Finishing this thesis with everything that transpired over the last year was achallenge, to say the least, but it would not have been possible without your ever-lasting love and support. I love you and I miss you everyday.Forever, your little Vee.xviChapter 1Introduction1.1 BackgroundAdmission requirements for most secondary mathematics teacher education pro-grams include a degree in mathematics or a related major. Consequently, univer-sity students enter such mathematics education programs with courses in Abstractand Linear Algebra, Number Theory, and Real and Complex Analysis, amongothers. This requirement does not exist without reason. Much of the content insecondary mathematics curricula is deeply connected to these advanced universitycourses, despite being well outside the content teachers would be expected to teachsecondary students. Some researchers claim that advanced subject matter experi-ence influences teachers to conceptualize the secondary mathematics curriculumat a deeper level and provide a richer learning experience for students, leading tohigher student achievement (Paige, 2002). However, there is competing evidencethat suggests no positive correlation exists between advanced mathematics course-work taken by teachers and their students’ achievement (Monk, 1994).1The post-secondary education of prospective secondary teachers could be con-ceptualized as two islands: mathematical content and mathematics pedagogy. Dur-ing a prospective secondary mathematics teachers’ mathematics education, math-ematics departments and professors are responsible for building the mathematicalknowledge of these future teachers, while education departments support futureteachers with subject-specific courses in pedagogy. Research in secondary mathe-matics teacher education follows a similar pattern, with studies focusing on eitherfuture teachers’ mathematical content knowledge (Almeida et al., 2016; Cofer,2015; Even, 1993; Knuth, 2002; Leung et al., 2016) or mathematics pedagogy(Dede, 2015; Fernández et al., 2016; Zazkis and Leikin, 2010).This divide between mathematics content and mathematics pedagogy remainswithin the structure of the university, as emphasized by Ted Eisenberg, who “lamentedprofoundly the growing divide between the mathematics community and the math-ematics education community” (Fried, 2014, p.3). In the Mathematical Associationof America’s (MAA) A Call for Change, they state that “the mathematical prepa-ration of teachers must provide experiences in which they develop an understand-ing of the interrelationships within mathematics and an appreciation of its unity”(Leitzel, 1991, p.3). Unfortunately, the divide between mathematics and education— in teacher education, the research literature, and the university — may not beconducive to such preparation. Do prospective teachers have the opportunity todevelop interrelationships between these two fields? That is, are they able to buildconnections between their advanced university mathematical knowledge and ped-agogical practice in secondary mathematics?2This is precisely where the research in this thesis becomes of importance.Rather than studying the islands of university mathematics content and mathe-matics pedagogy individually, I studied the bridge between the two and sought touncover the role that university mathematics coursework plays in secondary math-ematics teacher education. To date, there appears to be no empirical study whichexamines how prospective teachers bridge these two islands of knowledge. Wasser-man’s recent work (Wasserman, 2016; Wasserman et al., 2017) bridges the twoislands, but is theoretical in nature and indicates a need for empirical research inthe domain. This current study begins to shed light on the effect a post-secondarydegree in mathematics has on the ways prospective secondary teachers perceiveand understand secondary school concepts, as well as to what extent they drawupon connections between these two bodies of knowledge to inform their ped-agogy. Moreover, the investigation has implications for ways in which mathe-matics teacher educators might support future teachers’ MKT. Indeed, fosteringopportunities for the development of connections which integrate secondary andpost-secondary mathematics may be transformative to prospective teachers’ under-standing of the secondary curriculum and the ways in which it is taught.1.2 Research AimsThis study aims to observe, understand, and interpret the ways in which prospectivesecondary mathematics teachers draw upon knowledge from their post-secondarydegrees to enhance secondary mathematics learning. Despite the existence of deepconnections between secondary and post-secondary mathematics content (Cofer,32015; Wasserman and Stockton, 2013), research suggests that teachers do not per-ceive their advanced mathematical knowledge as playing an important role in ped-agogical practice (Zazkis and Leikin, 2010). With the majority of studies on sec-ondary teachers’ mathematical knowledge focusing explicitly on content in thesecondary curriculum, no such study exists that aims to understand the bridgethat connects secondary and post-secondary mathematics content. The primaryresearch questions are:1. What do prospective secondary mathematics teachers perceive as the role oftheir advanced mathematics knowledge in their development as teachers?2. In what ways do prospective secondary mathematics teachers relate advancedmathematics knowledge to a mathematics concept in the secondary curricu-lum?From the results of this question, I will shed light on the role of advanced math-ematical coursework in the development of future teachers, and point to ways inwhich mathematicians and mathematics teacher educators might foster the devel-opment of MKT for future teachers.1.3 SignificanceThis investigation will provide important information on how a post-secondary de-gree in Mathematics affects the way prospective secondary teachers understand in-struction of secondary school mathematics concepts and how the connections theyhave between these two bodies of knowledge influence their pedagogy. Indeed,as mentioned above, many secondary mathematics teacher education programs re-quire extensive mathematical coursework at the university level. However, this4knowledge is set aside once teacher education candidates enter their pedagogicalstudy. As Wasserman et al. (2017) argues, advanced mathematical knowledge hasthe potential to be transformative to a teacher’s pedagogy by connecting advancedmathematical knowledge to teaching practices. This study serves as a first step to-wards understanding the ways prospective teachers have connected their advancedmathematical knowledge to secondary content, without the intervention of mathe-matics teacher educators.Moreover, this investigation has implications for ways in which mathematicsteacher educators might explicitly support future teachers’ understanding of sec-ondary mathematics concepts and mathematics at the post-secondary level. In-deed, the interview topics discussed in this study could constitute rich mathemat-ical discussion in a mathematics pedagogy course, recognizing that teachers haveadvanced content knowledge that is relevant in a secondary school context and maybetter inform their pedagogy. As a consequence, the development of mathemati-cal content knowledge that is directly related to secondary mathematics instructioncould enrich the mathematical learning environment for secondary students. Suchan improvement in mathematics teacher education could result in teachers who areable to engage with secondary mathematics content with more depth and breadth,and in turn, provide secondary students with a rich mathematical learning experi-ence. This shift in mathematics education at the secondary level could improve stu-dents’ mathematics upon entry to university and in turn, increase entry into STEMfields for post-secondary education.Finally, this study may encourage mathematics departments to reconsider the5ways in which they support the mathematical development of future teachers. Howmuch are future teachers learning in their advanced mathematics courses? Whatare they learning and do they find it relevant to their future work? I hope that theresults of this study act as an impetus for mathematicians to interrogate the intent,goals, and pedagogies of advanced mathematics courses. In the same vein, but adifferent context, I hope that my research contributes to conversations about theways advanced mathematics expertise is drawn upon in teacher education. Withprospective teachers needing over 30 credits of mathematics coursework at the se-nior level, to me, it appears to be a missed opportunity to not draw upon, develop,and encourage reflection on the role of such courses by and for future teachers.1.4 MethodsIn an effort to investigate the research question, I utilized a qualitative researchmethodology. More specifically, the following research is a case study (Yin, 2013).Data collection and analysis was executed through one-on-one interviews, task-based interviews (Goldin, 2000) and document analysis. Sources of data includedtranscriptions of interviews, thematic coding of interview transcripts, participants’written work, and participants’ academic transcripts. Coding of the transcripts wasdone through a grounded theory approach, where codes were assigned to state-ments throughout the interview. Similar codes were then gathered into themes. Al-though a quantitative data focused methodology could provide insight into futureteachers’ mathematical knowledge for teaching, it would not provide the depth nec-essary to understand participants’ perceived role of advanced mathematics knowl-edge in teacher education. The qualitative case study approach was chosen to be6appropriate for this work, since case study allows the researcher to provide richdescription and provide insight into the knowledge, experience, and beliefs of thefuture secondary mathematics teachers in this study.Participants in this study were recruited on a volunteer basis from the sec-ondary teacher education program at my institution. Since all participants wereenrolled in the program with mathematics as a teachable subject, each of them metthe mathematics coursework requirement for entry into the teacher education. Intotal, five teacher candidates enrolled as participants in the study: Taylor, Jaime,Bailey, Adrian, and Casey. Each of the names assigned are pseudonyms and aregender neutral. Throughout the thesis, each participant will be referred to under thepronoun “they,” since gender was not part of my analysis of the interviews. As athank you for volunteering their time and experience in the study, each participantwas gifted a $25 VISA gift card.1.5 Theoretical FrameworkAs will be detailed in Chapter 2, teacher knowledge can be studied in many dif-ferent forms. Much of the empirical research on teacher knowledge falls eitherthe domain of content knowledge, pedagogical knowledge, or the intersection ofthese two conceptualizations. This intersection, defined as Pedagogical ContentKnowledge (PCK), was spearheaded by the work of Shulman (1986). Pedagogicalcontent knowledge is defined as “pedagogical knowledge that goes beyond knowl-edge of subject matter per se to the dimension of subject matter knowledge forteaching” (Shulman, 1986, p.9). This work was general to the context of teaching7and paved the way for future, subject specific conceptualizations. One such notableexample is that of Mathematical Knowledge for Teaching (MKT) (Ball and Bass,2002; Ball et al., 2008; Hill et al., 2005, 2008). Deborah Ball and her colleagueshave been pioneers in the study and measurement of mathematics teachers’ MKT,specifically at the elementary level.However, their framework for MKT may not be appropriate or transferable tothe context of secondary mathematics teachers, who have extensive mathematicalexpertise beyond the school curriculum. As such, the MKT framework of Silver-man and Thompson (2008)was utilized to understand the development of MKT forprospective teachers in this study. The use of this framework will be justified inChapter 2. This particular framework utilizes the work of Piaget and his conceptu-alization of reflective abstraction (Piaget, 1970, 1985).Given that my chosen MKT framework is built upon the work of Piaget, itis important to note that my work situates itself within a constructivist theory oflearning. The constructivist paradigm foundations itself on the assumption thatknowledge is constructed through experience and the processes of assimilation andaccommodation (Piaget, 1970). With much of Piaget’s later work landing in thedomain of mathematical knowledge, the distinction between physical knowledgebased on experience and non-physical knowledge (i.e. logical and mathematicalstructures) was of particular importance. He questioned how one’s knowledge ofthe abstract, that which cannot be directly experienced, is derived. Piaget suggestedthat knowledge of abstract logical-mathematical structures are acquired throughsimple abstraction and reflective abstraction (Piaget, 1970).8The constructivist paradigm being situated within experience aligns itself wellwith my chosen qualitative, case study methodology. Through engaging in one-on-one interviews, I wanted to understand the experience and knowledge of myparticipants by living an experience along side them. Moreover, the beliefs andvalues which I personally hold about teaching and learning land in the domain of aconstructivist theory. As such, I hold the belief that teachers of mathematics shouldbe “guides on the side,” facilitating rich discussions catered to the individual needsof their students.1.6 Personal StatementThroughout elementary and secondary school, I did not consider mathematics tobe a subject of personal interest. Growing up with both of my parents as artists, Ivalued creativity and imagination; I loved painting, dancing, choreographing, andcreating music, all of which challenged my creativity and imagination. My math-ematics learning, on the other hand, seemed to be devoid of these characteristics.The emphasis was on speed, not understanding; memorization, rather than creation.I can recall asking my teachers why we followed certain rules and algorithms, onlyto be told “that is just the way you do it.” These responses left me dissatisfied anduninterested, but my parents continued to emphasize the importance of a strongmathematics education. So, I stuck through it, followed the norms of rote memo-rization, associating my mathematics education with a pathway to success.At seventeen, I enrolled in courses at the community college. I had spent thelast eight years of my life pursuing a career in classical ballet, but my parents con-9tinued to emphasize the importance of an education, particularly if I wanted topursue a career in dance. The program I was enrolled in counted my communitycollege courses towards a high school diploma, as well as an Associate’s Degreefrom the college. In not knowing what I wanted to study, I took a mixed bag ofcourses; psychology, art history, foreign language, science, and mathematics. Themathematics course I took was Pre-Calculus. It was a class of about twenty-fivestudents with a professor who taught both at the community college and at the lo-cal public university. Being the keen student that I was, I sat myself in the frontrow of the classroom, assuming that my old ways of doing mathematics wouldget me through. However, I was surprised by the cultural shift in learning math-ematics that I experienced. Thinking mathematically was transformed from beingabout the destination to being about the journey; creativity and imagination werenow valued in learning and doing new mathematics. Some of my questions wereanswered with responses well beyond the context of the course, but it brought meto see mathematics as something so much bigger and more beautiful than what Iperceived it to be for so many years.This shift in learning and doing mathematics acted as the impetus for pursuingmy Bachelor’s and Master’s degrees in mathematics. When I taught my first coursein the fall of 2013, I was brought back to sitting in my Pre-Calculus class years agoand wondering what that professor did to help me become so interested in math-ematics. I had years of advanced mathematical coursework under my belt, so Iwondered how I could make that accessible to my students so that I might sparktheir curiosity, as my pre-calculus professor did for me. The culture in my class-room was one of inquiry, and my students would often ask questions beyond the10horizon of the course. However, I did not want to leave them with a reply similarto the ones I received in high school. I was challenged to think beyond the contentof the course to my advanced study, and think of how I could use that knowledgeto help them better understand the content of our course. This reflective processbrought me to have a deeper understanding of the mathematics I was teaching, aswell as building a classroom where creativity and imagination were of value.My interest in the relationship between school and advanced mathematicalknowledge stems from these experiences. It is my hope that my current researchwill begin to bridge this gap and create more dynamic relationships between fac-ulties of education and departments of mathematics. Both of these groups haveopportunities to learn and grow with each other. My hope is that one day, math-ematics educators could collaborate with mathematics departments to transformadvanced mathematics courses to better serve future teachers, while mathemati-cians could work with teacher educators to transform mathematics methods courseswhich draw upon and extend the mathematical expertise of future teachers. To-gether, we can work to enrich the mathematical knowledge of future teachers, andin turn, the mathematical education of students in both elementary and secondaryschool.1.7 OrganizationThe front matter of this paper, chapters 1, 2, and 3, shed light on the theoretical andmethodological considerations of the study. In chapter 2, I discuss existing litera-ture on teacher education. I examine the history of teacher education and certifi-cation, as well empirical studies measuring teachers’ mathematical knowledge for11teaching (MKT). Through this analysis, I elaborate on the gap of an accepted theo-retical framework for analyzing and measuring MKT, particularly in the context ofsecondary mathematics teachers. I provide justification for my chosen theoreticalframework of MKT (Silverman and Thompson, 2008).Chapter 3 examines the methodological considerations of this research, includ-ing extended background on all five participants. In Chapters 4, 5, 6, and 7, Iexamine results of the task-based interviews, through the categorization of mathe-matically related tasks. At the beginning of each of these chapters, I begin with anoverview of some of the connections I have made in my own studies between sec-ondary and post-secondary mathematics. Finally, Chapter 8 discusses the resultsfrom the previous chapters by interpreting them within the context of my theoret-ical framework. More specifically, this chapter will examine the development ofparticipants’ MKT in the context of their studies in mathematics and teacher edu-cation. The chapter concludes with a discussion on limitations, extensions for thefuture, and a discussion of the impact of this work for departments of mathematicsand faculties of education.12Chapter 2Literature Review andTheoretical FrameworkTeacher certification exams dating back to the mid-1800s reveal that there has beena long-term interest in teacher knowledge and quality (Shulman, 1986). Whatknowledge should teachers have? How deeply should they understand the con-tent they are to teach? How would a teacher handle a particular situation in theclassroom - pedagogical or otherwise? These are questions that are debated in theteacher education literature to this day. Although there is a great deal of interestin teachers’ general pedagogical knowledge, the past three decades have seen agrowing body of literature specifically dedicated to teachers of mathematics. Inparticular, researchers have found themselves concerned with the mathematicalknowledge used in teaching mathematics. To this end, the following chapter aimsto provide a comprehensive review of this subject: mathematics teachers’ Mathe-matical Knowledge for Teaching (MKT).13Through this chapter, we will explore the existing literature on secondary math-ematics teachers’ knowledge. As will be described, early studies focused on math-ematics knowledge, but more recent research has focused on frameworks for under-standing teachers’ knowledge. I will begin by going back centuries to understandand look at the historical development of teacher knowledge in general, notingthe pendulum effect through the years; that is, concerns in teachers’ knowledgehave continually swung back and forth between content knowledge and pedagogi-cal knowledge, creating confusion as to what knowledge is valuable for teachers tohave. Next, I will look at the combination of the two in Shulman’s (1986) conceptof Pedagogical Content Knowledge (PCK).Finally, I will turn my attention to the knowledge of mathematics teachers, ahot topic among education researchers due to concerns of declining North Amer-ican mathematics scores at the international level (No Child Left Behind Act of2001, 2019; Richards, 2014). I will examine various frameworks for “mathemati-cal knowledge for teaching” (MKT), developed as an extension of Shulman’s PCKfor mathematics teachers. However, as will be argued, the integration of advancedmathematical knowledge is not embedded in this theory, which has lead to alter-native theories of MKT, such as that of Silverman and Thompson (2008). I willdescribe three alternative frameworks for secondary mathematics teachers’ knowl-edge and justify the choice of my framework for this study.142.1 Teachers’ Content Knowledge: BeginningsThe historical discussion that follows is drawn from Shulman (1986). In his workRamus, Method and the Decay of Dialogue, Ong (1958) describes the importanceof pedagogy within the medieval university. The environment was one in which“content and pedagogy were part of one indistinguishable body of understanding”(Shulman, 1986, p. 3). Ong asserts that the defining characteristic of rich subjectmatter understanding was indicated by a students’ ability to teach via lecture anddiscussion. To this day, in order to receive the academic title of “doctor” or “mas-ter,” one must demonstrate their ability to lead a lecture and discussion during theirdefense. Even a millennia ago, Aristotle stated the following regarding the natureof knowledge:Broadly speaking, what distinguishes the man who knows from theignorant man is an ability to teach, and this is why we hold that artand not experience has the character of genuine knowledge (episteme)- namely, that artists can teach and others (i.e. those who have notacquired an art by study but have merely picked up some skill empiri-cally) cannot. (Shulman, 1986, p. 4)In the mid-1800s, the majority of examinations for school teachers focusedon subject matter knowledge (Shulman, 1986, p. 2). Excellence in teaching wasdefined by a teacher’s mastery of content, while pedagogical knowledge was a sec-ondary concern. Shinkfield and Stufflebeam (2012) provide a detailed account ofteacher evaluation in the first half of the 20th century, remarking that very fewschools engaged in the formal evaluation of their teachers. Despite limited devel-opments in evaluation, emergent teacher education policies of the 1980s were in15stark contrast to those of the 1870s. A teacher’s capacity to teach was then definedby their knowledge of pedagogical practices and basic subject matter knowledge.Shulman argues that the transition towards valuing pedagogical practice over sub-ject matter was partly due to policymakers’ decisions being based on educationalstudies, that themselves ignored subject matter (Shulman, 1986, p. 3). Thus, theevaluation of content knowledge receded, being overtaken by the evaluation ofeffective teaching practices as defined in various “process-product studies” (Shul-man, 1986, p. 3).The 1983 document A Nation at Risk painted a rather doom and gloom pictureof the existing status of the United States education system, claiming that “if anunfriendly foreign power had attempted to impose on America the mediocre edu-cational performance that exists today, we might well have viewed it as an act ofwar” (National Commission on Excellence in Education, 1983, p.5). The documentincluded an overview of the risks that the U.S. education system faced, evidencefrom various sources including declining SAT scores, as well as five recommenda-tions for improving the existing system. One of these pertained to improvement inthe quality of teacher preparation, with a substantial focus on subject matter knowl-edge. This recommendation stemmed from a criticism that “the teacher preparationcurriculum is weighted heavily with courses in “educational methods” at the ex-pense of courses in subjects to be taught” (National Commission on Excellence inEducation, 1983, p.22). With the release of Nation at Risk came reforms such as theNational Science Teachers Association’s 1984 recommendation that all secondaryschool science teachers have a minimum of 50 credit hours of science course workat the university level (Weiss, 1987, p.76). Almost two decades later, the No Child16Left Behind Act of 2001 called for teachers to be fully certified in their subject spe-cialization (No Child Left Behind Act of 2001, 2019).Canada has also experienced pressure in reforms of teacher education (Russelland Dillon, 2015; Sheehan and Fullan, 1995). Although teacher certification iscontrolled by individual provincial governments, each province requires that sec-ondary teachers have an undergraduate degree in a teachable subject. In Ontario,the 1960s brought forth a shift in which teacher education was moved from teachercolleges to Faculties of Education, initiating the present notion of secondary sub-ject specialization (Kitchen and Petrarca, 2013). More recently, the Ontario gov-ernment announced that certified teachers will require both an undergraduate de-gree and a 4-term teacher education program (Kitchen and Petrarca, 2013), furtheremphasizing the value they place on pedagogical knowledge. In Quebec, the gov-ernment mandates that candidates for a teaching diploma in general education atthe secondary level have at least 45 credit hours of university coursework in a basicschool subject, as well as a 4-year Bachelor of Education program with over 700hours of practicum (Gouvernement du Québec, 2011; Russell and Dillon, 2015).All of these policies encourage secondary teachers to have subject specific contentknowledge; however, once a teacher is hired by a school, they may be allowedto teach other subjects. This is particularly the case in British Columbia whereteaching certificates do not signify grade level or subject specialization (BritishColumbia Ministry of Education, 2016). Nonetheless, all of these reforms point tosustained political pressure to have teachers more educated in their subject area.Shulman’s (1986) notion of pedagogical content knowledge (PCK) acted as a17medium for reconciling concerns about teachers’ content knowledge and concernsof teachers’ pedagogical practices. For Shulman, the construct of PCK extendscontent knowledge to content knowledge for teaching. Content knowledge is stillat the forefront, but is placed in the context of how teachers navigate such knowl-edge throughout their teaching. Shulman (1986) emphasized that there was a lackof research focusing on the relationship between content and pedagogy and sug-gested that teacher education researchers should begin to explore this newly de-fined terrain. Following Shulman’s work, an entire body of work dedicated to PCK(Grossman, 1990; Wilson et al., 1987), extensions to particular subject areas, andmore recently the notion of technological pedagogical content knowledge (TPCK)has emerged (Koehler and Mishra, 2014). This general work in teacher knowledgelaid the groundwork for mathematics specific research on the relationship betweencontent and pedagogy (Depaepe et al., 2013).In particular, Ernest (1989) worked to extend Shulman’s notion of PCK to amore detailed framework for mathematics teachers. Similar to Shulman, Ernest(1989) argues that mathematics teachers should have both a curricular and ped-agogical understanding of mathematics. These constructs mirror that of Shul-man’s notions of curriculum knowledge and PCK, respectively. What distinguishesErnest’s framework from Shulman’s is the attention to the beliefs and attitudes ofteachers and the impact on their practice in the classroom. Ernest argues that thissystem is most likely unique to individual teachers and exists as a product of theindividual’s “view or conception of the nature of mathematics, model or view ofthe nature of mathematics teaching,” and “model or view of the process of learn-ing mathematics” (Ernest, 1989, p.250). In the context of secondary mathematics18teachers, one might ask the following question: what do prospective secondarymathematics teachers perceive as the role of their advanced mathematics knowl-edge in their development as teachers? This is a question we will investigatethrough this study.2.2 Mathematical Knowledge for TeachingNearly a decade before Shulman initiated work on PCK, Begle (1979) examinedsecondary mathematics teachers’ mathematical knowledge. His work was under-taken to provide “guidance to those interested in conducting comprehensive re-views of the factual information which exists about the effects of various variableson student learning of mathematics” (Begle, 1979, p. xv). In his review of the em-pirical literature on mathematics teachers between 1960-1976, Begle (1979) sug-gested that there was no direct correlation between student success and the numberof mathematics courses taken by their teachers. In an effort to support this, Monk(1994) examined secondary mathematics teachers and the effect that various uni-versity coursework had on their pupils’ improvement in mathematics. Using quan-titative measures, Monk found a minor positive relationship between the numberof mathematics courses taken and student improvement (Monk, 1994, p.130). Per-haps more interestingly, he also found that the number of courses in mathematicspedagogy had a more positive effect on student learning than increased undergrad-uate coursework in mathematics (Monk, 1994, p.130). Adding further murkinessto the water, the National Centre for Research on Teacher Education (NCRTE)claimed that teachers with undergraduate majors in the subject they teach did notoutperform other teachers in their explanations of fundamental concepts (NationalCenter for Research on Teacher Education, 1987). In their study of prospective19teachers’ mathematical content knowledge, Kahan et al. (2003) recognized thatthe most effective lesson plans from the prospective mathematics teachers in theirstudy were not necessarily those with the highest grade points averages in mathe-matical coursework.Indeed, the four decades of work following Begle (1979) have brought forth nocommon consensus as to what extent university mathematics coursework affectspupils’ learning. These inconsistencies have brought researchers to look beyondsubject matter knowledge and consider the interplay between content and peda-gogy in the mathematics classroom. In an effort to address this question, DeborahBall and her colleagues extend Shulman’s notion of PCK to the teaching of math-ematics (Ball, 1988, 1990; Ball and McDiarmid, 1990). Deborah Ball is arguablythe pioneer in research on the ways in which teachers of mathematics must know,understand, and teach the mathematical knowledge at stake in the school curricu-lum (Ball et al., 2005, 2001; Hill et al., 2008). Along with her colleagues, Ballhas conducted numerous studies in an attempt to describe what “teachers do inteaching mathematics” (Ball et al., 2005, p.17). In their later work, they defineMathematical Knowledge for Teaching (MKT) as mathematical knowledge that isused in teaching mathematics (Ball and Bass, 2002, p.5).In their most detailed description of MKT, Ball et al. (2008) take the cate-gories of PCK, content knowledge, and curricular knowledge as defined by Shul-man (1986) and subdivide them into more well-defined subcategories (see Figure2.1). They conceptualize MKT as having two dimensions: Subject matter knowl-edge and PCK. These dimensions break down further in their framework. Subject20Figure 2.1: Mathematical Knowledge for Teaching (MKT)matter knowledge is subdivided into Specialized Content Knowledge (SCK), Com-mon Content Knowledge (CCK), and Horizon Content Knowledge (HCK). The au-thors define CCK as mathematical knowledge possessed by the average adult. Thistype of mathematical knowledge may be held by parents of students to help themdetermine whether or not their child has solved the problem correctly. SCK, onthe other hand, is mathematical knowledge held by those beyond the average adult.This knowledge is independent of pedagogy. In order to have SCK, one must havea deeper understanding of mathematics so that they may make modifications orbuild connections between content.Finally, the notion of Horizon Content Knowledge (HCK) is a component ofteachers’ knowledge that recognizes how mathematical topics are interrelated acrossthe mathematics curriculum. Since HCK is a domain of MKT, HCK could be con-sidered an awareness of mathematics beyond the horizon of the curriculum that21impacts teachers’ practice (Jakobsen et al., 2013). Advanced mathematics is cer-tainly beyond the horizon of the secondary curriculum, so does taking a course inabstract algebra contribute to a teacher’s HCK?Taking a course in abstract algebra certainly increases a teacher’s mathematicalcontent knowledge. After taking such a course, students learn about the mathemat-ical structures of groups and rings, Galois Theory, and extend the notion of linearalgebra to arbitrary fields, among other concepts. However, for this knowledge tobe included in the domain of HCK, the teacher should be able to recognize the rela-tionships between the concepts in abstract algebra and the secondary curriculum ina way that influences their pedagogy. By recognizing the relationships between ad-vanced subject matter and their previously constructed knowledge of school math-ematics, the teacher has made a conceptual advance in their understanding of theelementary concept. For example, consider the case of irreducible polynomials. Inhigh school, students learn that real-valued polynomials can factor into linear andquadratic polynomials. In abstract algebra, however, students learn that these areirreducible elements in the polynomial ring R[x] and are then able to work withthem in a new, more abstract setting. This notion of a conceptual advance thatallows one to see a concept through a new lens is what Simon (2006) defines as akey developmental understanding Key Developmental Understandings (KDU). Wereturn to this construct later in this chapter.As with PCK, the MKT framework takes into account mathematical contentknowledge, as well as pedagogical knowledge, and seeks to gain insight into themathematical work of teachers. Their framework for MKT has been widely used22in empirical studies with elementary mathematics teachers, since the frameworkitself emerged from practice-based research with elementary teachers. At the ele-mentary level, Hill et al. (2005) found that teachers’ knowledge of teaching mathe-matics was the strongest predictor of student achievement in grades one and three.With these results in mind, Matthews et al. (2010) examined the effect of special-ized courses for teaching elementary mathematics to a group of pre-service ele-mentary teachers. Results showed that the teachers in the specialized courses hadhigher mathematical content knowledge than those who took standard mathematicscourses (Matthews et al., 2010, p.7). These results demonstrate that understandingteachers’ MKT is both a fruitful and practical endeavour.The work of Deborah Ball is exceptional in its focus and reach. Her researchhas transformed the landscape of literature and progress in mathematics teacherknowledge. Having an understanding of what is “in” MKT has served as a launchpad for numerous professional development initiatives for mathematics teachers(Clarke, 2007), as well as inspiring change in the pre-service education of futuremathematics teachers (Simon, 2008). Her substantial impact can be seen throughthe large number of citations of her work; one article, Ball et al. (2008), has beenreferenced nearly 6,000 times. While the work was initially done in the context ofelementary mathematics, it has reached beyond the elementary setting to secondaryand post-secondary mathematics (Artzt et al., 2012; Goos, 2013; Tchoshanov et al.,2017). One research direction which has stemmed from Ball’s MKT work is that ofAdvanced Mathematics Knowledge in Teaching, which we explore in the followingsection.232.3 Advanced Mathematics Knowledge in TeachingResearch on advanced mathematics knowledge of secondary teachers is a new andupcoming field of study. Before we begin a discussion on advanced mathematicalknowledge, it is important to define what is meant by it. I borrow the definition ofAdvanced Mathematical Knowledge (AMK) from Zazkis and Leikin (2010), whodefine AMK as “ knowledge of the subject matter acquired in mathematics coursestaken as part of a degree from a university or college” (Zazkis and Leikin, 2010,p.264). However, the roots of relationships between elementary and advancedmathematics date back to over a half a century ago. In his 1939 work Elemen-tary Mathematics from an Advanced Standpoint, mathematician Felix Klein wroteto the teacher who found themselves teaching in the “time honoured way” andwhose “university studies [in mathematics] remained only more or less pleasantmemory which had no influence upon his teaching” (Klein, 2004). The purpose ofthe book was to explore elementary mathematics from the school curriculum withthe assumption that the reader has extensive post-secondary mathematics expertise.Mathematicians and non-mathematicians align themselves with the perspectivethat advanced mathematics knowledge is of value for practicing secondary teach-ers. In their 2002 report, the U.S. Department of Education made the bold claimthat advanced subject matter experience influences teachers to conceptualize thesecondary mathematics curriculum at a deeper level and provide a richer learningexperience for students, leading to higher student achievement (Paige, 2002). Adecade later, this claim still held strong, when the Conference Board of the Math-ematical Sciences claimed that the knowledge of secondary mathematics teachers24should be well beyond the scope of the school curriculum and recommended thatall secondary mathematics teachers have coursework in single and multi-variablecalculus, introduction to linear algebra, statistics and probability, introduction toproof, abstract algebra, real analysis, modelling, differential equations, group the-ory, number theory, history of mathematics, geometry, complex analysis, and dis-crete mathematics (Conference Board of the Mathematical Sciences, 2012). Thisdocument provides detailed rationale for the inclusion of each of these topics inthe mathematics education of future teachers. This list is almost exhaustive ofsome university’s mathematics curriculum and suggests all secondary mathemat-ics teachers should have a full major in mathematics.Teacher education programs have taken strides to align themselves with suchrecommendations. Teacher education programs in Canada require that applicantswho wish to specialize in secondary mathematics have at least 30 credits of mathe-matics coursework at the upper level. Even though advanced mathematics course-work is seen to be essential by mathematicians, researchers, and professional or-ganizations, practicing teachers do not share the same sentiment. In their study ofmathematics teachers’ perceptions of AMK in teaching, Zazkis and Leikin (2010)found that teachers saw benefits from the skills learned in their undergraduate de-grees, but saw limited value in content specifics. That is, teachers in the studyvalued their undergraduate mathematics experience for building their persistencein problem solving, building connections within the curriculum, and overall confi-dence. However, they perceived content connections between AMK and secondarymathematics as being very limited and non-essential to their teaching. Through thisstudy, they conclude with a call for continued studies on the relationship between25AMK and MKT, as well as a more well-defined relationship between the two (Za-zkis and Leikin, 2010).In very recent work Wasserman and colleagues have been exploring the roleof AMK and HCK of secondary mathematics teachers, with the belief that knowl-edge of mathematics at the horizon can be “impetus for additions or alterations tothe teachers’ instructional plans” (Wasserman and Stockton, 2013, p.22). In theirinitial work, they share two vignettes that show how AMK and HCK influence ateachers’ pedagogy. For example, the second vignette demonstrates how extendedknowledge of abstract algebra and group axioms could influence a teachers’ lessonon linear equations. In particular, this knowledge may support lesson design so thatstudents have time to reflect on important mathematics such as the existence of aparticular identity or inverse, depending on the operation in question. Wassermanet al. (2017) follow up on this through an effort to make real analysis relevant toteachers, with the hope that such a model for teaching could help future teachersin “developing knowledge that is situated in professional practices and that theywill understand as valuable and be able to use in their daily work with students”(Wasserman et al., 2017, 574).As this section elaborates, the domain of understanding the role of AMK insecondary teachers’ practice is a new and developing field. In the section that fol-lows, we examine less recent literature on teacher knowledge, where the focus ison content knowledge at the secondary or post-secondary level, not the intersectionof the two.262.4 Studies on Teacher KnowledgeThe vast majority of research on teachers’ MKT exists at the elementary level.This is problematic since educational backgrounds of secondary and elementaryteachers typically differ substantially. Secondary mathematics teachers are oftenrequired to take advanced mathematics coursework during their post-secondarydegrees, while this is not a requirement of elementary teachers. To this day, thereare few studies on teachers’ MKT that investigate how this advanced mathematicalknowledge is used in their practice. One of the earliest such studies was that ofEven (1993), where she examined prospective secondary teachers’ understandingof the function concept. The participants in her study were 162 pre-service sec-ondary mathematics teachers who had completed the majority of the mathematicalcoursework required in their program. A questionnaire was distributed to the par-ticipants that included a variety of questions addressing subject matter knowledgeof functions, as well as pedagogically focused questions on functions. Further-more, a subset of the participants engaged in interviews regarding the functionconcept.Results revealed that the prospective teachers in Even’s study possessed a verylimited conception of function. For example, seven out of the ten subjects whoparticipated in the interview phase stated that all functions can be represented bya single symbolic formula, claiming that functions and equations “are the samething” (Even, 1993, p.105). After her analysis, Even boldly remarks that the re-sults of her study reveal “a situation in which secondary teachers at the end of the20th century have a limited concept image of function similar to the one of the 18th27century” (Even, 1993, p. 112). Even concludes her work with a call for an empha-sis on subject matter preparation in teacher education programs (Even, 1993, p.113). She states that prospective teachers need an environment that fosters power-ful mathematical understandings that can be useful in the teaching of mathematics.In a response to Even (1993), Wilson (1994) conducted a case study of a singleprospective secondary mathematics teacher and the impact that a ten week courseemphasizing mathematical content and pedagogy had on her understanding of thefunction concept. Throughout the study, the participant’s understandings of thefunction concept saw significant development, suggesting perhaps unsurprisinglythat courses which integrate content and pedagogy could be useful to prospectivesecondary mathematics teachers. Although it is concerning that some teachers havesuch limited understanding of secondary mathematics concepts, it is encouragingthat significant improvement is attainable.Following the work of Even (1993), Stump (1999) investigated prospective sec-ondary teachers’ understanding of slope. Slope is a fundamental concept in the sec-ondary curriculum and “challenges the distinction between ratio and rate” (Stump,1999, p.125), requiring that students have a solid understanding of proportionalreasoning. Stump (1999) questioned whether the secondary teachers (prospectiveand in-service) in her study understood the complexities of the slope concept andwhether secondary students’ difficulties with slope (Barr, 1980, 1981) were presentin these teachers. The study revealed that the teachers in question had a limitedunderstanding of the slope concept. Both the pre-service and in-service teachersdemonstrated misconceptions surrounding the concept of slope, were unable toanswer questions relevant to the secondary curriculum, and lacked connections be-28tween various representations of slope. The results of Stump (1999) bring one toquestion how secondary mathematics teachers are to provide rich learning expe-riences for students if they do not have a rich understanding of the mathematicsthemselves. If mathematics teachers have difficulty with concepts such as func-tions and slope, how do they fare when it comes to more advanced mathematics,such as those involving proof? Such concerns motivate my own interest in a welldeveloped framework for assessing MKT.Knuth (2002) sought to answer this question in his article examining secondaryschool mathematics teachers’ conceptions of proof. Through semistructured inter-views and proof-focused tasks, Knuth (2002) explored in-service secondary math-ematics teachers’ conceptions of proof. Through his analysis, he uncovered thatthe participating teachers recognized and acknowledged the importance of proof inmathematics, but not in mathematical pedagogy. Knuth (2002) argued that theseconceptions may exist due to teachers’ previous experiences with proof at the sec-ondary and tertiary level. In response to both instances, Knuth (2002) claimedthat proof is a mere tool for verification and yields no personal meaning for stu-dents (Knuth, 2002, p.400). Furthermore, Knuth (2002) observed that many ofthe teachers in the study did not have a solid understanding of what constitutes avalid proof (Knuth, 2002, p.401); that is, the teachers in his study were unable torecognize what features distinguish a correct proof from an incorrect one. Withteachers having such limited conceptions of proof, he suggests that “universitymathematics professors perhaps play the more significant role in shaping teachers’conceptions of proof” (Knuth, 2002, p. 403). Although it may be reasonable tostate that university mathematics courses and professors play a significant role in29the development of teachers’ conceptions of proof, is it reasonable to assume thatmathematics professors see “proof as a meaningful tool for studying and learningmathematics?” (Knuth, 2002, p. 403) What do university mathematics professorssee as the role of proof in their classrooms and in their pedagogy? Regardless, theissue of in-service teachers having limited conceptions of proof that Knuth (2002)unveils reiterates the potential post-secondary courses in mathematics could havefor secondary teachers’ practice.Following the work of Knuth (2002), Schwarz et al. (2008) conducted a com-parative case study on prospective secondary mathematics teachers’ knowledge ofproof in Germany, Hong Kong, and Australia. Within this study, the researcherswere concerned with future teachers’ “professional competencies” in the domain ofargumentation and proof (Schwarz et al., 2008, p. 792). Similar to Knuth (2002),these researchers were not only interested in participants’ ability to execute proofsrequiring only secondary level mathematics, but they also probed participants’positionality on the role of proofs in mathematics lessons at the secondary level(Schwarz et al., 2008, p. 793). An open-ended questionnaire was used to examinethe various facets and connections among participants’ knowledge and interviewswere conducted with selected volunteer student teachers afterwards. Overall, 186prospective teachers from the three countries completed the questionnaire.Results from the questionnaire revealed that the majority of prospective teach-ers from all three countries were unable to produce formal proofs from the sec-ondary curriculum and did not succeed at recognizing whether a given mathe-matical proof was valid or not (Schwarz et al., 2008, p. 807). Their analysis of30questions pertaining to beliefs about the nature of proof revealed that the major-ity of participants had a high affinity towards utilizing proof to understand moreadvanced mathematical content. These results directly contradict the finding fromKnuth (2002), that the teachers in his study did not mention proof having a sig-nificant role in promoting mathematical understanding (Knuth, 2002, p.400). Thisbrings one to question how each of these authors defines “affinity.” If the teachersin Knuth’s study had participated in the study of Schwarz et al. (2008), would wesee this same result? These contradictory results could be due in part to incon-sistencies in the theoretical framings of these studies, an issue which we turn ourattention to in the next section.In an effort to understand the dynamic between advanced mathematical knowl-edge and the school curriculum, Cofer (2015) examined prospective secondarymathematics teachers’ understanding of abstract algebra concepts which implicitlyappear in the secondary curriculum. Cofer (2015) utilized Ball and colleagues’theoretical conceptualization of MKT in order to identify the mathematical contentknowledge of her participants and how that knowledge affected their pedagogi-cal choices. She found that many of the prospective teachers in her study wereunable to make meaningful connections between school algebra and university al-gebra. For example, when asked questions regarding even numbers, participantswere unable to make any connections between the abstract definition of even num-bers being a subgroup of the integers and the elementary school definition of evennumber. In fact, there were multiple participants who were unable to provide anyaccurate definition for an even number. The results from Cofer (2015) indicate thatthere is a need for research that examines the connections between school and ter-31tiary mathematics and how that knowledge is visited in teacher education. Indeed,Suominen (2015) argues that undergraduate texts in abstract algebra are lackingin explicit connections between secondary and university mathematics. Studentsenrolled in abstract algebra courses often exit the course unable to comprehendthe concepts studied and find themselves unable to connect the concepts withintheir existing mathematical understandings (Zazkis and Leikin, 2010). In order tomove beyond this issue, Suominen contends that abstract algebra “can no longerbe considered simply as the generalization of school algebra but rather it shouldbe regarded as an extension of previous mathematical knowledge from algebra andgeometry” (Suominen, 2015, p.79).2.5 Frameworks for Teacher KnowledgeAlthough each of the studies outlined above examine the mathematical knowledgeof prospective secondary teachers, they are primarily concerned with mathematicalcontent knowledge. The role that content knowledge plays in pedagogy is a sec-ondary concern. To this end, there is very little empirical research. While most ofthese studies make mention of Shulman’s theoretical construct of PCK, only oneutilizes Ball’s framework of MKT. Why is this the case? One answer might arisefrom the how this particular MKT framework was developed. As elaborated uponin Section 2.2, Ball and colleagues’ framework originated in the context of ele-mentary school teachers’ mathematical work in practice. Based on this research,a distinction was made between common content knowledge (CCK) and special-ized content knowledge (SCK). Ball et al. (2008) define CCK as the mathematical“knowledge of a kind used in a wide variety of settings - in other words not unique32to teaching; these are not specialized understandings but are questions that typi-cally would be answerable by others who know mathematics” (Ball et al., 2008, p.399). Furthermore, they define SCK as “the mathematical knowledge ’entailed byteaching’ - in other words, mathematical knowledge needed to perform the recur-rent tasks of teaching mathematics to students” (Ball et al., 2008, p. 399). Thesedefinitions have been widely used and accepted by the elementary mathematics ed-ucation community and continue to influence a growing body of research (Austin,2015; Copur-Gencturk and Lubienski, 2013; Ottmar et al., 2015).It is important to ask whether these constructs are transferable to the contextof secondary mathematics teachers. Do these descriptions of content knowledgerepresent the knowledge of secondary mathematics teachers as well as they dofor elementary teachers? Is it reasonable to generalize the MKT framework fromone context to another? The actions of unpacking and connecting mathematicalconcepts are necessary tasks for mathematics teachers, but are not necessary forthe average mathematically literate person. For example, elementary mathematicsteachers need to know of various methods to motivate the idea of place value inarithmetic computations; this is knowledge that the average person does not have.Thus, within the MKT framework, such work falls under the umbrella of SCK.However, if one considers a secondary teacher who has a degree in mathematics,the work of connecting and unpacking mathematics may no longer be consideredunique to teaching. That is, the definition of SCK may need to be modified to ac-count for the mathematical experiences of secondary teachers. The work of Balland her colleagues is valuable because it extends Shulman’s framework of PCK tothe context of mathematics teachers and links pedagogy, mathematical knowledge,33and student success. However, extending this framework to secondary teachersmay require more care than is initially evident.There have been a number of efforts to evaluate MKT at the secondary level,with theoretical frameworks borrowing heavily from Ball et al. (2008). Among anumber of such efforts are the Knowledge of Algebra for Teaching (KAT) project(McCrory et al., 2012), the SimCalc rate and proportionality teaching survey (Shecht-man et al., 2006), and the High School Mathematics from an Advanced Standpoint(HSMPAS) Project (Usiskin et al., 2001). Although each of these projects seeksto broaden the understanding of MKT at the secondary level, the intent of eachproject differs. The KAT project seeks to refine the MKT framework of Ball andcolleagues, with KAT’s original purpose being to unravel the knowledge used inteaching algebra. Eventually, the KAT project evolved into an instrument usedto evaluate secondary teachers’ knowledge of teaching algebra (McCrory et al.,2012, p. 591). The SimCalc project is concerned with understanding the ways inwhich teachers integrate technology to help their students understand conceptuallydemanding mathematics. In contrast to the KAT project, the SimCalc project isnot concerned with the development of theory. Instead, it assumes Ball’s MKTframework and uses it as a basis to evaluate the effects of technology integrationon teachers’ understanding of the mathematics they teach. Finally, the HSMPASfocuses on curriculum development for prospective secondary teachers. Similar toSimCalc, HSMPAS uses Ball’s conceptualization of MKT as given, but assumesthat advanced mathematical knowledge from post-secondary degrees should havean impact on secondary teachers’ mathematical knowledge for teaching.34All of these projects recognize that the teaching of mathematics is a form of“mathematical work” (Ball and Bass, 2002, p.13) and that teachers’ knowledgea multi-dimensional, complex network interweaving content and pedagogy. Thelarge number of distinct projects that specifically examine secondary MKT revealsan extensive interest in the phenomena. Unfortunately, with all of these distinctprojects comes a disconnection with respect to theory:It is in the arena of underlying theory, however, that these instrumentsdiffer most. Despite claiming to cover roughly the same terrain, theseprojects have strikingly different approaches to specifying domains ofmeasurement - in essence, different approaches to organizing what is“in” mathematical knowledge for teaching. (Hill et al., 2007, p.131)The disparity among these theories makes it difficult to link various resultsin the research literature. If the goal among researchers is to pursue progress inthe domain of secondary MKT, I believe that one of the first steps should be tohave a theoretical framework for which researchers can agree upon. A more ambi-tious goal would be the development of an MKT framework which is accepted notjust within the secondary context, but in the elementary and tertiary as well. Onesuch framework that might fit these credentials is that of Silverman and Thompson(2008) and their framework for MKT.352.5.1 Framework of Silverman and ThompsonThe MKT conceptualization of Silverman and Thompson (2008) was developedwhile bearing in mind that while a great deal of research exists on the knowledgeof mathematics teachers, there is not a “commonly accepted theoretical frameworkfor research in mathematics teacher education” (Silverman and Thompson, 2008,p.501).In particular, existing frameworks focus on what is “in” MKT. Silverman andThompson (2008) distinguish their framework from others in its focus. Ratherthan understanding and quantifying the “what” of MKT, Silverman and Thomp-son (2008) interrogate how such knowledge is developed and what experiencesare necessary to develop understandings that can be pedagogically powerful. Theprimary inquiry of their work asks what cognitive processes and understandingscontribute to the development of MKT. The authors position their work and per-sonal epistemologies in a constructivist paradigm. As a foundation, they utilizeSimon’s notion of a key developmental understanding (KDU) (Simon, 2006), aswell as Piaget’s concept of reflective abstraction (Piaget, 1985). Both of these aretheoretical frameworks that, on their own, are disciplinary orientations for researchon the construction of mathematical understanding. Since Silverman and Thomp-son’s framework is not independent of other frames, it is necessary to have anunderstanding of KDUs and reflective abstraction in order to fully understand thisconceptualization of MKT.36Key Developmental UnderstandingsModern changes to the mathematics curriculum in Canada and the United Stateshave brought forth an increased focus to the conceptual nature of mathematics andtowards building student understanding of these concepts. However, the develop-ment of such understandings is difficult for students and for the teachers working tofoster them. Simon (2006) elaborates that the KDU construct is meant to serve asa way to identify “critical transitions that are essential for students’ mathematicaldevelopment” (Simon, 2006, p. 360). In an effort to distinguish these transitionsas “critical,” Simon remarks that KDUs have two major characteristics. First, theymust involve the students making a “conceptual advance” in their understanding.That is, once a KDU of a particular concept is constructed, students are then ableto make more mathematical connections and think about the concept in ways theyhave not before. Simon provides the example of students transitioning from un-derstanding a fraction, such as 15 , as a piece of a whole, to being an independentquantity; this is a “conceptual advance” which allows students to work with frac-tions in a new context. The second characteristic of a KDU is that students do notacquire it from another person’s explanation. The construction of a KDU is throughthe internal process of the learner whereby the student reflects on their own activityand experience.The KDU construct can play a major role in mathematics pedagogy, both ininstruction and assessment. If fostering the development of KDUs becomes a pri-ority for a teacher, they could plan their lessons in a way that encourages studentsto make the necessary conceptual advances. These lessons should be interactive,37inductive, and allow students the space to make their own realizations about themathematical knowledge at stake. Since one must coordinate assessment with in-struction, how might one assess whether a student has achieved particular KDU?Turning once again to the KDU of fraction as quantity, Simon presents Figure 2.2as an example of a question that assesses whether or not a student has made thisconceptual advance. This diagram could be associated to a question in which astudent must identify if the shaded quantity represents 14 of the whole.Figure 2.2: Assessment item for fraction as quantity (Simon, 2006, p.368)The notion of a KDU in the context of AMK is an evolving domain of research.Wasserman and colleagues have been examining the role of such personally power-ful understandings in the MKT of prospective and in-service mathematics teachers(Wasserman, 2016; Wasserman et al., 2017). To give the reader a better sense ofwhat a KDU in this domain looks like, I present the following example. Exponen-tial functions are a fundamental concept in the upper secondary curriculum, dueto their value in modelling of various real-world phenomena. The notion of anexponent is first studied as repeated multiplication, when the exponent is a wholenumber and then easily extended to the context of rational exponents. However,the extension to irrational exponents is never explicitly discussed until a univer-38sity course in real analysis, while the use of irrational exponents is used in thesecondary curriculum. In order to fully understand how to work with irrationalexponents, a learner must have an understanding of irrational numbers beyond thedefinition of “not rational.” Indeed, irrational numbers can be thought of as thelimit of a sequence of rationals. Such an understanding and conceptualization ofirrational numbers would allow the learner to work with irrational exponents be-yond mere approximations.Reflective AbstractionJean Piaget’s work in explicating the possible mechanisms behind children’s think-ing often found itself grounded in mathematically oriented tasks (Beth and Piaget,2013). In his later work on the theory of genetic epistemology, Piaget’s focus wason understanding the formation of logical structures in children (Piaget, 1970).The distinction between physical knowledge based on experience and non-physicalknowledge (i.e. logical and mathematical structures) was of particular importanceto Piaget, as he questioned how one’s knowledge of the abstract, that which can-not be directly experienced, is derived. Piaget suggested that knowledge of abstractlogical-mathematical structures are acquired through simple abstraction and reflec-tive abstraction (Piaget, 1970).Piaget defined simple abstraction, which he later termed empirical abstraction,as generalizations “drawn directly from external objects” (Piaget, 1980, p. 89). Forexample, a child may abstract a relationship between weight and size by holdinga different object in each hand. They may realize that larger objects imply heav-39ier weight, but may also realize that a larger object need not be heavier. Theseabstractions are based solely on experience with physical objects, and thus repre-sents an empirical abstraction. Although some mathematical knowledge may beconstructed in this way, Piaget argued that the majority of logical-mathematicalknowledge is not constructed through experience with physical objects. Rather,it is derived through the coordination of actions performed on physical objects.Piaget defines abstract knowledge constructed in this manner to be a product ofreflective abstraction.Piaget (1980) considered reflective abstraction to be the mental process inwhich humans construct new knowledge without having a direct interaction withphysical objects. The “reflective” aspect of this abstraction has two-dimensions,both based off separate meanings of the word “reflection.” Firstly, through reflec-tive abstraction, the knower is projecting their knowledge at one level to a level ofincreased abstraction, just as light projects off a mirror. The second dimension isthe reorganization of knowledge from reflexive thought. Since the knowledge pro-jected originates from a lower level, the knower must reconstruct the abstractionsfrom the lower level so that their knowledge connects within the structure of thehigher level of abstraction.Numerous scholars have recognized the importance of reflective abstraction inthe study of mathematics teaching and learning. Although the notion of reflectiveabstraction was developed in the context of the logical development in children,Piaget himself observed that reflective abstraction could be the logical mechanismthat has influenced the historical development of mathematics as a field (Piaget,401985). Ed Dubinsky bases his theoretical framework for mathematics learning atthe postsecondary level on reflective abstraction, arguing that “Piaget’s ideas can beextended and reorganized to form a general theory of mathematical knowledge andits acquisition which is applicable to those mathematical ideas that begin to appearat the postsecondary level” (Dubinsky, 2002, p.96). In his thesis on intellectualdevelopment in mathematics education and instruction, Brun (1975) remarked thatthe primary goal of instruction in mathematics should be the fostering of oppor-tunities for reflective abstraction. Despite the significance of reflective abstractionand its accompanying constructivist paradigm in literature on mathematics teach-ing and learning, critiques do exist. Those who align themselves more with thecultural psychology of Vygotsky (Kozulin, 1990) claim that reflective abstractionand the constructivist paradigm provides a limited view of mathematics learning(Cobb et al., 1992), disregards intersubjectivity (Lerman, 1996), and does not takeinto account sociocultural theories of teaching and learning (Lerman, 2000).2.5.2 Mathematical Knowledge for TeachingSilverman and Thompson’s conceptualization of MKT is based on the argumentthat, although a great deal of research exists on the knowledge of mathematicsteachers, there is not a “commonly accepted theoretical framework for research inmathematics teacher education” (Silverman and Thompson, 2008, p. 501). The au-thors contend that the majority of research surrounding mathematics teachers hasbeen centred around what mathematics teachers need to know to teach mathemat-ics (Silverman and Thompson, 2008, p.500). As such, the previous frameworksfor MKT focus on the mathematical knowledge that allows teachers to interact41with both students and mathematics on a meaningful level. Although Silvermanand Thompson (2008) agree that it is valuable to recognize the attributes of exem-plary teaching, they question how teachers develop such knowledge. Thus, in theirwork, Silverman and Thompson (2008) transfer the focus of “mathematical reason-ing, insight, understanding, and skill needed in teaching mathematics” (Silvermanand Thompson, 2008, p.500) towards offering experiences that could lead to thetransformation of a mathematical understanding “having pedagogical potential toan understanding that does have pedagogical power” (Silverman and Thompson,2008, p.502).The work in Silverman and Thompson (2008) situates itself in elementary andsecondary mathematics. Contrary to the argument made in Section 2.5 of otherMKT frameworks, the developmental MKT framework presented by Silvermanand Thompson (2008) can be extended to post-secondary mathematical knowledge.In the context of my study, the goal is to understand how advanced mathematicalknowledge contributes to MKT. The post-secondary mathematics curriculum isgrounded in abstraction and generality, which as the literature suggests, can be asource of confusion for many students (Suominen, 2015). While the perspectivetaken in these courses is abstract and general, this does not imply that the contentis irrelevant in the context of secondary mathematics. Indeed, Suominen (2015)argues that the content of post-secondary abstract algebra is an extension of sec-ondary school algebra.However, building connections between the abstract generalizations of post-secondary courses to the concrete context of secondary mathematics requires that42the learner develop their own, personal understandings between the content areas.For mathematics majors intending to continue into teacher education, it is possiblethat the content knowledge developed in advanced mathematics classes could beconstructed in the lofty domain of abstraction and generalization, without ground-ing in existing content knowledge. If this is the case, the future teacher may beunable to communicate the relevance of this content in secondary mathematics.If connections are made between the content of advanced mathematics courses tothe secondary curriculum, this knowledge has the potential to impact the way ateacher approaches particular topics in secondary mathematics. That is, the knowl-edge constructed may have the potential to have a powerful impact on pedagogicalpractice.To give the reader context of how advanced mathematical knowledge couldimpact the pedagogical practice of future teachers, I present the following exam-ple. The notion of a Euclidean Domain is central to the study of rings in abstractalgebra. Put simply, a Euclidean Domain is a structure where one can do divi-sion, with the familiar constructs of quotients and remainders. I refer the readerto Aluffi (2009) for a detailed description of Euclidean Domains, but in short, thestructure of the division algorithm for the integers (Z) and the ring of polynomialswith coefficients in the real numbers (R[x]) is identical. That is, dividing inte-gers, like 786÷ 37, is similar in process to the division of polynomials, such as(x3+√2x2−4x+ 32)÷ (x2+1).Division of integers and the long division algorithm is a topic in the elementarycurriculum, while the division of polynomials is a topic in upper secondary math-43ematics. If a secondary teacher has this understanding of the relationship betweendivision algorithms in Z and R[x], it may impact their approach to teaching poly-nomial division. Indeed, one could rethink of 786 and 37 as 7 · 102 + 8 · 101 + 6and 3 · 10+ 7, respectively. This could provide a nice context for discussing theprocedure of the division algorithm of polynomials, by taking into account whatstudents already know about the division algorithm for integers. In this example, ateacher has personally powerful understanding (KDU) of the division of polynomi-als, which relates to their prior knowledge of integer long division, which in turn,could potentially impact their pedagogical practice.The MKT framework of Silverman and Thompson (2008) presents how math-ematical understandings with pedagogical potential (KDUs) transform into under-standings with pedagogical power via reflective abstraction. This framework con-tinually emphasizes “transformation” and “development,” with the intention of un-derstanding how teachers develop the exemplary teaching practices noted in Balland Bass (2002) and Kahan et al. (2003). Rather than identifying the “what” inteaching, Silverman and Thompson hope to lead other researchers towards inter-rogating how prospective and in-service teachers develop MKT throughout theircareers. In turn, their framework of MKT is intended to encourage and guideteacher educators towards designing teacher education programs that encourageteachers, regardless of whether they are in their first or thirtieth year of teaching,to think critically about their MKT. Thus, the development of MKT becomes thedevelopment of habits which examine one’s mathematical knowledge, pedagogicalpractices, and the interplay between them.44Cognitive Reliance?Although Silverman and Thompson (2008) rely on the cognitive perspectives ofPiaget (1985), I argue that their framework also offers space for sociocultural per-spectives of mathematics instruction. Their developmental MKT framework offersteachers room to create their own classroom environments that may explore alter-native ways of knowing and allow students to develop rich mathematical under-standings. This is indicated by the intended classroom being a “dynamical space,one that will be propitious for individual growth in some intended direction, butwill also allow for a variety of understandings that will fit with where individualstudents are at that moment of time” (Silverman and Thompson, 2008, p.507). InSimon’s definition of key developmental understandings (KDU), he states that al-though KDUs and reflective abstraction derive from a cognitive perspective, “theydo not conflict with social constructs such as negotiation of meaning and social andsociomathematical norms” (Simon, 2006, p. 364). He argues that the KDU con-struct coordinates cognitive and social perspectives of learning, so that researchprogress might be made on problematic questions such as internalization (Bereiter,1985). Since Silverman and Thompson base their MKT framework on Simon’s no-tion of a KDU, the case for their MKT framework leaving space for social theoriesis further justified.However, one should be cautious in the coordination of social and cognitivetheories of mathematics learning. One of the first assumptions in Piagetian the-ory is that language is a product of thought. Piaget (1970) himself argued thatthought and logical structures exist in those who are without language. He used45the case of children who could not hear or speak to justify that there exists “well-developed logical thinking in these children even without language” (Piaget, 1970,p.46). However, Vygotskian theorists would argue the exact opposite; to them,language is the mechanism which forms thought. Indeed, Vygotsky believed that“a word is the microcosm of human consciousness” (Vygotsky, 1986, p. 256). Therole of language in thought is central to both theories, but they stand on completelyopposite grounds in regards to what role it plays. If one is to develop a philoso-phy of learning mathematics that coordinates these two perspectives, as many re-searchers have done (Burr, 2015; Confrey, 2002; Ernest, 1998), one must be awareof the underlying theoretical assumptions of each. Indeed, attempts at integratingthese two theories together may bring researchers to face theoretical roadblockssimilar to the ones that physicists face in their attempts to combine quantum the-ory and general relativity. Although each of these theories are extremely powerfulin their particular contexts, there are major obstacles when attempting to integrateone into the other (Lerman, 1996). However, with so many opposing argumentson this issue, I find it impossible to fully refute the integration of a social elementinto Piagetian constructivism, nor can I claim it as an unviable framework for thelearning of mathematics. Thus, I am still able to justify Thompson and Silverman(2008) as a viable theoretical framework for my intended research.2.6 SummaryDespite common goals, mathematicians and mathematics educators divide them-selves not only within the confines of the university, but in research endeavours aswell (Fried, 2014). In the Mathematical Association of America’s (MAA) A Call46for Change, they state that “the mathematical preparation of teachers must pro-vide experiences in which they develop an understanding of the interrelationshipswithin mathematics and an appreciation of its unity” (Leitzel, 1991, p.3). Un-fortunately, the divide between mathematics and education may not be conduciveto such preparation. Mathematics departments and professors are responsible forbuilding the mathematical knowledge of future teachers, while education depart-ments support future teachers with subject-specific courses in pedagogy. When,how and where can prospective teachers build connections between their advancedmathematical knowledge and the secondary curriculum?In a desire to bring these fields together, my research of the mathematics teachereducation literature has revealed that there are a number of studies examiningsecondary teachers’ knowledge of secondary mathematics, as well as studies ontheir knowledge of post-secondary mathematics. However, there is no such studywhich examines the way that a post-secondary degree in mathematics influencesa teacher’s understanding of the secondary mathematics curriculum. To fill thisgap, I plan to employ the MKT framework of Silverman and Thompson (2008)to determine how prospective secondary mathematics teachers come to developtheir MKT through reflective abstraction and key developmental understandingsconstructed during their post-secondary mathematics degrees. My work aims toanswer this question and understand the role that advanced coursework plays inmathematics teacher education by placing advanced coursework in the context ofsecondary mathematics pedagogy.Although there is a significant body of research pertaining to mathematical47knowledge for teaching (MKT), the terrain is difficult to navigate. Researchersin the field each have their own definition of what is entailed by MKT and thusdifferent ways to examine and evaluate it. As mentioned above, the terrain of el-ementary teachers’ MKT is a single piece of land with its foundation in Ball andcolleagues’ conceptualization of MKT, which has allowed researchers to make sig-nificant developments in the field. Unfortunately, no such common theory existsfor secondary MKT, causing the field’s development to progress at a pace that isslower than research at the elementary level. Even though there have been a numberof significant findings on secondary teachers’ knowledge in teaching mathematics,the theoretical bases of each result are different. Indeed, the lack of a commonlyaccepted framework is causing the research community to miss opportunities tobetter understand the knowledge used in teaching mathematics and make connec-tions within the literature as a whole. It is my personal hope that an acceptedframework for secondary MKT comes to the forefront in the coming years, so thatprogress can be made in the future. With my own work concerning secondaryMKT, I look forward to being a part of this conversation and making strides toimprove the education of secondary mathematics teachers in the future.The presented analysis of Silverman and Thompson (2008) argues that theirframework provides researchers with a useful lens for understanding the develop-ment of MKT, while also leaving space for sociocultural perspectives on mathemat-ics education. In light of my extended research, I maintain my stance on this facetof Silverman and Thompson (2008), but argue further that their conceptualizationfor MKT could be the bridge for unifying the research community’s understandingof the knowledge of elementary teachers, secondary teachers, and even university48professors. Their use of key developmental understandings (KDUs) and reflec-tive abstraction provide a basis for understanding teachers’ knowledge in a waythat transfers the focus from “mathematical reasoning, insight, understanding, andskill needed in teaching mathematics” (Silverman and Thompson, 2008, p.500) toa transformation of mathematical understandings “having pedagogical potential toan understanding that does have pedagogical power” (Silverman and Thompson,2008, p.502). Transitioning to this perspective allows researchers to move awayfrom the specifics of teacher knowledge and towards building “professional prac-tices that would support teachers’ ability to continually develop MKT throughouttheir careers” (Silverman and Thompson, 2008, p.509). In turn, this is a transitionfrom a framework that is dependent on the level being taught, to one which focuseson practices that are conducive to the continued development of MKT for teachersat any level.As reviewed in this chapter, many educational movements have recognized theimportant role that mathematics teachers play in supporting a quality education inschool mathematics. Beyond the issue of theoretical framing, we have seen thatthere is a divide between the mathematical knowledge of teachers and how thatknowledge impacts their pedagogy. Despite requiring prospective teachers to haveadvanced mathematical coursework at the tertiary level, it is unclear as to whatrole AMK plays in their MKT. It is my hope that this study will be able to begin tobridge this gap and create more dynamic relationships between faculties of educa-tion and departments of mathematics.This review has revealed gaps in the literature, which point to a need for a study49which examines what prospective secondary mathematics teachers perceive as therole of their advanced mathematics knowledge and the ways in which they relatetheir advanced mathematics knowledge to concepts in the secondary curriculum.In the following chapter, I detail the methodological considerations for my studywhich aims to answer these important research questions.50Chapter 3MethodologyThis chapter describes the methodological considerations of this study. I beginwith a justification of my chosen research methodology (case study), with respectto the literature on social science research, as well as research in mathematics ed-ucation. Following this, I describe the setting of the study and the backgroundsof the individual participants who contributed to the study. I then will detail thedata that was collected and provide rationale for obtaining such data. The chapterconcludes with details on how data collected through the study was analyzed.3.1 Research Methodology: Case Study and InterviewUnlike many studies of secondary mathematics teachers’ mathematical knowl-edge, this study was not intended to examine participants mathematical knowledgeof secondary or post-secondary mathematics through standard qualitative tests.Rather, this study aims to explore the connections participants make between sec-ondary and post-secondary mathematics. These connections are dependent on theindividual experiences of each participant, which vary due to coursework taken at51the university, teaching experience, and perceptions of mathematics. In order to in-vestigate these connections, a qualitative approach to data collection and analysishad to be employed to explore the mathematical knowledge, connections, and ex-periences of the prospective teachers in the study. Since the study would be draw-ing data from a small number of prospective secondary mathematics teachers withan array of experiences in mathematics and pedagogy, a case study was deemedthe most appropriate methodology for investigating the complexities of the roleadvanced mathematical knowledge plays in mathematical knowledge for teaching.In terms of data collection methods, the semi-structured research interview (Kvale,1996) was chosen as the primary source of data collection. The following sectionswill argue for the use of case study and the interview as research methods for thisstudy.3.1.1 Case StudyCase study is a flexible research methodology that provides researchers with a richpicture of a phenomena and can be “characterized as being particularistic, descrip-tive, and heuristic" (Merriam, 1998, p. 29). The definition of what necessarilyconstitutes a case study varies from author to author (Stoecker, 1991), but somekey features remain, including that the heart of any case study is the case. Milesand Huberman (1994) define the case as existing within the bounds of a particularcontext; that is, the case is a real-life phenomenon, which manifests itself in a re-stricted domain. For example, studying the general population’s understanding offractions does not constitute a valid case. Firstly, there is no limit in terms of whocan participate. The sample would need to be condensed to a specific person orgroup of people with a unique characteristic (i.e. grade 4 mathematics teachers).52Secondly, the topic of fractions is far too broad; the subject should be condensedto a more concrete problem - such as division of fractions - for it to align with theparticularistic nature of case study research.After defining the case, the flexibility of case study allows the researcher toengage with multiple modes of data collection and to cross reference data from thevarious sources. As Merriam (1998) remarks, most qualitative studies in educationutilize one, at best two, of the three widely used methods of qualitative data col-lection - interview, observation, and document analysis. Researchers employing acase study methodology, however, often engage with all three of these modes ofdata collection and triangulate resultant data to converge on research results andtheoretical propositions (Yin, 2013).The case study methodology has been widely used in the field of mathematicseducation research, particularly in the context of prospective secondary mathemat-ics teachers. Conner et al. (2011) conducted a case study with six prospectivesecondary mathematics teachers and investigated the change in their beliefs aboutmathematical reasoning and proof over a two-semester course sequence. Theircase study utilized survey, interview, observation, and written work to come to arich description of changes in the student teachers’ belief systems. Kinuthia et al.(2010) investigated pre-service teachers’ development of the use of technology inthe mathematics classroom by conducting a qualitative case study in which theytriangulated focus group interviews and various reflections by the student teach-ers over the course of a technology integration class. On a more quantitative note,Buchholtz et al. (2008) examined prospective secondary mathematics teachers’ ad-53vanced mathematics knowledge at universities in Germany, Hong Kong, China(Hangzhou), and South Korea through a case study. Their study was quantitativein nature, but based off their data, they suggest that prospective teachers are unableto connect advanced knowledge to the secondary curriculum.In qualitative education research, where many research studies are used as jus-tification for intervention and changes of existing education programs, the credibil-ity of research must be examined in some way. Unlike quantitative research wherecredibility is associated with the appropriate use of statistical tests, the nature ofqualitative research calls for different criteria to judge trustworthiness, credibility,confirmability, and data dependability (Yin, 2013, p. 45); namely external validity,internal validity, and reliability. The issues of internal validity, external validity,and reliability in qualitative research are key criteria in evaluating the rigour andtrustworthiness of a case study. These constructs will be addressed within the con-text of this study in Section 3.5.3.1.2 Interviews as DataHow might one come to understand another’s thought, experience, story, or cul-ture? To interview is to question. It is the process of asking questions, aimed at adeeper understanding of the interviewee and the topic at hand. How one goes aboutasking questions, acquiring and interpreting answers is a question of method andmay take a wide variety of forms. The flexibility that the interview offers as a modeof inquiry positions it as one of the hallmark research methods within social scienceresearch. Holstein and Gubrium (2000) approximate that nearly ninety percent ofpublished social science research utilizes the interview in some form. The technol-54ogy of the twenty-first century has made it easier than ever for researchers to utilizeinterview data. These technologies include: audio and video recorders which al-low researchers to revisit their conversations; transcription machines which greatlyreduce days of labour into a mere hours; and coding programs that are able to effi-ciently manage and analyze massive amounts of data.Despite these technological advances, the interview itself remains the same; itis a conversation that leads to an “inner view" of the respondent (Chirban, 1996;Kvale, 1996; Kvale and Brinkmann, 2009). Interviews serve as a method to under-stand the other: the “hows," “whats," and “whys" of their lives. How social scienceresearch interviewers come to understand these constructs, as we will see, is non-uniform. The methods and practices of researchers who utilize the interview asa research method depends greatly on epistemological commitments, the researchquestion, and context. Thus, as a researcher engaging with interview as a potentialresearch method, I found myself engulfed in a massive body of methodologicalliterature.As Kvale (1996) asks in the opening of his book, “if you want to know howpeople understand their world and their life, why not talk with them?" (Kvale, 1996,p. 1) The research interview is a conversation (Burgess, 2003; Lofland and Lofland,1984) where participants elaborate on their life experiences, in their own words.For the researcher, the purpose of the interview is to inquire about the perspectives,or views, of an individual. Kvale and Brinkmann (2009) provide two metaphors forhow a researcher might conduct an interview: that of a miner or a traveler. Withinthe traveler metaphor, the interview is a journey where the researcher wanders,55engages in conversation, and perhaps follows a method through their exploration.In contrast, the miner arrives to the interview with a defined goal in mind. Theconversation in the interview is directed toward uncovering knowledge which isembedded within the participant.3.1.3 Variations on the InterviewVariations on the ways in which interviews are conducted vary across disciplinesand epistemological positions. Stemming from modernist social science tradition,the structured interview is an interview in which all questions are predeterminedby the researcher, both in terms of wording and order; multiple respondents willreceive the same questions in the same order (Clifford et al., 2016). Fontana andProkos (2007) remark that the structured interview requires that the interviewer“play a neutral role, never interjecting their opinion of a respondent’s answer"(Fontana and Prokos, 2007, p. 20). Thus, structured interviews are intended forobtaining an objective account of another’s experience, borrowing from the rig-orous practices of the scientific method. On the opposite end of the spectrum isthe unstructured interview. With origins in ethnographic methods (Bruner, 1986),the unstructured interview sees the interviewee as a narrator of their experienceand life history (Sandelowski, 1991). This type of interview gives the intervieweethe ability to adjust the direction of the interview, elaborating on points that are ofsignificance to them through stories that need not follow a particular progression(Denzin, 2001).Someplace in between these two lies the semistructured interview, which isarguably the most common variation of the interview among researchers in the56social sciences (Kvale and Brinkmann, 2009). Similar to the structured interview,some questions may be prepared and tested ahead of implementation, but unlike thestructured interview, there is the freedom for the interviewer to probe and exploreresponses. The semistructured interview is subjective; different respondents willprovide the interviewer with different responses, thus altering the overall courseof the interview. Many authors see the subjective nature as a benefit, rather thanobstacle (Ginsburg, 1997; Kvale and Brinkmann, 2009; Qu and Dumay, 2011),arguing that variations in responses help one to better understand the complexitiesof a particular phenomena.3.1.4 Debates on the Interview as DataResearchers who associate themselves to a postmodern school of thought oftendismiss the structured interview, claiming it as an ineffective research tool. Forexample, in his critique of the research interview from a postmodern standpoint,Scheurich (1997) states the following:The researcher uses the dead, decontextualized monads of meaning,the tightly boundaried containers, the numbing objectifications, to con-struct generalizations which are, in the modernist dream, used to pre-dict, control and reform, as in educational practice. (Scheurich, 1997,p. 64)For Scheurich (1997), the acknowledgement of subjectivity should be at theforefront of research interviewing, not the prospect of generalizability. Proponentsof the structured interview have a similar distaste for the unstructured interview ofthe postmodern tradition. Critics of postmodernism remark that it does not lead to57any “true" understanding (Spiro, 1996) and that its effects are “relativism; nihilism;solipsism; fragmentation, pathos, hopelessness" (Hill et al., 2002, p. 5). Since thepostmodern interview does not provide the scientific certainty modernists desire inorder to justify changes in policy and practice, it is viewed as a fruitless researchtool.The critiques of those who thoroughly oppose unstructured interviews andthose who oppose structured interviews are, in fact, very similar. Each group dis-misses research on the basis of an asserted a priori philosophical position. Criticismof these methods and their underlying philosophies are based off characteristicsthat the philosophies under scrutiny simply do not have. As Rosenau (1991) re-marks, the anti-theoretical position that postmodernism foundations itself on, is infact a theoretical stand. Further, Eagleton (1996) observes that “for all its vauntedopenness to the Other, postmodernism can be quite as exclusive and as censori-ous as the orthodoxies is opposes" (Eagleton, 1996, p. 26). Philosophical beliefsaside, there are obvious benefits to utilizing structured interviews in social scienceresearch. Similarly, there is a time, a place, and a purpose for postmodern researchinterviewing. What is troubling is the explicit rejection of valuable research basedon epistemological beliefs. I do not reject the postmodernist interview, nor do Ireject the structured survey interview. Rather, I perceive these variations on theinterview as having their place in the social science literature and research commu-nity. The decision of which variant to use should be dependent on the investigator’sresearch question and what they wish to uncover in their work.These criticisms are from researchers who are proponents of the interview as a58research method. Despite its widespread use in social science research, criticismsof the interview as a research method still ensue. Critics often claim they are “notscientific, but only [reflect] common sense" (Kvale, 1996, p. 285). I concur withKvale (1996) that perhaps one must carefully define “science," before definingsomething as “not science." Merriam Webster has multiple definitions of science;“the state of knowing: knowledge as distinguished from ignorance or misunder-standing," “a department of systematized knowledge as an object of study," “asystem of knowledge concerned with the physical world and its phenomena" (Sci-ence, 2019). The definition we take to mean “scientific" will change whether thequalitative research interview is “scientific" or not. Unfortunately, just as we havemultiple definitions of science, terms such as “knowledge" and “systematized" mayalso have alternative meanings. With so many variations on what constitutes sci-ence, it does not seem as though one can state that the qualitative interview is notscience. As Kvale (1996) puts it so well, “the automatic rejection of qualitativeresearch as unscientific reflects a specific, limited conception of science, insteadof seeing science as the topic of continual clarification and discussion" (Kvale,1996, p. 61). The qualitative research interview is capable of providing system-atized knowledge, provided that the researcher has a rigorous understanding of theinterview as a research method, taking into consideration the issues of validity andreliability.Scientific practices, assumptions, inferences, and mathematics must be basedon solid arguments which are logically sound. The research community of thephysical sciences has an understanding of what constitutes “good science." Namely,that it is valid, reliable, and generalizable. The shift of this holy trinity of scien-59tific research to social science research, particularly to the method of interview-ing in qualitative social science research, is one of the largest battle grounds forresearchers in the field. Some qualitative researchers completely dismiss the con-structs of validity, reliability, and generalizability as outdated modernist constructsthat are irrelevant to the study of human experience and psychology (Constas,1998; Scheurich, 1997), while others argue that there exists a definitive reality,thus valuing these constructs (Denzin and Lincoln, 2000). However, these argu-ments are once again based on asserted a priori philosophical beliefs and defi-nitions of the constructs in question. A characteristic of research that these twocamps can agree on, however, is the mutual hope that their research will build theunderstanding of the phenomenon of study; that their research is trusted and willinform future research. Regardless of your philosophical stance, there are waysto ensure that qualitative interview research responds to validity, reliability, andgeneralizability.3.1.5 SummaryRegardless of the methodological and philosophical debates surrounding the in-terview as a research method, the interview still maintains its position as a hall-mark research method in the social sciences. The research interview combinesstructure with flexibility (Legard et al., 2003), positioning it as a method that isavailable to any researcher, regardless of their epistemological position. Whetherone’s research relies on descriptive statistics and mathematical analysis (Schwarzet al., 2008) or on narrative and poetic responses (Richardson, 2000), there existsa variation of the interview that will both complement and enhance one’s research.Through interviews, we can come to discover not only the phenomena in question,60but connections related to that phenomena in various contexts.This thesis explores the impact that post-secondary mathematical content knowl-edge has on prospective secondary mathematics teachers’ understanding of sec-ondary mathematics concepts. Although it will be valuable to understand whatmathematical content knowledge the participants in my study have, my interestslie in how teachers relate their advanced mathematical knowledge to the secondarycurriculum. I hope to go beyond mere content, and explore the meaning that myparticipants have constructed. Each participant will have unique educational ex-periences in mathematics and pedagogy, resulting in mathematical knowledge forteaching that is specific to them. For this research, the semi-structured interviewoffers itself as a methodological tool that will allow me to tackle and explore thesubtleties behind mathematical understanding, to treat each participant as an in-dividual with their own unique experience, and as Spradley (1979) noted, to be alearner; to listen and learn of pedagogically powerful mathematics that I myself,may not have thought of before.3.2 SettingData for this study was collected from five prospective secondary mathematicsteachers. At the time of data collection, each participant was enrolled in the Bach-elor of Education program in Secondary Mathematics at the Vancouver campusof the University of British Columbia. As per the requirements of the SecondaryMathematics B.Ed. program at UBC, all participants obtained degrees in mathe-matics, or a related subject. More specifically, the UBC Teacher Education Of-fice states that potential teacher candidates have at least 30 credits of mathematics61coursework, 18 of which must have been at the senior level (3rd year or higher).Furthermore, there is a “breadth requirement" for potential teacher candidates, stat-ing that a candidate must have at least three credits from at least three of five“core" topics, those being algebra, probability/statistics, geometry, discrete math,and number theory. The participants that were interviewed in this study had math-ematical coursework and experience well beyond the secondary curriculum.Each participant provided their undergraduate transcripts with coursework takenand grades obtained. The subject GPA of each participant was calculated out of100. This was done by multiplying the number of credits for the courses taken bythe grade obtained and summing over all mathematics courses taken. This numberwas then divided by the total number of credits taken, yielding a score out of 100.3.2.1 ParticipantsIn this section, I will outline the backgrounds and experiences of each participantin the study. The names used in this thesis are pseudonyms, to protect the identitiesof the participants. All participants completed their undergraduate mathematicalcoursework at large, Canadian research institutions.Taylor had their Bachelor of Science in Mathematics. They completed theirBSc in 2017 and transitioned immediately to the Bachelor of Education programin Secondary Mathematics at UBC in September of 2017. In their BSc, Taylor took62 credits of mathematics coursework, with a final subject GPA of 79.6.Jaime had their Bachelor of Engineering in Engineering Physics. Their BEng62was completed in 2000, making them the participant farthest removed from post-secondary mathematics coursework. They also completed a Master’s of BusinessAdministration, prior to their entrance into the UBC BEd program. Jamie took 28credits in mathematics coursework, with a subject GPA of 72.1.Bailey completed their Bachelor of Arts with a double major in Mathematicsand English in 2015. Their degree included 45 credits of mathematics coursework,with a subject GPA of 88.7. Bailey had extensive TA experience in mathematicsand computer science, prior to entering the teacher education program. Bailey hadEnglish and Mathematics as subject specializations in their BEd.Adrian completed a Bachelor of Science with Honours in Mathematical Physicsin 2015 and a Master’s degree in Theoretical Physics in 2017, before enteringthe Bachelor of Education program, with teachable subjects of mathematics andphysics. In their BSc, Adrian completed 39 credits of mathematics coursework,with a GPA of 95. Even before starting their BSc, Adrian was committed to theidea of a career in academia as a research physicist. However, throughout theirstudies, Adrian was involved in a number of activities involving teaching and learn-ing of mathematics and science, including extensive tutoring, TA experience, andsummer science camps for youth. It was during their Master’s degree that Adriandecided to pursue a career in education, after realizing their passion was in teach-ing and learning, rather than academic physics research. Adrian hopes to teachmathematics and physics for International Baccalaureate (IB) students, and was inthe IB cohort throughout their BEd studies.63Casey completed their Bachelor of Arts in Mathematics in 2013, prior to enter-ing the Bachelor of Education program. In their degree, Casey completed 61 creditsof mathematics coursework, with a GPA of 72.2. Additionally, Casey audited somegraduate level mathematics courses offered through their institution. Casey statedan interest and specialization in algebraic structures, which they hoped to bringinto their teaching.The experience of the participants in this study is summarized in Table 3.1.Participant Degrees Math Credits Math GPATaylor BSc Math 62 80Jaime BEng Physics 28 72Bailey BA Math, English 45 89AdrianBSc Mathematical PhysicsMSc Physics39 95Casey BA Math 61 72Table 3.1: Summary of participant backgrounds643.3 Data CollectionFor the data collection, all five participants provided the researcher with their un-dergraduate academic transcripts. These transcripts provided the researcher withbackground knowledge of the participants’ mathematical coursework. Addition-ally, each participant filled out a short survey outlining their relevant work experi-ence. Once the researcher obtained this information, one-on-one semi-structuredinterviews (Seidman, 2013) with the participants and the researcher were sched-uled. The interviews were conducted in a conference room in the Faculty of Ed-ucation building, with the interviewer and participant sitting next to each other.Scrap paper, pens, and pencils were provided to each participant. Interviews audiorecorded and any written work produced was scanned. The interviews consisted ofthree parts, each of which will be elaborated below.3.3.1 Experiences in Post-Secondary MathematicsTo begin the interview, participants were asked to reflect on their post-secondarymathematics courses and the role this knowledge plays in their pedagogical prac-tice (Zazkis and Leikin, 2010). In this portion of the data collection, I was inter-ested in understanding whether participants perceived their advanced courses inmathematics to have an impact on their practice as secondary mathematics teach-ers. Leading questions in this portion are included below and were presented orallyto the participant:• How do you conceptualize mathematics as a field of study?• Do you think it is important for secondary mathematics teachers to knowadvanced mathematics?65• What roles do you see those four years of learning playing in your next yearof study?• Do you see your post-secondary degree in mathematics having an impact onyour teaching?3.3.2 Connections Between Secondary and Post-SecondaryMathematicsThe second portion of the interview was a task-based, semi-structured interview(Goldin, 2000), intended to explore the ways in which participants connected theiradvanced mathematical knowledge to problems in the secondary curriculum. Par-ticipants were provided with a list of seven tasks written on a sheet of paper. Par-ticipants were then asked to choose four of the tasks to discuss with the researcher.The tasks I chose to engage participants with are inspired by previous studiesof secondary mathematics teachers’ PCK and MKT which have drawn from ad-vanced mathematical knowledge. In taking Silverman and Thompson (2008) asa framework for the development of MKT, finding tasks that had the potential toreveal KDUs were chosen. As a gentle reminder for the reader, recall that a KDUis defined to be an understanding which transforms the way a learner understands aparticular mathematical concept, allowing them to work with the concept in waysunfamiliar to them previously. In the context of this study, I hoped for partici-pants to share post-secondary mathematical understandings that transformed theirunderstanding of the secondary curriculum. The tasks used in this study needed tohave the potential to reveal participants’ understandings that connected secondarymathematical knowledge to advanced mathematical knowledge. Furthermore, I66wanted to use tasks which had been used or examined in previous research stud-ies on teachers’ secondary and/or post-secondary mathematics content knowledge.Thus, I converged on problems that readily bridged these two content levels.For example, Task C on the factorization of polynomials was chosen becausepolynomials are a major concept of the secondary curriculum, as well as the studyof abstract algebra in post-secondary mathematics. This task had the potential toreveal KDUs which bridged post-secondary abstract algebra and secondary schoolalgebra. A similar argument can be made for Task A. Indeed, exponents and ex-ponential functions are a major topic of discussion in secondary mathematics, butthe proof that this notion can be extended to any “type” of power is not examineduntil a course in real analysis. Such a construction requires an understanding ofirrational numbers as a limit of a sequence of rationals, an understanding that al-lows the learner to work with irrational numbers in ways they were unable to whenirrationals were simply “not rational.”In an effort to dig deeper into the understandings and mathematical knowledgefor teaching of the participants, leading questions for each of the tasks are included.Note that many of the questions begin with “what advanced mathematics is relevanthere?" This is to recognize whether the participants have developed a personallypowerful understanding (KDU) of the particular secondary mathematics concept.Then, the question of “how could you make this relevant/accessible to your stu-dents?" explored the participants’ efforts to make a KDU pedagogically powerful;that is, to reflect on their advanced mathematical knowledge and to recognize itsrelevance in the secondary curriculum. This is precisely the process of reflective67abstraction: to reflect on and connect knowledge constructed at various levels ofabstraction. Initially, participants were asked how they would respond to the sit-uation. If the response did not clearly make a connection to their post-secondarymathematics work, teachers were asked, quite explicitly, to make connections be-tween advanced mathematics and the secondary curriculum. This approach waschosen to ensure that participants were clear in the types of mathematical connec-tions that I was interested in. Below is the list of mathematical tasks provided tothe participants to choose from, along with follow-up questions. The reader maynote that each task was inspired by previous work in the field. In comparison to thepast work, the focus of the discussion was on the connections participants madebetween secondary and post-secondary mathematics in the context of the giventasks.• Task A: Your students are confused as to why we can define and calculate2√3.– How would you respond to your students? What mathematics is rele-vant here? How can you make it accessible?– Inspired by work from Wasserman et al. (2017)• Task B: A student is working through a problem and asks if 0.999 . . .= 1.– How would you respond? What knowledge from your post-secondarymath classes could you use to explain it? How could you make yourresponse more accessible to your student?– Inspired by work from Krauss et al. (2008)68• Task C: You are teaching a week on factoring polynomials and you havefound that your students are struggling to recognize when they should stoptrying to factor.– How would you respond? Can you think of anything you learned inyour post-secondary mathematics courses that might help your stu-dents? How could you make it accessible? Is there anything froman advanced course that might provide motivation for this topic?– Inspired from the researcher’s teaching experience.• Task D: Your students are learning about inverse functions. What would youinclude in your lesson plan?– What knowledge from your post-secondary mathematics work mightbe relevant in this context? How could you make it accessible/useful inyour pedagogy?– Inspired by work from Leung et al. (2016) and Zazkis and Kontorovich(2016)• Task E: You are teaching a week on symmetry to your students. What wouldyou include in your lesson plan?– What knowledge from your post-secondary mathematics courses mightbe relevant here? How could you make it accessible?– Inspired by Sultan and Artzt (2010)• Task F: You have been teaching a unit on quadratic functions for a fewweeks and one of your students asks you why they need to know about them.69– How would you respond? Did you talk about quadratic functions inany of your university mathematics courses? Could you make it acces-sible/useful in your pedagogy of this topic?– Inspired by the researcher’s own teaching experience.• Task G: A student is confused as to whether √2 is an irrational or rationalnumber, especially after realizing it is the length of the diagonal of a squareof side length 1.– How would you respond? In what contexts did rational and irrationalnumbers appear in your university mathematics courses? How wouldyou use that knowledge in your teaching?– Inspired by Sirotic and Zazkis (2007)I have summarized the tasks which participants chose to engage with in Table3.2. As noted above, participants were asked to choose four tasks with which toengage. However, Adrian, Bailey, and Taylor engaged with five of the tasks. In thecase of Adrian, this was due to their desire to continue the conversation, whereaswith Bailey and Taylor, the discussion naturally emerged from the discussion inTask G.70ParticipantTask A(2√3)Task B(0.999 . . .)Task C(Factoring)Task D(Inverses)Task E(Symmetry)Task F(Quadratics)Task G(√2)TaylorJaimeBaileyAdrianCaseyTable 3.2: Tasks chosen by participants3.3.3 Secondary Mathematics InstrumentThe third, and final portion of the interview consisted of a mathematical instrumentcontaining problems drawn from the BC curriculum which correspond to tasksfrom the second portion of the interview. As the literature suggests a degree inmathematics does not necessarily imply a thorough understanding of secondarymathematics (Cofer, 2015; Even, 1993; Stump, 1999), this portion of this interviewwas intended to respond to such claims. With only a few exceptions, participantssuccessfully responded to all questions in the instrument. Bearing this in mind andafter reviewing participants’ responses to these problems, I concluded that theirresponses did not substantially contribute to the emergent themes of connectionsbetween secondary and post-secondary mathematics observed through the task-based interviews. Thus, this portion of the data was withdrawn from my analysis.713.4 Data AnalysisThe primary mode of data analysis was the transcription of the audio-recordedinterviews. After each interview had been completed, the audio-recordings weretranscribed. Once all interviews were complete, the portions of the transcripts as-sociated to each task were grouped. The grouped transcripts were analyzed throughan emergent coding process. Key phrases, adjectives, and statements from the par-ticipants were underlined and associated to a code which summarized the themeof the underlined portion of the transcript. In coding, I was particularly interestedin statements that shed light on what the participants understood mathematically inthis task, both at the secondary and post secondary level. I was also concerned withparticipants’ remarks that connected mathematical content to pedagogical choicesin their future teaching. For example, in the task exploring inverse functions, aparticipant’s statement on the importance of domain restrictions in teaching inversefunctions would be associated to the code domain restrictions, while a participant’scomments on the general nature of inverses being dependent on operations wouldbe given the code operations. This coding procedure was completed for all thetasks, as well as the initial interview on perceptions.Common codes were grouped together into emergent themes that are represen-tative of participants’ understandings in relation to the relevant task or question.This was done for both the one-on-one interview exploring participants’ percep-tions of their degrees and for the task-based interviews. Since many participantschose to write out some mathematics during their task-based interviews, the writtenwork produced was used to support the dialogue from the task-based interviews.72Reference to relevant written work was documented in the coded transcripts. Fi-nally, by utilizing the academic transcripts provided by participants, statementswere cross-matched to the advanced mathematical coursework they had taken. Thiscross-matching provided context for the origin of mathematical concepts discussed.The focus of this study is on the links made between secondary and post-secondary mathematics content. In Chapters 1 and 2, I argue that both universitymathematics courses and mathematics methods courses play a role in developingsuch knowledge. With respect to data analysis, the MKT framework of Silvermanand Thompson (2008) and associated framework of KDUs in Simon (2006) wasused to focus my analysis on expressed understandings of the participants, partic-ularly ones which drew upon knowledge from their advanced degrees. Silvermanand Thompson (2008) centre the development of MKT in terms of mathemati-cal understandings with pedagogical potential, becoming ones with pedagogicalpower.In my coding and development of emergent themes, I was particularly inter-ested in how participants’ mathematical understandings bridged secondary andpost-secondary content and pedagogical practice. Such comprehension would beexamples of personally powerful understandings with the power to change the waythey teach secondary mathematics. In Chapter 2, I explain that reflective abstrac-tion may be the mechanism for the development of such knowledge. This theoryallowed me to contextualize the expressed experiences of participants’ own mathe-matics education and whether these experiences were conducive to the integrationof advanced knowledge into their MKT.73The emergent themes are explored in more detail in the following chapters, butTable 3.3 on the following page summarizes the emergent codes and themes fromthe tasks. In this table, I have provided all codes that emerged from the transcripts,as well as the themes which were generated from the codes. The rich descriptionsof the emergent themes of participants’ understandings and mathematical knowl-edge for teaching will be examined and detailed in Chapters 4, 5, 6, and 7.74Task Codes ThemesPerceptionsprocedural, theory, problem solving,reasoning, facts, skepticism,connections, application, elegant,content, concepts, pedagogy,real world, proof, practical,pragmatic, support, enrichment,exposure, rote, understand,rigour, pure, applied, abstracttwo facets of mathematics,content versus beliefs and valuesproblem solving skillspragmatic teacher educationdisconnect of post-secondaryTask A(Value of 2√3)rational, rigour, faith,continuity, exact, series,infinity, inverse,concrete, definecomplexity of exponents,approximation vs. exact,existence versus definitionTask B(Value of 0.999 . . .)memorized, thirds, rational,sequence, convergence, intuition,approaching, limit, concrete,asymptote, faith, cardinalityreliance on memorized proof,concrete conceptualizationslimits and infinityTask C(Factoring)recipe, approach, graphs,factoring, quadratic, roots,complex, cubics, proof,application, Galois, zero,abstract algebra, calculusrelating graphing & factoring,extend quadratic/cubic behaviour,complex factors & roots,disconnect from abstract algebraTask D(Inverse functions)reciprocal, operation, domain,sine, opposite, exponents,application, restriction, test,function, mapping, reflectionInverse w.r.t operation,domain restrictionsreflection vs. functional inverse,“undoing” of functionsTask E(Symmetry)picture, nature, reflection,rotation, physics, tactile,visual, graphs, geometry,groups, triangle, inversesymmetry, nature and art,types of symmetry,geometric symmetry,example drivenTask F(Quadratics)application, curve, formulafactor, basic, complex, vertex,roots, interest, memorize,velocity, extrema, concrete,meaning, visual, graph, bridge,physics, model, accessiblerelating graphing & factoring,beginning access point for learning,limited applications impact interestTask G(Irrationality of√2)memorize, proof, contradiction,logic, confidence, intuition,definition, zoom, limitnumericallyproof in secondary math,procedural logic,decimal versus exactTable 3.3: Interview transcript codes and themes75As stated above, common codes were grouped into themes. However, codes fora particular task may be grouped into multiple themes. To help the reader betterunderstand the coding and thematic generation, consider the following quote fromAdrian, in response to Task D:Adrian: Yeah, these are reciprocal functions, which is not what we aretalking about, but some people call them inverses because youare taking the, technically this is the multiplicative inverse of thefunction, but that’s not what we are talking about. What we aretalking about is inverse functions, which is taking the opposite ofthe function.The codes reciprocal, operation, and opposite were used to interpret this quote.While both codes of reciprocal and opposite explicitly appeared in this quote, thecode operation was used to refer to Adrian’s mention of multiplicative inverses,versus functional ones. These codes were grouped together into the relevant themeof recognizing inverses with respect to operations, as well as the theme of “undo-ing” of a function, which Adrian refers to as the “opposite.”As a second example of the coding process, consider the following transcriptbetween the interviewer and Bailey:Interviewer: Can you think of a limit that would be equal to one but also havesomething to do with 0.999 . . .?Bailey: Like we would want an equation that is approaching one from the76bottom, so that would be a 1 over x plus something. Here’s mygraph. I’m rusty on all my graphs look like. It would just be 1over x plus 1. Would that be right? Yeah.Interviewer: And so as we take this limit we are getting one, but what doesthis have to do with point 999?Bailey: Because 1 minus a tiny tiny piece is .9999, right? So that asymp-tote is approaching it.The codes limits, approaching, and asymptote were used to understand howBailey understood the equality of 0.999 . . . = 1. The appearance of these codes inBailey’s execution of this task revealed that the connections and understandingsBailey had made between secondary and post-secondary mathematical knowledgeexisted in the content domain of Calculus and Real Analysis. However, this tran-script also reveals that Bailey may not have a personally powerful understanding(KDU) of limits in the context of number, since the use of a continuous function isnot appropriate for the justification of 0.999 . . . = 1. This transcript suggests thatBailey may have benefited from further learning experiences that would help builda more robust understanding of number by drawing upon their advanced contentknowledge and engaging in the process of reflective abstraction.Further, a single code may fit into multiple themes. This can be seen throughBailey’s quote from Task C, where they stated:77Bailey: You could probably factor if the coefficients were complex be-cause you can have i2 and you can get some negative ones. Somy whole argument of being able to get zero becomes untrue.x2 + 1 has no roots in the real plane, but it does in the complexplane.The codes zero, roots, and complex emerged from this quote and contributedto the theme of relating graphing and factoring, as well as the relationship betweencomplex factors and real-valued intercepts. All of the themes which are relatedto the tasks are representative of the expressed understandings and mathematicalknowledge for teaching of the participants, as individuals and as a group.A second phase of study was performed on the emergent themes. Since the goalof this research is to better understand the role of advanced mathematics knowledgefor future teachers, I wanted to have a better understanding of how these themes fitin the framework of the academic journey to become a secondary teacher. Commonthemes from all tasks were gathered together and associated to either “advancedmathematics” or “teacher education.” The intent of this coding was to help betterunderstand where the development of MKT could be supported. Rather than plac-ing the focus on the knowledge, understanding, and experience of the participants,the study of the themes in the context of a teacher’s academic journey will help putinto perspective what mathematicians and mathematics teacher educators can doto enhance and build upon the mathematical understandings of their students. Weexamine this in Chapter 8.783.5 Validity and ReliabilityParticularly in qualitative education research, where many research studies are usedas justification for intervention and changes of existing education programs, thecredibility of research must be examined in some way. Unlike quantitative re-search where credibility is associated with the appropriate use of statistical tests,the nature of qualitative research calls for different criteria to judge trustworthi-ness, credibility, confirmability, and data dependability (Yin, 2013, p. 45); namelyexternal validity, internal validity, and reliability. The issues of internal validity,external validity, and reliability in qualitative research are key criteria in evaluatingthe rigour and trustworthiness of a case study.The concept of external validity aligns with generalizability; that is, to whatextent can the results of the study be applied to other studies? The issue of gen-eralizability has “plagued qualitative researchers for some time” (Merriam, 1998,p. 207). To reconcile this matter, I find it useful to reiterate the purpose of casestudy. For Merriam (1998) and myself, the priority of case study is the case. Myintent as a case study researcher is to understand my case in depth, not necessarilyto generalize to what is true of many. Although I research “the particular” in casestudy, if research of a similar case is conducted, the results obtained from my workmay serve as a flag for themes in their data.Guba (1981) suggests the analog of external validity in a qualitative paradigmmay be considered to be “transferability.” He offers thick description as a modeto “permit comparison of this context to other possible contexts to which transfer79might be contemplated” (Guba, 1981, p. 86). The utilization of multiple sourcesof data in this study is conducive to thick description, as each of my data sourcescontributes uniquely to the individual participants. Furthermore, my sampling ofparticipants from the teacher education program at UBC is meant to maximizeresultant data of the study. Despite representing only a small fraction of their co-hort, their varied experience in advanced mathematical coursework yields multipleperspectives for the ways in which future teachers make connections between sec-ondary and post-secondary mathematics.Validity is not a construct only relevant to the final results or method, but aconstruct that consistently informs the research interviewer. During the interviewportion of the proposed study, the researcher should be attentive to whether thequestions and responses in the interview are guiding the respondent in a particulardirection. To address this, I followed Ginsburg (1997) and his multi-phase ap-proach in clinical interviews. He advises researchers to return to the same questionat various moments in the interview, phrasing the question in a different way eachtime. In this sense, the researcher can cross-check statements made by the respon-dent, strengthening the viability of any conclusions made by the researcher. Thistechnique was employed throughout the interview process, to ensure that I was notmisinterpreting participants’ remarks.Thomas Schwandt defines “triangulation” as a means of “checking the integrityof the inferences one draws” (Schwandt, 2007, p.298). Triangulation will be theprimary mode of increasing validity and reliability in this study, and will be donethrough cross referencing data from the various interview transcripts The utiliza-80tion of triangulation, as well as detailed accounts of methods throughout the re-search process are ways in which the I can assure that results align with the datacollected. By implementing multiple research methods, I will obtain more data tocompare and contrast, to support inferences, and to enrich the research.3.6 SummaryIn this chapter, I have detailed the methodological considerations of this study. Idiscussed the details of the study’s setting, participants, data collection and analy-sis. Finally, I described the ways in which I handled the issues of generalizability,validity, and reliability. After the completion of this chapter, the groundwork formy study is complete. As a reminder to the reader, the work that follows aims toanswer the questions:1. What do prospective secondary mathematics teachers perceive as the role oftheir advanced mathematics knowledge in their development as teachers?2. In what ways do prospective secondary mathematics teachers relate advancedmathematics knowledge to a mathematics concept in the secondary curricu-lum?In the following chapter, I begin the presentation of results through partici-pants’ perceptions of the role of advanced mathematics knowledge in their growthas a teacher.81Chapter 4Perceptions of the Role ofAdvanced Mathematics inPedagogical DevelopmentIn this chapter, we discuss the portion of the interview exploring participants’ per-ceptions of the role of post-secondary mathematics education in their growth assecondary mathematics teachers. The primary prompts for this portion of the in-terviews included:1. How do you perceive mathematics as a field of study?2. Is a degree in mathematics needed to teach secondary mathematics?3. What do you perceive as the role of advanced mathematics knowledge inyour growth as a teacher?4. In what ways is your advanced knowledge being drawn upon in your teacher82education?4.1 Perceptions of MathematicsTo begin the interview with the participants, I wanted to gather a sense of the per-ceptions and values participants held about mathematics as a field of study. Indeed,as remarked in the literature, teachers’ perceptions of mathematics can play a sig-nificant role in the pedagogy of teachers. This first interview question of “how doyou perceive mathematics as a field of study?” was intended to connect partici-pants’ responses to later interview questions and to further understand their futuregoals in teaching mathematics. Overall, participants’ responses to this questionrevealed two distinct conceptions of mathematics: as a tool for understanding theworld and as a pure, abstract, self-contained knowledge system. This distinctionwas expressed by all participants. Additionally, the relationship between mathe-matics and problem solving was a common theme expressed by four out of five ofthe participants.All participants expressed mathematics as having two facets. This was ex-pressed succinctly by Jaime:Jaime: I see math as kind of having two sides to it. Part of it is definitelya way of explaining the world. And definitely on the science side,it’s a way to explain how things work and kind of simplify themto build a model. The other side though is more pure, abstractmath, where it’s the realm of a lot of logic. This idea will leadto this idea which may not have an obvious or maybe any direct83relevance to the actual world. So I kind of see those two sides ofit.Jaime and Bailey both expressed mathematics having a characteristic of beauty.This beauty was conceptualized as a product of the abstract and independent natureof mathematics. As Bailey stated, mathematics can be conceptualized as “a systemwe have constructed that explains things outside of itself” and even with this char-acteristic, it additionally “has perfection within the system.” Beauty was conceptu-alized by Jaime through a relationship between mathematics and art. Jaime viewedthe act of doing mathematics as an artistic endeavour, where creativity was a keycomponent to success. However, they noted the complex nature of conceptualizingmathematics as art, in that mathematicians are governed by “rules” different thanthat of a painter. Jaime later shifted to conceptualizing creativity in mathematics assimilar to creativity in music, in the sense that “everyone can respect someone whocan freely improvise and be creative on the piano, but it’s a lot of work to get there.”Jaime came to appreciate this relationship later in their undergraduate work,remarking that through their education in engineering, “it got to the point in myeducation where physics and math were inseparable because everything in physicswas explained through mathematics.” Jaime later conceptualized mathematics notjust as a way of describing, but as a self-contained puzzle, where various theo-ries and structures were brought together to uncover new ideas and solve unknownquestions. Casey brought forth a similar conception of mathematics as a way ofbuilding new structures and ideas out of existing ones, through the process of rigourand proof.84Although Taylor saw a distinction between applied and pure mathematics, theyremarked that one can not exist without the other. Indeed, Taylor mentioned in theirinterview that “mathematics is about understanding what is in the real world butalso in the abstract. Abstractly, you have to theorize everything and whether thattheory also fits in the physical world.” This was a unique statement from Taylor,with respect to other participants’ responses and points to the value of understand-ing both facets of mathematics. However, Taylor followed up this comment withanother, stating that sometime mathematics may exist without application to thephysical world and might exist as a “brain exercise.” This conception is in linewith other participants’ perspectives on mathematics as a self-contained knowl-edge system. Indeed, as Adrian remarked, “we have created rules in a space andwe want to see what those rules produce. It’s almost like a little game that we’veplayed, but with incredible, far reaching consequence with what we can do with it.”The use of mathematics as a tool to understand the physical world was valuedby Jaime and Adrian, who both had their undergraduate work in physics. Althoughboth of these participants saw value in the abstract side, their expertise broughtthem to value the applied side and looked forward to bringing this into their futureteaching. Adrian noted that “with my focus in physics, I looked a lot at math as away of describing things quantitatively and drawing out patterns and sort of seeingthe world in a very structured way.” Adrian looked forward to bringing their sci-entific and mathematical expertise into the classroom, so that future students couldsee value in mathematics for solving applied problems and understanding why suchproblems and questions are important. They hoped that in doing this, their number85one goal of making class “interesting, relevant, and engaging” for students couldbe achieved.Regardless of these two facets, all participants remarked on the relationshipbetween mathematics and problem solving. Participants saw value in an educationin mathematics for building skills in problem solving:Taylor: Mathematics is the study of the thinking process of logical think-ing.Bailey: Math is a way of working, with problems as a way of learning,rather than problems as a way to reinforce learning.With their undergraduate mathematics being focused in pure mathematics, Tay-lor, Bailey and Casey each saw value in mathematics as a way of building anddeveloping critical thinking skills through problem solving. As Casey noted, puremathematics distinguishes itself with a “purification in proof and theory.” Eachof these participants saw an intrinsic value in studying mathematics for the sakeof mathematics for their future students. Their hope was that the study of mathe-matics in a self-contained system would help their students learn “how to logicallysolve problems and work through things” even if it is not their intended field of fu-ture study, according to Bailey. This sentiment was echoed by Taylor who viewedthe learning of mathematics as an opportunity to develop reasoning and “sophisti-cated thinking skills” inside and outside of mathematics. With an increased focuson critical thinking in modern curricula, these remarks are not without merit.86Similar to their conceptions of mathematics, Adrian and Jaime took a moreapplication based approach to their values of mathematics and problem solving,viewing mathematics as a powerful tool to explain how the world works. Adrianmentioned the “Math Matters” movement and how powerful mathematics couldbe in helping students understand local and global issues affecting society today.Casey took a more research based approach to their response, viewing the power ofmathematics to “uncover or discover a problem that hasn’t been researched beforeand find a way to progress that problem up to a certain point in our field.” Thiscomment aligned well with Adrian’s hope of bringing in research and extensionprojects for their advanced students, so that they might be able to have an idea ofcurrent research questions in science.Overall, participants remarked that their undergraduate experience in mathe-matics changed their perspective on what constitutes mathematics. Bailey men-tioned that mathematics in high school seemed to be a “series of things” with anend and no purpose. Bailey felt that their university experience changed this per-spective and that they looked forward to bringing it into their future teaching. Theyhoped that their university experience would help students see that “if you thinkyou don’t like math, maybe you don’t like one part of math because there is somuch to it.” Taylor shared their struggle in shifting from thinking of mathematicsas a tool to thinking of it as a form of argumentation. This shift was difficult forTaylor in first and second year mathematics, reflecting on a feeling of “why can’t Ijust understand this?” As remarked above, Jaime experienced a similar shift, nowviewing mathematics in an artistic light.87These remarks from participants are an appropriate segue to the followingsection on participants’ perceptions of the role advanced mathematics knowledgeplays in their identity as a future secondary teacher. Even though all participantsperceived intrinsic value in learning mathematics, these sentiments are not echoedas strongly in the questions that follow.4.2 Role of Advanced Knowledge for TeachersIn this section, we will explore participants’ responses to questions 2 and 3; that is,what do participants perceive as the role of their advanced knowledge as a teacherand do they think advanced mathematics knowledge is important for secondaryteachers to have?Participants answered these questions with varied responses and degrees ofstrength in their beliefs. Overall, all participants expressed that a major degree inmathematics is not necessary to teach secondary mathematics. The view of whatextent of post-secondary mathematics training is necessary varied from participantto participant.Four participants expressed the value of post-secondary content expertise beingof value in the classroom. In particular, Bailey, Taylor, Adrian, and Casey statedthe importance of having content knowledge beyond their students’. As Baileymentioned, “you need more experience than where your grade 12 students are go-ing to be.” Adrian agreed with this remark from Bailey, stating “I think I couldget by with first-year university knowledge.” Adrian perceived this as important so88that “you’re more advanced than your students and have a perspective on where itcan take them in an academic sense.”The view of advanced knowledge being of value for building connections wasexpressed by all participants, except for Jaime. Adrian, Taylor, and Casey all sawvalue in post-secondary mathematics degrees. They each expressed that this ad-vanced knowledge is important for being able to field students’ questions, answer-ing students’ questions in different ways, having an understanding of conceptualbackground, and for being able provide context for what mathematics is on thehorizon.Taylor expressed a unique perspective on the value of a mathematics degree forsecondary teachers, in that it helped them to learn to think like a mathematician.Taylor: How are you sure that this statement is true? The humblenessof seeing the nature and making sophisticated thinking skills tohow much we do not know about the world in general. That’swhat mathematics taught me and I want students to know that sortof aspect of mathematics. Mathematicians don’t make randomstatements about things. They try to formulate a right questionand try to develop in a certain way that the question they pose ishelping the big question they originally posed.Taylor also mentioned the value of their mathematics education in buildingskepticism, reasoning skills, and knowing connections between different fields of89math. Similar to Taylor, Adrian saw great value in the ability to provide opportu-nities for enrichment. Adrian remarked that university specialization is of value inbuilding “research and extension projects” and giving advice to students who aregenuinely interested in mathematics and university studies.These remarks focus primarily on skills and practices learned through post-secondary studies in mathematics, as compared to content. When content wasthe focus of the conversation, Jaime and Bailey expressed seeing little value inadvanced content. For them, the notion of advanced content was “too distant”and they saw “limited connection threads” between secondary and post-secondarymathematics. Jaime held the strongest view on this position:Jaime: I think that a lot of what I did in university math was so distantfrom what I did in high school, I don’t think it was essential. Weare doing stuff in three dimensions and all this weird stuff. It’sso far away from what high school kids are doing. I think there’sa downside of taking a lot of advanced math, that you go prettydeep down the rabbit hole and then you can get out of touch.Jaime supported this by noting their belief that pedagogical skills are sepa-rate from content knowledge and that content knowledge isn’t necessary to be anexcellent teacher and that advanced degrees might just be “screening tools” for be-coming teachers. They shared during the interview that “if someone knew highschool math well, they could turn around and teach that well.”90Bailey was in partial agreement with Jaime on this, mentioning that “prag-matically, we need more math teachers.” Bailey elaborated on this, remarkingthat “having a love, understanding, and interest" in mathematics is more importantthan extensive university coursework. They did, however, mention a collectionof courses of which it would be useful for a secondary teacher to have. Baileythought that coursework in calculus, linear algebra, proofs, number theory, andgeometry could act as alternative lenses to view the secondary mathematics cur-riculum through and could offer fun problem solving opportunities.Even though all of the participants saw at least some value in teachers havingpost-secondary degrees in mathematics, the participants perceived their degrees ashaving value to them, personally. The recurring codes in participants’ responseswere connections, problem solving, and application.The skill to “build connections” between concepts was seen as valuable to allthe participants. Adrian, Casey, and Jaime saw their advanced knowledge havinggreat value in being able to say where content goes later in the curriculum. Jaimesummarized this well in saying that their advanced knowledge “gives me a sense ofwhat all this can mean in the end.” Adrian took a more research oriented approachto their response, remarking that their advanced degree allowed them to “appreci-ate the immensity of knowledge that is out there in terms of math and physics andhaving an idea of what is actually being researched.” Pragmatically, Casey viewedthat a teacher with a mathematics degree might have conceptual understanding of aparticular area of expertise, which they might be able to bring into their classroomfor enrichment.91Taylor found value in their advanced degree expanding the scope on what con-stitutes mathematics. They noted that they did not have a good idea of what math-ematics was in high school, but that “in university, I started to realize that it’ssomething very different from what I learned in high school.” Since Taylor wentthrough a personal revolution of their views of mathematics, Taylor wanted to sharethis with their students, with the hope that students could see “that computation isnot everything mathematics does, but more about why certain things work the waythey do.” Similarly, Bailey remarked on how extended content knowledge yields a“bigger sense of how things fit together” and followed this in saying it would helpthem include some “fun math little tidbits.” However, they were unsure on how todo this within the curriculum, as university mathematics “really is diverging fromwhat is taught in high school math.”Concerns of the restrictions of the curriculum were common in participants’views of bringing their advanced content knowledge into their teaching; Adrianelaborated that the role of their advanced degree in their teaching is different intheory and in practice. They remarked on how, even though they see personalvalue in it, that it might not make a difference in practice:Adrian: It feels very much like this is what you need to know and it’sour job [as a teacher] to get you [the students] to know it. Asopposed to this is an interesting field of study, we want to exploreit, what kinds of questions can we ask, and leading them on this92inquiry process where we get them to explore ideas beyond theirconventional grade level. That just seems like not at all what’shappening.Taylor mirrored this concern, that even if they wanted to include content be-yond the curriculum, teaching secondary school is not completely autonomous.They worried that their hopes and goals in teaching might not be achievable in areal classroom.Experience in problem solving was viewed as a benefit of advanced mathemat-ical coursework by Taylor, Jaime, Adrian, and Bailey. For Taylor and Bailey inparticular, experience in proof was viewed as a benefit they both wanted to bringinto their classrooms. After stating that their perception of what constituted math-ematics changed from high school to university, Taylor remarked that their devel-opment to think logically and prove rigourously was a contributing factor. Theyreflected on the role of computational thinking being heavy in high school and thatthey did not have a conception of proof. In their university mathematics courses,they admitted that their professors “didn’t really explain to me” what it meant toprove something and they struggled in courses where proof was a component. Tay-lor noted that they wanted to share this experience and university knowledge withtheir students so that they might not be “totally embarrassed when they go to uni-versity.”Bailey also saw immense value of proof in secondary mathematics education,remarking that the habit of teaching mathematics in a “do this, do that” manner93“really seems to kill math.” They worried that this mentality makes mathematics“an exercise of just doing steps instead of solving problems.” Their hope is that in-troducing proof techniques, through they might be beyond the curriculum, wouldbe a valuable tool to bring context and explanation to the question of why particulartechniques and strategies work or are used.As Jaime did not have coursework in proof, the value they saw in problem solv-ing was experience in learning how to approach a problem. Jaime remarked on thevalue of understanding limitations, assumptions, and context of problems, partic-ularly with respect to modelling in problem solving. They viewed their advancedcoursework as helping them understand that “creating a model is useful becauseit tells us something, but we have to remember that it doesn’t tell us everything.That’s one of the things that I got through my degree is limitations on things thatyou do and do not know.” Adrian saw similar value in problem solving, partic-ularly in applying mathematics to “real world” problems and understanding whythese models are useful. However, they feared that their ideas for extensions mightnot be well-received by students, in which case “everything after first or secondyear undergrad gets thrown away as not very important.”4.3 Advanced Knowledge in Teacher EducationAs elaborated above, all of the participants saw some value in their own advancedcoursework experience for their future work as teachers, as well as benefits ofadvanced mathematics courses for secondary teachers in general. However, partic-ipants felt that their extended expertise in mathematics was not being drawn upon94in their mathematics teacher education. This concern was succinctly summarizedby Bailey, who remarked: “I need my math to teach, but the teacher education pro-gram isn’t requiring me to have any knowledge of math.”Jaime, Adrian, and Bailey each wanted a more practical and pragmatic ap-proach to their teacher education. Each of these participants expressed a desire formore content focused education with respect to what they perceived as the goals oftheir methods courses in mathematics:Jaime: I found them a little bit scattered. I think we are seeing a lotof bits and pieces of here are some neat little ideas, but I find itreally hard to pull them together. I think I would have liked to domore or at least see more ideas of how to specifically bring thisinto teaching the curriculum. I’m more interested in the teachingside. Like how would you introduce a concept? I have found thatwe got a few neat ideas. One week we did a math art project. Andyeah, that’s cool for all of us because we like it, but it might notbe so relevant for a math teacher.Adrian: My math methods feel nebulous in terms of what the focus is.It’s more like where are some nice connections in math, here aresome nice ideas of what a math teacher should be, here are somemathematical related activities, here are some projects, some pa-pers, some analyses. I think it would have been really nice tofocus on how am I going to teach this [concept]?Bailey: We’ve read a lot of like, theory, theoretical papers, about methods95of teaching math, but not in any concrete way. We didn’t go into itwith enough meat to do anything with it. It’s just this grab basketof oh, you can teach through movement or you can teach maththrough art. But you can’t really. There is a whole curriculum.Yeah you can add that in, but there is a whole curriculum that youneed to find a better way to teach.Adrian and Bailey brought forth interesting perspectives on this end, since theyboth had two teachable subjects and were taking methods courses in those subjectareas.Bailey: In my other methods courses, we do a lot of really applicablestuff. Like, we look at a paper someone wrote about teachingcritical theory about Shakespeare and we look at the actual teach-ing methods for teaching that topic and teaching different types ofwriting and book suggestions and like what you teach and all thisstuff. And I think it’s really building on both our knowledge ofEnglish as a discipline and like actually giving us practical waysto teach it. In my math methods, you can do all them knowingliterally grade 10 math.Adrian: I really like the way my other course was structured. It was verymuch structured around showing cool experiments and how toconnect it to the curriculum and give you a chance to teach. Inmath, I think having a focus on “how am I going to use this?”would have been nice. How do I make all of these teaching ideasand concepts effective and relevant? How do I design an effective96activity that hits all these points, and is engaging, and assess thecurriculum?Adrian and Bailey found that they were building subject expertise through theirother methods courses, but found this to be lacking in their mathematics methodscourses. They continued to express a desire for “practical, actual math in the class-room” and worry that after their teacher education “we are going to teach math theway we probably would have before. I’ve come out with no concrete examples forsecondary math.”With the exploration of concrete ways to teach content, comes the questionof whether or not it is possible in methods courses. Adrian enjoyed opportunitiesto explore teaching math through social justice, but felt as though “our professorsgive us an idea that is not enough, but don’t really follow up with how to make itenough [in the classroom],” while Jaime remarked that “from grade to grade, thecontent is going to be different and it’s unrealistic to try to cover all that. I’m moreinterested in the teaching side and how you would introduce a concept.” Baileysuggested a pragmatic approach:Bailey: I think even learning the process of looking at how somethingis often taught, thinking about how it’s taught, thinking of newideas, doing that for a few topics will probably help you practicedoing it for other topics. In my other teachable, I have all thesetouch points of jumping off new ideas and ways of teaching thatI don’t feel like I really have from math.To navigate these struggles, Taylor and Adrian felt that they needed to prompt97themselves through self-guided reflection on how they could use their advanced de-gree experience, but this was not being prompted through their courses. As Adrianlamented, “I’ve done it on my own because it’s something I’m interested in andsomething I’m good at, but not something that the instructors have encouraged.”Taylor expressed similar experience on their end, remarking that “I think of thematerials that I learned in university when I see the material here in the educationprogram and how I can advance that material.” However, the feeling of being ableto integrate post-secondary expertise into secondary teaching was not held by allparticipants.Bailey shared that they felt as though it may not be possible to build subject ex-pertise in their courses, saying that in math “you really are diverging from what istaught in high school math. So there is not use in talking about group theory in aneducation program for high school math because that’s never gonna come, it’s a dif-ferent discipline almost.” This is in contrast to their other methods courses, wherethey said “you’re becoming more proficient in writing, which is what your studentsare doing.” Is there such a great divide between secondary and post-secondarymathematics content?Taylor and Casey perceived some degree of “doing mathematics” in their math-ematics methods courses. Interestingly, both Taylor and Casey made mention ofthe history of mathematics as an example. Taylor took a course in the history ofmathematics during their mathematics degree and some of this material was ex-plored in a course in their teacher education program. They saw this extendedknowledge being useful in providing alternative proofs and ways of understanding98the Pythagorean Theorem. Casey expressed concern, however, that the content intheir teacher education course might not have been as relevant as it could have beenfor teaching of the secondary curriculum:Casey: We did cover a lot about the history of zero and the history of one,the stepping stones of math. And I didn’t know about Babyloniantablets. But I felt we stopped off at around the year 400 AD. Wedidn’t cover anywhere from the year 1000 to 1900 mathematics.And then I thought what would be relevant for high school math-ematics.The relevance of the history of mathematics was the only direct relationshipbetween the secondary and post-secondary curricula brought forth by the partic-ipants. Although this connection existed in their teacher education courses, theparticipants’ responses suggest that this connection may be limited. Indeed, thehistory explored in the teacher education context is more in line with an ethno-mathematical perspective on mathematics education, while the courses offered inmathematics departments tend to be more centred on European perspectives. Re-gardless, one must ask how this fits in to participants’ concerns of “practical andpragmatic” responses to teaching existing mathematics curricula.4.4 SummaryIn this chapter, we discussed participants’ responses to interview questions whichexplored their perceptions of the role of advanced mathematics knowledge in theirgrowth as teachers. Through this discussion, a disconnect was observed between99the content versus the beliefs and values learned in post-secondary mathematicscoursework. Furthermore, an even greater disconnect was expressed by partici-pants in regard to the ways in which their mathematical expertise was being drawnupon during their teacher education.All participants expressed a love and affinity for mathematics as a subjectand looked forward to bringing this into their teaching. Many of the participantsviewed their education in post-secondary mathematics having a personal impact onwhat they understood to count as mathematics, which they did not experience inhigh school. Through the skills and values gained in their advanced mathematicscourses — such as proof, logic, rigour, and application — participants hoped theywould be able to give their future students an opportunity to see the “two-faceted”nature of mathematics as both a way to read the world and as a self-containedknowledge system. The inclusion of problem solving was mentioned by all partic-ipants as a skill learned through their post-secondary coursework which they hopedto extend to their secondary teaching to develop critical thinking skills and contex-tualize mathematical content.The attributes of their post-secondary mathematics education which partici-pants were excited to bring forward in their classroom were primarily skill based,rather than content based. When the focus of the conversation became about math-ematics content, participants did not see much value. Indeed, many participantsexpressed a great disconnect between the secondary and post-secondary curricu-lum, considering that university mathematics is too far removed from what studentslearn in secondary school. While some advanced courses such as number theory100were mentioned as being useful for enrichment, overall, participants felt that theconnections between secondary and post-secondary mathematics dropped off aftersecond year university mathematics.Following this line of thought, participants unanimously agreed that extensivepost-secondary mathematics coursework (beyond the second year) need not be apre-requisite to teach mathematics. While some participants took a more pragmaticapproach to this question, addressing the demand for more mathematics teachers,others expressed that advanced content knowledge may not imply better teachingof the curriculum. Participants did not see value in their own content expertise fortheir teaching, and in turn, did not see the value in requiring such content expertisefor other mathematics teachers.These opinions may have been exacerbated by participants’ perceptions of theways in which their content expertise was being drawn upon in their teacher ed-ucation, as well as their desire for a more pragmatic teacher education program.In sum, participants did not feel as though they needed the extensive mathematicscoursework that was necessary for entrance into their teacher education program.They perceived the amount of mathematics they were doing in their program tobe minimal and expressed a desire for critically examining the content of the cur-riculum. This is in contrast to what they believed to be the focus of their methodscourses, which was bringing in fun and interesting connections between mathemat-ics and other disciplines, such as art or social studies. Although participants sawvalue in this, they expressed concern in not having the expertise to critically exam-ine the existing curricula, current ways of teaching it, and finding better ways to do101so. Overall, participants expressed a disconnect between “the curriculum they haveto teach” and the techniques they were learning in their methods courses. Of allthe participants, Taylor and Adrian expressed some value of advanced courseworkbeyond the second year, but their remarks were more based in enrichment for thecurious student.Some participants who had chosen two teaching specialities shared very differ-ent experiences in their other methods courses, where they felt as though they werebuilding upon their content expertise to enhance their classroom pedagogy. Theyfelt that their advanced content expertise was of importance, while being drawnupon and extended in these courses.The initial remarks from participants in this first portion of the interview sug-gest that the participants do not perceive post-secondary mathematics as an ex-tension of the secondary curriculum. In the following chapters, we examine theaforementioned claim more closely in the context of mathematics questions thatdo have extensions to the post-secondary curriculum, while also examining con-tent knowledge in more depth.102Chapter 5The Overextension of FamiliarMathematical Ideas: A Case ofPolynomialsThe following chapter elaborates on participants’ engagement with tasks C and F:Task C: You are teaching a week on factoring polynomials and you have foundthat your students are struggling to recognize when they should stop trying tofactor. How would you respond?Task F: You have been teaching a unit on quadratic functions for a few weeks andone of your students asks you why they need to know about them. How would yourespond?Since the mathematics of these tasks are intimately related, the analysis of re-sponses to these tasks will be combined in this chapter. As will be the structure103for the results chapters which follow, I begin with a “mathematical background,”which will set the mathematical context for participants’ responses. This back-ground is in no way comprehensive, but covers some of the major connectionswhich I have made in my own studies, as well as connections mentioned by partic-ipants who engaged with tasks C and F. Next, I examine participants’ responses tothe tasks and the higher level connections made to advanced mathematics content.5.1 Mathematical BackgroundPolynomials are a fundamental concept in the secondary mathematics curriculum.Linear graphs are some of the first graphs that students encounter in their math-ematical studies and are often used as one of the first examples for the study offunctions. Polynomial functions are widely used in many fields outside mathemat-ics in modelling various social and physical phenomena.Generally, a polynomial of degree n over R is a function which may be writtenas f (x) = anxn + an−1xn−1 + . . .+ a1x+ a0, where an,an−1, . . . ,a0 are elements ina set R and x is a variable that takes values over R.In this portion of the study, I was interested in exploring the understandingsparticipants’ held in regards to the factoring of polynomials. Knowledge of fac-toring and finding zeros of polynomial functions are key skills in the secondarycurriculum, as well as in university mathematics courses. In the following section,I outline some of the major places where polynomial functions appear in the sec-ondary and post-secondary mathematics curricula, along with how and why they104are studied.In the BC curriculum, students first encounter the notion of lines in grade 9and continue working with polynomials up to and including Calculus 12. The stan-dard progression of study begins with linear functions (degree one polynomials),quadratics (degree two), and higher degree polynomials. Typically, factoring ofpolynomials is explored first, before moving to the graphing of polynomial func-tions, where the relationship between graphing and zeros is particularly useful.Quadratic functions act as a first introduction to a “curved” function, after studentshave gained confidence working with linear functions. Quadratic functions can beused for modelling phenomena that obtain extreme values. Examples include, butare not limited to: modelling revenue and/or profit, maximizing areas, object tra-jectories, and scenarios involving time, distance and velocity. Even if quadraticsdo not precisely describe a particular phenomenon, they act as a welcoming entrypoint to modelling with functions.In university, polynomials are central to helping students build an understand-ing of calculus, both in differential and integral calculus. In differential calculus,polynomial functions are often utilized as examples in nearly all topics, becausestudents are familiar with them. In particular, students’ pre-existing knowledge ofthe existence of roots and knowledge of finding zeros is central to using them ascommon examples throughout the course. In integral calculus, the use of polyno-mials is central to the concept of Taylor approximations and Taylor series, whichconstitutes the latter half of most integral calculus courses. Furthermore, determin-ing anti-derivatives involving polynomial and rational functions constitute a large105portion of the examples students encounter in integral calculus. The most promi-nent appearance of polynomials in determining anti-derivatives is through the con-cept of partial fraction decomposition. In this technique, when encountered witha rational function R(x) =p(x)q(x), where p(x) and q(x) are polynomials, studentsdecompose the single rational function into a sum of simpler, rational functions.The key in this technique is decomposing the denominator into irreduciblepieces. For a simpler example, consider 16 . When factoring natural numbers, theirreducible components are prime numbers, so we factor 6 as 2 ·3. And indeed, 16can be decomposed as 12 − 13 . So the question is: What are the irreducible compo-nents in the context of polynomials?Consider the following example of∫ 10x+1x2+5x+6dx.The denominator of the integrandx+1x2+5x+6, can be factored as (x+3)(x+2).Since the denominator decomposed as a product of two linear functions, we needto find constants A and B such thatx+1x2+5x+6=Ax+2+Bx+3.After a bit of algebra, one can determine that A = −1 and B = 2. Thus, theintegral can be rewritten as ∫ 10−1x+2+2x+3dx.Now, an antiderivative of the integrand is − ln |x+2|+2ln |x+3|, so, the Fun-damental Theorem of Calculus may be applied to determine the area under the106curve.These problems become more technical as the degree of the integrand’s denom-inator increases. If the degree of the denominator is 2, one of two things happens:one, the denominator decomposes as two linear terms, in which case partial frac-tion decomposition as it was done above is the technique of choice. Or two, thedenominator does not factor, remains as an irreducible quadratic (such as x2 + 1),so that the anti-derivative may involve the inverse tangent function arctan(x). Stu-dents often encounter a table such as Table 5.1 when learning about the techniqueof partial fraction decomposition in Integral Calculus.Type of factor Example DecompositionLinear factor x−a Ax−aRepeated linear factor (x−a)n A1x−a + A2(x−a)2 + · · ·+ An(x−a)nIrreducible quadratic factor x2+bx+ c Ax+Bx2+bx+cRepeated irreducible quadratic (x2+bx+ c)n A1x+B1x2+bx+c +A2x+B2(x2+bx+c)2 + · · ·+ Anx+Bn(x2+bx+c)nTable 5.1: Partial fraction decomposition guidelinesThis table is dependent on the fact that all real-valued polynomials can be de-composed into a product of linear and irreducible quadratic terms. Indeed, linearterms of the form ax+b and irreducible quadratics of the form ax2+bx+c are the107irreducible, non-factorable pieces in the context of decomposing polynomials, justas prime numbers are the irreducible elements in the context of factoring naturalnumbers.Another course in which the factoring of polynomials is of great importance isin Linear Algebra. At its core, Linear Algebra is the study of linear functions andvectors in multidimensional space. Vectors, which have magnitude and direction,are added and multiplied by scalars, while linear functions take vectors as inputsand abide to the rules of vector addition. Matricies, which are the core of study inLinear Algebra, are a way to organize information about linear functions. The ma-jority of introductory Linear Algebra courses in post-secondary restrict their studyof vector spaces to “real-valued” space, that is, Rn.The study of Linear Algebra in such courses often culminates with the study ofeigenvalues, eigenvectors, and eigenspaces. Many real-world phenomena may bemodelled with linear functions and eigenvalues make solving such problems mucheasier. A classic example of the use of eigenvalues and eigenvectors is the predator-prey phenomenon over time t. Suppose that species x, wolves, are a predator ofspecies y, bunnies. A higher population of wolves will result in a lower the popula-tion of bunnies that are available to reproduce. At the same time though, a smallerbunny population means that the wolves have less food available. So, a smallerbunny population will affect the reproduction rate of the wolves. Then, a smallerwolf population makes reproduction easier for the bunnies. This cycle continueswith these two populations directly affecting one another and can be representedby a differential equation of the form:108dxdt= ax+bydydt= cx+dyAs is, this is a complicated system of differential equations to solve, since thesystem is “coupled”. That is, the two pieces are dependent on each other. However,using linear algebra allows one to “de-couple” the system into independent piecesthat can be studied independently. Determining what these independent pieces areis the goal of the study of eigenvectors and eigenvalues.So what do polynomials have to do with this? The answer lies in the deter-mination of eigenvalues of a given matrix A. By definition, x is an eigenvector ofA if Ax = λx. Bringing everything to one side yields (A− λ I)x = 0, where I isthe identity matrix. If a non-zero solution to this equation exists, this means that(A−λ I) is not invertible. So, the determinant of the matrix A−λ I must be zero.The solutions to the characteristic equation det(A−λ I) = 0 represent the eigen-values of the matrix A. Since the characteristic equation is a polynomial, studentsmust apply their understanding of factoring and polynomial solutions in order todetermine eigenvalues of various matrices.These ideas are expanded upon in-depth if and when students take a course inAbstract Algebra. This course is normally only taken by mathematics majors, withsome students interested in theoretical physics enrolling in the course. Abstractalgebra is concerned with the study of general algebraic structures. The notions ofgrade-school arithmetic and high school algebra are placed within a larger, moregeneral theoretical structure. The structures studied abide to pre-defined axioms109and are studied generally. Most universities offer a two course sequence in abstractalgebra: one on groups and another on rings and fields. Groups, rings, and fieldsconstitute three categories of algebraic structures with far reaching extensions inmathematics, with groups having relevance in chemistry and physics.Before trying to understand what a group, ring, or field is abstractly, studentsoften encounter sets that satisfy the axioms of one or more structures. Often, thefirst example of a group might be the integers, Z, under the addition operation,while the first example of a ring might be the real numbers, R, under the opera-tions of multiplication and addition. These familiar structures are extended to moredynamic ones, including various polynomial rings.The types of questions one might ask about various rings and field include:What are the irreducible elements? That is, what are the elements of the ring orfield in question that cannot be decomposed into less “complicated” pieces? ForR[x], the ring of polynomials whose coefficients are real, the irreducible elementsare linear and irreducible quadratics. When one extends R to include the complexnumber i =√−1, the polynomial ring C[x] has only linear polynomials as its ir-reducible elements. This implies the famous Fundamental Theorem of Algebra,which states that all complex valued polynomials in C[x] may be factored into aproduct of linear polynomials. Another question might be: How can we create newrings from existing ones? To answer this question, the answer to the previous mustfirst be understood so that the notion of a “prime ideal” may be understood. Then,the notion of a “quotient ring” may be constructed as a way to generate new ringswith new algebras. Finally, one might ask whether it is possible to understand the110algebra of one ring by understanding the algebra of another, more familiar one?The answer to this question is yes. One such example is that of polynomial ringswhose coefficients belong to a field (such as R or C). In this case, all of the famil-iar theorems that are true in Z have analogs in the polynomial ring, including theDivision Algorithm, Euclidean Algorithm, and Unique Factorization. Even moreinformation may be gathered if two rings happen to be “isomorphic” to one an-other, meaning that a perfect bijection exists between the two rings.5.2 Participant UnderstandingsAll five participants engaged with the researcher with discussion of the factoriza-tion of polynomials, while all but Bailey engaged with the tasks on quadratics. Thepopularity of participants choosing these tasks should not be a surprise, due to theimportance that is placed on the study of polynomials, both in secondary and post-secondary mathematics courses.Among the participants, all had taken calculus and linear algebra, with Taylor,Bailey, and Casey having taken courses in Abstract Algebra. Taylor and Baileytook two algebra courses (one on groups and another on rings and fields) at theirrespective universities. Casey took one course officially, as well as auditing twocourses without credit. Adrian admitted to knowing very little about Abstract Al-gebra, having acquired an understanding of its utility through brief mention in sometheoretical physics courses. As an engineering major, Jaime had no exposure to theconcepts of Abstract Algebra in their mathematics courses, but did have numerouscourses in differential equations, where the techniques of determining eigenvectors111and eigenvalues are of significance.In their responses to justifying the study of quadratics in secondary school,many participants viewed the study as an access point to studying curved functionsand modelling progressively complex phenomena:Taylor: Quadratic functions incorporate the concept of a curved functionand are the basis for how a lot of mechanisms work.Jaime: For modelling, you would be using lots of more complicatedfunctions and all the work with quadratics is setting up for that,so you can get them into a form where you can do things mean-ingfully, or so you can find where the zero points are.Adrian: A lot of things can be modelled by quadratics, as it turns out.They are a pretty good basis for a lot of modelling. Also, it’saccessible, it builds on stuff they already know.Although each of these participants expressed value in teaching quadratic func-tions as an entry point to modelling complex phenomena, Adrian and Taylor ex-pressed concern about making the concept sufficiently interesting to students. Adrianwas concerned that “polynomials can describe certain systems, but outside of sci-ence and math, I don’t think it’s super useful,” while Taylor lamented that “myknowledge of applications of quadratics is very limited and I need to find out howI can make students more interested.” Taylor expressed that they appreciated mathfor the sake of math and that a focus on “real world application” takes away fromstudying mathematics independent of application.112On the other hand, Jaime viewed applications as important to “get the mys-tique” out of quadratics. They noted that “a lot of the time we just see these equa-tions and they don’t really mean anything.” Jaime’s concern in teaching quadraticswas to put meaning to “a formula that just exists almost for the sake of existing.”Graphing was considered to be a valuable entry point for building meaning, since“you can just look and see behaviour.”While Taylor, Jaime, and Adrian’s responses focused more broadly on the fea-tures of quadratics, such as zeros, vertex/extrema, and applications to physics,Casey’s responses were example driven:Casey: You could talk about quadratics and engineering the construc-tion of a bridge. Quadratics have a very symmetric property toit, because the arch is the most economically sound method ofconnecting to points across a body of water. Speaking of speedequals distance over time, you could talk about quadratics withthat and flying aircraft.When prompted to explain how the second example related to quadratics, Caseyemphasized a skill based focus to this problem. Casey stated they would use ratio-nal functions as a means to teach the importance of quadratics. However, in theirresponse, it was evident that quadratics were used as a means to solve the problem,rather than motivating the concept independently:113Figure 5.1: Casey’s work on quadratics and air travelCasey: We say a plane flies, the way I learned it was Glasgow to Hali-fax, with umm, it goes faster going from Glasgow to Halifax, sothe airwind is a couple more kilometres per minute, but comingback we get delayed, presuming there is the same wind, becauseof course, presuming the plane is just pushing you in the samedirection, you go faster from G to H than from H to G. So thisis the distance, that’s some kind of time and it’s delayed by twohours and this is two hours. But to solve this quadratic here, whatwould you get? You would get a 48s and these two would cancelout, but then you would have this problem here and calculate theaverage speed of the plane. That’s what I would explain why isit useful, the arc of a bridge kind of problem and also the planeproblem. But I might start with the plane one first.It is unclear as to why Casey considered this to be a valuable introduction to theuse of quadratic functions. While quadratics are useful in modelling phenomenon114with distance, rate, and time, using them in conjunction with rational functions,factoring, and physics might be a difficult entry point for many students. As wewill see below, the factorization of polynomials is not as trivial as some may think.Indeed, all participants in the study demonstrated misconceptions regarding thefactoring of polynomials, in both their secondary and post-secondary knowledge.Of all the participants, Jaime was the least confident in discussing the factoring ofpolynomials. Immediately, however, they recognized the importance of this typeof open ended question for students.Jaime: We are developing habits of how do you approach a problem.And part of this is that we have these different techniques wehave used and it’s going to be a mental checklist of does it looklike this or that? You may need to use one technique, two, maybeyou can’t use any.Although Jaime recognized the complexity of this problem, when asked towork through some problems, Jaime struggled.Interviewer: So if we started off with a simple case of factoring a quadratic,how would you respond?Jaime: It’s been a long time since I factored quadratics, but I rememberthis type, the question and having this in my mind and being notquite sure if I’m done or not.Interviewer: So what could it look like?115Jaime: I have no idea. It’s been a long time. I’m not sure where you aregoing with that.Interviewer: What about something like x2 + 2x+ 1 versus x2 + 1? For thesetwo, how would you approach factoring these? If you can factoror if you can’t?Jaime: I feel really embarrassed. I know there are specific things andpatterns, but it’s been so long I can’t remember what they were.Sorry, I have a vague recollection of where this goes, but I don’tknow.Interviewer: Let’s talk about this one (x2 + 1). Is there a way to graph thisfunction that would tell you what it looks like? Do you remembera way to relate factoring to graphing?Jaime: So we are dealing with parabolas and shifting. Yeah, I rememberthe graphing side of it. I like the idea of bringing that in. How thatwas factored out though, I don’t know. It’s been so very long. Theway I look at this, the idea of shifting it, being symmetric aboutx, these are the things I would tap into. I honestly have no ideaabout factoring or the point of it, other than finding a differenceof squares. So yeah, I see that your zero is going to be here (aty = 1).Interviewer: So where is your zero in this case?Jaime: It’s at 1. So that’s the stuff I remember, but in terms of factoring,I don’t know. But I chose it (this task) because I remember that116feeling of not knowing. There is an uncertainty and I rememberit getting worse in university.This dialogue with Jaime brings up two major concerns. The first being a lim-ited understanding of a fundamental concept from the secondary curriculum. Jaimeadmitted earlier in the interview that they were a bit rusty on their mathematicsand needed to review, but the concept of factoring quadratics is not an advancedsecondary concept. However, at the time of the interview, Jaime was half waythrough their teacher education program and preparing to go on practicum, wherethey would be teaching in a classroom. If tasked with teaching this fundamentalconcept in their practicum, Jaime would not only be working on developing thepedagogical aspects of their teaching, but the mathematical as well.Secondly, the difficulties Jaime experienced in this task raise questions aboutthe degree of review Jaime would have to do before teaching such a concept. Thetwo examples posed to Jaime are two of the simplest examples for quadratic factor-ization, but the extent to which Jaime was able to talk about these two quadraticswas limited. One should question whether the depth of mathematical knowledgefor teaching developed through such a review would include HCK and to whatextent KDUs would be developed. As a new teacher entering into a career in edu-cation, how much time will Jaime have to dedicate to reviewing mathematics to adepth and breadth beyond the context of the prescribed curriculum and text?Of the remaining four participants, a common thread was shared. This com-mon thread, which points to a major misconception of participants’ understanding117of polynomial functions, factoring, and roots, was explicitly observed with eachindividual participant. At some point of engaging with the task, each participantstated a variation of the following:A real-valued polynomial can be factored if and only if it has a root.Although this is true when the polynomial in question is of degree two, thisassumption fails for any polynomial of a higher degree. However, participants eachhad strongly held misconceptions to this end. Consider the following dialogue withBailey:Interviewer: So with the idea of roots and factoring, if a polynomial doesn’thave any roots, does that mean it can’t be factored?Bailey: Yes. Yes. And if it doesn’t have any nice roots, it can’t be factorednicely.Interviewer: What would you encourage your students to do in factoring x5−1?Bailey: I would have them graph it. I can’t think now of what it lookslike. It’s going to be a squiggly-ish kind of thing. It’s gonna haveone root. You know that 15−1 = 0, right? So you can find that.Interviewer: So what would be a factor of x5−1?Bailey: x−1 would be a factorBailey was correct in their description that if a polynomial has a root at x = a,then x− a is a factor of the polynomial. Indeed, connecting graphing of polyno-mials to factoring could help students build deeper connections between geometric118and algebraic representations of polynomials. However, Bailey later generalizedthis logic in the opposite direction:Interviewer: And what would be leftover? It doesn’t have to be exact.Bailey: Let’s do polynomial division. Wow, I don’t remember how. Weare multiplying the x4, gonna multiply it in, ummm, x4− 1 mi-nus....x...it’s gonna be x4 + x3 + x2 + x+ 1....something like that.Maybe there will be a negative somewhere.Interviewer: So can this be factored?Bailey: No, no. So since we have a one at the end, things get nice and Ithink you could do a fun, you could talk about powers of 1 andhow if it’s negative, if it’s an odd power, we could have a negativeone, but it’s even. We have an x3 and an x, so if we have negativenumbers. We have 1− 1+ 1− 1+ 1 and we have two of thesame thing. This is always going to cancel itself out, so we can’tpossibly get zero.119Figure 5.2: Bailey’s factoring of x5−1Bailey seems to suggest here that in order to determine whether a polynomialcan be factored, one just needs to see if it has a root. As mentioned above, thisclaim is false. Bailey has generalized the logical implication that a root impliesfactoring to factoring implies a root. Taylor chose the same polynomial of x5− 1as Bailey to work with. After finding the factor of x−1 and performing long divi-sion, Taylor claimed that the leftover quartic factor could not be decomposed anyfurther, as is seen in Figure 5.3.Figure 5.3: Taylor’s factoring of x5−1Adrian demonstrated a similar misconception and was very explicit in their un-derstanding. Adrian demonstrated a robust understanding in relating the graphs of120quadratic and cubic functions to the existence of roots. The written work associatedto this dialogue excerpt may be seen in Figure 5.4:Adrian: My first instinct is always drawing a picture. If I draw this picture,and I’m going to assume they have learned vertex form, but at theleast I can plug it into Desmos. Maybe they don’t understand whyit looks like that, but you can get it. So this one (quadratic withtwo roots) looks like this (draws graph). And what we are doingis that we are able to break it down into these two points. Theseare the points that go to zero and allow us to break it up (factor).The fact that this crosses the x axis at these points allows us tofactor it like this. But for this one (x2+1) it looks like this (abovethe x-axis). It does not have those pieces that we can break it upinto, so we are not able to factor it. These are the two possibilities.Well, no. I guess the other possibility is that you have it whereit just touches the x-axis, in which case, there are two, it’s gonnalook something like (x−a)2. The point being that there are threedifferent possibilities. It touches once, it touches twice, it doesn’ttouch at all. And that tells us it’s going to look like this, can’tfactor it and it will look like this (above the x-axis)Interviewer: And what about a cubic? What are the possibilities in breaking itdown into cases if you had a cubic as well?Adrian: Well, there is going to be at least one solution guaranteed, sincethe mean value theorem says there will be one solution. Thereis a case where you get one solution, two solutions where it will121just touch like that, and a case where you have three solutions,and that’s it.Interviewer: And so for factoring, what would that entail?Adrian: I think for this one (a single root) you’re going to end up with anx−1 and a quadratic without any real roots. This one (two roots)I think you’re gonna end up with one root and a double root on theother side, and this one, you’ll have three roots and three linearfactors, something like that.Figure 5.4: Adrian’s factoring approach122In this portion of the interview, Adrian demonstrated a deep understandingof the relationship between graphing and roots of polynomial, in the context ofcubic and quadratic functions. Adrian was able to connect understandings of thegeneral algebraic form to the impact on the shape and location of the graph inthe Cartesian plane and did not need explicit examples in order to demonstratethis understanding. However, when extending the polynomial to a higher degree,Adrian generalized their understanding of irreducible quadratics to polynomials ofhigher degree.Interviewer: So we are seeing a relationship between roots and being able tofactor. Does that apply generally?Adrian: I suppose, yeah.Interviewer: If we move to something like a quartic, something like x4 + 1,what about that? Can this be factored?Adrian: No, because it doesn’t have any x-intercepts. So it’s just goingto sit like that. The factors link to whether there is a solutionto it being equal to zero, because if you let x be one of thosevalues, zero times something that isn’t zero is still zero. Wholething is zero. These are connected to this idea of where does ittouch the x-axis. So if your function touches the x-axis and it’s apolynomial, usually, in almost any, I can’t think of a case whereyou can’t break it up like this.Here, Adrian explains why x4 + 1 cannot be factored, by connecting it to theshape of the graph. Indeed, the graph of y = x4+1 is a translation of the graph of123y = x4 up the x-axis by one unit. Thus, the graph of y = x4 +1 does not have anyx-intercepts. That is, there are no values of x for which x4 +1 = 0. However, thisdoes not imply that x4+1 cannot be factored. The roots do tell us something aboutits factorization, but it does not tell us everything. Since the curve does not haveany real roots, we know that it will not have any linear factors of the form x− a.Indeed, if it did, we would have that:y = x4+1 = (x−a)(x3+bx2+ cx+d) (5.1)According to Adrian’s earlier explanation of cubic polynomials, the polyno-mial of x3+bx2+cx+d must have at least one root, so x4+1 would then have tworoots, with x = a being the second (substituting x = a into Equation 5.1 yields anoutput of zero). So, we could have that x4+1 factors into two irreducible quadrat-ics. We can write x4+1 as a difference of squares by observing thatx4+1 = x4+2x2+1−2x2= (x2+1)2− (√2x)2= (x2+√2x+1)(x2−√2x+1)Indeed, the quadratics of x2 +√2x+ 1 and x2−√2x+ 1 are both irreducibleas their graphs lie above the x-axis and are concave up, as per Adrian’s earlier jus-tifications for the irreducibility of quadratics.124Adrian reconfirmed the misconception again later in the interview:Adrian: If you can graph that (the polynomial) and it doesn’t touch thex-axis, you are done factoring.Interviewer: And so we are saying that if we are up at degree five (for a poly-nomial), we don’t....Adrian: We know there is one (root).Interviewer: Right, we know there is one root. But we don’t necessarily knowwe can factor the rest? Is that true?Adrian: Yes.Casey held multiple misconceptions about the factoring of polynomials androots at the secondary level. Casey began their discussion with mention of theevaluation of the discriminant of a quadratic.Casey: I would start with saying well what is the value of 2a? And wecan get different examples of 2a and they can plug it into theircalculator. And then we can kind of work and reverse engineera little and say that ok, well, for any given a here, and we dothis separately on a page and do the top, the top bit here (thediscriminant of b2−4ac). So I would start, I don’t even know. 36and 4 and the 6 and then 9.Interviewer: Let’s not think of particular examples. In what ways is this goingto factor?125Casey: Well, it should look like x minus number, x minus number.Interviewer: So is it (the polynomial) always going to look like this?Casey: Yeah, basically.In this excerpt, Casey claims that all quadratics can be factored into two linearterms. While this is true of some quadratics, it is not true for all. This was some-what contradicted later in the interview when discussing x4 + 1 and through theirdescription of the relationship between roots and factoring of polynomials.Interviewer: Let’s say something like x4+1. Can this be factored?Casey: No.Interviewer: Ok, why not?Casey: Well, because, ok, well, if we could break it down here into sayx2+1. Then what I would do is make the substitution y = x2 andsay that’s y2 +1 and then square root of negative one. They (thestudents) understand square root of negative one. That’s what Iwould do, I would go back to grade 9 kind of material and explainthat.Interviewer: What would you say is the relationship between roots and factor-ing?Casey: Well, factoring is just like a visual representation of what thepolynomial looks like. And we know that. Umm, I mean, that’show I would start. That is really the very definition of what fac-toring really becomes.126Figure 5.5: Casey’s factoring of x4+1In this excerpt, Casey seems to combine the technique of substitution for thereduction of quartics to quadratics and claims that because the quartic reduces toan irreducible quadratic, that the quartic does not factor. Their work can be seen inFigure 5.5.5.3 Post-secondary ConnectionsWhen asked to discuss polynomials in a post-secondary context, Adrian, Bailey,and Taylor were able to speak to the behaviour of polynomials when complex num-bers are considered. Each of these participants was familiar with and mentionedthe Fundamental Theorem of Algebra, in one way or another.Theorem 1. THE FUNDAMENTAL THEOREM OF ALGEBRA: Let f (z) be a degreen polynomial with coefficients in C. Then, there are exactly n+1 complex numbersw0,w1, . . . ,wn (not necessarily distinct) such thatf (z) = w0(z−w1)(z−w2) · · ·(z−wn).That is, every polynomial function over C can be factored into linear factors overC.Below are excerpts from these participants regarding the extension of polyno-mials to include complex roots.Bailey: You could probably factor if the coefficients were complex be-cause you can have i2 and you can get some negative ones. So127my whole argument of being able to get zero becomes untrue.x2 + 1 has no roots in the real plane, but it does in the complexplane.Adrian: There are n factors and they will either be real or complex. If youwant to know how many real ones, graph it and see how manytimes it touches the x-axis, that’s about it. I mean, there is al-ways n solutions to it, they are just usually complex. You can talkabout these (polynomials) having complex roots and that meansdifferent things in different situations. Especially in quantum me-chanics and any sort of periodic, ahh, what’s it called, FourierAnalysis. Anything with periodic functions, Fourier decomposi-tions, all that sort of stuff, complex numbers and roots are reallyimportant.Taylor: A quartic polynomial can be factored, but we would have to in-corporate the complex number, the imaginary number. It’s a Fun-damental Theorem of Algebra that when you have a degree of4, degree of n, there are n solutions including the solution forcomplex numbers, but then it doesn’t tell us how we can write itsroots. The proof wasn’t very easy to understand, but now that Isee those number of roots, how many roots can be there, that theylet me know that there are four solutions, four complex solutionsto the polynomial equation, not necessarily telling us how to doit, but telling us there are four solutions.These participants knew the extension of this problem to the context of com-128plex analysis. Indeed, some of the concepts in complex analysis are deeply rootedin problems in the secondary curriculum. However, based on the dialogue in theinterviews, their understanding of the significance of this statement may not havebeen fully developed. The power of the above theorem is that any polynomial maybe factored into linear terms that are dependent on its roots. While participantsknew the statement of the theorem and recognized its significance in the contextof complex numbers and the factorization of polynomials, their dialogue suggeststhat their understanding on the matter was restricted to the course they took in com-plex analysis or abstract algebra. That is, participants held limited links betweenthe complex analysis and secondary mathematics content. In what ways could theFundamental Theorem of Algebra be motivated so that it builds their understand-ing of Complex Analysis, while simultaneously drawing upon their knowledge(and misconceptions) of related concepts in the context of real numbers?Casey frequently connected their understanding of post-secondary content. How-ever, the depth in which they were able to do so was limited. In particular, Caseymade frequent mention of concepts from Galois Theory, but it seems as thoughtheir understanding of these advanced concepts may have negatively impacted theirunderstanding of the material in the secondary context. When discussing the fac-torization of cubic polynomials, these misconceptions became evident.Interviewer: Ok, so let’s say we have a cubic then, let’s start out with ax3 +bx2+cx+d. If you were to factor this, what are the possibilities?What would it look like?Casey: Well that would be something that goes back to here (to the alter-129nating group).Interviewer: And what do you mean by that?Casey: If it’s the 1 then you’re going to get the whole number here, if it’sA3, it’s the rotational groupa, if it’s the S3 it’s the, well if it’s A3it’s all three rational numbers and if it’s the S3 one then you’vegot that and two imaginary numbers.Interviewer: And are you always going to be able to factor this?Casey: No.Interviewer: And why not?Casey: Because you can’t. Because you just have to make some exam-ples that end up being something that, well, the one that I had inthis lesson plan was like, I guess going back to your question, itwas a really really, I guess if you want to call it squirly cubic, butanyway, ultimately one of the x’s happened to be something likethis and another x happened to be that. It was something, but Igraphed that but then I graphed the inverse.Interviewer: So focusing on this, so you’re saying that it is possible that a cubiccannot be factored?Casey: Very much so. And in how to factor the nth polynomial, and sortof how beyond the cubic, it’s not always umm, well no, beyondthe quartic, it’s not always possible. Like we talked about ax3.This is an odd polynomial and it might not be factorable. There130are specific cases depending on what relation a, b, and c have, butif it works out that this relation happens, it’s a lot more likely thatthis is factorable.Figure 5.6: Casey’s factoring of cubicsIn discussing the factorization of cubics, Casey tries to relate the cubic polyno-mials to their Galois groups of the alternating and symmetric groups of three ele-ments (A3 and S3, respectively). Indeed, one of the major points of study in a coursein Galois Theory is the factorization of cubics and the existence of a “quadratic for-mula equivalent” for higher degree polynomials, which is more commonly knownas “solvability by radicals”. In the early 1800s, Paolo Ruffini of Italy and NielsHenrik Abel of Norway proved that, given a polynomial of degree five, there is noalgebraic formula to solve for its roots. Evariste Galois later refined these ideas inwhat was later defined as Galois Theory.The precise statement to which Casey was probably referring, with respect tothe factorization of cubics, is the following:Theorem 2. Let f (X) be a separable, irreducible cubic in Q[X ] with discriminant∆. If ∆ is a perfect square in Q then the Galois group of f (X) over Q is A3. If ∆ isnot a perfect square in Q then the Galois group of f (X) over Q is S3.Casey was trying to draw upon their post-secondary understanding of the fac-torization of cubics and corresponding Galois groups, but it appears as though the131exposure to these concepts may have confused their understanding of them at thesecondary level. The connection between roots and splitting fields (i.e. where apolynomial can be fully factored) is detailed in the following theorem.Theorem 3. Let f (X) ∈Q[X ] be a separable cubic with discriminant ∆. If r is oneroot of f (X) then a splitting field of f (X) overQ is K(r,√∆). In particular, if f (X)is a reducible cubic then its splitting field over Q is Q(√∆).With all of the claims that exposure to advanced mathematics content helpsdeepen future teachers’ understanding of the secondary curriculum, this excerptwith Casey suggests an instance of the opposite occurring. Casey also makes theclaim that beyond the quartic, you may not be able to factor. Once again, this isan instance of mixing theorems from Abstract Algebra with content from the sec-ondary curriculum. Indeed, it is true that there is no “quadratic formula equivalent”for polynomials above degree 5 and Taylor described this in their interview throughthe following excerpt:Taylor: Quadratics pop up quite frequently in many, many areas. Calcu-lus for sure and Galois theory, as well. Quadratic function hasformula for its solution. So what we know of as quadratic for-mula. Even cubic equation has a solution, based on the coeffi-cients of that cubic function, but once you increase the degreeof the function to, from five and bigger, you no longer have aformula expressible by coefficients, and isn’t it amazing that youdon’t have the formula out of this coefficients? Because we havea solution for quadratic, cubic, and fourth degree, but why notfifth? And how do we know? How did mathematicians know132this and how did they figure it out? Yeah. But then again, maybequadratic equation is the most famous for having, deriving the so-lution, finding the root of that equation, it’s really the most basicone. Why does it have it while the fifth degree function doesn’thave it?5.4 An Experience of Abstract AlgebraAs laid out in Chapter 4, many of the participants did not see their advanced mathe-matics courses as having a significant impact on their mathematical knowledge forteaching, besides the development of their problem solving skills and understand-ing of the importance of proof and rigour.With respect to abstract algebra and the factoring of polynomials in this partic-ular task, Bailey shared a profound reflection of some of their advanced, abstractmathematics courses. Recall that Bailey was one year out of their undergraduatedegree at the time of this interview.Interviewer: In thinking about all the courses that you’ve taken, when did thestudy of polynomials come up?Bailey: Calculus. In calculus, you talk a ton about polynomials and youneed to understand how they work. Math like really does moveaway from numbers or functions. You’re really just talking aboutconcepts and ideas. Especially in something like abstract algebra.You’re really out there....Interviewer: Did you talk polynomials in abstract algebra at all?133Bailey: Ummm, I feel like no, it’s been a long time. That course is a blur.Interviewer: Why was it a blur?Bailey: I had Professor X. Do you know him? That was my professor. Ibarely survived. So, I took group theory with him and I also tookhonours linear algebra with him in my second year. And he isobviously a genius but he doesn’t teach in a way that’s accessible.Once you lose the train of the class you’re just gone for the restof it. So you kind of have to teach yourself the whole thing,especially if you’re not able to keep up with the class, which Ireally was not. So, you can tell he’s a genius, but in terms ofteaching, I think it was really good for the couple of students thatwere also geniuses and could keep up with that.Interviewer: And did you take another abstract algebra course after that?Bailey: Yeah, I took ring theory after.Interviewer: And did you, what did you talk about in that class?Bailey: I really forget ring theory, I gotta say. Yeah, it’s a total blur. It’sinteresting because I was supposed to have someone who wassupposed to be very good and then she got sick and had to leave.So I had a sub kind of thing, a grad student, and he was not verygood either. But it moved at a slower pace and I could followit more. I feel like there are some polynomials in ring theory. Ireally, I actually don’t remember those classes. Which is strange.134It was only 2 years ago. I never applied it to anything else. Theywere just isolated things and I never looked at them at all.Interviewer: When you took those courses, who did you think the professorswere seeing as the target audience?Bailey: Like, people who were going to do math research, probably.Interviewer: Do you think that’s what the, who the target audience should be?Bailey: Umm, I don’t know. It’s interesting at that level. Because thatmath is pretty advanced and it’s applicable, but not really beyondmath. So I guess it’s fair that’s who the audience is.The examination of most abstract algebra texts would yield significant contentassociated to polynomials. One could argue that claiming polynomials were nota topic of study in abstract algebra would be equivalent to never examining theRenaissance in an art history course. The study of these structures occupies a sig-nificant portion of many courses in this realm.Rather than focusing on the fact that Bailey did not recall learning this contentin their courses, I turn the reader’s attention to the experience Bailey had as a stu-dent in these courses. Bailey shares that their experience in this course may nothave been the most conducive for learning. Recall that Bailey was a strong stu-dent, who graduated with an 89% average in their university mathematics courses.This leads one to question what advanced university mathematics courses are of-fering future teachers, or more broadly, students who do not plan to take careers inacademia? I will return to this theme in the discussion in Chapter 8.1355.5 SummaryIn this chapter, I examined participants’ engagement with tasks C and F, both ofwhich explored the content domain of polynomial functions. All five participantsengaged with task C, with Bailey being the only participant who did not engagewith task F. The analysis of the dialogue from these tasks revealed that partici-pants, overall, did not hold conceptualizations of polynomial functions that bridgedsecondary and post-secondary mathematics. Indeed, as was elaborated in Section5.4, Bailey perceived the content of abstract algebra to be very “out there” andnot relevant to the content of the secondary curriculum. While Bailey was theonly participant to explicitly state such a disconnect, the lack of connections ex-pressed by other participants prompts further questioning. Even if participantsdo recognize content connections between post-secondary and secondary mathe-matics, these connections may not be substantial enough to impact their practice.Indeed, Taylor recognized that the work in Galois theory was related to Task C, butthey did not perceive it as having potential in their pedagogical practice with thetask.I argue that the study of polynomials in abstract algebra does have potentialto impact a teacher’s mathematical knowledge for teaching. All participants whoresponded to the task recognized a substantial connection between graphing, theexistence of roots, and factoring. As many participants expressed, the graphingof a quadratic can be easily extended to understand the factorization of the poly-nomial in terms of its roots. This is also true in the case of cubics. However,these understandings were overextended to all polynomials, with the claim that “a136polynomial can be factored if and only if it has a root.” This misconception couldbe adjusted with exposure to the ideas of irreducible polynomials, as studied in acourse in abstract algebra. However, the participants who did have this courseworkexperience held the same misconception. The only secondary to post-secondaryconnection that was expressed by multiple participants was the connection of fac-toring to complex valued roots and the Fundamental Theorem of Algebra. That is,if a quadratic has no x-intercepts, it has a complex root. However, this conceptual-ization was extended to quartics and higher-order polynomials.Even though the behaviour and mathematics of quadratics was overextendedto all polynomials, the ideas and concepts of quadratics are a central component ofthe secondary curriculum. Many participants expressed the study of quadratics asa beginning access point for ideas such as graphing, factoring, and mathematicalmodelling. Participants expressed significant value towards problem solving, ap-plication, and mathematical modelling in Chapter 4 and viewed quadratic functionsas an accessible context for exploration and the development of problem solvingskills. At the time, however, participants also expressed concerns regarding thelimitations of applications involving quadratic functions. While participants wereaware of some commonly used applications, such as kinematics, they were unsureon how to make the concept meaningful and interesting for a wide variety of stu-dents. They were further concerned that repeating the same types of applicationsof quadratics might actually hinder student’s interest and perpetuate the notion thatmathematics has limited “real-world” applications.Participants’ engagement with these two tasks is a valuable first look into the137connections future teacher’s have constructed between their advanced mathematicsknowledge and the content they are to teach. The forthcoming chapters will furtherthe exploration of such connections in other content domains.138Chapter 6The Role of Limits, Infinity, andFormal Definitions in SecondaryMathematics6.1 IntroductionMany degrees in pure mathematics are characterized by two courses: Abstract Al-gebra and Real Analysis. These courses constitute the basis for more advancedcoursework, graduate studies, and mathematical research. Prior to taking thesecourses, many undergraduates enrol in a course on Mathematical Proof, in whichthe principles of proof, abstraction, and common definitions in many mathemat-ical domains are studied. In Chapter 5, I discussed Abstract Algebra at length,with respect to polynomial functions and factoring. In this chapter, I will exam-ine participants’ engagement with the tasks that relate to courses in Real Analysis,139Calculus, and the introduction of advanced mathematics. The interview tasks thatdraw on these subjects include the tasks on inverse functions, limits, and exponents.6.2 Inverse FunctionsI remind the reader that the task regarding inverse functions was the following:You are teaching a unit on inverse functions. What would you include in yourlesson plan?This task was partially inspired by the work of Zazkis and Kontorovich (2016)and Leung et al. (2016), where they examined teachers’ understanding of inversesand their associated notations. Among the five participants, Adrian and Casey en-gaged in this task. Before exploring their understandings of this topic, let us diveinto some of the mathematics of inverse functions to provide context for the inter-view data in Sections 6.2.2 and 6.2.3. The responses from these two participantsvary drastically in their appropriateness and depth of understanding.6.2.1 Mathematical BackgroundAmong all the concepts covered in this study, the notion of inverses has the earliestappearance in the school curriculum. The term “inverse” is very general, mathe-matically. Indeed, the first place in which an inverse is studied in school is withrespect to addition and subtraction. When the idea of 0 is presented to students,you could see questions like “if you have two cookies and I take two away, howmany cookies do you have?” The student would reply that they have no cookiesand that we denote the idea of “none” by the number 0. Similarly, one could ask“if you have two cookies and I share none with you, how many do you have?” The140number 0 is known as the “additive identity” in the real numbers. That is, if youtake a real number a and add 0, you still have a. Then, one would say that −a isthe additive inverse of a, since a+(−a) = 0.A similar idea is explored when students gain the key developmental under-standing that a fraction is a kind of number. Initially, when fractions are learnedin school. they are understood as a piece of a whole. For example, 12 would beviewed as one half of a whole of something, whether that something be the area ofa square, a pizza, or a collection of cookies. Later on, students learn that you cantreat 12 as a number. Just as you can add 2 to 1, you can add12 to 1 through theproperties of fraction addition, so that the mixed number 1 12 is equivalent to32 .Furthermore, fractions can be multiplied, just as whole numbers may be mul-tiplied, once additional properties of fraction multiplication are explored. Then,students can come to realize that 12 · 2 = 1, 32 · 23 = 1, and 12 · 1 = 12 . By “flipping”a number and finding its “reciprocal,” the product of the original number and thereciprocal equals 1. That is, given a real number a, a · 1a = 1. In formal language,we say 1 is the multiplicative identify element of the reals and that 1a is the multi-plicative inverse of a.The notion of inverse is then generalized to concepts outside of numbers, thefirst being the notion of an inverse function. Just as additive and multiplicativeinverses return a number to the identity, functional inverses undo the output ofa function and return an identity element. The identify element in the realm offunctions is the function that returns exactly what was put in: that is, f (x) = x.141However, in the previous examples, the identity was followed by an operation. Inthe instance of functions, the operation is function composition. The following twodefinitions are that of the identity function and inverse functions. In these cases,we take X to be an arbitrary set of elements.Definition 1. The function f : X → X defined by f (x) = x for all x ∈ X is theidentity function on X. We use Ix to denote the identity function on X.Definition 2. Let f and g be functions. f and g are inverse functions if and only iff ◦g = IDg and g◦ f = ID f , where D f and Dg are the domains of f and g, respec-tively. We say that a function f is invertible if and only if an inverse exists.This is the final appearance of inverses in the school curriculum, but not inthe post-secondary curriculum. Any student who takes a course in Linear Algebrawill see the notion of inverses for matricies. Similar to the earlier instances of in-verses and identity elements, these must be defined in the case of matricies. Special“matrix multiplication” and “matrix addition” are defined in the first week of anylinear algebra course, along with “additive” and “multiplicative” identity matricies.The notion of inverses is fundamental in all fields of mathematics. These con-cepts are studied generally in courses such as an introduction to proof, abstractalgebra, real and complex analysis, topology, differential equations, among manyother courses. Essentially, all mathematics majors will encounter the study of in-verse functions multiple times through the course of their mathematical studies.1426.2.2 Participant UnderstandingsAs mentioned above, Adrian and Casey were the two participants to engage withthis task. The responses given were very different between these two participants,with Adrian demonstrating a depth of understanding of inverse functions, whileCasey demonstrated multiple misconceptions at the secondary and post-secondarylevels.Adrian’s initial response to the question was immediately indicative of theirdepth of understanding:Adrian: I did a little bit about this awhile ago. So when you say in-verse functions, you mean inverse, not reciprocal functions? Likearcsin, arccos, log?Interviewer: Well, what’s the difference?Adrian: Well there is a huge difference!Interviewer: Tell me about it!Adrian: Well in one case, you have a function, you can talk about thereciprocal of that function which is 1 over the function. For ev-ery value, take 1 over that value. And typically that’s done in agraphing sense and if I want to graph the reciprocal of that func-tion, basically anywhere it is zero, I’ll have an asymptote andanywhere in between, if it’s big and positive, it will be small andpositive. If it’s big and negative, the reciprocal will be small andnegative. And you go through this whole sort of, usually whatyou’ll end up with, if you have some sort of polynomial curve,143you’re going to end up with something with big dips in places.It’s never zero because 1 over something can never be zero. Itcan be very small, but not 0, unless it goes to infinity, in whichcase I guess it would be zero, technically sort of.Interviewer: So this is reciprocal functions?Adrian: Yeah, these are reciprocal functions, which is not what we aretalking about, but some people call them inverses because youare taking the, technically this is the multiplicative inverse of thefunction, but that’s not what we are talking about. What we aretalking about is inverse functions, which is taking the oppositeof the function. So if I have f (x) = y, what I am talking aboutis f inverse of y and basically swapping x and y, which is whatis taught in school. You’re taking the opposite of that function,which is a weird idea in and of itself. And sometimes it’s eas-iest to look at how you would go about doing this in practice.Classically, what usually gets done is that you can get betweenthese two by simply swapping x and y, and that’s equivalent toreflecting over the line y = x and this usually gets paired aftertransformations so that they know how to do reflections over acertain axis. Some stuff I like to do is if you have a piece of paperthat is sufficiently thin and you can draw a thick black line, wecan actually flip it around like this and look through the piece ofpaper and see what it would look like and physically perform themanipulation. Which is kind of valuable, especially if students144are struggling with the actual mental how do I flip that? And Ithink beyond that, examples are really awesome. logs and expo-nents. You can talk about undoing functions. If they have alreadygone through lessons on exponents, they have probably encoun-tered logs, so the idea that you could undo this, going backwards,doing the opposite of this. sin, arcsin. I don’t think there areany other common inverse functions I can think of. You can talkabout domain restrictions, so in order for this to still be a func-tion you still need all your function things, which is more of agrade 10 topic, but it’s still brought in as to what it means to bean inverse function. And you can define your inverse functionsdifferently, depending on what range you want to talk about.In this excerpt, Adrian immediately spoke to detailed understandings on thedifferences between inverses and reciprocals, with respect to functions. Adrianwas later able to formalize their understanding of inverse functions via his claimthat “if g(x) is the inverse function of f (x), then f (g(x)) = g( f (x)) = x. In theirdescription of inverse functions, Adrian began to speak to domain restrictions, butthis was not explored further in the interview.Adrian brought up an interesting understanding with respect to logarithms.Other participants, when engaging with the task on exponents, conceptualizedthe logarithmic function as being independent of exponential functions. However,Adrian shared his conceptualization of the logarithmic function as follows:Interviewer: And so for the case of y = ex, how do we get the inverse function145log?Adrian: This reflection, by swapping x and y and defining something. Re-ally, that’s the way logarithms are defined, is what is the thing thatundoes an exponential? We call it log. This is what it looks likebecause we know this whole idea that if we have an inverse, weswap the x and y coordinates and we insist that it has these prop-erties and it’s logically consistent and now we have logarithms inour mathematical construction.Adrian shares their understanding that the construction of the logarithm is aconsequence of the properties of exponential functions. Historically, the construc-tion of logarithms came well before exponential functions, due to their properties ofmapping multiplication to addition. Regardless, the construction that Adrian sharesis the one that is most commonly seen in modern school mathematics. Overall,Adrian demonstrated a full-bodied understanding of functional inverses in this por-tion of the interview, sharing some common misconceptions students may make,as well as in-depth applications, which will be described in the following sectionon post-secondary understandings.When Casey started to respond to this task, they immediately conveyed mis-conceptions about inverse functions.Casey: You could have something to do with the very fact that these twocircles are reflected in the line y = x. Now, I mean there is a lotmore umm, you know, I guess inherent with inverse functions.146Interviewer: Such as?Casey: Well, if we want to, it depends because if it’s grade 11 then itwould be algebraic inverses.Interviewer: What is an algebraic inverse?Casey: It’s sort of flipping the parabola around (draws graph of y =±√x).Interviewer: So the inverse of this (graph of quadratic) is that (graph of y =±√x)?Casey: Yes, but then we start to talk about one-to-one and onto, right?But then we start talking about real number domain and you beginto ask why can’t it be the U here and just over here?In this portion, Casey demonstrates two misconceptions, but also alludes toa detailed understanding of domain restrictions with respect to inverse functions.The first misconception is in their first claim of reflecting circles being an inverse.The notion of “swapping x and y” corresponding to a reflection over the x-axis isrelevant to functions, but a circle is not a function. Immediately, Casey provides anexample that is relevant to a unit on symmetry and reflections, but not with respectto inverse functions.This same misconception is demonstrated when Casey draws the inverse ofa quadratic function in the shape of a sideways “U”. Casey begins to clean upthis misconception by mentioning domain restrictions. Indeed, if we consider the147function f (x) = x2, its inverse function is g(x) =√x. The relation h(x) = ±√x,as Casey drew, is not a function (it has two outputs for every input). Even thoughy= x2 is an easy example in most mathematical contexts, it is not a simple examplein the case of inverse functions. This is due to the following theorem:Theorem 4. A function is invertible if and only if it is one-to-one and onto.Casey had this vocabulary and used it throughout the interview, but also demon-strated misunderstandings while using it:Casey: It was in mathematical proofs, it was actually the first time I eversaw what an inverse really means. To be one-to-one and onto andif both those criteria are satisfied, then it is a true inverse.Interviewer: What do you mean by that? If a function is one-to-one and onto,then what does that mean?Casey: Well, injective and surjective.Interviewer: Then it is...?Casey: Then it’s like a real inverse, I thinkInterviewer: And what does that mean?Casey: Well, oh I know, it means the vertical line test. In grade 10, youtake just a ruler and go across the graph here and of course, justthe regular U (parabola) it’s all one-to-one but when you turn it90 degrees, no it’s not one to one, there are two points, and it’sthe inverse of the left hand side of the y graph there. So there’s aproblem there, it’s not one-to-one.148Interviewer: The sideways one is not one-to-one?Casey: Yeah, the sideways horseshoe.Interviewer: And what does it mean to be one-to-one exactly?Casey: That every x coordinate maps to every y coordinate exactly once.There is a unique point. But what I feel an inverse functions, andthat’s why I chose this task, is there’s a real, there’s a real problemhere, especially with this cubic thing happening here that I foundstudents had trouble with, is that well, the vertical line test, thisis the first time they have ever seen an S shape, but the one-to-one problem here is they kind of understand taking a ruler andseeing like you just go across the graph and see that verticallyevery point is one-to-oneInterviewer: So the vertical line test is for showing that something is one toone?Casey: Yeah, I should call this actually the “C” parabola shape, well, theletter “C”, no, you can just stick the ruler right there and you havetwo points. But the biggest disconnect is realizing that the lowerpoint cannot map to the one that’s over there.149Figure 6.1: Casey’s work for the “inverse” of y = x2In this dialogue excerpt, Casey starts off well by noting that inverse functionshave something to do with being one-to-one and onto. In order to clarify the math-ematics here, we say that f is a surjective (onto) function if and only if the range(i.e. the possible outputs of the function) of f is equal to the codomain (i.e. the setf sends its inputs to). For example, f (x) = x3, where f :R→R, would be an ontofunction, since the range of f (x) is all real numbers, which is equal to the codomainof R. However, the function g(x) = x2, where g : R→ R, is not one-to-one, sincethe range of g(x) is R≥0, which is not equivalent to R. One could construct g to beonto, by changing the codomain. However, this is not customary in practice.Similarly, a function is one-to-one if every output comes from a single input.For example, f (x) as above would be one-to-one since there are no two inputs thatyield the same output. However, g(x) as defined above is not one to one, sinceevery output (other than 0) comes from exactly two inputs.From this, we can observe Casey’s understanding of what it means for a func-150tion to be one-to-one is flawed. In fact, one does not need these more advanceddefinitions to see this. The vertical line test is presented in high school as a test todetermine whether or not a graph is a function. However, Casey uses the verticalline test to justify that f (x) = x2 is one-to-one (which it’s not) and that the graphof ±√x is not one-to-one. There are two major issues with this second remark.Firstly,±√x is not a function, since the various inputs go to two outputs. As a con-sequence, trying to remark whether this graph is one-to-one function is impossible.The function of g(x) = x2 is often utilized as a counterexample to the notions ofone-to-one and onto, due to students’ familiarity with it. The use of it as a primaryexample for introducing inverse functions could be more confusing for students,rather than helpful. As a teacher myself, this is an example I would avoid untillater in a lesson plan.When describing inverse trigonometric functions, Casey brought in the notionof one-to-one, but without the misconceptions of earlier:Figure 6.2: Casey’s written work for the inverse of sin(x)Casey: This would have to be restricted one-to-one, this we can callsin(x) and you can see that this looks like a [graph of] sin(x)151but we would need to know to stop here at this point and then justmake sure it’s back at this point to make arcsin(x).Interviewer: What do you mean of this point back to that point?Casey: Well, to make it one-to-one, so it does not overlap. Because theminute you cross this root here, you’re in trouble. You get a repeatwith the first one because it’s periodic.In this excerpt, Casey demonstrates a correct conception of one-to-one, with aminor flaw. Casey was able to effectively communicate that in order to define theinverse function arcsin(x), we needed to restrict the domain of sin(x). This, how-ever, was the only instance throughout the interview where Casey demonstrated acorrect conceptualization. Later on, Casey returned to their generalization of in-verses “switching x and y” with respect to their example with circles (see Figure6.3):Casey: I wanted them to do this concept here [of (x,y) to (y,x)]. Anyold way they wanted to show me, as I walked. around that oh,yeah, (8,1) is now (1,8). If you have the Euclidean, because theinversion here is y = x, you just flip the image up here and youcan do that.152Figure 6.3: Casey’s work for circles and inversesCasey seems to have generalized the notion of a reflection to that of inversefunctions. Although there is a connection between real valued functions and re-flections over the line y = x, this can not be generalized in the way Casey outlinesabove. Through this dialogue with Casey, it is evident that they have extensiveknowledge of inverse functions, but also hold some misconceptions that could im-pact their pedagogical practice.6.2.3 Post-Secondary ConnectionsThe analysis above documents participants’ understandings of inverses that is rele-vant to their teaching in a secondary classroom. In the following section, I describethe understandings participants held in the context of post-secondary mathematics,and the ways (if any) they thought this knowledge could be brought into their sec-153ondary classrooms.Adrian continually mentioned exponential and logarithmic functions through-out their description of inverses. Through their discussion, they thought that dis-cussion of logarithmic scales could be of great value to secondary students:Adrian: Well, even if they heard of scales that are logarithmic, but havenever thought of what it is to be a logarithmic scale. It’s weird,even now, doing the mental acrobatics to actually figure out thatif it is increased by a factor of 10, it really is increasing by 1. It’sbizarre, I think, for a lot of students.In response to the number e, I inquired as to why Adrian thought it was soimportant for students to know and care about e:Adrian: Honestly, I did not understand why e was so important until I didcalculus. If y = ex and I take the derivative of y, it’s also equalto ex. And this is the only function for which it is true. That’skind of neat. This is the only function for which its derivativeand all of its subsequent derivatives are equal to the value of thefunction.Interviewer: And how do you define e? When you think of e, what is the firstdefinition you go to?Adrian: This one [the differential equation], 100%.Interviewer: Any others?154Adrian: I know there is one definition where it is like,(1+ 1n)n, limit as ngoes to infinity, where this comes back to compounding interest.I guess the other I can think of is just Taylor series, but that’s theonly other one I can think of.Figure 6.4: Adrian’s work for the value of eIn this excerpt, Adrian shares three different definitions of e that could beused in teaching (see Figure 6.4). Each of these definitions was covered in post-secondary mathematics, but as is mentioned at the beginning, Adrian did not reallylearn the importance of the number e until university. This is an explicit example ofadvanced mathematics contributing to a secondary teachers’ mathematical knowl-edge for teaching in the secondary classroom.Casey’s connections of inverses to post-secondary mathematics were based incoursework. Casey’s immediate mention of inverses was with respect to a coursein mathematical proof, which was elaborated in an excerpt in the previous section.When prompted to discuss. inverses in a post-secondary context, Casey continuedto talk about mathematics without accurately describing what was being discussed.For example, Casey discussed the field of algebraic topology as being relevant to155inverse functions, but was unable to articulate how: “Sort of the topological senseof these inverse functions go into more in the realm of the hypercube kind of thing.Because it’s a lot more topological wise where you can deform a shape to ulti-mately create an inverse rather than just taking the parabola and going, you needthe Euclidean plane.”At this point, Casey was on the verge of developing a very interesting exam-ple of inverses, which aligns with the notion of “undoing,” as Adrian mentions isimportant with inverses. Indeed, Casey is referring to functions which will deformone space into another, while the “inverse” will bring us back to where we started.Precisely, what Casey is referring to is known rigourously as a homeomorphism.Abstractly, homeomorphisms are continuous, bijective functions from one spaceto another, which have a continuous inverse. The classic example of a homeomor-phism would be the deformation of a coffee cup into a donut. Indeed, if a coffeecup were made out of moldable clay, one could “deform” and manipulate the clayto transform it into a donut. This can be done without tearing or poking holes intothe clay.While Casey had this example in their mind, they were unable to articulatethe ways in which it was connected to the earlier discussions on inverses (throughundoing a function). Rather, Casey attempted to relate it to the example of theparabola, which as was discussed earlier, is an inappropriate example for thisprompt. Although Casey held this more advanced knowledge, they were unableto link it to what they already knew. Once again, this example is a demonstrationof the power of post-secondary mathematics knowledge in providing examples and156contexts for building a deeper understanding of secondary mathematics concepts.6.3 LimitsThe task on number sense and limits which participants responded to was the fol-lowing:A student is working through a problem and asks you whether 0.99999 . . .= 1.How would you respond?Of the five participants, Adrian, Bailey, Jaime, and Taylor shared their under-standings and the pedagogical choices they would make in the classroom. This taskinterrogates participants’ understanding of number, limits, and approximations.6.3.1 Mathematical BackgroundStudents explicitly encounter the notion of a limit if and when they take a mathcourse that examines the notion of asymptotes. Asymptotes are first seen in rela-tionship to rational functions, which may be studied in grades 10 and 11. Asymp-totes have significant applications to the physical and social sciences, through themodelling of various phenomena that “level out” over time.In high school, asymptotes are often introduced as features of graphs. However,the precise definition of an asymptote requires the notion of a limit. These may beobserved through the following definitions.Definition 3. The line y = L is a horizontal asymptote of the graph of f (x) iflimx→∞ f (x) = L or limx→−∞ f (x) = L.157Definition 4. The line x= a is a vertical asymptote of the graph of f (x) if limx→a−f (x)=±∞ or limx→a+f (x) =±∞.The avoidance of the limit definition of asymptotes is challenging, as it oftencauses a misconception that asymptotes are lines that graphs “cannot cross,” result-ing in a less than full-bodied understanding of what a limit represents.The notion of a limit is the foundation of all calculus. Indeed, the derivative andthe integral are both limits, at their core. All courses in Calculus begin by definingthe notion of a limit. However, the level of rigour in which the limit is definedvaries. In Stewart’s Calculus, which is probably the most widely used Calculustext in North America, the definition of the limit is as follows:Definition 5. We write limx→a f (x) = L if we can take the values of f (x) arbitrarilyclose to L (as close to L as we like) by taking x sufficiently close to a (on either sideof a), but not equal to a.This definition uses non-rigourous language to define a limit. Computations oflimits are then relegated to a "plug and chug" method, where most of the functionsstudied are ones which are continuous, have a removable discontinuity at a, or havea jump discontinuity at a. This restricts the types of functions that students haveexposure to in first-year calculus, while simultaneously simplifying the complexityof a limit.The proper definition of a limit is seen once a student enters a course in RealAnalysis. Rudin’s Principles of Mathematical Analysis is a widely used text inNorth American undergraduate real analysis. The definition of limit is given as158follows:Definition 6. We write limx→a f (x) = L if there is a point L with the property that forevery ε > 0 there exists a δ > 0 such that | f (x)− L| < ε for all points x in thedomain of f for which 0 < |x−a|< δ .This definition provides a mathematical context for the words "sufficiently"and "arbitrarily" that are used in Stewart’s definition of limit. Through this use ofrigourous language, the family of function in which one can study limits of func-tions opens up dramatically.This is not the only instance of the notion of a limit that students run into intheir mathematical careers. Indeed, the notion of a limit is observed in the con-cept of sequences, which is a concept that may be studied in school at a veryearly age. For example, the Fibonacci numbers are an example of a sequence.This sequence is commonly used with school age children to connect mathemat-ics with nature and art, as this sequence has connections to the organization ofpine cones, the nautilus shell, the golden ratio, and other natural phenomenon.Mathematically, the Fibonacci sequence is the sequence of numbers {Fn}n≥0 ={1,1,2,3,5,8,11,19,30,49, . . .}. Each new term in the Fibonacci sequence is gen-erated by adding the two terms prior. That is, Fn = Fn−1+Fn−2.When students first encounter sequences in Calculus, the definition of the limitof a sequence is colloquially simplified, as it was for functions. The followingdefinition is seen in Stewart’s Calculus:Definition 7. A sequence {an}n≥0 has a limit L if we can take the terms an to be as159close as we like to L by choosing n to be sufficiently large. Then, we say limn→∞an = Land that the sequence converges to L.The use of colloquial language in this definition may lead the learner to makecertain generalizations about the convergence of sequences due to a limited collec-tion of sequences which may be studied. Again, the formal definition of a conver-gent sequence may be viewed in Körner’s Companion to Analysis:Definition 8. We say that a sequence a1,a2, . . . tends to a limit L as n tends toinfinity if, given any ε > 0, we can find an integer n0(ε) such that |an−L|< ε forall n≥ n0(ε).Once again, one can observe the rigour given to the words “sufficiently” and“arbitrarily” when the context is changed from Calculus to Real Analysis. Manylearners come into courses in real analysis with pre-existing conceptions of whatit means for a limit to exist, based on their experiences and definitions in calculus.In all honesty, I was one of these students. When I entered my first course in realanalysis, it was as though I had learned nothing at all in my calculus courses. Ifanything, I would argue that my studies in calculus hindered my initial learning inreal analysis, since I came in with certain ideas of what was the “right” way to dealwith limits, which was invalid in this new context.The final instance of limits which will be discussed in this section, is that ofseries. A series, at its core, is an infinite sum of a sequence.Definition 9. The series ∑n≥0an converges if and only if the sequence of partial sumsgiven by {Sn}n≥0 = {a0,a0+a1,a0+a1+a2, . . .} is a convergent sequence.160The convergence of a series is dependent on the notion of a limit, since wewant to show that the sum of the various terms of the sequence have a limit, as ngoes to infinity. As will be described in the next section, one participant used theirunderstanding of sums, limits, and series to describe the equivalence of 0.999 . . .=1 in their interview.6.3.2 Participant UnderstandingsUpon first glance of this problem, three out of four of the participants went to thesame example: the equivalence of 13 and 0.333 . . ..Taylor: Starting with the fraction 1 over 3, cause that leaves us point 333that’s going to be infinitely long and if I multiply by 1 over 3 by 3,that gives us 1. But if you are on the decimal side, that’s gonna bepoint 9999, so yeah. I would say they are equal. Yeah, because ifyou look at the fraction part, it clearly gives us 1, but the decimalpoint gives us infinitely long repeated decimal. Yeah.Bailey: You can start with a fraction. We know that a third equals 0.333repeating forever and then from that it’s clear that 3 thirds is 1.Because one third plus one third plus one third is point 999 for-ever and no matter where we start, we are just gonna get nines.We can take as many threes as we want for the dots, if we take itto infinity, it’s just gonna be 9999 forever.Adrian: This one is an easy one! 1 over 3 is 0.333 dot dot dot. And if wemultiply by three, over here we get 1, but on the other side we get.999 dot dot dot.161Figure 6.5: Taylor’s work for 0.999 . . .= 1These quotes are interesting in that they use a seemingly less obvious equiva-lence to prove another equivalence. Indeed, it seems more intuitive that 0.999 . . .=1, rather than13= 0.333 . . .. Each of these participants used an equivalence thatholds the same amount of uncertainty as the initial problem, if not more.All participants felt the need to “concretize” this problem for students. Thiswas particularly relevant to Jaime, who believed that 0.999 . . . 6= 1.Interviewer: So what would be your mathematical justification to a grade 9student about this problem?Jaime: Just figuring out why they are different? I would take the ap-proach of using an analogy. Like 12 and 1 are not the same. But ifyou are looking at half a centimetre versus one centimetre froma metre away or ten metres away, there is a technical difference.Well, what if we make it .75 and .8? Then would you say they arethe same? Like, could you come up with a definitive rule on that?And I think that would be the way I angle that, is that any kindof rule of making them the same is arbitrary. I’m thinking again162about analogies with measuring. These two things are a little bitdifferent.Interviewer: If a student wanted an arithmetic argument as to why they weredifferent, what would you say?Jaime: Like nine ninety nine pricing. If you have it as two digits, ifyou pay with coins, you get nothing back, but a bill, you getsomething. So they are fundamentally different.Jaime wanted to provide students with a concrete context for the abstract issueof infinity, which is an excellent way to build a more robust understanding. How-ever, it seems that through these concrete examples, Jaime compromised their ownunderstanding of infinity. As Bailey noted, “I would need to see a lot of evidenceas to why that’s true in different ways. Cause it just feels unnatural” and as Taylorstates, “this tells us that certain things are true even though they are not intuitive.”Jaime continually discussed the idea of understanding this problem from a “prac-tical point of view” and “eventually having to truncate and make a compromise.”However, in compromising and truncating, the conceptual complexity of the prob-lem may be lost and may promote the development of misconceptions in students’understanding of infinity. Adrian captures this challenge in the following excerpt,with an excellent example for building understanding of infinity:Adrian: I think the biggest issue is infinity. The idea of what it means tobe truly infinite. Maybe introducing the idea of the infinite hotelor the fact that there is the same number of even numbers as thereare all positive numbers. I think this sort of strikes at the core163of what it means to be infinite, which I think is a really toughproblem to wrap your head around. I would have them look athow many numbers there are, different types of infinities, or whatit means to be infinite. Not just something that’s very large.6.3.3 Post-Secondary ConnectionsThe mention of limits and post-secondary mathematics was prevalent throughoutthis task. All participants made mention of limits, in one way or another. Throughthe interview, it became clear that the problem hinged on the conceptualizationof a number as a limit. This will be explored further through the analysis of theexponents task, but we first focus our attention on the concept of number as limitthrough participants’ engagement with this task.Of all the participants, Adrian shared the most robust understanding of thisproblem in terms of limits:Interviewer: How could you prove that 0.999 with infinitely many digits isequal to one?Adrian: Hmm, my first instinct is limits, but I’m not sure how to employlimits here.Interviewer: So what is your instinct to go with limits?Adrian: Cause it looks like a limit here. In my mind, whenever I amthinking about infinities or large lists of things, I start thinkingabout limits. Umm, I mean, I could maybe do a limit of....ok,164how could I set this up like a limit? I could do a magnitude of1 minus the sum as i goes from 1 to n 0.9 times 0.1. Actually,I’ll do this from 0 to n and 0.1 to the power of i and look at thelimit as n goes to infinity here. Now, I can set up this as a limitand maybe try to convince myself that this limit should be zero.That the distance between these two numbers should be 0 andthen these two numbers would be the same.Figure 6.6: Adrian’s work for 0.999 . . .= 1Adrian was able to construct 0.9999 . . . as limn→∞n∑i=00.9(0.1)n (see Figure 6.6),by writing the number as the limit of a sum of a sequence. In this case, whatAdrian constructed is equivalent to the geometric series∞∑i=00.9(0.1)n. Using con-cepts from Calculus and Real Analysis, one can prove that∞∑i=00.9(0.1)n convergesto0.91−0.1 = 1. Thus, the limit that Adrian constructed proves the equivalence of0.999 . . .= 1. This construction is in contrast to the construction Bailey presented,using the concepts of asymptotes of real-valued functions, as observed in Figure6.7 and the following dialogue:165Figure 6.7: Bailey’s work for the limiting behaviour of 0.999 . . .Interviewer: Is there any material from your math degree to help you answerthis question.Bailey: I can’t think of a simple answerInterviewer: And that’s ok. You can be as complicated as you want it to be.Bailey: I guess I can’t, it really seems, I don’t know. I can’t even verbalizewhy that’s true other than it’s a system we created and in oursystem it clearly is true that .99999 . . . equals 1. I assume thereis some reason that is much more eloquent than that, that I’mnot thinking about. But talking about approaching infinity andinfinite number of digits.Interviewer: So what mathematical ideas do you need for that?Bailey: Like Calculus, right? Like asymptotes and approaching infinity.Like some stuff from proofs about different sizes of infinity is166good knowledge to back up your understanding of it.Interviewer: What’s a fundamental idea needed to define an asymptote?Bailey: Approaching infinity? Approaching some number, like limits? Isthat what we are looking for?Interviewer: So how could we define using a limit that .999 is equal to one?What kind of limit could we look at for this?Bailey: We could just look at a function that, you know...Interviewer: What kind of function would you look at?Bailey: Like 1 over x would probably work? Actually, that one wouldn’treally work. You could set up a graph that has that asymptote,like plus one. And talk about limits. But that wouldn’t help meexplain it to a grade 10 student.Interviewer: Can you think of a limit that would be equal to one but also havesomething to do with this .999 infinite?Bailey: Like we would want an equation that is approaching one from thebottom, so that would be a 1 over x plus something. Here’s mygraph. I’m rusty on all my graphs look like. It would just be 1over x plus 1. Would that be right? Yeah.Interviewer: And so as we take this limit we are getting one, but what doesthis have to do with .9999?167Bailey: Because 1 minus a tiny tiny piece is .9999, right? So that asymp-tote is approaching it.Initially, Bailey was unsure of what advanced mathematics could be used tosolve this problem, but eventually landed on the notion of asymptote. In the pro-cess, Bailey revealed some misconceptions of the asymptote concept. This is evi-dent in the following excerpt:Interviewer: Let’s say you had a really talented student in grade 11. How couldyou take these ideas and make it accessible to push them beyondwhat they are doing in the curriculum?Bailey: Like you could draw a graph for them that has an asymptote ap-proaching one, from the bottom and say, let’s look at differentvalues and calculate a table and see that it is clearly going to one.But I don’t know if that would really explain it, but that kind ofshows that it’s never getting to oneInterviewer: And why’s that? Could you elaborate?Bailey: Because an asymptote never gets there, it gets close, but nevergets there. So even though with this example, the limit is 1, it’snot, I mean, I think it might be counterintuitive. I don’t know if itexplains that it’s true. It might almost be an argument as to whyit’s not true. Cause like it’s not actually ever getting there.The primary misconception here is that “an asymptote never gets there.” In-deed, many of the examples students see in school mathematics suggest this. How-168ever, if one examines the limit definition of an asymptote, this is not implicit. In-deed, the function f (x) = 1 has a horizontal asymptote at y = 1 since limx→∞1 = 1.Similarly, there are functions such as g(x) = sin(x)x , which cross the asymptote ofy = 1 infinitely many times. This misconception of asymptote is well-documentedin the literature (Kajander and Lovric, 2009).However, this observation is somewhat tangential to the task. The primaryobservation from these excerpts is that Bailey does not appear to conceptualize0.999 . . . as a limit. Indeed, this could be inferred from comments such as “Because1 minus a tiny tiny piece is .9999, right?” Bailey seems to think of 0.999 . . . asbeing independent from this problem. Rather, Bailey wants to look at concreteinstances of 0.999 . . ., that is, instances such as 0.999 . . .9︸ ︷︷ ︸n digits.6.4 ExponentialsThe task on numbers which participants responded to was the following:Your students are confused as to why they can define and compute 2√3. Howwould you respond?At its core, this question attends to participants’ understanding of exponen-tial functions and their extension to non-integral powers. Of the five participants,Adrian, Bailey, and Taylor responded to this scenario.1696.4.1 Mathematical BackgroundThe notion of an exponent is first seen in school with integral exponents. That is,an = a ·a · · ·a︸ ︷︷ ︸n times. That is, an is a times itself n times. We say that a is the base andn is the exponent. In this definition, n must be a counting number. Otherwise, thenotation of “n times” does not make much sense. Indeed, what would it mean tomultiply something by itself 1.2 times?This notion is later extended to negative exponents by defining a−n =1anandis extended further to rational numbers by examining connections between radicalsand exponents, where n√a = a1/n and ( n√a)m = amn .Exponents are examined from a mostly computational basis, until the conceptof exponential functions is studied in late secondary school. The exponential func-tion with base a is defined as f (x) = ax. The graphs of these functions are oftenmotivated by having students create a table of values for a given base a with vari-ous inputs of x (normally integers) and connecting the dots to create a continuousline.However, prior to this “connecting of the dots,” learners had only been exposedto evaluating powers which were integral or rational. What about all of those num-bers in between (i.e. irrational numbers)? The exponential function is defined forx values which are irrational, but this idea is never explored until a student takes acourse in real analysis. How does one define numbers like 3pi , 2√3, and ee?170The theorem for rational exponents is dealt with in detail in the first 10 pages ofRudin’s Principles of Mathematical Analysis, with the problem of extending to allreal numbers being an exercise in the first chapter. Indeed, the study of the numberssystem that provides the basis for the remainder of the study in real analysis isnecessary before progressing to complex mathematical notions that depend on thissystem. Included in this study is the recognition of irrational numbers as the limitof a sequence of rationals. This is an instance of concepts which implicitly appearin secondary mathematics, but cannot be properly dealt with until a learner takesa course in real analysis. Even if this concept cannot be covered in secondaryschool, the teacher with the awareness of the complexity of extending exponentialpowers to rational and irrational powers could bring context and take extra carewhen teaching related material with their learners.6.4.2 Participant UnderstandingsAs mentioned above, the participants who chose to engage with this question wereAdrian, Bailey, and Taylor. Overall, all participants were able to engage with thisproblem at the secondary level, but did not have rigourous mathematical explana-tions to justify the construction of irrational exponents. Despite this, there were anumber of commonalities amongst the responses. In particular, the use of rationalexponents and inverse functions.At the very beginning of the interview, Taylor made the observation that “hav-ing the exponent that’s not a whole number is just weird to [students], I guess.”All participants mentioned the “concrete” nature of whole number exponents: theability to multiply a number by itself a finite number of times. From this, partici-171pants shifted their attention to rational exponents and square roots, but recognizeda large conceptual shift for students:Taylor: Well, the square root of 2 is actually the same thing as 2 to thepower of a half. And this is true by definition, because whenyou square the square root of 2, that the student may know ornot know, but this is by the definition of square root and you’resquaring it again, which means that you’re multiplying the num-ber by itself, then you get the square root of 4 which is 2. That’swhy it’s 1 over 2 because when you have 2 to the power of a halfand square it again, you get 2 to the power of 2 over 2, which isjust 2. So, square root of 2 and 2 to the power of a half are thesame thing. So, the exponent doesn’t always have to be a wholenumber.Bailey: A student I tutored had a lot of trouble conceptualizing what pow-ers to a half are. Because all of the sudden we go from expo-nents meaning times itself so many times and then somethingelse. Like, negative one doesn’t mean we are timesing negativeone times, it means something else. I conceptualize it as the in-verse functions. Instead of multiplying by itself twice, we had tomultiply the previous thing to get 2.172Figure 6.8: Bailey’s work explaining rational exponentsAdrian: You can start talking about fractional powers, like the square root,cube roots, and so on and so forth. I would probably do thisfrom the inverse. If this is, if I have x = 21/2, I can square bothsides and get this idea. Something squared is equal to 2, what isthat something? You’re throwing square roots in, which studentsdon’t like as it is.All participants instantly used the relationship between radicals and roots tojustify rational exponents. Furthermore, relating the rational exponents (in partic-ular, exponents of the form 1n ) allowed participants to relate non-integral powersback to the more concrete territory of integral powers. However, participants lostthis convenience when the conversation shifted to irrational powers:Taylor: I could approximate it with a rational number. I want to be carefulas to not exceed the value of√3. But also it has to be smallerthan 2, so tightening the range of what potential value can be. Wedon’t know that this value exists. Where does it lie on that graph[of the exponential function] would be the question to students. Ithink that narrowing down the range would be able to help themunderstand the value that it takes.173Bailey: If we think of 2√3 as 2 to the 1.7 something, I would say it’s like2 to the power of 1 and then some fraction. Obviously we can’twrite it as a fraction, but we could write 1.75 as 1 and three quar-ters. Like, we understand what that means. We can understandthat this is root four to the power of 3. And from there, I feel likewe could elaborate to say that even though we can’t put it into apower and that it’s very close to this.Adrian: If you can understand 2 to the 1.7, I don’t think it’s too much toextend it 2√3. Like, get students to understand that these thingscan exist on their own. And maybe this is a good way of break-ing it down. it’s 21 times 20.5 times all these things multipliedtogether and you can find each of those separately and estimatethem in different ways. Maybe that’s a good way of doing it? It’stwo to the 1.7 ish, we don’t really care.Figure 6.9: Adrian’s work finding a value of 2√3174The idea of relating an irrational exponent to a rational exponent is a veryreasonable approach and is certainly a way to define irrational exponents. However,no participant was familiar with the formal definition and drew upon the notion ofapproximations and other mathematical ideas (such as continuity) in order to justifythe concept of an irrational power:Taylor: I mean, I would say that the exponent can be continuous thatby that I would say that it doesn’t have to be a whole numberor rational value. But something that falls out of range of thatwhole number and rational numbers. So irrational, since it lieson that number line from zero to infinity, it lies on that numbercontinuum. I would say that square root of 3 is also somewhereon that, since I know the value, somewhere between 1.7 and 1.8.So you can have the value as an exponent too. And as you sawthrough the graph of it, the exponent function is continuous, sothere’s a value for 2 square root of 3 as well.Figure 6.10: Taylor’s work for work defining 2√3175Bailey: So yeah, I would just take this and make it a fraction over a mil-lion, hundred million. So essentially we are taking the hundredmillionth root and taking the 7 hundred whatever power. I thinkit would be pretty easy to show it does exist, but where I think itwould be hard would to have some reason as to why it exists. Wecan come up for something as to what it means for fraction. Likesquare root, cube root. That has a definition that makes sense.Whereas once it becomes an irrational number, I don’t see anykind of concrete definition. I guess you could do something likeI’m looking here. If you take 2 to the root three and take it to theroot three and then you get 2 to the 3. Maybe you can get some-where with that? You’ve obviously got lots of irrational powersbut maybe if you played around with that maybe you could workit to someplace that shows something.Adrian: Plug it into Desmos and see what happens. You get this graph.What can you recognize? You recognize two to the 0 is one, twoto the 1 is two, two to the 2 is four. You can even recognize thepoints down here...how about the points in the middle, how do weaccess these points? And giving them this idea that you can workin between, the idea that there is something between these twonumbers, that there is a continuous pattern that exists. Umm, thatmight be a good way to do it, but it’s still not satisfying. It’s notas, like the fact that you can plug this into Desmos doesn’t proveit’s a thing. For some students it will be fine, for some students it176won’t be. And as a mathematician, I’m not satisfied with it. AndI’m trying to remember how this was actually introduced to mebecause it’s been so long.Figure 6.11: Adrian’s work extending exponents to irrationalsAdrian and Taylor had a similar idea of bringing the graph of exponential func-tions in to help convince students that they could define and compute 2√3. How-ever, as Adrian notes, this is not sufficient. Both Taylor and Adrian thought theycould bring in the notion of continuity in order to justify the existence of 2√3, butthis is not mathematically sound, as Adrian notes. In fact, existence is a necessarycondition for continuity.Definition 10. A function f (x) is continuous at x = a if the following three condi-tions are all satisfied:1. f (a) exists1772. limx→a f (x) exists.3. limx→a f (x) = f (a)If continuity is to be used as a justification, existence is necessary. This is an in-teresting example that demonstrates how advanced mathematical knowledge couldnegatively impact a teacher’s understanding of secondary mathematics. Indeed, thenotion of continuity is a concept that can be explained non-rigourously and is oftenassociated to the idea of drawing a line and never lifting your pen up. However,the rigourous definition requires an understanding of what happens at a particularpoint, where the notion of a point in Euclidean space is abstract in and of itself. AsAdrian remarked, “the fact that you can plug this into Desmos doesn’t prove it’s athing.”Of all the participants, Bailey came the closest to a rigourous definition:Interviewer: So how would you conceptualize an irrational power?Bailey: I feel like we could elaborate to say that even though we can’t putit into a power, that it’s very close to this. And you could almostdo the whole approaching from both sides thing, using rationalnumbers. But yeah, I don’t actually have a good explanation forthat and I don’t know that there is one. It’s gotten so abstract. Idon’t know of a concrete explanation for how it exists.Bailey was very focused on concretizing the idea of exponents, seeing as thefirst conceptualization of integral powers is very concrete. As Bailey notes, evenrational powers can be concretized “because we have a concrete representation of178where square roots exist in nature and there are cool origami proofs. And if youcan do that, you can convince them [students] that square root two exists.” As willbe elaborated below, this dialogue brings forward an interesting perspective on theissue of “exactness” and “existence.”6.4.3 Post-Secondary ConnectionsThe notions of “defining,” “exactness,” and “existence” were problems for Taylorduring this task, as demonstrated in the following excerpt. As will be seen, Taylorcame very close to the proper definition, but their progress may have been hinderedby confusing these three notions.Taylor: I’m pretty sure that there is a way to approximate this...ummm, Ithink it involves some series. But yeah, I would have to googleit. I’m pretty sure there is a way to do itInterviewer: What would you use the series for? Just kind of roughly?Taylor: So it’s not square root of 3 for the one I remember but finding thevalue of pi, there is a way to know the exact value of pi using ei-ther the Taylor series or the Newton’s method. I can’t quite recallwhat the name was, but yeah. But the process is quite beautiful.Yeah. But yeah, so in terms of finding square root of 3, I thinkthere would be a way to find it using Taylor series, which I’m notquite sure now, but yeah.Here, Taylor wants to use the idea of series to deal with√3. This is not far offfrom what Adrian and Bailey suggested in decomposing√3 into rational pieces.179However, the issue of defining the irrational exponent still exists. Taylor’s goal atthis point it to find the value, rather than justify existence. In order to try to seeif Taylor could use this idea to get to the definition, the interviewer hoped Taylorwould recognize√3 as the limit of a sequence of rationals.Interviewer: So if you were to associate a series to the square root of three,what kind of series would you associate it to? What would theseries be made up of?Taylor: Mmmmm, I’m not so sure.Interviewer: No?Taylor: No.Interviewer: Could you associate a sequence to it?Taylor: Ummmm, yeah, I think, you can associate a sequence, it’s notvery specific but let’s say we have a value that is approachingfrom, I don’t even, I don’t know if I’m using the right term.Interviewer: That’s okay don’t worry.Taylor: Approaching from above and approaching from downward thenthere is some value that is in between. Then finding a correctfunction of these two would allow us to find the value of squareroot of 3.Taylor shifts attention from defining to finding an “exact” value. This is inter-esting because pi and√3 are exact values, while other methods, such as a series,180are alternative representations which converge to√3. This dialogue continued to apoint in which a misconception of irrational numbers was unveiled:Interviewer: So what you are saying is you would want to define two se-quences that would approach square root of three?Taylor: Right, yeah.Interviewer: And generally can you think of a....is there an association gener-ally between irrational numbers and sequences?Taylor: Yeah, umm, I would think so, yeah.Interviewer: And what kind of relationship would there be?Taylor: Because infinite series can incorporate the problem of infinitedecimal point and sometimes it’s not very repeating per se, I meannot repeating as decimals, so when those series are incorporatedthey would also involve a value of irrational numbers such assquare root of two or three.It is in the last remark from Taylor where they claim that a series representationof an irrational number should involve irrational numbers. This is not the case. In-deed, as mentioned in 6.4.1, all irrational numbers may be the limit of a sequenceof rationals. Although Taylor refers to series in their justification, it is not truethat series which converge to irrational numbers contain irrational terms. Indeed,the famous series ∑n≥11n2= 1+14+19+125+ . . . is a series with all rational termsconverging topi26, which is irrational. This is one example of many.181Overall, from these excerpts comes forward key developmental understandingof exactness and irrational numbers. The “exact” form of pi is pi . By not writingout the decimal expansion of pi , “exactness" is maintained. The case is similarfor√3;√3, e, pi , and√2 are all exact values. At this moment, super computersare searching for the next digits of pi and this process will never stop. Through adecimal expansion, information about the number is lost, as is “exactness.” Tay-lor’s remarks are mirrored in Bailey’s lament of wanting “concrete representationsand explanations” for irrational numbers. Indeed, one might suggest that irrationalnumbers are concretely irrational.One final path that both Taylor and Adrian saw as being relevant for this taskwas the relationship between exponentials and logarithms. Through this discus-sion came an interesting conception of the logarithm function. In the high schoolcurriculum, exponentials are introduced first, partly due to the "concrete" nature ofthe definition of ax, when x is a natural number. Much later in the curriculum, thelogarithm base a is defined as the inverse of the exponential function f (x) = ax.That is, g(x) = loga(x) is the function such that ( f ◦g)(x) = (g◦ f )(x) = x, as dis-cussed in 6.2.1.While Adrian tried to bring logarithms into their strategy, they quickly aban-doned that route, recognizing that it did not simplify the situation in any way:Adrian: Something squared is equal to two, what is that something? Iguess you can sort of use the same idea here, but it’s not as neat.Interviewer: And why is it not as neat? Could you elaborate on that?182Adrian: I think if you can connect this to fractional powers, they can un-derstand it as a cube root, or even if you have two to the two overthree, you can think of it as two squared, cube root. But this onedoes not have as nice of an interpretation. You’re now thinkingof this as, yeah.....uhh, you can’t think of it in terms of multiplieda certain number of times or taking an integer form. Umm, yeah,and I mean even if you use this argument, this rationale, you cantake the square root three root (i.e.√3 x) of both sides. Some-thing to the root of square root three (√3 x) equals two, which isprobably not helpful and makes it worse.Although Taylor did not respond to the task which explicitly explores under-standings of inverse functions (see 6.2), the dialogue that follows unveils an inter-esting conception:Interviewer: Do you have anything you would want to add to this? Any ideasfrom your math courses that you took that are coming up thatcome to mind with this problem?Taylor: Since I am dealing with the exponent function I would introducethe idea of the inverse of two to the power of x.Interviewer: And how would that be useful?Taylor: That would be useful because, not really useful, but again howinverse function the exponent function relate.Interviewer: And how do they relate?183Taylor: They relate because it gives identity function?Interviewer: Go ahead, you’re fine!Taylor: Let’s say this is an exponent function, y = 2x. To find the in-verse function there is logarithmic function: y becomes x and xbecomes y, then we are trying to figure out what y is in terms ofx here. So I would bring log on both side to get log(x) equal tolog(2) to the power of y, which is y times log(2) so that againwould be log(x) over log(2) which is why that would imply thatby the logarithmic rule, would be log2(x) which is y. This is aninverse function of 2x. Then it’s inverse, how do you check it?We just, let’s say this exponent 2x is f (x) and let this be, let theinverse function log2(x) be g(x). And if we make it a compositefunction f ◦g, would be by definition f at g(x), that means two tothe power of, and in the place of x, we substitute log, so 2log2(x),and that gives us a value of x, by definition that proves that thelogarithmic function is inverse function of 2x, because when youcomposite it gives identity function.184Figure 6.12: Taylor’s work on exponents and logarithmsThe interest in this excerpt lies in Taylor’s use of logic. At the end, Taylorstates that “by definition that proves that the logarithmic function is inverse func-tion of 2x, because when you composite it gives identity function.” However, inorder to prove that log2(x) is the inverse function of 2x, Taylor must use the factthat it is an inverse. This relates to Taylor’s previous dialogues in the sense thatit brings forward some confusion between what constitutes a definition and/or ex-istence. Indeed, the logarithmic function may be defined as the inverse functionto the exponential. One can show that an inverse of f (x) = 2x exists (since 2x isinjective), but this does not tell one what the inverse is.A post-secondary education in mathematics is where prospective teachers en-counter such logical notions such as definitions, axioms, existence, and exactness.The results from this task reveal that mathematics educators might want to recon-sider the ways in which these logical pillars are addressed in the post secondarycurriculum, so that future mathematics teachers may have a more robust under-standing of the mathematics they teach.1856.5 SummaryIn this chapter, I have explored participants’ responses to tasks involving conceptsfrom Calculus and Real Analysis. Results from these tasks reveal that overall,advanced mathematics coursework in calculus and real analysis did not have a sig-nificant impact on teachers’ mathematical knowledge for teaching, with respect tothese tasks.Participants remarked that their advanced mathematics coursework offered valu-able experiences which helped them build a more robust understanding of the na-ture of infinity. However, while engaging in the interview tasks, some misunder-standings about infinity were revealed. Indeed, with respect to both the task involv-ing 0.999 . . ., as well as 2√3, participants were unable to conceptualize these twonumbers as limits. Particularly in the case of 2√3, this limited participants in theirjustification of “what” this number was and how to define it. Approximation ver-sus exactness and existence versus the definition were two tensions exhibited by theparticipants who engaged with this task. In the case of 0.999 . . ., all participantswho engaged in this task presented a memorized justification, which is arguablyless intuitive. Rather than focusing on a limiting notion of this number, partici-pants wished to justify the equivalence concretely, using metaphors and graphs.While metaphors and examples can be excellent tools for developing understand-ing of abstract concepts, some participants drew upon examples that did not fullycapture the concepts they were hoping to concretize.186While only two participants engaged in the task on inverses, analysis of thetranscripts revealed that the notions of operation and the “undoing” of these oper-ations were important facets of their mathematical knowledge for teaching in thisdomain. However, this revealed a tension with respect to what it means to “undo”in a functional sense. Casey revealed an overextension of the idea of inverses withrespect to the notion of reflecting the graph of a function over the line y = x. Thissame misconception also brought forth the issue of domain restrictions and therole that these play in the existence and finding of inverse functions. While Caseymentioned that domain restrictions should be studied when teaching about inversefunctions, it was clear that Casey held numerous misconceptions regarding the me-chanics and use of domain restrictions.Overall, results from this chapter suggest that participants’ understanding ofthese tasks were primarily held in the domain of secondary content. While someconnections were made to notions in calculus and real analysis, these understand-ings did not appear to impact the ways in which participants would approach themin a classroom. That is, the understandings developed in their university mathe-matics coursework were not built up from their existing mathematics knowledge.While participants held understandings of limits, sequences, exactness, and conti-nuity, these understandings seem to only exist in the domain of “university math”which, as detailed in Chapter 4, was perceived to be disconnected from secondarymathematics content. Results from this chapter suggest that university coursesin calculus and real analysis might benefit from a reconsideration of the ways inwhich they support the construction of connections between secondary and post-secondary mathematics, so they might better foster the development of MKT for187future teachers. This will be discussed more in-depth in Chapter 8.188Chapter 7The Tensions of Proof andApplications Observed ThroughGeometric Tasks7.1 The Square Root of TwoIn this section, I will discuss participants’ responses to Task G, which was thefollowing:A student is confused as to whether√2 is an irrational or rational number,especially after realizing it is the length of the diagonal of a square of side length1. How would you respond?The purpose of this task was to explore participants’ understandings of the realnumber system. Of the five participants in the study, Taylor and Bailey chose to en-gage in this task. Overall, responses to this task revealed a reliance on memorized189proofs, but analysis of the transcripts revealed more general misunderstanding inthe context of proofs in secondary and post-secondary mathematics.7.1.1 Mathematical BackgroundIrrational numbers could be considered one of the most abstract concepts of thesecondary curriculum. Without them, the real number system is vastly incomplete,but the theoretical jump from rational to irrational numbers can be challenging forstudents and teachers alike (Fischbein et al., 1995). These difficulties are due inpart to the nature of irrational numbers. One such aspect of their nature that mighthinder theoretical understanding is the fact that, despite the rationals being a denseset, they do not cover the entire real number line. That is, any interval of the realline, no matter how small, is guaranteed to have a rational number in it. However,there are “holes” which need to be filled. These numbers are the irrationals. Asecond piece which may hinder understanding of irrationals, is their relationship toincommensurability. As noted by Fischbein et al. (1995), this difficulty can evenbe observed through the historical development of understanding irrational num-bers, as the discovery of incommensurable segments was a result of early Greekmathematicians, while the fully theory of irrationals was not developed until thenineteenth century.The challenges associated with irrational numbers are exacerbated by subsetsof numbers such as constructible, algebraic, and transcendental numbers. The firstdefinition of irrational numbers that students might encounter is that irrational num-bers are decimals which do not terminate and do not repeat. This definition is vagueand leaves space for interpretation. Indeed, a number like 0.10100100010000 . . . is190irrational, but it somewhat follows a repeating pattern.Some irrational numbers are constructible; that is, they can be constructed us-ing a finite number of arithmetic operations (including the square root) on the inte-gers.√2 is an example of an irrational constructible number, while irrationals suchas e and pi are not. Although the definition of “constructible” is purely geometric,there is a deep relationship between constructible numbers and field extensions inabstract algebra. I turn the reader to Aluffi (2009) for an extended mathematicaldiscussion. Irrational numbers such as e and pi are examples of transcendentalnumbers, who are not roots to any polynomial equation, with integer coefficients.Numbers which are roots to such equations are called algebraic. The relationshipbetween these subsets of irrational numbers grows further, with any constructiblenumber being algebraic.A course in elementary number theory may be one of the first places under-graduate students begin to rigorously look at the real number system, despite thefocus of most elementary number theory classes being the integers. One of the firstsuch exposures would be the proof of√2 as irrational. This proof utilizes proper-ties of integers and the rationals, in conjunction with a proof by contradiction. Theproof requires some degree of abstraction, but does not involve the use of unfamil-iar definitions and constructions. Indeed, the most complicated piece of this proofmay be the technique of proof by contradiction, as will be observed by participants’responses in 7.1.2. I present the proof of√2 being irrational to provide context forthe section that follows:191Proof. Suppose that√2 is rational. Then,√2 =mn, for some m,n ∈ Z, where mand n are relatively prime. That is, m and n have no common factors. Squaringboth sides, we have that 2 =m2n2, which is equivalent to 2n2 = m2. This means thatm2 is an even number, which implies that m must be even. Thus, we may writem= 2k, for some k ∈Z. So, 2n2 = 4k2, which after dividing both sides by 2, yieldsn2 = 2k2. Under the same argument as before, this implies that n is also even. Thisis a contradiction to our assumption that m and n are relatively prime, since wehave just shown that both m and n are even numbers. Thus,√2 is not rational.7.1.2 Participant Understandings and Post-Secondary ConnectionsAs mentioned above, of the five participants, Taylor and Bailey engaged in thistask. Interestingly, both participants initially responded in an almost identical man-ner, making reference to the elementary number theoretic proof that the square rootof two is irrational, which was outlined in 7.1.1. The interview dialogue was brief,due to both participants immediately drawing from post-secondary mathematicsknowledge, so I have combined sections where they have previously been writtenseparately.Taylor: I would first ask the student what is a rational number. Theymight say it is a fraction, but by fraction, what do you mean?Well, all fractions have a numerator and denominator which areintegers or whole numbers. So ok, I would proceed with sayinglet’s suppose square root of 2 is rational and argue by contradic-tion.Bailey: So, first I would talk about rational and irrational numbers, be-192cause it seems the student is confusing what that means. If I hadstudents who understood what that means, I might actually showthem the proof of root 2 being irrational. Proof by contradictionis a little confusing, but it’s a cool proof.Figure 7.1: Taylor’s proof that√2 is irrationalTaylor chose to talk though the entirety of the proof during their interview (seeFigure 7.1), while Bailey made reference only to the strategy of using the proof.Although the intent of this task was to explore participants’ understanding of thereal number system, these remarks opened an opportunity to discuss methods ofproof. With this in mind, I was curious to understand how Taylor and Bailey con-ceptualized the method of proof by contradiction.193Taylor: Assuming from what is absurd, you can derive the truth. Thatmeans that if you assume a stupid thing, that you can, then youwill in the end derive something that doesn’t make sense fromwhat you assumed.Bailey: I would say that we know that this first thing is true and if wecan take steps where we know all of them are valid and we get tosomething that is clearly untrue, then your first premise had to beflawed by logic.Taylor and Bailey both demonstrated an understanding of the utility and strat-egy behind proof by contradiction. That is, if you are trying to prove that a state-ment P is true, assume that ¬P (read “not P”) is true. From this, a successful proofby contradiction will yield that if ¬P is true, then a statement Q and ¬Q are bothtrue. Since Q and ¬Q are opposite statements, both cannot be true at the same time.This will always be false. The truth or falsity of P is actually equivalent to the truthor falsity of the conditional statement if P then Q∧¬Q. This may be observed inTable 7.1.P ¬P Q∧¬Q ¬P =⇒ Q∧¬QT F F TT F F TF T F FF T F FTable 7.1: Truth table for logical equivalence of P and ¬P =⇒ Q∧¬Q194Notice that the far-left and far-right columns are equivalent. Thus, proving Pis equivalent to proving ¬P =⇒ Q∧¬Q. The latter conditional statement may beproven using a direct proof method. When participants were asked why proof bycontradiction is a valid method of proof for the problem they wanted to use it for,dialogue revealed that they were able to use it as a tool, but did not have a rigorousjustification for its utility:Taylor: Umm, it works because, umm, I think it works because when youmake an argument that is not true, by assuming that certain thingthen you see a contradiction because of that assumption. Thenwe have to meet that the statement doesn’t work. So you haveto go to the other assumption and start with that. Yeah, uh, thestatement. We make a statement that is not true and if you assumethat as a true then we get, we might, get a statement that is directlycontradicting the statement we assumed.Bailey: The deductive logic is hard. Saying that if all these steps arevalid and this is untrue then the only option is that the premisewas untrue.From this dialogue, it is unclear whether or not Taylor and Bailey understandthe reasoning behind the proof by contradiction strategy. Although both make noteof the strategy of P versus ¬P, they make reference to the premise of the statementbeing the fundamental component to the argument, rather than the logical equiva-lence of P to ¬P =⇒ Q∧¬Q.195When prompted to reflect on the logic behind the strategy, Taylor seemed torecognize that they did not fully understand the reasoning behind the method ofproof by contradiction:Taylor: For me, I would say the logic was always what was taken forgranted. In my undergrad, it was a rule of law, for making argu-ments. To make my students understand why it’s making sense, Ihave to study it for myself first.While Taylor and Bailey were both very familiar with the proof by contradic-tion strategy, the dialogue suggests that they may have never had the opportunityto develop a personally powerful understanding of this proof method. Taylor rec-ognizes that they used this method regularly in their mathematics studies, but thatthey never questioned how or why it worked. In order to have a pedagogicallypowerful understanding, one needs to have a personally powerful understanding— a KDU.This dialogue brings forth an interesting connection between post-secondarymathematics education and mathematical knowledge for teaching. Both Taylor andBailey saw value in bringing strategies of proof into their future teaching. Baileyremarked on how they found it strange that proof did not have a more significantrole in the curriculum, since it holds such an important position in post-secondarymathematics. However, as Taylor notes, methods of proof may not be intuitiveand a deep understanding of why such methods of argumentation are valid requiresdeep understanding of logic. Even so, Taylor felt as though the methods of proofwere taken for granted in their studies. This resulted in Taylor recognizing a tension196in their MKT, stating “it’s ironic because I argue an important aspect of mathemat-ics is asking why, but I’m sort of reinforcing students to just accept this process asa legitimate solution.”Taylor chose to conclude this task with the number theoretic proof, since theyviewed it as a simple but rigourous argument that would be accessible to highschool students. Bailey, on the other hand, presented an alternative method focusedon “zooming in” on the number line (see Figure 7.2):Bailey: I would probably “zoom in.” Ok, maybe√2 is like 1.41 some-thing. So I would say, well, here’s 1.5. And we would zoom inand have a new number line where this is 1.4 and this is 1.5 andok, it’s in between here. And do a few of those to show no matterhow far deep we go, we have more, we are closer to the numberon either side.Figure 7.2: Bailey’s picture of “zooming in” to√2In this excerpt, Bailey uses the decimal expansion form of√2, so I was curiousto see how Bailey would respond without this assumption in place:Interviewer: So numerically, how would you justify the decimal form of√2?197Bailey: We could do 1 squared is 1, and 2 squared is 4. So like, let’s try1.5. That’s too much. Let’s try 1.3. That’s too little. And just getcloser and closer to it. It would be like an asymptote of a graph.We are approaching√2. But I feel like it might convince mystudents the opposite. Like, it’s so clear that it never gets there,so it makes it seem like the number doesn’t exist.Bailey’s remarks point to the understanding of irrational numbers being thelimit of a sequence of rationals, as outlined in 6.4.1. However, based on Bailey’sdialogue in 6.4.2, it is unclear as to whether or not this understanding was fullydeveloped. Recall that Bailey had difficulty in providing a justification for theequality of 0.999 . . .= 1 and wanted to relate the number 0.999 . . . to the functionf (x) = 1x + 1. During this task, Bailey was unable to conceptualize 0.999 . . . as alimit of a discrete sequence. While their work with√2 points to some conceptionof number as limit, remarks from 6.4.2 leave room for interpretation.7.2 SymmetryIn this section, I will explore participants’ responses to the task on what they wouldinclude in a lesson plan unit on symmetry. The task was phrased as follows:You are teaching a week on symmetry to your students. What would you include inyour lesson plan?Of the five participants, Jaime, Bailey, and Casey engaged with this task. Over-all, connections to the post-secondary curriculum were limited, with only Caseybringing forward an explicit example of symmetry in the post-secondary curricu-lum. The mathematics of symmetry discussed was focused at the secondary or ele-198mentary level. Even for discussion generated at the post-secondary level, the con-versation was primarily in the context of particular examples or problems, ratherthan a general extension to the post-secondary curriculum.7.2.1 Mathematical BackgroundAt its core, symmetry is a notion of balance and proportion. Mathematically, thedefinition becomes more complicated. For our purposes, I define a geometric ob-ject to be symmetric if it is invariant to particular geometric transformations. Thesetransformations include reflection, rotation, scaling, and translation.Symmetry has the possibility of appearing early in the elementary mathemat-ics curriculum, through the examination of plants, animals, and other symmetricobjects in nature. Later in the secondary curriculum, some formality can come tosymmetry through the language of functions: Even functions, where f (x) = f (−x)are symmetric across the y-axis, odd functions who are symmetric to the line y = xand satisfy f (−x) = − f (x). The absolute value of a function, | f (x)| is an addi-tional context where symmetry can be explored, particularly if all outputs of thefunction f (x) are negative. Overall, translations and reflections of functions in theplane are a context where the familiar notions of symmetry are made rigourous.Translations, reflections, and rotations of geometric shapes can be combinedto create mathematical works of art, and tessellations can be an exciting way forstudents to explore various types of symmetry. By definition, a tessellation is atiling of the Euclidean plane using one or more geometric shapes. The key inconstructing a tessellation is that no gaps should exist and no geometric shapes199should overlay. Tessellations have a rich history in ancient architecture, design,and art. One of the most famous examples of bridging tessellations, mathematics,and art is the work of the artist M.C. Escher. As was noted by mathematician DorisSchattschneider, although the relationship to mathematics was evident in his art, thework of Escher was heavily mathematical Schattschneider (2010). The mathemat-ics in his work was non-trivial and required a rich understanding of geometry andsymmetry. Such a context could be a rich avenue for students and future teachersof mathematics to extend and expand their mathematical horizons, while drawingupon content they already know.In the context of advanced mathematics, a significant appearance of symmetryin the post-secondary curriculum would be the symmetric and alternating groups,S3 and A3, respectively. Initially, one may define these groups in the followingway: Consider the set X = {1,2,3}. We apply a bijective function from X to Xthat rearranges the elements of X . Such a function f : X → X may be as follows:f (1) = 2, f (2) = 3, and f (3) = 1. For short hand, we would represent this asf =1 2 32 3 1. The set S3 is the set of all permutations of these three objects. Intotal, there are six permutations of the set X = {1,2,3}. They are:f1 =1 2 31 2 3 , f2 =1 2 31 3 2 , f3 =1 2 32 1 3,f4 =1 2 33 2 1 , f5 =1 2 32 3 1 , f6 =1 2 33 1 2200This definition can be extended to construct the symmetric group on n ele-ments, denoted Sn. Now, with this first introduction to the symmetric group, onemight question why it has the name “symmetric” group. Indeed, the constructionof this group, at a surface level, does not seem to have anything to do with symme-try, but of permutations. The direct relationship between the symmetric group andgeometric symmetry appears when one introduces the notion of a dihedreal group.A dihedral group is the group of symmetries on a regular polygon. For a regularpolygon with n sides, the dihedral group on that n-gon would be denoted by Dn.Given an equilateral triangle, the group of symmetries would be D3. How manyelements does this group have? That is, how many symmetries are there to anequilateral triangle? The symmetries we consider are rotational and reflectionalsymmetries and they exist as follows:d1 =12 3d2 =13 2d3 =21 3d4 =32 1d5 =23 1d6 =31 2Observing the pictures above, note that the element d2 is a reflection along thevertical bisector, which flips the vertices 2 and 3. The element d3, on the otherhand, is a rotational symmetry of 60 degrees, in the counterclockwise direction.Returning to the example of S3 and permutations of the set {1,2,3}, notice thateach element of S3 corresponds to an element of D3. With a little bit more work,one can actually show that these two groups are in fact isomorphic. That is, thealgebra of the dihedral and symmetric groups of three elements are equivalent.201The examples presented above only scratch the surface of possibilities for ex-ploring symmetry in secondary and post-secondary mathematics. The implicationsof symmetry in physics, chemistry, biology, and art are far-reaching and could takeup the entirety of a book. For those looking for extended literature on symmetry,mathematics and applications, please refer to Field and Golubitsky (2009).7.2.2 Participant UnderstandingsOf the five participants, Jaime, Bailey, and Casey engaged with this task. WhileCasey immediately jumped to connections in the post-secondary curriculum, Jaimeand Bailey kept most of their examples to lie within the secondary context. In theirdialogue, the emergent theme was a focus on visual and tactile symmetry.The relationship between nature and mathematics was expressed as an inter-esting avenue to explore symmetry, by both Jaime and Bailey. Jaime suggestedhaving students find and suggest places that they see symmetry in their daily lives,while Bailey thought it would be valuable to bring in interesting examples likesnowflakes and plant growth to have a more interdisciplinary discussion on therole of symmetry and nature. Although a connection was not explicitly mentionedby Jaime, they remarked that “facial symmetry is very tied to people’s perceptionof beauty.” While Jaime did not mention this, their connection between nature andsymmetry had the potential to open up an interesting lesson exploring the GoldenRatio, symmetry, art, and biology.Both Jaime and Bailey recognized that there are two types of symmetry within202the secondary curriculum, rotational and reflective. Jaime and Bailey saw these asbeing particularly important, since both of these symmetries appear frequently innature, yielding multiple places of entry for “students to build their own workingdefinition” of symmetry, as Jaime suggested.When considering the mathematical content of the secondary curriculum, Jaimeand Bailey both mentioned graphing and transformations as being a major compo-nent of symmetry in the curriculum. Bailey mentioned “the only symmetry we talkabout in high school is the graph, like here’s the axis of symmetry and that’s reallystraightforward.” Jaime mirrored this comment when mentioning the graphing ofparabolic functions and axes of symmetry, but went a little further:Jaime: I think that the idea of symmetry with respect to time is this ideathat we are talking about not just folding these papers and seeinga reflection, it’s a more general concept. Like, if you were re-versing time, the same thing would happen but it would go back-wards. We see a ball fall down, but according to Newton’s lawsof motion, it has the exact same motion coming up.Jaime used their experience in physics to give context to the symmetric shapeof parabolic functions, which extended the notion of symmetry as a visual/tactileconcept, to something more abstract. Although all of the examples Bailey sug-gested were purely visual, they considered bringing in mathematical objects suchas fractals, to explore a more abstract conceptualization of symmetry. The keypoint for Bailey was “bringing in cool visualizations, cool stuff they can see.” Bai-ley thought that students would have difficulty understanding the abstract terms of203symmetry and thought that visual and tactile representations of symmetry wouldbe an optimal “entry way” for students to begin exploring the concept.Interestingly, none of the participants made mention of the relationship be-tween mathematics, art, and symmetry. This is particularly interesting consideringthat Jaime and Bailey both made mention of the math and art projects completed intheir math methods courses during their teacher education. Both participants foundthe assignment to be fun and productive, but did mention concern about how to fitthese types of projects into the stated curriculum.7.2.3 Post-Secondary ConnectionsOverall, connections made between the secondary curriculum and post-secondarymathematics by participants were limited. Jaime made no explicit connections topost-secondary content, Casey brought forth one example, and Bailey mentionedsome examples at a surface level, but the general view was captured in the follow-ing statement:Bailey: I can’t think of ways that my math degree would help them under-stand the stuff that’s in the curriculum, like symmetry of graphs.As has been the case for the other tasks, connections made between symmetry,secondary, and post-secondary mathematics were primarily example driven ratherthan conceptually driven. As mentioned above, Casey started the interview by onceagain discussing the alternating and symmetric groups of three elements (A3 andS3, respectively). Although Casey had these groups in mind as a post-secondary204example of symmetry, when discussing how it would be used in their pedagogy, itbecame clear that Casey’s understanding was primarily process driven.Interviewer: What would you include in a lesson on symmetry?Casey: Given a 60-60-60 triangle, we could take the bisector and labelthe corners A, B, and C. The bottom line is if we flip the triangle,it’s the same. But then, also given the rotations of the triangle,we could also explain that it’s the same as if you flipped it alongone of the bisectors. I would just draw a fancy squiggle line andshow that the verticies of the triangle have flipped. It’s related toGalois theory.Interviewer: Can you elaborate on the relationship to Galois theory?Casey: Well, we are taking this whole 60 degrees and we are flipping it,but the point is that in the Euclidean plane, it is symmetric, so...Interviewer: Could you be a bit more specific as to how this relates to Galoistheory?Casey: Well, it’s the alternating group. And this one is the symmetricgroup. Alternating group is how you rotate the triangle, but thesymmetric group is not just rotations, but reflections. What Iwould do is start with a dotted line (bisector) and then it flipsand I think that it would ultimately, it would be something likeCBA because we, I think we are flipping it here.Interviewer: And so what are you trying to do exactly?205Casey: Well, we are trying to explain symmetry in a triangle, but this Idon’t think would be understandable to students.Figure 7.3: Casey’s work on symmetries of an equilateral triangleCasey recognized that the various symmetries of a triangle would be an ad-vanced connection between symmetry in the secondary curriculum and advancedmathematics content, as is evident by the dialogue and Figure 7.3. However, basedon the dialogue, it appears as though the connective threads were limited. Caseytried to walk through the example and provided accurate definitions, but the dis-cussion of these connective threads was limited in depth. At the end of the excerpt,Casey mentions that this concept would not be accessible to secondary students.When prompted as to whether or not it could be made accessible, Casey stated thatthe point of the triangle was to explain symmetry. Once again, this is an exampleof advanced mathematical knowledge that has pedagogical potential, but has notbeen developed enough to impact pedagogy in a classroom setting.206This excerpt of dialogue is particularly interesting. The alternating and sym-metric groups are a tactile example of a very abstract concept. Bringing this struc-ture into a unit in the secondary curriculum could be a rich opportunity to engagestudents with some advanced mathematics beyond the horizon. However, in orderto do so, teachers must have a rich understanding of the mathematics in order tomake it accessible to a novice. Even though Casey had the content example, theutility of it in the classroom would be limited, due to a somewhat isolated under-standing of the groups in question.As is evident from the quote at the beginning of this section, Bailey was notentirely sure how post-secondary mathematics could be helpful in helping studentsunderstand the content of the prescribed, secondary curriculum. This quote mirrorsBailey’s concern from 4.2, where they expressed that post-secondary mathematicscontent was “too distant” from the secondary curriculum.Despite this view, Bailey did bring forward a number of examples of sym-metry in the post-secondary context. In particular, Bailey mentioned symmetricmatrices, Euclidean and non-Euclidean geometry. Bailey viewed the conceptsfrom Euclidean geometry being of value, particularly when considering the useof symmetry in various geometric proofs. They remarked that “in the Euclideangeometry course I took, symmetry obviously comes up in the proof of geomet-ric shapes. It’s valuable to see that two things are the same, but opposite.” Inregard to non-Euclidean geometry, Bailey admitted to not knowing an extensiveamount, but thought that considering geometry on different types of surfaces, such207as a parabaloid or sphere, could be an interesting avenue to explore how “differentsymmetries exist” on those and how “we have this one system and there are othertotally different systems which have totally different systems in there.” They latermentioned that they would enjoy finding some resources on that to bring into theclassroom. Once again, this is an example of how teacher educators might havean opportunity to draw upon and develop the extensive mathematical expertise offuture teachers.7.3 SummaryIn this chapter, I examined participants’ responses to the mathematical tasks thatwere geometric in nature. While participants did draw upon post-secondary math-ematics to provide context to how they would approach tasks in the classroom (i.ethe proof that√2 is irrational), most of the discussion was focused on individualexamples, rather than conceptual connections between content areas and levels.This was particularly evident in the context of symmetry, where connectionswere example driven and did not contribute to an understanding of symmetry dif-ferent from a secondary context. Participants who engaged in this task were quickto mention applications of symmetry in art and nature, but the depth of mathemat-ics mentioned did not extend past the secondary level. Indeed, all of the exam-ples brought forth by participants were visual examples of reflectional or rotationalsymmetry. Casey was the only participant who engaged in this task that brought anexplicit connection to post-secondary with dihedral and symmetric groups, but theconnections were once again example driven. When prompted to elaborate on their208understanding of these mathematical objects and their role in higher mathematics,Casey did not provide further context beyond a worked example.The task which discussed the irrationality of√2 revealed an unexpected butfruitful result regarding the role of proof in the secondary and post-secondary math-ematics classroom. The dialogue revealed that perhaps post-secondary mathemat-ics courses should be more conscious of students’ understanding of proof concepts,so that they do not simply become techniques without an understanding behindthem. Indeed, there has been major pushback towards technique driven mathe-matics teaching at the elementary and secondary level and nearly all participantsmentioned their hope of bringing their knowledge of proof into their future sec-ondary mathematics classroom. This result leads one to question whether despite amathematician’s desire for students to have deep, conceptual understanding, manytechniques may be “taken for granted,” as they were for Taylor. Finally, the com-plicated nature of irrational numbers was mentioned, similar to the task involving2√3. Participants mentioned that justifying√2 simply as a number with the valueof√2 could be a challenging conceptualization for many students and turned todecimal representations to help understand “the value” of√2.In this chapter of results, the common thread of limited conceptual connectionsbetween secondary and post-secondary mathematics continues to be woven. Whilesome connections were made through explicit proofs and examples, transcriptsrevealed that these connections did not run deep. As I move into the discussion,I prompt the reader to consider how to facilitate the construction of mathematicalknowledge for teaching with depth and breadth in content connections.209Chapter 8Conclusion and Implications8.1 ConclusionsIn this study, I sought to examine the ways in which prospective secondary mathe-matics teachers drew upon advanced mathematics in their practice. My work wasmotivated partly by my own practice and experience, but primarily by the claimsthat coursework in advanced mathematics helps build connections to the secondarycurriculum that can be transformative to teachers’ practice and deepen their under-standing of the secondary curriculum. After examining the literature, it was clearto me that a gap existed between empirical studies on teachers’ secondary contentknowledge and advanced content knowledge. One of the goals of this researchwas to bridge these two content areas and begin to understand the ways in whichprospective secondary mathematics teachers build their own bridges and connec-tive threads.As a refresher, the research questions for this study were:2101. What do prospective secondary mathematics teachers perceive as the role oftheir advanced mathematics knowledge in their development as teachers?2. In what ways do prospective secondary mathematics teachers relate advancedmathematics knowledge to a mathematics concept in the secondary curricu-lum?I utilized a qualitative, case study methodology to examine these questions.This methodology allowed me to gain rich descriptions of the understandings, be-liefs, and experiences of the participants in my study, as expressed through theone-on-one, task-based interviews with participants. The five participants engagedin their choice of four tasks, from a list of seven pre-chosen tasks which embeddedconnections between secondary and post-secondary mathematics. The interviewswere transcribed and coded, as described in Chapter 3. The themes that emergedfrom the data suggest that the prospective mathematics teachers in this study hadlimited opportunities to build content connections between secondary and post-secondary mathematics. In the discussion that follows, I examine the themes inthe context of improving the education of future teachers, both in university math-ematics courses and teacher education.8.1.1 The Role of University Mathematics in Teacher DevelopmentThe first research question stated above aimed to extend the work of Zazkis andLeikin (2010) and understand what prospective teachers perceived as the role oftheir advanced knowledge in their development as teachers. One of the majorthemes developed in my analysis of participants’ perceptions of the role of ad-211vanced mathematical knowledge (AMK) in their work as a teacher was the notionthat much of the content in post-secondary mathematics is disconnected from whatis taught in secondary school. Although all participants perceived value in havingan advanced degree for reasons such as having experience beyond the students, be-ing able to field questions, and having an increased awareness of what mathematicsis, the majority of the benefits mentioned were focused on skills and beliefs, ratherthan content.The participants in this study lamented that although admission to the teachereducation program required advanced mathematical coursework, their mathemat-ics methods courses did not require them to use their extensive mathematical ex-pertise. Participants with two teachable subjects found this to be unique to theirmathematics methods, since their content expertise was being extended and drawnupon in their other methods courses. In particular, in Section 4.3, Bailey remarkedon how they felt their expertise in literature was used and extended during theirEnglish methods courses, while their mathematics courses required no more thangrade 10 mathematics. As a caveat, I must once again remark that I did not observeor obtain syllabi for the methods courses taken by participants. My remarks andanalysis are based solely on the shared perceptions of participants in my study.Bailey’s remarks on the differences between their two methods courses con-nects well with other participants’ commentary on the structure of their teachereducation program. Many participants remarked that they wanted a more “prag-matic” approach to their mathematics methods courses. Since I am unaware of theprecise nature of the methods courses, I am unable to discuss their structure in any212specific way; however, I can speak to one of the sub-themes revealed in the partic-ipants’ remarks. The desire for a more pragmatic program was often followed bya remark that the mathematics methods courses did not require any knowledge ofmathematics.Indeed, teaching mathematics and building MKT is mathematical work (Balland Bass, 2002). Generating appropriate activities, understanding where contentextends to and develops from within the curriculum, and evaluating where studentsmight be confused in a particular lesson are all elements of MKT and require notonly pedagogical expertise, but mathematical expertise. KDUs are understand-ings that transform the ways in which one understands and works with a concept.In turn, such an understanding could transform and impact MKT, as understoodthrough the framework of developing MKT (Silverman and Thompson, 2008). Al-though the notion of a KDU was developed to help teachers understand key learn-ing moments in the curriculum, the construct could be used to help future teacherssee how their advanced mathematical knowledge is connected to the mathematicsthey plan to teach. However, in order to develop KDUs, prospective teachers needto experience opportunities for learning that foster the development of such under-standings. My research indicates that prospective secondary mathematics teachersperceive that there were few opportunities to draw upon and little need for ad-vanced mathematical knowledge in their mathematics methods courses.To summarize, participants in this study perceived their expertise as unneces-sary in their teacher education program, as well as in their future work as teach-ers. This view was evidenced by participants’ perceived disconnect between sec-213ondary and post-secondary mathematics content. It is entirely possible that despitea perceived disconnect between these content domains, participants may have heldtransformed understandings of secondary mathematics concepts due to their con-tent expertise. The task-based interviews served as a context for exploring thispossibility. As will be elaborated below, the perceived disconnect was explicitlyobserved through participants’ engagement in the tasks.8.1.2 Content Connections Between University and SecondaryMathematicsThe second research question above aimed to understand the ways prospectivesecondary teachers’ related advanced mathematics knowledge to secondary math-ematics content. Overall, I found that participants had limited content connec-tions between secondary and post-secondary mathematics content. Participants ex-pressed a perceived disconnect between secondary and post-secondary content andthis disconnect was observed in discussing connections between post-secondarymathematics and specific problems at the secondary level.Participants in the study demonstrated multiple misconceptions about the be-haviour of real-valued polynomial functions, both at the secondary and post-secondarylevel (Chapter 5). While depth of understanding of base cases, such as quadraticsand cubics, were demonstrated by participants such as Adrian and Taylor, theseunderstandings remained at the secondary level. The content knowledge gained incourses, such as abstract or linear algebra, did not seem to have an impact on theirunderstanding of polynomials, beyond the use of specific examples. Participants’214discussion of number and limits, while accurate in a secondary context, did notgo beyond the horizon of the secondary curriculum (Chapter 6). Even participantssuch as Bailey, Taylor, and Adrian who had taken advanced courses in real analy-sis did not hold conceptualizations of irrational numbers beyond the definition of“not rational.” With respect to examining prospective teachers’ connections withingeometric tasks, results were somewhat tangential to what was expected (Chapter7). Indeed, results indicate that participants had a limited understanding of the useof proof methods, particularly proof by contradiction. Taylor, for example, rec-ognized this gap in their understanding and and connected it to their desire to useproof as a means for building understanding in their future classrooms (Chapter4). Through our discussion of the irrationality of√2, Taylor remarked that theymay have taken their understanding of proof strategies for granted and that theywould need to teach themselves again, before teaching others. Taken together,these results reveal that participants’ did not hold personally powerful understand-ings of post-secondary mathematics that had the potential to impact their pedagog-ical practice.The process of reflective abstraction is understood to be the mechanism inwhich learners build new knowledge, with the construction of new knowledgebeing based on past knowledge and experience. So, according to Piaget, the con-struction of new understandings should be built by extending already existing ones.Stakeholders and educators alike have continually emphasized the importance ofbuilding mathematical connections, to help build a more robust understanding ofthe secondary curriculum (Conference Board of the Mathematical Sciences, 2012).Indeed, as elaborated in the review of the literature, this is one of the major reasons215for requiring mathematics teacher to have extensive experience in advanced math-ematics. However, taking university mathematics does not necessarily imply thatreflective abstraction is happening.Bailey remarked that university mathematics “really is diverging from what istaught in high school.” In the interviews, Bailey expressed a love and passion formathematics and teaching, but based on their remarks throughout the interview,their university mathematics education may have not offered the meaningful op-portunities necessary to engage in reflective abstraction and build KDUs. As notedin the analysis of participants’ responses during the polynomials tasks, Bailey ex-pressed that in the abstract algebra courses they took, the content was “out there,”far removed from secondary mathematics, and the structure of the course was use-ful for the students who would pursue research careers in mathematics (Chapter 5).Suominen (2015) followed on the work of Cofer (2015), who found that “un-dergraduate abstract algebra students are not recognizing mathematical connec-tions between abstract algebra and secondary school mathematics” (Suominen,2015, p. 75). Through examining the connections to secondary mathematics ex-plicitly stated in abstract algebra texts, she argues that the teaching of abstractalgebra should be reconceptualized as an extension of secondary algebra and ge-ometry, rather than a generalization. The work of Suominen (2015) dove tails withthe laments of Bailey. Possibly due to Bailey’s experiences in advanced mathemat-ics, they came to the belief that secondary and post-secondary mathematics sharelimited connection threads. Simon (2006) argues that KDUs are not constructed byseeing examples or being relayed information, but through a personal process of216reflective abstraction. If advanced mathematics courses are taught in the traditionalmanner, are there explicit opportunities for students to build meaningful KDUs andengage in reflective abstraction? One should consider how the work of Suominen(2015) in the context of abstract algebra extends to other advanced mathematicscourses.One such example, is the conceptualization of number as limit. This concep-tion appeared in the task on 0.999 . . . = 1, as well as the task on 2√3. Building arobust understanding of number has been stated to be of immense value and thor-oughly relevant content knowledge for mathematics teachers (Conference Boardof the Mathematical Sciences, 2012). Past work suggests that real analysis canbe a rich context for future teachers to develop their MKT, as it opens opportuni-ties to gain a robust understanding of certain mathematical concepts, such as mystudy’s tasks about 2√3 and 0.999 . . . (Wasserman et al., 2017). Despite the ma-jority of the participants having coursework in analysis, the results of these taskssuggest missed opportunities to extend the content of the secondary curriculum to amore robust and rigorous post-secondary context. Wasserman et al. (2017) presenta framework for teaching real analysis that may be a suitable for mathematicianswho are interested in teaching a course that will help future teachers build connec-tions and KDUs to related content of the secondary curriculum.In their framework, they argue that future teachers would benefit from a coursein real analysis that is “building up from practice and stepping down from practice”(Wasserman et al., 2017, p.562), so that fundamental mathematical ideas that areburied in the secondary curriculum resurface in real analysis courses. I argue that217the results of my study further support a need for the utilization of Wasserman’sframework, which may prove useful for professors of real analysis. Gauging bythe extensive post-secondary mathematical coursework taken by the participantsof this study, they appeared to have the mathematical skill necessary to build con-nections between secondary and post-secondary content. However, based both onthe limited connections drawn during the task-based interviews, as well as the per-ception that advanced courses are more relevant for gaining skills than extendingcontent, their university mathematics coursework may have provided limited op-portunities to engage in reflective abstraction and develop KDUs. Employing theframework of Wasserman et al. (2017) might be an appropriate first step in consid-ering how mathematicians can build real-analysis courses that will help build theMKT of future teachers.While some students, such as Taylor and Adrian, exited their advanced courseswith a sense of a bigger picture, others, like Bailey and Jaime, left their advancedcourses feeling a disconnect between university mathematics and the curriculumthey are to teach. I am in no way arguing that university mathematics pedagogyshould be completely transformed or that the traditional pedagogical methods ofuniversity mathematics courses should be disposed; indeed, as was noted by manyparticipants, there are benefits to learning how to “think like a mathematician,” asTaylor put it. Problem solving skills, rigour, proof, and understanding are funda-mental to a quality university mathematics education. Unfortunately, based on theresults of this study, a focus on the first three elements may undermine understand-ing.218Although the traditional definition-theorem-proof approach used in many upper-year mathematics courses (Thurston, 1998) may be working for some, it may leavejust as many (if not more) in the dark. University mathematics courses do havethe opportunity to provide future secondary mathematics teachers with knowledgethat could impact their MKT. With many advanced courses being requirements forfuture teachers, the results here suggest that advanced courses may not be provid-ing sufficient opportunities for teacher candidates to build connections between thecontent they know, the content they are learning, and the content they will eventu-ally teach. All of the participants in this work were successful university students.Indeed, many of the participants completed their undergraduate coursework withvery high GPAs and course marks. Most courses in advanced mathematics fo-cus heavily on theory and rigour, but the results of this study suggest that sucha focus on theory may hinder understanding. Bailey, who was a very successfulmathematics student, lamented that some of their classes felt like “a total blur,”only two years later. How might mathematicians adjust their pedagogical prac-tice so that successful students, like Bailey, complete their coursework seeing therelevance and connections of advanced coursework to mathematics they alreadyknow? However, I feel that it is necessary to note that I am not entirely aware ofthe lived experiences and lived curriculum of the participants in my study. Indeed,participants may have experienced connections in their coursework, but these ex-periences were not recalled during interviews in the study.Additionally, some participants’ demonstrated misconceptions of secondarymathematics content, which is in line with previous work in the field (Even, 1993;Leung et al., 2016; Stump, 1999). Indeed, misconceptions at the secondary level219were observed in every task. In the context of polynomials, all participants overextended their knowledge of quadratics and cubics to the cases of higher degreepolynomials. With exponential functions, irrational numbers, and decimals, mis-conceptions were held by multiple participants with respect to approximation ver-sus exactness. The notion of reflection symmetry was overextended in the contextof inverses by Casey. While all of these misconceptions exist in the domain ofsecondary mathematics, they have the potential to be corrected by drawing uponparticipants’ advanced mathematical expertise.The results of this thesis shed light on the content links future teachers make be-tween secondary and post-secondary mathematics. The links observed in this workwere few and did not extend beyond first-year mathematics, for the most part. Ifthere were connections to content beyond the first-year mathematics curriculum,the depth and power of these connections to teaching secondary mathematics waslimited to singular examples. Despite extensive mathematics coursework at thepost-secondary level, results suggest that their knowledge of advanced mathemat-ics did not transform their understanding of secondary mathematics content. In thefollowing section on implications, I will detail suggested actions educators and re-searchers may want to take to support the development of MKT that integrates ad-vanced mathematical knowledge into future teachers’ understanding of secondarymathematics content.2208.1.3 LimitationsAs was elaborated in Chapter 3, the qualitative, case study approach was utilizedto help provide rich descriptions of the ways participants perceive and draw uponadvanced mathematics knowledge to inform their teaching. The sample size of thisstudy, at only five participants, allowed for me to deeply engage with the quali-tative data obtained during the one-on-one interviews. However, it is importantto note that these results are not generalizable, and were not intended to be. Thedata, results, and description in this study are unique to the participants of thisstudy. The results do not necessarily extend to all prospective secondary mathe-matics teachers with similar backgrounds. Rather, the intent of this study was toexamine the ways post-secondary mathematics knowledge informs mathematicalknowledge for teaching (MKT), to provide initial insights that mathematicians andmathematics educators might consider as they reevaluate the ways they support thedevelopment of MKT for future teachers. The results of this study suggest the needfor future work which examines faculty members’ (in mathematics and mathemat-ics education) perceptions of the role of their courses in the development of futureteachers’ MKT.Another limitation of the study comes from the tasks included in the task-basedinterviews. Although all of the tasks offered to participants were inspired and de-veloped from past literature on MKT, they constitute a very limited collection oftasks that could be used to understand the role of advanced mathematics knowl-edge (AMK) in developing MKT. It is entirely possible that participants may havebeen able to draw upon post-secondary mathematics knowledge in other tasks, but221this work would require an additional study where the tasks allow participants todraw upon their mathematical expertise more generally. However, results from theone-on-one interviews about participants’ perceptions of the role of AMK in theirgrowth as teachers suggests connections may still be limited, due to the perceiveddisconnect between content in secondary and post-secondary mathematics.Regardless of these limitations, the results of this study are of value and haveimplications for the ways mathematicians, mathematics teacher educators, and sec-ondary mathematics teachers (practicing or pre-service) build connections betweensecondary and post-secondary mathematics.8.2 ImplicationsWhen I entered into this work, I hoped that this study could be a step in helping bothmathematicians and teacher educators to consider the ways in which they supportconnections between advanced mathematics and the development of mathematicalknowledge for teaching and key developmental understandings. In the sectionsthat follow, I outline the prospective implications of this work in both research andpractice.8.2.1 Implications in ResearchThe results of this work have opened up a number of new questions and avenuesfor exploration. The first portion of the study examined prospective teachers’ per-ceptions of the role of advanced content for their teaching, but I am now curious toexplore mathematicians’ perspectives. In particular, what role do mathematiciansperceive advanced mathematics plays in the development of secondary teachers?222In what ways do they support the development of MKT in their courses? Whenthey are teaching advanced courses, who is their target audience? To whom arethey teaching? These are all questions that I believe should be explored, so that wehave context for the ways advanced mathematics courses are currently taught.Another extension area for future research into AMK and MKT would be class-room observations in university mathematics courses. These observations wouldexplore the explicit connections being made between secondary content and thecontent of the course being observed, by both the professor and the students. Thiswork could occur in any advanced mathematics course, but courses in abstract al-gebra, real analysis, geometry, and proof appear to be important, due to their directand rich relationship to the secondary curriculum. Furthermore, such observationswould provide depth and context to some of the current study’s participants’ claimsthat there are limited connections being made in university mathematics classes. Inunderstanding how and where mathematicians already build connections betweensecondary and post-secondary mathematics, we may gain a sense of how and whereto make such opportunities more frequent.Additionally, an interesting domain to investigate is the analog of the abovestudy, but in teacher education. That is, in what ways do teacher educators andfuture teachers build connective threads between post-secondary and secondarymathematics content in math methods courses? Again, classroom observationswould provide some support or defence to the claims made by participants in thisstudy, such as “not needing math beyond grade 10” and feeling as though therewas not a focus on “how am I going to teach a concept”? Finally, these classroom223observations could provide context on the ways in which mathematics methodscourses help build the various facets of MKT for future teachers, including con-nections between secondary and university mathematics.8.2.2 Implications in PracticeThe results of this study suggest that prospective teachers may need more explicitopportunities for reflective abstraction in their advanced mathematics courses. Thisis an important, but ambitious endeavour, that requires mathematicians to thinkdeeply about the content they teach, where the construction of KDUs might oc-cur, and would involve reconceptualizing some courses as extensions, rather thangeneralizations (Suominen, 2015). Furthermore, it would require professors to in-clude explicit opportunities inside or outside of class that encourages students togo through the process of reflective abstraction, relate new content to what theyalready know, and build new meaning and inferences.The notion of a KDU may be a helpful context for mathematicians to considerthe ways they are supporting the development of MKT. Although Simon (2006)focuses on how elementary mathematics teachers could identify critical learningpoints in the elementary curriculum, this construct could be equally as beneficialfor mathematicians to consider in university mathematics. In particular, mathe-maticians may want to question what moments in the curriculum are a “trans-formation” of concepts previously studied by students. Once these moments areidentified, Simon (2006) argues there must be transformation of the instructor’spractice, if the focus is to be on the development of KDUs. Indeed, as mentioned224in Chapter 2, students are the ones who have to go through the process of reflec-tive abstraction to develop their own KDU. Identifying these key moments in thepost-secondary mathematics curriculum may provide a basis for the developmentof connections between what advanced mathematics and the mathematics they al-ready know and plan to teach.Alternatively, mathematics departments might want to consider the develop-ment of a new course that would explicitly examine secondary mathematics froman advanced perspective. Many departments offer mathematics courses for fu-ture elementary school teachers, but few offer a mathematics course specificallyaimed at future secondary teachers. If it is indeed the case that limited connectivethreads are being developed in advanced mathematics courses, prospective sec-ondary mathematics teachers might benefit from a course that takes the contentfrom advanced courses — considered to be “disconnected” and “out-there” — andrelates it retrospectively through the content of the secondary curriculum. Fol-lowing a framework similar to Wasserman et al. (2017), such a course could be atransformative course for prospective teachers that changes their perspective on therole of advanced mathematics in the development of MKT. The results of my studycould provide a start for this kind of mapping of connected threads and concepts inadvanced mathematics courses.Mathematicians, particularly those in positions of course construction and de-velopment, may not be the only ones who should consider the development ofopportunities to build connections between secondary and post-secondary mathe-matics content. Even if prospective teachers exit their undergraduate degrees hold-225ing the perception that their advanced mathematics knowledge is not relevant totheir future work as teachers, teacher educators are in a prime position to disruptthis belief. Participants’ responses to the task on symmetry is an excellent ex-ample. Although most participants’ responses landed in the domain of secondarycontent, there were connective threads to post-secondary mathematics. With theexistence of these threads, such as symmetry groups and alternative geometries,comes an opportunity for teacher candidates to build upon and explore these con-nections with respect to the curriculum. While some may argue that such advancedknowledge isn’t necessary, the argument I make in this work is that it has the po-tential to have a positive effect on future teachers’ understanding and pedagogyof secondary mathematics. Teacher educators can provide meaningful, curriculumcentred learning moments for future teachers to develop their MKT.The task on applications of quadratic functions could be another content areafor teacher educators to build the MKT of prospective teachers. Despite recogniz-ing the importance of the concept as an “entry point” for more advanced mathe-matical modelling, some of the participants communicated that they had limitedexamples for how to motivate this central concept of the secondary curriculum.This dovetails with some participants’ concern that they are afraid they are goingto “teach the way they were taught,” due to the view that they have limited con-crete examples to bring into the classroom. These same participants lamented pro-foundly at the prospect of students loosing interest in mathematics due to limitedand contrived examples. The exploration of new applications of concepts (such asquadratic functions) could be an opportunity for teacher educators to develop thepedagogical and content expertise of future teachers. The inclusion of more math-226ematical work in math methods courses may be an interesting avenue for teachereducators to explore.Adrian mentioned that they tried to reflect on how their advanced content ex-pertise connected the secondary curriculum in their own practice, but did not findit being emphasized in their methods courses. All the participants in this study hadextensive mathematical expertise, which was unique to the coursework they hadtaken in their undergraduate studies. While Adrian had post-secondary expertise inphysics and applied mathematics, Casey had more experience in pure mathemat-ics. Both of the participants, being in the shared space of a mathematics methodscourse, had much to learn from each other.Indeed, the shared space of a mathematics methods course has the potential toyield opportunities for teacher candidates to share content extensions and mathe-matical knowledge that could impact their future teaching practice. In order forthis to happen, there need to be incentives and opportunities to do so. Teacher can-didates are content experts and teacher education is a space to build MKT, of whichpedagogical and subject matter knowledge are elements. Even if the focus is onpedagogical knowledge, the results of this study could encourage teacher educatorsto consider the ways in which they are building, extending, and drawing upon themathematical content knowledge and expertise of future teachers.All the above future work remains in the context of prospective secondarymathematics teachers. Another extension of this work would be in the domainof practicing teachers. Zazkis and Leikin (2010) examined practicing teachers’227perceptions of the role of their advanced mathematics knowledge (AMK) in theirteaching and found that the participating teachers held many of the same viewsas participants in this study. In particular, they observed that participants viewedAMK as valuable in building skill and confidence, but not necessarily in regardto content. Combining the insights from Zazkis and Leikin (2010) and the cur-rent study leaves me interested to follow-up with the current study’s participantsto explore whether they maintain the same perceptions of the role AMK in theirpedagogy, after several years of teaching experience. Due to limited research inthis area, both task-based interviews and/or classroom observations, would provefruitful.Overall, I foresee numerous opportunities to develop resources for mathemati-cians and mathematics teacher educators to help them in developing connectionsbetween secondary and post-secondary content. Before doing this, however, itwould be prudent to investigate how these connections are already being devel-oped in university mathematics and teacher education. I look forward to the re-search community engaging with the results of this study through related work thatwill one day benefit the mathematical learning of future teachers, while simultane-ously building meaningful relationships between mathematicians and mathematicseducators.8.3 Closing RemarksThis study has examined the role of advanced mathematics knowledge in the math-ematical knowledge for teaching of future secondary mathematics teachers. Througha qualitative case study, I examined the ways in which participants explicitly drew228upon advanced knowledge to inform their teaching. This was done through task-based interviews, which were composed of potential classroom situations whereadvanced mathematics knowledge could be used to enhance their pedagogy. I sup-ported the results from these task-based interviews with interviews that exploredwhat participants perceived more generally as the role of their advanced mathemat-ics knowledge in their growth as teachers.Results from this study suggest a perceived disconnect between mathematicsstudied at the university level and mathematics taught and studied in secondaryschool, which has been observed in existing literature (McGuffey et al., 2019; Za-zkis and Leikin, 2010). This was observed through participants’ remarks of therole of their advanced knowledge in their teaching. Although beliefs and values de-veloped in university mathematics were viewed as valuable by many participants,connections between mathematics content were viewed as limited and irrelevantto their future work. This was supported through the task-based interviews, whereparticipants demonstrated content misconceptions at the secondary level, on top ofproviding a limited number of content connections between secondary and post-secondary mathematics.Previous literature suggests that advanced mathematics knowledge has the po-tential to transform teachers’ understanding of the secondary curriculum throughthe expansion of one’s horizon content knowledge (Wasserman and Stockton, 2013).However, past literature also suggests that secondary mathematics teachers do notperceive advanced mathematics to play an important role in their teaching (Wasser-man et al., 2015; Zazkis and Leikin, 2010). The results of this study are in line with229previous literature and further support the need for building connections betweensecondary and post-secondary content. My study extends previous literature by ex-plicitly examining prospective secondary teachers’ perceptions of their advancedcontent expertise and the connective threads of this expertise to the content theywill eventually teach. The development of such connections to impact pedagogicalpractice may require a reconceptualization of advanced mathematics courses andmathematics methods courses, so that they more frequently engage students in theprocess of reflective abstraction (Piaget, 1970) and the construction of key devel-opmental understandings (Simon, 2006) between secondary and post-secondarymathematics.It is my hope that the results of this study will encourage mathematiciansand mathematics educators to find common ground in the domain of secondaryteacher education. Both of these parties play fundamental roles in the educationand training of future mathematics teachers, who in turn prepare prospective stu-dents for university mathematics. This study could inspire further collaborationbetween mathematicians and mathematics education scholars in the academy, asthe results are of importance to both of these academic departments. Althoughsome strides are being made increase cross-departmental research and collabora-tion (Fried, 2014), I believe that the domain of secondary mathematics teachereducation has the potential to be a mutual investment for these two groups to fur-ther collaborate in enhancing the mathematical knowledge for teaching of futureteachers, and in turn, the mathematics learning of students in secondary school.230ReferencesAlmeida, R., Bruno, A., and Perdomo-Díaz, J. (2016). Strategies of number sensein pre-service secondary mathematics teachers. International Journal ofScience and Mathematics Education, 14(5):959–978. → page 2Aluffi, P. (2009). Algebra: chapter 0, volume 104. American Mathematical Soc.→ pages 43, 191Artzt, A. F., Sultan, A., Curcio, F. R., and Gurl, T. (2012). 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MathematicalThinking and Learning, 12(4):263–281. → pages2, 4, 24, 25, 26, 32, 65, 211, 227, 228, 229242Appendix AInterview Materials243Please choose four tasks from the following list: • TASK B: Your students are confused as to why we can define and calculate 2 ".  • TASK C: A student is working through a problem and asks you if 0.99999…= 1. • TASK D: You are teaching a week on factoring polynomials and you have found that your students are struggling to recognize when they should stop trying to factor.  • TASK E: Your students are learning about inverse functions. What would you include in your lesson plan?  • TASK F: You are teaching a week on symmetry to your students. What would you include in your lesson plan?  • TASK G: You have been teaching a unit on quadratic functions for a few weeks and one of your students asks you why they need to know about them.  • TASK H: A student is confused as to whether 2 is an irrational or rational number, especially after realizing it is the length of the diagonal of a square of side length 1.  2441. One of the factors of 3𝑥% − 16𝑥 + 𝑘 is 𝑥 − 7. Determine the value of k. Justify your answer.  2. Which of the following is not a function? Justify your answer.  3.  When a polynomial P(x) is divided by x+3, the remainder is 2. Which point must be on the curve y = P(x)? Justify your answer. a. (3, -2) b. (-3,0) c. (-3, 2) d. (3,2)  4. For which of the following functions is 𝑓 𝑓 𝑥 = 	𝑥 for all x in the domain of 𝑓 𝑥 ? I: f(x) = x, II: f(x) = -x, III: f(x) = 1/x a. I and II only b. I and III only c. II and III only d. I, II, and III 5. Write 0. 99998 as a fraction. Justify your answer. 6. Find the graph of 𝑦 = 35 − 5. What is the x-intercept? Justify your answers.  245 • How do you conceptualize mathematics as a field of study?  •  • Do you think it is important for secondary mathematics teachers to know advanced mathematics? Why or why not?  •  • What roles do you see those four years of learning playing in your next year of study? What grade(s) do you hope to teach?  •  • Do you see your post-secondary degree in mathematics having an impact on your teaching? In what ways?   246Appendix BRecruitment emails anddocuments247     Prospective Secondary Mathematics Teachers’ Knowledge for Teaching  Dear Teacher Candidate,   We are writing to invite you to take part in the doctoral dissertation research of Vanessa Radzimski in the Department of Curriculum and Pedagogy at the University of British Columbia, Vancouver Campus. The primary investigator for this study is Dr. Cynthia Nicol and the co-investigator is Vanessa Radzimski. Through this research study, we hope to better understand the relationships between university level mathematics knowledge and knowledge for teaching in the secondary mathematics classroom. As a future secondary mathematics teacher with a post-secondary degree in mathematics you are in an ideal position to provide valuable first-hand information from your own perspective.  As a participant, we invite you to participate in an interview component (one-on-one or focus group). We’ll ask you to complete a short survey so that we may assign you to either the one-on-one or focus group interviews. This will ensure that we have a diverse group of participants in each. We ask all participants to provide the researchers with their university transcripts, clearly indicating mathematical courses taken, in order to provide background information for the interviews. For those in the one-on-one interviews, we’ll ask you to think about math problems of the type found in the BC Grade 8-12 math curriculum. All interviews will be audio recorded and transcribed. Your responses to the questions and academic transcripts will be kept confidential. You will be assigned a false name for all written reports and publications. Total time for participation will be around 2 – 2.5 hours. For those in the focus group interviews, we will discuss your experiences in your mathematics degrees. The interview will be transcribed and you will be assigned a false name for all written reports and publications. Total time for participation in the focus group will be around 1 – 1.5 hours.    As a token of appreciation for your time, you will receive a $25 VISA gift card as compensation for your participation in the study. Your participation will be a valuable addition to our research and findings could lead to greater public understanding of mathematics teacher education and influence a higher level of communication between departments of mathematics and departments of teacher education at the university. It could also serve as an opportunity for you to think deeply about the ways in which your advanced mathematical knowledge might inform your pedagogy.  If you are willing to participate please contact Vanessa Radzimski by E-mail or phone within ten days of receipt of this letter, suggesting a day and time that suits you and we will do our best to arrange a meeting to your availability. The purpose of this meeting will be to discuss the study in more depth and sign the consent form, if you agree to participate. If you have any questions please do not hesitate to ask.  Thank you again for considering this research opportunity! Cynthia Nicol and Vanessa Radzimski 248  E-mail to Course Instructor for Recruitment  Dear XXX,  Under the supervision of Dr. Cynthia Nicol of the University of British Columbia, I am conducting a study entitled “Prospective Secondary Mathematics Teachers’ Knowledge for Teaching.” Through this research study, we hope to begin to understand how a post-secondary degree (minor or major) in mathematics influences teachers’ pedagogy in the secondary mathematics classroom. As the course instructor for EDCP 342: Mathematics - Secondary: Curriculum and Pedagogy at the University of British Columbia, we would like to request a time to visit to your course so that we can recruit potential participants for this study.  We would like to arrange a time, at your convenience, to visit a session of your class and outline the research to your students. The co-researcher (myself) will provide consent forms to students in the class and return at the end of the course to answer any questions and complete the consent process with any students who wish to consent that day. Additionally, students will be able to e-mail Dr. Nicol, or myself, with an interest to participate within ten days of the date of initial contact. In total, we will only need approximately ten minutes at the beginning of your class.  Precautions will be taken to protect your confidentiality. I am the only one who will know your identity and all identifying information will be masked.  If you have any questions please contact me (by phone or e-mail) or my supervisor, Dr. Cynthia Nicol (by e-mail). Our contact information is provided on the attached consent form. Please respond by e-mail within seven days of the date this e-mail was sent to express your approval of my visit. If you would like me to provide you with a hard-copy of the consent form in advance, please let me know and I will make arrangements.  Best wishes,    Vanessa Radzimski PhD Candidate University of British Columbia  249 Page 1 of 1 Version: October 30, 2017 Verification of Payment for Participation in “Prospective Secondary Teachers’ Mathematical Knowledge for Teaching”  By signing this document, you are verifying that you, __________________________________, received the $25 VISA gift card that you would receive for participating in the study entitled “Prospective Secondary Teachers’ Mathematical Knowledge for Teaching” with Cynthia Nicol (primary investigator) and Vanessa Radzimski (co-investigator).  Signature: ____________________________________    Date: _________________________   250

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