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Teaching and learning mathematics using Wall Math in a grade 1 classroom Bateson, D. Lynne 1998

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Teaching and Learning Mathematics using Wall Math in a Grade 1 Classroom by  D . Lynne Bateson B.Ed., University of British Columbia, 1971  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in  T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Curriculum Studies  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A A U G U S T , 1998 © D . Lynne Bateson, 1998  In  presenting this  degree at the  thesis  in  partial  fulfilment  of  the  requirements  University  of  British  Columbia,  I agree that the  freely available for reference and study. I further agree that  for  an  advanced  Library shall make it  permission for extensive  copying of this thesis for scholarly purposes may be  granted by  the head of my  department  understood  that  or  by  his  or  her  representatives.  It  is  publication of this thesis for financial gain shall not be allowed without permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  copying  or  my written  ABSTRACT  This study explored students' interactions with a primary mathematics programme, Wall Math, as it operated i n a single Grade 1 classroom. The programme, based on constructivist principles, as exemplified by Cognitively Guided Instruction, and conforming to both the British Columbia Ministry o f Education's instructional resources package for Kindergarten to Grade 7 and the National C o u n c i l of Teachers o f Mathematics' standards, employs a structured set of questions which are posted on a w a l l of the classroom for students to work on over several class periods. The programme is designed to provide students with meaningful learning opportunities for the intended learning outcomes of the primary programme, while also providing constant review and repetition of important skills. In addition, the programme provides students with opportunities to develop personally and socially as they interact with their teacher and fellow students.  This study followed primarily six  Grade 1 students (three boys and three girls o f varying mathematical abilities: high, average, and low) through a series o f five sets ("boards") o f problems over a period o f approximately three months.  A narrative style is used to  illustrate how the students dealt with both new and review material, and how the students interacted with the questions, their teacher, and each other.  TABLE OF CONTENTS  Abstract  ii  Table of Contents  iii  List of Figures  v  Acknowledgements  vi  Chapter 1 Introduction to the Study Purposes of the Study Organization of the Thesis Chapter 2 The Wall Math Programme i n Context What Does Wall Math L o o k L i k e ? The I R P , Wall Math, and the N C T M Chapter 3 Review of the Literature Wall Math and Cognitively Guided Instruction What is "Math"? Problems and Problem Solving Connecting Mathematics with Real Life "Whole M a t h " Chapter 4 Methodology of the Study Practitioner Researcher The Collection of Data Data Analyses The School, the Teacher, and the Students o f the Study The School The Teacher The Children Ethical Review and Consent Chapter 5 Wall Math Board Diaries Diary #1 — January 8 - 14,1998 Day 1 — Wed., Jan. 8 Day 2 — Thurs., Jan. 9 Day 3 — F r i . , Jan. 10 Day 4 — Tues., Jan. 14 Diary #2 — January 20 - February 2, 1998 Day 1 — F r i . , Jan. 23 Day 2 — T u e s . , Jan. 27 Day 3 — Wed., Jan. 28 Day 4 — Fri., Jan. 30 Day 5 — M o n . , Feb. 2 Diary #3 — February 17 - 23, 1998 Day 1 — Tues., Feb. 17 Day 2 — W e d . , Feb. 18 Day 3 — T h u r s . , Feb. 19 Day 4 — M o n . , Feb. 23 Diary #4 — M a r c h 3 - 11,1998 — iii —  .  1 1 3 4 6 19 30 30 36 39 41 42 44 45 46 50 51 51 51 52 55 57 57 58 63 64 68 71 72 73 75 79 83 87 88 92 95 96 99  Day 1 — Tues., March 3 Day 2 — Wed., M a r c h 4 Day 3 — M o n . , M a r c h 9 Day 4 — Wed., M a r c h 11 Diary #5 — March 24 - A p r i l 1, 1998 Day 1 — Tues., March 24 Day 2 — Wed., M a r c h 25 Day 3 — Tues., March 31 Day 4 — Wed., A p r i l 1 Chapter 6 Summary and conclusions Problems Arising i n the Study Suggestions for Further Research Conclusion  100 102 105 109 114 115 119 124 129  References  144  Appendix Ethical Review and Consent Materials University Ethics Authorization Parent(s)/Guardian(s) Consent Letter Parentis)/Guardian(s) Consent Form  149 149 150 151  — iv —  131 141 142 143  LIST OF FIGURES  Figure 2-1. A n example of a Wall Math board Figure 2-2. A n example of the type of problem that is used for the first question of Wall Math Figure 2-3. A n example of a question from the second set of problems that apply to the I R P Learning Outcome of "Number (Number Concepts)" i.e., place value Figure 2-4. Another example of a question from the second set of problems that apply to the I R P Learning Outcome of "Number (Number Concepts)" i.e., reading numbers Figure 2-5. A n example of a question from the third set of problems that apply to the I R P Leaning Outcome of "Shape and Space (Measurement)" i.e., linear measurement Figure 2-6. The 20 question types for each board of Wall Math Figure 2-7. Front side of the generic answer sheet designed for Wall Math Figure 2-8. One version of the back side of the generic answer sheet designed for Wall Math Figure 2-9. Year-round pictograph of students' tooth loses Figure 3-1. Cognitively Guided Instruction (CGI) model Figure 5-1. Counting the candies from the estimation jar on a counting tray Figure 5-2. The chalkboard and Unifix cubes that Douglas used to solve the "Stumper" Figure 5-3. D o n stringing his St. Patrick's Day 'bead' jewelry pattern  7 8 8 9 10 11 12 13 27 33 83 85 102  ACKNOWLEDGMENTS I am grateful to m y thesis committee o f D r . L i n d a Peterat. and D r . Heather Kelleher, and particularly my research supervisor, D r , A n n Anderson, for their insightful comments, constructive criticism, support of the study as a whole, and for the narrative style that I have employed in this thesis, and their trust in me to carry out the necessary work. I am eternally appreciative o f m y fellow teacher, Jane O n o , who was my codeveloper of Wall Math , not only for her sincere and dear friendship over many years, but also for her ideas and the hours and hours of work that went into making Wall Math  a reality.  I would like to sincerely thank both School District 43 (Coquitlam) for endorsing the study, and my principal, M r s . Jane Rothmund, for her support and for trusting me enough to allow me to conduct the study in m y classroom. M y thanks go to m y best friend, partner, and husband, D a v i d , for his secretarial skills, his editing ability, and his moral support, but mosdy for his having enough faith i n me to leave me alone and allow this thesis to be "mine". I should also thank his health care practitioners for enabling him to physically do whatever he can, and the Curriculum Studies Department of the Faculty of Education for allowing him to remain professionally active. Finally, I would like to thank a l l the students i n my class for just being themselves: lovable, curious, varied human beings who are attempting to make meaning of the world i n which they live. They are the study.  — vi —  CHAPTER  1  INTRODUCTION TO T H E STUDY  What is the teacher? A guide, not a guard. What is learning? A journey, not a destination. What is discovery? Questioning the answers, not answering the questions. What is the process? Discovering ideas, not covering content. What is the goal? Open minds, not closed issues. What is the test? Being and becoming, not remembering and reviewing. What is school? Whatever we choose to make it. A l a n A . Glatthorn (Ministry of Education, 1989, p. 19) Purposes of the Study In British C o l u m b i a ( B . C . ) , control o f the intended curriculum lies with the provincial Ministry o f Education ( B C M E ) . The B . C . elementary school mathematics curriculum is articulated and presented to teachers in the form o f an Integrated Resource Package (IRP), published by the B C M E , and distributed to every practicing teacher o f Kindergarten (K) to Grade 7 students in the province. However, as is the case everywhere, control over the implementation o f the applied curriculum rests with the individual teacher; when classes begin, teachers apply the mandated, intended curricula i n whatever way they deem appropriate for the particular group of children they have before them at that particular time. W h i l e the B C M E mandates the Intended Learning Outcomes (ILOs), how they are applied is the responsibility of the classroom teacher, and whether these I L O s are attained by the students depends i n part on both the materials and methods used i n that implementation by the teacher.  This study is about the efforts of one classroom teacher to use a locally  developed programme, called Wall Math, to assist the students i n her class to attain the I L O s of the provincial Grade 1 mathematics curriculum. The major purpose of the study is to  explore the ways that the programme can facilitate the understanding and retention o f learned mathematics skills by Grade 1 children.  The programme, dubbed Wall Math  by the first group of children to use it simply because it  is a series o f questions and activities posted on a w a l l o f the classroom, was originally developed as a system for the maintenance and practice o f the mathematics skills, strategies, and knowledge learned during mathematics lessons.  The children used these skills,  strategies, and knowledge to complete activities and solve problems, posed by the teacher or, at times, posed by the children themselves, that are similar to, or taken from real life situations, or related to other classroom studies in a variety of curricular areas. The goals of using Wall Math  are many.  They include: improving students' skills o f cooperative  learning, listening to and learning from their peers; keeping learned skills current by providing frequent review of their use; developing and practicing problem solving abilities using a variety o f skills and strategies that can lead to a greater understanding and ease o f use; setting challenges for students, thus encouraging risk taking; using real world problems, data, situations, and experiences that w i l l help the children understand that mathematics is not just a school subject, but is something that they w i l l need and use i n their every day lives outside of school; and finally, to make mathematics fun and enjoyable.  These goals are  congruent with the goals of both the Ministry o f Education i n the I R P , and, as I discovered when reading the literature, the National Council of Teachers of Mathematics ( N C T M ) i n the Curriculum and Evaluation Standards for School Mathematics ( N C T M , 1989).  A s part of the I R P , the B C M E provides a list of recommended learning resources that have each undergone an extensive evaluation, by at least the developers and the Ministry, prior to being included on the list. Materials and methods which have been developed by individual classroom teachers usually have not undergone such evaluation, and thus do not have the luxury of being defensible on the grounds that they are tried, tested, and proven programmes.  — 2 —  In order that a teacher can defend the use o f a locally developed programme, such a programme should undergo rigorous, systematic examination to ensure that it simultaneously assists i n the development o f the I L O s and does not lead to any unintended, less-thandesirable outcomes. A second purpose of this study, therefore, is to submit Wall Math  to a  systematic inquiry into its outcomes with students when used with Grade 1 children.  This study w i l l describe, in detail, Wall Math, and document how and why Wall Math  was  developed. It w i l l link the programme with the B . C . primary mathematics curriculum and the N C T M Standards,  and principles of, and beliefs about learning derived from constructivist  research and illustrated in Cognitively Guided Instruction. M o s t importantly, it w i l l provide examples from student learning that support these linkages. In doing so, it is hoped that the reader w i l l develop a better understanding of the ways children think and learn about mathematics and develop mathematical literacy, and ways that this teacher and her programme promoted such development.  O r g a n i z a t i o n of the Thesis This study is organized into six chapters: Chapter 1 has placed the problem i n its setting and provided initial background for the study; Chapter 2 provides the reader with a description of Wall Math  and how it is situated within the context o f the B C M E ' s I R P and the N C T M  Standards; Chapter 3 w i l l explore the educational, psychological, and mathematical research and literature which provide a framework for the study; Chapter 4 contains a brief description of the methodology used in the study; the diaries in Chapter 5 provide the "raw data" for the study in the form of narrative descriptions of classroom interactions with five sets ("boards") of Wall Math  questions; and Chapter 6 presents a summary of the study, discusses some of  the benefits and limitations of the study, and finally proposes some future directions for additional research in this area.  — 3—  CHAPTER 2 T H E WALL MATH  P R O G R A M M E IN C O N T E X T  Wall Math is a system for maintaining the mathematics skills, strategies, and knowledge, as set out i n the I R P , learned during mathematics lessons. It is also a system i n which these skills, strategies, and knowledge are used to solve problems posed by the teacher or children that are similar to, or taken from real life situations. It is called W a l l Math simply because the tasks and problems are posted on the wall, thus making mathematics a visible part of the curriculum, and each set of twenty tasks and problems, which is usually changed every two weeks or so, is termed a "board". F o r each board, each student's work is recorded on his or her own individual, generic answer sheet.  The Wall Math  system is not a completely original idea, but is a customized version of a  commercially produced programme called "Mountain M a t h " which was first seen by myself and several colleagues at a N C T M conference about six years ago i n Seattle. This commercial programme was very " A m e r i c a n " and quite expensive; it was not usable in our classrooms i n the form in which we first saw it. However, the ideas behind the programme were interesting, fit with our philosophies of child-centered and cooperative learning, and intrigued us. The concept caught our attention, and it stimulated lengthy discussions as each of us began developing our own version of the system which the children quickly dubbed "Wall Math".  Although we used a common framework i n our development of the new  programme, we each developed the system to fit our own teaching grade(s) and our own children's needs. The system has now been in the process o f development for five years, and is, or has been used at the Grades 6/7, 3/4, 2/3, 1/2, and K levels at Oldtown School. The system is also being used by teachers in several other schools who have used our ideas, which we conveyed to them in the form of after-school workshops, as a starting point to develop their own relevant sets of Wall Math  problems and procedures. W h i l e there are  many commonalties among the several versions of Wall Math  that originated i n our school,  and the several others that are operating in other schools, there are sufficient differences that, for the purposes o f this thesis, I w i l l discuss only my own version of the system.  I believe, as do many researchers (Bebout, 1993; Carpenter, A n s e l l , Franke, Fennema, & Weisbeck, 1993; Carpenter & Fennema, 1988; Carpenter, Fennema, Peterson, Chaing, & Loef, 1989; Cobb & Steffe, 1983; Cobb, Y a c k e l , & W o o d , 1988; Folkson, 1995; Harper, Malette, Maheady, & Brennan, 1993; Jitendra & K a m e e n u i , 1993; Jordan, L e v i n e , & Huttenlocker, 1995; Lampert, 1988; Putnam, Lampert, & Peterson, 1990; W i l l i a m s , McPherson, Mackintosh, & Williams, 1991; W i l s o n , 1991; Y a c k e l , Cobb, W o o d , Wheatley, & M e r k e l , 1990), that children come to school with an understanding o f many of the mathematics concepts that they w i l l need to solve many of the everyday problems that they w i l l and do encounter in their lives. They know how to divide their candy into two equal groups to share with their friend, and they know that i f they put two sets o f " L e g o " together, then they can build something much larger than they could by using only one set.  As a  teacher, my task is to use this inherent knowledge to build upon, so that the children w i l l learn the standard language, procedures, and written forms o f their mathematical knowledge. However, because every child is unique,— they each have different learning styles, different rates of learning, different knowledge and different ways of knowing — I must present the required curriculum i n a variety of ways using both m y o w n experience and that o f researchers, so that all the children may learn conventional mathematics.  I also believe that children learn best in an environment that is safe and supportive; one that encourages risk-taking and learning from one's own mistakes without danger to one's selfesteem. A s I developed my Wall Math  system, I had these beliefs i n mind, along with the  B C M E ' s ILOs and the belief that without practice, the conventional mathematics knowledge, strategies, and skills learned during classes and lessons can quickly disappear. Use it, or lose  it! I believe it was Alfred Noyes Whitehead who is credited with saying that "Knowledge doesn't keep any better than dead fish!"  Wall Math  was developed to use classroom learning i n a wide variety of circumstances that  are similar to, or actually taken from, real life situations, so that the problems or questions are both interesting and relevant to my particular class at the particular point in time. A s the children get used to working on the Wall Math problems, and become familiar with the types of questions posed, they often make suggestions for problems to be used in the future. This then makes the questions even more relevant and interesting.  What Does Wall Math L o o k L i k e ? M y Wall Math  is located on one of the magnetic chalkboards in my classroom, making it  easy to change the questions on a weekly or bi-weekly cycle. The chalkboard is almost three metres long, and has a small (approximately one metre) bulletin board at one end. There is also a table located under the chalkboard on which are placed the various manipulatives (measuring and weighing devices, stamps for making patterns or displaying place value, the estimation jar, etc.) with which the students w i l l need to work to answer this board's problems. The boards have sufficient space to have a l l twenty o f the questions displayed at one time. A picture o f one o f the W a l l Math question sets/boards is displayed i n Figure 2-1.  Figure 2-1. A n example of a Wall Math board. The twenty questions are divided into three separate sections. The first section consists o f only one question, which covers the I R P learning outcome o f "Patterns and Relations (Patterns)". Figure 2-2 presents an example of one type of activity which would be presented to the students in this first section. It usually requires that the students look at a pattern which has been presented, copy that pattern, recognize what the pattern is, and then demonstrate that recognition. Students may demonstrate this recognition and understanding of the pattern by continuing the pattern, by using the appropriate manipulative blocks which are identical i n size, colour, and shape to the pattern presented. The method o f recording their answer on the answer sheet is usually a choice left for the children to make, but might include pasting pre-cut shapes which are identical to those presented in the question, which they would then colour; using inked rubber stamps which are again identical to those in the question; or using their own drawings.  — 7 —  1. Copy and continue  A\ 7A\  Figure 2-2. A n example o f the type of problem that is used for the first question of Wall Math. A s students' patterning skills and reading skills develop, this question might take the form of challenging the students to construct a pattern that complies with a written description.  The second section consists of nine questions which fall within the I R P learning outcomes o f "Number (Number Concepts)" and "Number (Number Operations)", and includes a variety of questions that involve counting, completing number lines, addition, subtraction, place value, printing numerals, and reading number words. Figures 2-3 and 2-4 present examples of these types of questions.  2. Draw and print as a number two tens and three Figure 2-3. A n example of a question from the second set o f problems that apply to the I R P Learning Outcome of "Number (Number Concepts)" i.e., place value.  — 8—  3. Words please  17 Figure 2-4. Another example of a question from the second set o f problems that apply to the IRP Learning Outcome of "Number (Number Concepts)" i.e., reading numbers. Figure 2-3 requires the students to represent, in a graphical manner, place value concepts. The c h i l d w o u l d use place value blocks, or some other manipulative, to show his/her understanding o f these words, then use tally marks, place value stamps, or some other graphic form to record on their answer sheet some acceptable representation o f the number 23 and also to write the numeral "23". The question illustrated i n Figure 2-4 requires the children to read the numeral 17 and then print on their generic answer sheet the word(s) which represents that number, "seventeen".  The remaining ten questions, in the third section, include a variety o f problem solving exercises that cover the I R P learning outcomes of "Shape and Space (Measurement)", "Shape and Space (3-D Objects and 2 - D Space)", "Shape and Space (Transformations)", and "Statistics and Probability (Data Analyses)" and "Statistics and Probability (Chance and Uncertainty)".  Figure 2-5  displays an item which might be included to reinforce the  mathematics skills under the I R P learning outcome o f "Shape and Space (Measurement)". It requires students to select the requested measuring device to measure their own arm. They then record their answer on their answer sheet. This example also shows how a l l students w i l l do the main part o f the question, but for students who wish to do further work, or for more advanced students (in Grade 2 rather than Grade 1) a " T r y i t " addendum to the question w i l l provide a further challenge. In this case, because the Centicubes are smaller than the Unifix cubes, the challenge is to count more objects and compare these numbers, possibly as a ratio. W e have a rule that i f there are two parts to a question, and you are i n Grade 1, you  only have to do the first part, but i f you are in Grade 2, you must do both parts of any twopart questions. This rule is altered for special needs students. Figure 2-6 presents a listing of all the 20 question types.  4. Measure The length of your arm with unifix cubes * do it again with centicubes Figure 2-5. A n example o f a question from the third set of problems that apply to the I R P Leaning Outcome o f "Shape and Space (Measurement)" i.e., linear measurement.  Figures 2-7 and 2-8 present the front and back sides respectively of the generic answer sheet that students use to record their answers to each question on a given Wall Math board. The back side o f the generic answer sheet has several versions which differ for Question 19 — Measure.  The sheet displayed in Figure 2-8 has a picture o f a thermometer, but others  contain pictures of various measuring instruments. A supply of these answer/response sheets is always available to students.  — 10 —  The following is a description of a typical set o f questions for a board with no special occasions (such as Halloween, Christmas, Valentine's D a y , etc.) 1. Copy and continue a pattern, or create and record a pattern according to a set of rules or instructions composed by the teacher or another student. 2. C o p y and continue the number line segment. 3. Read the number and record it using words. 4. Read the number word(s) and record it using numbers. 5. Print the numbers that come before and after. 6. Print the number that comes between. 7. Place value: show V tens and ' y ' as a picture and as a number. 8. A n addition computation. 9. A subtraction computation. 10. Print an equation from a picture or story problem. 11. Order objects by attributes and explain the order (use mathematics vocabulary o f comparison). 12. Count the tallies (skip counting by 5s and 10s) or keep track of counted objects by using tallies, and print the number. 13. Draw a picture to demonstrate an understanding of an equation or the action i n a story problem. 14. Observe a set o f pictures or objects and explain why one does not belong. 15. Count objects: often a real-life count using money or needing drawings or tally counting to keep track (trees on the playground, cars going by in five minutes, etc.), often for the purpose of comparing (Are there more cars today than yesterday?, D o we have enough money to buy this?, etc.) 16. Geometry: includes a wide variety o f tasks from naming shapes, finding real objects o f a specific shape, building shapes, etc. 17. Use a number wheel to practice computation. 18. Estimate and count objects in a jar, a picture, or outside (birds i n a tree, cars i n the parking lot, etc.) 19. Measurement o f time, temperature, length, mass, volume, with some results used for graphing (Measure the length o f your foot with " U n i f i x " cubes and use the results to make a class graph). 20. A "Stumper": a problem that requires students to stretch their thinking, a fun number activity, or a logic problem. These are often presented after reading a story such as "The Panda Party" or "The Twelve Days o f Christmas". The focus o f the questions can be related to other areas o f study i n the classroom (for integration purposes), or to a special holiday or event. I have many sets o f questions which I developed around most traditional holidays and events such as " H o t D o g Day", sports days, etc. Figure 2-6. The 20 question types for each board of Wall Math.  WALL MATH  1. Pattern  2. Number Line "  o  3. Word please 4. Number please  8. Add  5. Before and After «  o  o —  9. Subtract  6. Between _ft  A  O  7. Draw and print as a number  10. Equation please  11. Order please  o  o  o o  Figure 2-7. Front side of the generic answer sheet designed for Wall Math..  12. Tally Count  17. 1  AJO  /  13. Picture please  A  \  VTT  7  2V/  \  18. Estimate then Count  does not belong because  WW 1 1 1 1 1 1 1 1 1 1 1 1  19. Measure 15. How many? C  40 30 20 10 0 -10  }  (  14.  5  °c  20. Stumper! 16. Geometry  Figure 2-8. One version of the back side of the generic answer sheet designed for Wall Math..  —  13 —  In order to have available a year's worth o f activities that would cover all of the learning outcomes set out in the I R P , I initially produced a series o f 40 items of increasing difficulty for each o f the 20 main question types. Most o f the activities use manipulatives o f some sort, and cooperative work and learning is encouraged.  The activities in the last set o f 10  questions (Questions 11 to 20) often require input from home and/or the use o f the outdoors; use o f real life knowledge or skills; and the use o f higher level thinking skills.  Although  creating these questions and activities was a time-consuming task, I sincerely believed, and I continue to believe that the activities must be pre-made, and then presented on succeeding Wall Math boards in an order of ascending difficulty. The questions must also be at least tentatively pre-made at the start of the year i f the system is to become an effective part of the regular mathematics routine of the classroom. A teacher does not have time to sit once every week, or even every other week, to produce 20 quality questions covering all areas of the mathematics curriculum. I found that once I got started on a topic, producing 40 questions of varying degrees of difficulty was not that hard a task to accomplish. However, because each class is so different, the rate o f increasing level of difficulty from board to board has been, and I suspect w i l l continue to be an on-going problem. Frequently I have had to "parachute" new questions into the original set of questions in order to provide more practice in a specific area, or I have skipped questions because a particular class did not need further practice of that particular skill. I have also created many questions that are 'holiday' or 'theme' related, that can be used in place of the original questions. These special activities have been made each year so that they w i l l fit with the needs or interests of the class at the particular time. A t times I have quickly made up a new question immediately prior to posting a board in order to take advantage of some current event that has been unanticipated, but has captured the attention or interest of the students.  Because I usually have a dual-age class, many o f m y question sets must be done at two levels. Some questions just require an added sub-question that challenges the older students,  and possibly the more 'mathematics-wise' younger students to think of another way of doing the activity, or to complete the activity i n more detail. Figure 2-5, which was previously shown, presents a question which is an example o f one o f these dual level activities. Other questions, such as many of the computation questions (#8 and #9) simply have two separate questions o f increasing difficulty or complexity. In this case, as previously mentioned, all of the Grade Is do the first card/question which has been designed to be at their level o f practice, while the Grade 2s and the more able Grade Is do the question on the first card, as well as a second card/question which has a Grade 2 level question. A third card/question is sometimes added to provide a challenge for the Grade 2 students.  Once a l l o f the activities have been completed, assessed (which I do as the children are working and I am helping individuals), and discussed as a group, the record sheets are placed in a special binder that is used to collect all o f their mathematics recording sheets, writings, problems, and drill sheets. When I first started using this system, I sent the record sheets home, but quickly discovered that because the children could not usually adequately explain to their parents what this sheet was all about, the parents paid no attention to it. This resulted in some waning o f interest on the part o f the children, and i n parents frequently not understanding the importance of providing the information asked for as homework so that data collection for some o f the problems could be completed. Questions such as, What time do you go to bed on a school night? are used for both graphing and recording the time on two kinds o f clocks, and they also provide an important real-life connection. Without this information from parents, the children are not as interested in the activities as they are i f their own personal information is part of the product.  However, keeping the papers together and  sharing them on parent conference night, with the wall of questions right there i n front of us, appears to have solved this problem. K n o w i n g they are going to share their work with a parent seems to help maintain the interest of the children, and parents begin to understand  — 15 —  how important their participation and assistance is for the collection of data for many of the time, measurement, geometry, and graphing questions.  On Wall Math days, which are normally Tuesday and Wednesday, we usually go over the entire set of questions as a class before I send the children off to work. This is done in order to ensure that all students have the appropriate vocabulary to read the questions, to clarify misunderstandings the students might have about any of the questions, and to give the students an opportunity to ask any general questions they might have. During the early part of the year, these discussions frequently result in the presentation of possible solutions. However, in order to encourage risk taking, to stimulate them to look for more than one solution to a problem, and to help the less able students answer more o f the questions, I encourage these preliminary discussions. A s the year progresses and the children become more familiar with the process and my expectations, these preliminary discussions become shorter and are basically used only for the purpose of clarifying questions, and the post-work discussions become more lengthy.  When I first started Wall Math, I encouraged everyone to work in cooperative groups, or with a partner. However, I found that certain children only did the first two parts of the board because those particular questions are straight-forward number questions and, unless helped by me, those students never got to the problem solving questions. Over the last two years, I have, at times, been using a different strategy that has resulted in these 'non-workers' accomplishing more, which, in turn, seems to help build their self-esteem, which then allows them to begin working more, either on their own or with a carefully chosen "buddy". M y strategy is simply to work with this group as a unit to discuss and find solutions to the questions, starting with Question 19 rather than with Question 2 (Question 1 often has a limited number of materials to use, so the children must take turns completing this question), while the other children are encouraged to start with something that interests them. Then,  once the last nine questions are completed together, the group is encouraged to finish the board by themselves since these are generally the children who do not work well as part of even a carefully chosen team as yet. Because Question 2 0 — Stumper is always discussed as a whole group, it is not done with this small group.  The goals and objectives of using Wall Math  with my students are many. I hope that the  frequent use of the mathematics knowledge, skills, and strategies learned in lessons w i l l help to make them become automatic.  1 hope that students' skills of cooperative learning w i l l  grow by students sharing and listening to, and learning from their peers.  I hope that the  practice o f problem solving using a variety o f skills and strategies w i l l lead to a greater understanding and ease o f use.  I hope to set challenges for students to attempt, thus  encouraging risk taking. I hope that by using real world problems, data, situations, and experiences the children w i l l understand that mathematics is not just a school subject, but something that they w i l l need and use in their everyday lives outside o f school. Finally, I hope to make mathematics fun and enjoyable for my children so that their future interactions with mathematics w i l l not be as painful and intimidating as they were for me.  Wall Math,  as it appears in my classroom, was developed by me to reflect my philosophy of  teaching and the needs and interests o f the 6- and 7-year o l d children in my classroom. However, as discussed at the beginning of this chapter, several versions of the system have been developed within a common framework for use by other teachers at a variety o f grade levels in their own classrooms. While there are many commonalties among the versions that are in use, there are sufficient differences that, for the purpose of this study, only my version has been investigated.  As I developed the twenty categories of questions for my Wall Math  board, I kept in mind  the Ministry mandated mathematics curriculum and its ILOs, the needs of the children in my  class, my knowledge and beliefs about how children learn, and my own, personal teaching style. The mandated curriculum is divided into four general categories, eight sub-categories, and a variety of specific ILOs within each sub-category. Within the twenty questions on the Wall Math board, every I L O can be addressed at some time during the year, either directly, or indirectly through a follow-up activity or discussion initiated by a specific question.  The N C T M suggests that, in many classrooms, mathematics is taught as units of study, rather than an integrated programme.  "The mathematics curriculum is generally viewed as  consisting of several discrete strands. A s a result, computation, geometry, measurement, and problem solving tend to be taught in isolation." ( N C T M , 1989, p. 32). Wall Math creates integration within the various mathematics strands, as well as the integration of mathematics with other curricular areas and real life situations. In the typical Grade 1 classroom, and as I used to do, children are taught about number, then taught to add and subtract, usually by learning the procedures and algorithms. This unit o f knowledge (Number Operations: Addition and Subtraction) is probably the only one that is practiced throughout the school year.  W h i l e continuing to practice these procedures, children are then taught units o f  knowledge on such things as place value, geometry, time, measurement, problem solving (i.e., word problems), and money. However, once each of these units has been completed, the skills learned are then usually used only occasionally, i f ever, during the remaining part of the school year. Then, past experience has shown me that when I, as the next year's teacher, decide to teach the Grade 2 curriculum requirements for that unit, the children have either completely forgotten, or at least require a thorough review of the Grade 1 skills before they are ready for something new to be added to their knowledge. Wall Math can be used not only to provide practice for these units of study throughout the year, but also can be used to introduce and practice small parts of these units throughout the year so that familiarity and success can help lead to a deeper, more lasting understanding of the whole concept.  —  18 —  Money is a difficult concept for many Grade 1 children, particularly for those who have had little or no prior experience with it. However, by introducing the use of pennies into the Wall Math  programme early in the school year, the children become familiar with using them for  counting by ones, making patterns, sorting, or problem solving. A n early Question 14 — Which doesn't belong? Why? on the Wall Math board presents three Canadian pennies and an American one. The children are to look at them, decide which one does not belong with the others, and explain their decision, both orally and in writing. A s they learn to skip count by tens and fives, the children are then introduced to using, and counting, dimes and nickels. Counting mixed groups of coins then becomes.just a case of remembering the 'counting money rule' as my children call it, and "creating equivalent sets of coins up to 100 in value" (Ministry of Education, 1995, p. 144) becomes a snap. The I L O of "use money as a form of exchange" (Ministry of Education, 1995, p. 144), which has always been a difficult concept for me to get across to children who do not have regular access to money, becomes easier when the children have already been made familiar with the coins, their values, and how to count them by using coins as manipulatives for patterning, counting, adding, subtracting, and problem solving within a variety of Wall Math  questions.  The I R P , Wall Math,  a n d the N C T M  Although the B C M E ' s I R P does not specifically state that it's underlying philosophy o f teaching mathematics and it's general overall I L O s are similar to those o f the N C T M , I found, as I read through the N C T M publications, that they did, in fact, match quite well. A n unintended, but, I believe, important outcome of this study then became to link the overall goals of the IRP to those of the N C T M , and to place the Wall Math  activities within this  context.  Both the B C M E and the N C T M stress that the outcome of a mathematics programme should be "mathematical literacy" for all students. In order to promote the achievement of this goal,  — 19 —  the N C T M has hypothesized "three features of mathematics [that] are embedded in the Standards." (1989, p. 7), and, although not precisely stated, these three features also are contained in the M i n i s t r y ' s I R P . The first is that " ' k n o w i n g ' mathematics is ' d o i n g ' mathematics" ( N C T M , 1989, p. 7). That is, a student is more likely to learn a new concept with greater understanding i f s/he is actually using it for a purpose that is o f interest to him/her. "This active process is different from mastering concepts and procedures. W e do not assert that informational knowledge has no value, only that its value lies in the extent to which it is useful in the course of some purposeful activity" ( N C T M , 1989, p. 7).  The second feature specifies the expected student activities associated with doing mathematics, and includes two general principles. "First, activities should grow out of problem situations; and second, learning occurs through active as w e l l as passive involvement with mathematics" ( N C T M , 1989, p. 9).  The third feature "sees teachers encouraging students, probing for ideas, and carefully judging the maturity of a student's thoughts and expressions", and, "that problem situations must keep pace with the maturity — both mathematical and cultural — and experience of the students" ( N C T M , 1989, p. 10). The teacher must be a facilitator, who assists the children in thinking about their work and constructing their own knowledge, rather than being a transmitter of units of knowledge.  These features reflect the work of the many constructivist researchers whose findings indicate that children come to school with prior knowledge o f mathematics onto which new knowledge is built (Bebout, 1993; Carpenter, et al., 1993; Carpenter & Fennema, 1988; Carpenter, et al., 1989; Cobb & Steffe, 1983; Cobb, et al., 1988; Folkson, 1995; Harper, et al., 1993; Jitendra & Kameenui, 1993; Jordan, et al., 1995; Lampbert, 1988; Putnam, et al., 1990; Williams, et al., 1991; Wilson, 1991; Yackel, et al., 1990). Cobb et al. (1991) explain  that "from the constructivist perspective, mathematical learning is not a process of internalizing carefully packaged knowledge but is instead a matter o f reorganizing activity, where activity is interpreted broadly to include conceptual activity or thought" (p. 5). Koehler and Grouws (1992) go on to say that the constructivist assumptions about how students learn changes the assumptions about what teacher actions or behaviors might be desirable. If one subscribes to a constructivist view of learning, then the goal is no longer one of developing pedagogical strategies to help students receive or acquire mathematical knowledge, but rather to structure, monitor, and adjust activities for students to engage in (p. 119).  These features also reflect the numerous research articles and reports of Carpenter, Fennema and others on Cognitively Guided Instruction ( C G I ) that show that children can learn computation skills by solving word problems, and that real mathematical knowledge and understanding comes by doing mathematics, not by memorizing procedures and algorithms.  In order to achieve mathematical literacy, the B C M E and the N C T M have set out several overall goals that illustrate and explicate their philosophy about how the knowledge and skills should be taught. A general goal of both the I R P and the N C T M is that children should develop a positive attitude toward mathematics and become confident i n their mathematics abilities. The I R P states that "Classroom practice and teaching strategies should promote positive attitudes towards mathematics for all students" (Ministry o f Education, 1995 p. 2) (emphasis i n original). The N C T M states that one of it's " N e w Goals for Students", is "becoming confident in one's own ability," and that "students need to view themselves as capable of using their growing mathematical power to make sense of new problem situations in the world around them" ( N C T M 1989, p.6).  One o f the features o f Wall Math  is that it  can provide a wide variety of questions at many levels o f knowledge so that every child in the class is capable of completing most, i f not all of the questions at some level, thus, hopefully creating an T can do it' attitude in most children. M a n y o f the questions are set up with a basic question and then a challenge. For example, Question 19 — Measure — 21 —  may ask  the children to measure the length of their foot using " U n i f i x " cubes.  Then a challenge  would be set for them to find out how long both their feet are, or to find out how long their foot and a friend's foot are together. If an even greater challenge is needed, a child or small group of children could be asked to find out how long everyone's feet are i f they were all put together. This challenge would require the use of a survey and a calculator to complete it, but it is within the capabilities of some Grade 1 children. The basic activity, however, is within the capabilities of all Grade 1 children, and I have found that those not capable or interested in completing the challenges do not seem to notice or care that others are doing something different; they appear to be completely satisfied and happy with successfully completing the basic question. The results of the challenge are shared with the whole group so that those who did not participate in the challenge this time are at least exposed to the more complex ideas and strategies being used by their classmates, with the hope that some w i l l respond to a challenge question, or work on one with a partner, on another board.  A t the end of the school year, I asked the children in my class to tell me how they felt about Wall Math ; to tell me what they liked and didn't like about it. Quotations from their writing can be found in the description of the students in Chapter 4. Most of the children wrote that it was fun and easy, along with telling which questions were their favorites. Although we always do Wall Math  after recess, many of the children examine a new board, particularly  Questions 14, 15, 16, 18, 19, and 20 as soon as they come into the classroom in the morning. They enjoy Wall Math and the problem solving within Wall Math  enough that some of the  children have some of the answers ready to record as soon as the answer sheets are handed out!  A second overall goal of both the IRP and the N C T M is that o f problem solving. The I R P states that "Problem solving is the cornerstone of mathematics instruction. Students must learn the skills of effective problem solving, including the ability to communicate solutions,  — 22 —  so that they w i l l become reasoning, thinking individuals able to contribute to society." (Ministry of Education, 1995, p.2) The N C T M (1989) states that "problem solving should be the central focus of the mathematics curriculum. A s such, it is a primary goal o f all mathematics instruction and an integral part of all mathematical activity" (p. 23). They also state that "the development of each student's ability to solve problems is essential i f he or she is to be a productive citizen" ( N C T M , 1989, p.6). Wall Math is set up so that there are many problems to be solved within each board, and sometimes within an activity. Question 1 — Pattern, often begins with a problem such as, given these materials, make a pattern that is x units long, make your product y patterns long.  For some children, creating their own pattern  to these specifications is enough of a challenge. Others, however, might be encouraged to tell how many of each unit w i l l be needed to complete their pattern and/or how many units w i l l be used all together.  See Diaries #3 and #4 (Chapter 5) for descriptions of using this  type of a problem solving activity. Question 20 — Stumper is always a problem of some sort; a logic problem, a counting problem, using a code (words become telephone numbers), and ' i f - then' problems are a few of the types that are used here. The diaries (see Chapter 5) contain the descriptions of five of these "Stumper" problems and how the children solved them. Many of the questions that are not counting or computation involve problem solving in the sense that the children do not readily know the answer and must actively look for, or think about a solution, and then find some way to communicate this solution to their classmates and teacher. Even the computation questions sometimes pose problems. A redcoloured " T r y It" tag on an equation indicates to the children that this is something new and that I'm looking to see i f they can discover and explain how to do it. This sign w i l l appear on a new type or level of equation when some of the children have indicated to me during group activities and private discussions that they are ready to try the next step in computation without my assistance. Strategies for solutions are discussed individually, or in small groups, until explanations are clear and precise enough to share with the whole group. According to the N C T M , "problem solving should be the central focus of the mathematics curriculum"  ( N C T M 1989, p. 23). This type of problem solving is a central focus of Wall Math . Although problem solving, in the sense of finding a solutions to word or story problems, is not a major component of Wall Math, word problems are included among the questions, sometimes as part of Question 15 — How Many,  but most frequently in Question 10 —  Equation please. More commonly, however, the children are asked to create a word problem as a challenge.  Question 10 — Equation  Please  and Question 13 — Picture  Please  frequently set this challenge, once the children have become familiar with reading and solving a variety o f word problems done during small and whole group activities and discussions as part of my normal mathematics programme.  A third overall goal o f both the I R P and the N C T M is that o f communication. The I R P states that "Mathematics is a language, a way of communicating ideas...[and that] all activities that involve students in exploring, investigating, describing, justifying decisions, and explaining promote the development o f communication skills." (Ministry o f Education, 1995, p.3) In the discussion about "Mathematics as Communication" ( N C T M , 1989, p. 26) the N C T M suggests that it is important to the development of mathematical understanding that young children be encouraged to talk about their solutions and to interact with others so that they can construct their mathematical knowledge, learn other ways of thinking about solutions, and clarify their own thinking. They also suggest that writing about solutions helps clarify thinking. It has been my experience that, at the Grade 1 level, talking about their thinking works very well, but writing does not.  I have found that, at this level, the children are  expending so much of their writing energy actually putting the words on paper, with some semblance o f correct spelling, grammar, and sentence structure, that they do not get the mathematical thinking and language down as well as they can talk about it. When asked to explain, i n writing, how they got an answer, I usually get sentences such as, "I counted them," (Cathy), or "I added the 12 and the 5 and the 10 and got 27. There are 27 balls" (Don). Yet when I ask the children to explain verbally, I get explanations such as, " W e l l ,  first I added the 12 and the 10 to get 22, then I added the 5 on to get 27," (Don), or "First I added the two groups of 10 to get 20 then I added the 2 and the 5 to get 7. Then I added the 20 and the 7 to get 27. So there's 27 balls on the s h e l f (Mike).  I have also found that it is  much easier to get children of this age to alter their explanations, while I question them in order to delve deeper into their thinking, i f they are talking than i f they are writing, because they are usually very reluctant to re-do or change their writing.  Communication in the form of working with classmates in a cooperative manner, and sharing thinking or talking with individuals or within a small or large group is a central focus o f Wall Math.  During Wall Math  work sessions the classroom often appears to be in chaos, as the  children move from place to place to look at the w a l l , to get and use manipulatives or equipment, to talk to each other, or to talk to me.  However, i f one watches and listens  carefully, most children are doing mathematics, and this doing, as suggested by the N C T M ( N C T M , 1989), w i l l result in a better knowledge of mathematics. A l s o , each work session generally ends with a whole group discussion of the day's activities and their results. These sharing times are set up with two purposes in mind.  The first is to assist children in  clarifying their own thinking by sharing it with others; listening to the ideas of others may help clarify a child's thinking or understanding, and asking and answering questions can help clarify the thinking of many children. The second purpose is to collect assessment data for both planning future instruction and for reporting to students and parents. "Attending to students' communications about their thinking also gives teachers a rich information base from which they can make sound instructional decisions." ( N C T M , 1989, p.28) F o r me, these sharing sessions provide a wealth of information about many children in a frequently short, but intensive set of discussions. These times also allow me an opportunity to share new ideas or formal methods of computation without the expectation from the children that they must remember and/or use it the very next time a problem of this type comes up. These  — 25 —  work sessions and group discussions where students "socialize" serve a variety of additional educational and developmental functions. Y a c k e l , Cobb, W o o d , Wheatley, and M e r k e l (1990) point out that, in addition to giving students problems to resolve, much learning or construction of knowledge takes place through social interactions, with the teacher and peers as part of problem solving. When children are given the opportunity to interact with each other and the teacher, they can "verbalize their thinking, explain or justify their solutions, and ask for clarifications. Attempts to resolve conflicts lead to opportunities for children to reconceptualize a problem and to extend their conceptual framework to incorporate the alternative solution methods" (p. 19). (Koehler & Grouws, 1992, p. 119) A fourth overall goal of both the I R P and the N C T M is that o f connecting mathematics concepts to each other and applying mathematics to real life experiences. The I R P states that "Students become aware of the usefulness of mathematics when mathematical ideas are connected to everyday experiences.  Learning activities should help students to relate  mathematical concepts to real-world situations and allow them to see how one mathematical idea can help them understand others." (Ministry of Education, 1995 p. 3) The N C T M states that, It is important that children connect ideas both among and within areas of mathematics. Without such connections, children must learn and remember too many isolated concepts and skills rather than r e c o g n i z i n g general principles relevant to several areas. W h e n mathematical ideas are also connected to everyday experiences, both i n and out o f school, children become aware o f the usefulness o f mathematics ( N C T M , 1989, p. 32). Wall Math  is a way of helping children make the connections among mathematics strands,  between mathematics and other curricular areas, and most importantly, between mathematics and real life. Diary #4 contains a description of a Question 15 — How Many,  that asks the  children to tell how many teeth the children in our class have lost this year. In order to complete this task the children needed to count the paper teeth on a hallway graph that was originally started in September, is displayed throughout the year, and is updated every time a child loses a tooth. Figure 2-9 shows a picture of this graph.  — 26 —  Figure 2-9. Year-round pictograph of students' tooth losses. However, the question then has two challenges. The first one asks the children to tell in which month the most teeth were lost. In order to do this the children must understand how a bar graph works. The second challenge asks the students to tell how many children  have  lost teeth so far this year. This answer, of course, is different from the number of teeth lost, and requires the children to do some problem solving using reasoning and/or logic skills. The children are then asked to explain orally why the answers are not the same. In order to do this activity, the children must be able to read and interpret the information presented on the bar graph.  Question 16 — Geometry, always asks the children to connect their geometric knowledge to real life experiences. Diary #1 (see Chapter 5) contains the descriptions o f the children's answers to the question, Name the shape of a traffic light. This was not a question which had a single, correct  answer, I was to discover.  D i a r y #5 (see Chapter 5) contains the  descriptions of several discussions that happened when the question asked the children to name the shape of a volcano. These descriptions, and many others, contain examples of the children making not only the connections I want them to make, but, more importantly, the  — 27 —  personal connections that should lead to a greater understanding of the concept and the construction of further knowledge which is based on these personal understandings and connections.  A fifth overall goal of both the IRP and the N C T M is that of developing mathematical reasoning. The IRP states that, "Students should understand that mathematics is not simply memorizing rules. Mathematics should make sense, be logical and be enjoyable" (Ministry of Education, 1995, p.3). It then goes on to say that, "Students require the freedom to explore, conjecture, validate, and convince others i f they are to develop mathematical reasoning skills. A n d it is important that their ability to reason well is valued as much as their ability to find correct answers" (p.3). The N C T M makes an even stronger statement in its discussion of mathematical reasoning when it states that, "making conjectures, gathering evidence, and building an argument to support such notions are fundamental to doing mathematics. In fact, a demonstration of good reasoning should be rewarded even more than students' ability to find correct answers" (p. 6).  Wall Math  encourages mathematical  reasoning in that it requires the children to solve problems using their own methods, thinking, and ideas, and by asking them to share and validate these with both their classmates and their teacher. Question 14 —Which doesn't belong? Why?  requires the children not only to  explain the reason(s) for their own choice, but also to listen to, and accept the choices of, and reasons for a classmates' decision. The descriptions in Diaries #2 and #5 (see Chapter 5) are examples o f this type of discourse. Question 20 — Stumper  always requires the children to  use their skills of mathematical reasoning to find solutions to, and explanations for such "Stumpers" as the logic problems described in Diary #4, the counting problems described in Diary #5, or the number puzzles described in Diary #2.  Question 16 — Geometry  often  requires the use of mathematical reasoning to explain why a particular answer was given. The descriptions in Diaries #1 and #5 provide examples of situations where explanations  — 28 —  about the shape names chosen were required in order to convince classmates that the choice made was legitimate and appropriate.  This chapter has provided a fairly detailed description of the Wall Math programme as as used in my classroom and implemented for this study. It has also situated Wall Math the context of the B C M E ' s I R P for primary mathematics and the N C T M  Standards.  within  CHAPTER  3  REVIEW OF T H E LITERATURE  Traditionally, a thesis such as this does a formal review of the literature, primarily including studies that are similar to the study being reported. However, i f one attempts to locate studies where participant observers or teacher researchers study a mathematics programme with the goal o f coming to understand how students interact with the programme, their teacher, and their peers, little can be found. O n the other hand, the body of literature having to do with the foundations of the Wall Math  programme is more extensive. This chapter  attempts to familiarize the reader with current thinking about the nature of mathematics, how students learn mathematics, and how mathematics might best be taught, and situates Wall Math within that current thinking.  Wall Math and Cognitively Guided Instruction M a n y teachers who continue to use the traditions of rote learning, teacher-centered education, and view mathematics teaching as drill and practice using operation algorithms, need to begin to change their style of, and beliefs about teaching, as well as their beliefs about how children learn mathematics, i f they are to assist the children i n their care to achieve the goals of the I R P and the N C T M standards. W h i l e reading the literature published during the ten years since the N C T M Standards  were released, at least three major programmes o f research  have focused on helping teachers make these changes i n their beliefs and instruction. Although a l l three share the assumptions that students construct knowledge rather than simply absorbing it, and "that significant changes i n practice depend on teachers fundamentally altering their epistemological perspectives so that they appreciate that students construct knowledge and that they must adapt instruction accordingly" (Carpenter, Fennema, & Franke, 1996, p. 5), each group has gone about relaying their message to teachers in different ways. A l l three "groups of scholars who have studied mathematics teachers have  — 30 —  reported that as their knowledge of their own students' thinking grew, teachers' knowledge of mathematics increased, their beliefs about mathematics and its instruction were modified, and instructional change occurred" (Fennema, Carpenter, Franke, L e v i , Jacobs, & Empson, 1996, p.404).  The group o f Fosnot, Schifter, and Simon, in their Summer Math for Teachers  Project  (Schifter & Fosnot, 1993; Simon, 1995; Simon & Schifter, 1991), "encouraged teachers to develop mathematical knowledge and a constructivist pedagogy by having them participate in workshops that reflected the pedagogy the researchers wanted the teachers to emulate" (Fennema, et al., 1996, p.404). Results showed that as teachers learned new ideas about mathematics, they changed their beliefs about the importance o f making instructional decisions based on children's understanding. These changed beliefs then led the teachers to make instructional decisions based on their knowledge of children's understanding, and change their instructional practices to reflect constructivist principles more strongly.  A second group, Cobb, Y a c k e l , and W o o d , from the Purdue Problem-Centered Project  Mathematics  (Cobb, W o o d , & Y a c k e l , 1990; Cobb, W o o d , Y a c k e l , & M c N e a l , 1993; C o b b ,  W o o d , Y a c k e l , N i c h o l s , Wheatley, Trigatti, & Perlwitz, 1991; C o b b , Y a c k e l , & W o o d , 1992), designed instructional activities that provided "opportunities for children to construct their own mathematical knowledge" (Fennema, et al., 1996, p.404). Teachers were then given workshops on the use o f these activities i n which they were encouraged to reflect "about their instruction and students' thinking" (Fennema, et al., 1996, p. 404), and "to ask students to describe their thought processes as they engaged i n the instructional activities provided" (Fennema, et al., 1996, p. 404). The researchers then "attributed the changes they found in instruction to teachers' opportunities to consider how their students learned" (Fennema, et al., 1996, p.404). The results of this study were reported to be that children in  — 31 —  the classes o f project teachers had higher levels o f conceptual understanding and more positive attitudes toward mathematics than children i n the classes of non-project teachers.  The third group, Carpenter, Fennema, Peterson, Chiang, and L o e f (1989), in a project called Cognitively Guided Instruction  (CGI), set out to "help teachers build relationships between  an explicit research-based model of children's thinking and their own students' thinking by encouraging reflection on how the model can be interpreted in light of their own students and classrooms" (Fennema, et al., 1996, p.405). Teachers i n the project attended workshops that were "designed to familiarize them [the teachers] with the findings o f research on the learning and development o f addition and subtraction concepts i n young c h i l d r e n " (Carpenter, et al., 1989, p.503). The teachers were then given time, encouragement, and assistance to design a mathematics programme for their own classroom that would reflect their new knowledge o f children's learning, and their own teaching style. The researchers then studied these teachers i n their own classrooms throughout the following school year. Results showed that C G I teachers agreed more than did control group teachers with the belief that instruction should facilitate children's construction o f knowledge, C G I teachers spent significantly more instructional time interacting with students about problems, and C G I teachers allowed students to use multiple strategies more frequently (Carpenter, et al., 1989). "Differences i n students' achievement on both problem solving and recall o f number facts favored the C G I group" (Carpenter, et al., 1989, p.526).  A similar study was carried out by Villasenor and Kepner (1993), using 12 Grade 1 classes with over 55% "disadvantaged minorities" (p. 62), l i v i n g i n an urban centre. indicated that students i n the C G I classrooms "demonstrated  This study  significantly greater  achievement i n solving arithmetic word problems and i n using advanced strategies i n their solutions. C G I students also showed a significantly superior achievement on completion of number facts and use o f advanced strategies i n completing number facts" (p. 67).  The  authors and the results of this study challenge the "belief, attributed to many who represent minority or disadvantaged populations, that students, especially those who are minorities or disadvantaged, must learn the basics, often in a rote manner, before moving on to problemsolving and process-oriented mathematics" (p. 68). The authors also concluded that the study "provides evidence that teachers can [and should] modify their instructional behaviors to teach in a more conceptually based, process-oriented manner" (p. 68) i n order to assist these children in attaining greater achievement in mathematics.  The authors o f C G I have designed a model for curriculum development shown in Figure 3-1.  Teachers' Knowledge  Students' Learning  Behaviour  Figure 3-1. Cognitively Guided Instruction (CGI) model (Fennema, Carpenter, & Peterson, 1989, p.204). For me, this model not only makes sense, but it formalizes the way I have been teaching for many years. A l s o , the results of the Villasenor and Kepner (1993) study verify my beliefs that disadvantaged children learn in much the same manner as do other children. The teacher must first discover what the children know, and how they know it by listening to them, and then begin instruction from there. Assume nothing, listen to and question students first, and then begin instruction. Although the model does not even mention "listening", the process of listening to students and observing their behaviours is critical to the teacher understanding their behaviours and their learning, and thus making appropriate instructional decisions.  The model presented above is also consistent with the way the B C M E and the N C T M suggest all primary teachers should teach.  A discussion o f the overall goals o f both the  B C M E and the N C T M can be found in Chapter 2. Programme  Foundation  Document  In the rationale o f the  Primary  (Ministry o f Education, 1989), the authors suggest that  the curriculum i n general, not just the mathematics curriculum, "begins from where the child is, and builds on the child's interests and natural sense o f wonder" (p. 20); that it "engages the c h i l d i n meaningful activities and experiences w h i c h provide a context for the development o f thinking processes" (p.20); that it "builds on, extends and enhances successful experiences" (p. 20); that it "invites children to cooperate and collaborate with each other" (p.20); and "is integrated wherever possible" (p.20). A l l o f these characteristics are found within the process of C G I .  C G I originally started in 1986, at the University o f Wisconsin, as a research programme to investigate the impact o f research-based knowledge about children's thinking on teachers and their students. The focus of this research programme was to educate teachers about  how  children think about mathematics, and how to use this information to learn about their own children i n their o w n classes.  " E a c h teacher [then] creates a teaching and learning  environment that is structured to fit his or her teaching style, knowledge, beliefs, and children" (Fennema, Carpenter, & Frank, 1992, p.2). T w o major themes are reflected i n the guiding principles of C G I .  The first is that instruction should develop understanding by  stressing the relationships between skills and problem solving, with problem solving serving as the organizing focus of instruction. The second, based on the knowledge that a l l children come to school with some ability to mathematically solve problems, is "that instruction should build on students' existing knowledge" (Carpenter, et al., 1989, p.525). This second theme is based on the research that was carried out with 40 Grade 1 and 1/2 classroom teachers who received special training on the use of C G I , and a control group. The results of this study showed that "providing teachers access to explicit knowledge derived from  research on children's thinking d i d influence their instruction and their  students'  achievement" positively (Carpenter, et al., 1989, p. 529).  Penelope Peterson and her colleagues (Peterson, Fennema, & Carpenter, 1991b) conclude a report on C G I by stating that a C G I classroom is characterized by: • teachers who have a knowledge base for understanding their c h i l d r e n ' s mathematical thinking; • teachers who listen to their students' mathematical thinking and who build on the knowledge they get by listening; • teachers who use their knowledge of students' mathematical thinking to think about and develop their mathematics instruction; • teachers who place increased emphasis on mathematical problem solving and decreased emphasis on drill and practice of routine mathematics skills; • teachers who provide their students with opportunities to talk about how they solve mathematics problems and to solve problems in a variety of ways; • classrooms in which students do a lot of mathematics problem solving and describe the processes they use to solve problems; and • classrooms in which students demonstrate increased levels of mathematics problemsolving abilities while maintaining high levels of computational performance (p.126).  Five other characteristics, four of which can be inferred from the C G I literature, seem to me to be additional, necessary characteristics o f a successful C G I classroom and teacher. Teachers in a C G I classroom must accept the fact that children enter school knowing a lot about mathematics and problem solving (Carpenter, et al., 1989).  They must be able to  create and keep the interest o f their students by providing problem situations that are of interest to, and within the capabilities of the children in their classroom. They must also create problems from everyday occurrences within the classroom, from literature read to or by the children, from the children's out-of-school lives, and from other curriculum areas as they arise in discussions. Experience has shown me that taking advantage of potential problem solving situations in other curriculum areas becomes almost ingrained and a way of thinking and acting once begun. The learning environment of a C G I classroom must provide the children with the security and understanding needed for them to feel safe about taking a risk. Cooperative learning, and 'on task' talk and movement are encouraged.  — 35 —  A s I read this research and reflected on the way that I use Wall Math  in my classroom, I  began to see many o f the C G I characteristics in my mathematics classroom and in my teaching style. I also found that many of the C G I characteristics were present throughout my whole curriculum.  Reading the diaries found in Chapter 5 w i l l give the reader an  understanding of how the C G I characteristics and Wall Math fit together.  W h a t is " M a t h " ? What is math?  This question can pose a real dilemma for children, their parents, and  teachers. Many teachers, the N C T M , and the B C M E view mathematics as a tool for solving problems that occur in our daily lives. Our students, the citizens o f tomorrow, need to learn not only more mathematics but also mathematics that is broader in scope. They must have a strong academic foundation to enable them to expand their knowledge, to interpret information, to make reasonable decisions, and to solve increasingly complex problems using various approaches and tools, including calculators and computers ( N C T M , 1991, p. iv). The rationale o f the B C M E ' s  I R P for elementary  school mathematics  states that  mathematical literacy is a requirement of all citizens, and that becoming mathematically literate involves developing the ability to explore, to conjecture, to reason logically, and to use a variety of mathematical methods to solve problems. It involves the development of self-confidence and the ability to use quantitative and spatial information to solve problems and make decisions (Ministry of Education, 1995, p. 2). However, many parents, and therefore their children, even at the Grade 1 level, think o f mathematics as simply memorizing procedures, or algorithms, and facts. These children, almost without exception, have already learned that mathematics is finding the "right" answer to a computation question, and to look to their supervising adult for confirmation that their answer is indeed the "right" one. I frequently hear, "When are we going to do some real math?" from some of my children every year. "Real math", to them, appears to be doing sheets of computation, since they usually ask this question as they show me a practice sheet picked up from a box of leftover drill worksheets given to me by a teacher who retired. "I  — 36 —  want to do this k i n d o f math," they tell me. However, although this box o f sheets o f "real math" has been i n plain sight throughout the year, the children have only asked to take them home to work on, they have rarely chosen to do them on their o w n i n the classroom. The only time they do their "real math" at school is once a week when they are given a computation sheet to complete as a morning activity. A morning activity is one that is picked up by the children on their way into the classroom each morning, and is to be completed either by themselves or with a partner, but requires no instruction from the teacher. These computation sheets usually have names printed on them so that I can individualize the practice. Or, o f course, they get some "real math" when they are working with me to learn the formal methods o f working on equations.  This, however, is only done after much  informal work has been shared and discussed as we, as a whole group, solve story problems or the many, impromptu problems involving number that occur while we listen to stories and work on other areas of the curriculum.  Parents frequently ask me for worksheets or flashcards for their children to take home so they can practice their math. When I suggest that they use dice or card games for computation practice, I usually get questioning or even disbelieving looks. A n explanation seems to help them understand why I make these suggestions, but very few actually follow up on them. However, I have had some of the children of these parents tell me about doing mathematics workbooks or flashcards at home. What they don't get from me, they eventually go out and buy! W i t h this attitude, and its corresponding way of thinking about what mathematics is, already instilled i n a six year old's mind, it is very difficult to convince them that mathematics is not only computation, but also is the many strategies and processes that are used to solve problems, and that the use of computation is simply a method that can be used to find a solution. F o r these children, computation can usually be completed quickly, with little or no thinking involved, because they have already developed the strategies for finding the correct answers.  The result of all these correct answers is immediate gratification,  — 37 —  escalation of self-esteem, and the opportunity to be off doing another activity of their own choosing. It has been my experience that children who do not have this "real math" attitude toward mathematics are more open and accepting of multiple strategies for finding solutions to problems, and more able to accept that there may be more than one answer to a problem.  In an article about young children's perceptions of mathematics i n problem-solving environments, Franke and Carey (1997) cite several researchers' views on the consequences of this idea of "real math" being held b y many people. Because children perceive mathematics as a 'given,' it is not likely that they feel compelled to made judgememts about their strategies or solutions to problems (Schoenfeld, 1983), and it is difficult to engage them in a discussion of their mathematical thinking. If children believe that to know mathematics is to produce correct answers using efficient procedures, then they are less likely to place value on engaging in discussions of problem interpretation and alternative solution strategies (Cobb, 1987; Garofalo, 1989a; Lampert, 1990). Similarly, this traditional view of mathematics may fundamentally influence the nature of children's participation in meaningful mathematics learning and essentially inhibit them from engaging in the p r o b l e m - s o l v i n g tasks envisioned by the current reform movement (Garofalo, 1989a; Lampert, 1990; Schoenfeld, 1983, 1992; Underhill, 1988). (Franke & Carey, 1997, p. 9) According to the N C T M (1989), One way to dispel this incorrect notion [that mathematics is computation] is to offer them [the children] more experiences with other topics; even so, unless connections are made, children w i l l see mathematics as a collection of isolated topics. O n l y through extended exposure to integrated topics w i l l children have a better chance of retaining the concepts and skills they are taught, (p. 33) Further, Schoenfeld (1992) suggests that students develop their sense of mathematics — and thus how they use mathematics — from their experiences with mathematics (largely i n the classroom). It follows that classroom mathematics must mirror this sense o f mathematics as a sense-making activity, i f students are to come to understand and use mathematics in meaningful ways. (p. 339) Through the use o f Wall Math  in the classroom, and by the assignment o f a variety o f  homework questions that emphasize the use o f mathematics in real life situations, the children and parents o f my class soon begin to realize that, although the ability to do computation quickly and accurately is an important mathematical skill, mathematics is much more than merely computing. The fact that "research in the field of student perceptions  — 38 —  suggests that students' views of mathematics can be determined or affected, at least in part, by the classroom teacher (Cobb 1987; Schoenfeld, 1983)" (Franke & Carey 1997, p. 9) strengthens the idea that mathematics is best taught by integrating it, as frequently as possible, into other areas of the curriculum, and by using real life connections.  Problems a n d P r o b l e m Solving The terms "problem" and "problem solving" can cause the same type o f understanding dilemma as did that of the question, "What is math?" According to both the N C T M and the B C M E , problem solving should be the "central focus" ( N C T M , 1989, p.23) or "cornerstone" (Ministry of Education, 1995, p.2) of the mathematics programme. But, what is considered a "problem", and what is meant by "problem s o l v i n g " depends on the i n d i v i d u a l ' s interpretation.  Even researchers interpret these terms i n a variety o f ways (Schoenfeld,  1992). Schoenfeld discusses two types o f problems and the processes used to solve them. The first type are "problems as routine exercises" (Schoenfeld, 1992, p.337). These are the traditional word or story type problems, that require a student to use their computation skills for finding the correct answer.  The methods for finding the solution have generally been  demonstrated to the students many times before they are assigned a series of similar problems that w i l l allow them to practice the particular problem solving or computation skill that the teacher wants them to learn. According to Schoenfeld (1992), the ultimate goal o f these types of problems is usually not to learn problem solving skills, but to review or perfect new processes or computation skills. Schoenfeld's second type of problem are those "that are problematic" (p. 338). These are the non-standard, non-traditional types of problems that generally require creative thinking and mathematical understanding in order to find a solution.  "The mathematician best known for his conceptualization o f mathematics as  problem solving and for his work in making problem solving the focus o f mathematics instruction is P l o y a " (Schoenfeld, 1992, p. 339).  He convincingly argues that problem  solving in mathematics depends on guessing, insight, and discovery (Schoenfeld, 1992).  — 39 —  "Problems" and "problem solving", in the context of this thesis, are used in both of the ways Schoenfeld describes them.  I use traditional story problems as a means of teaching (in a  manner similar to that espoused by C G I ) the formal computation skills and problem solving processes the children must learn in an interesting and, for them, problematic way. However, the children in my class are not assigned a series of practice problems. Instead, they are then encouraged to use these skills to solve both traditional and non-traditional, or non-standard types of problems that are posed as part of the Wall Math board.  Worth (1990) describes a problem as any task set for the children in which they "do not immediately know the solution" (p. 52). "Inherent in the use of a problem-solving approach is that the teacher answers questions with questions, encouraging children to think" (Worth, 1990, p. 293), and that the children "are involved in actively exploring ways to reach a solution" (Worth, 1990, p. 293) to the problem. Carpenter, Carey, and K o u b a (1990) suggest, in a discussion about creating word problems, that "the best problem-solving activities are those that the teacher develops for students in her or his class. The best problems are those generated cooperatively with students" (p. 129). They also suggest that classroom "themes can be a source of problems as well as a way to integrate mathematics, reading, language arts, social studies, and science" (Carpenter et. al. 1990, p. 129). "It is interesting enough to note," reports Suydam (1990), "that although we talk about problem solving in mathematics, we make more use of problem solving in other content areas" (p. 293). In some classrooms, this is probably true, since many teachers tend to teach mathematics as individual skills to be learned by rote. However, in my classroom, the children solve problems in all areas of the curriculum. Wall Math helps integrate the total curriculum, and has many diverse problems for the children to solve.  — 40 —  According to Fennema, Carpenter, and Peterson (1989), "word problems are the major organizing activity in C G I " (p. 213). Their research has shown that children, i n classrooms where the C G I philosophy of instruction is practiced, not only learn how to solve word problems using a variety of strategies, but also have shown a better understanding of the computation skills required to solve these problems, and improved computation skills. (Fennema, et. al. 1989). A n integral part of the C G I process is that children are encouraged to talk about their solutions and to listen to others talk about their solutions, and that the teacher not only listens to, and values the ideas presented by the children, but also uses these discussions as a means o f knowing what the children know and how they know it. This knowledge is then used to help plan the next problem situation.  The questions on each Wall Math problem solving.  board reflect the ideas discussed about problems and  They pose a variety o f problems for which the children "do not  immediately know the solution" (Worth, 1990, p. 52), which may have more than one solution, or which may have several ways of finding the solution. The problems come from many sources within both the curriculum and the children's world, and although most of the problems are made up by me for the particular group in my classroom at the particular time in question, the children sometimes make suggestions for changes to the existing problems, or suggest ideas for future problems.  Connecting M a t h e m a t i c s with Real Life The idea of integrating classroom mathematics with real life situations is an underlying theme that runs throughout both the provincial curriculum and the N C T M Standards, and is generally accepted as a positive factor by researchers. Constructivist research has shown that new concepts should be built on what the child already knows, and that this is best done by presenting the children with problems "requiring mathematical thinking [that] should evolve naturally out of day-to-day activities in the classroom" (Ministry of Education, 1995, p. 2), or  — 41 —  "from situations generated within the context of everyday experiences" ( N C T M , 1991, p. v), In an article on developing problem-solving abilities and attitudes, W o r t h (1990) suggests that teachers as often as possible, base problem situations on c h i l d r e n ' s everyday experiences in school and at home. Children are interested in problems that they can relate to their lives and deal with on a personal level. This connection helps them apply the mathematical ideas they are learning as well as helping them interpret their world (p. 42). The Wall Math  problems presented to children in my classroom use a wide variety of  questions and situations that relate to the children's home life, their outside world, classroom units, literature, special days, or holiday celebrations. I believe that integrating these types of questions into Wall Math  helps create a "developmentally appropriate curriculum [that]  encourages the exploration o f a wide variety o f mathematical ideas in such a way that children retain their enjoyment of, and curiosity about, mathematics" ( N C T M , 1998, p. 16).  "Whole Math" The term " W h o l e Language" has been used for years to describe a philosophical approach and method of teaching beginning reading and writing. " W h o l e M a t h " (Baker, Semple, & Stead, 1990), though not a term frequently used, applies the same philosophical approach to the teaching of mathematics. The philosophy of "Whole M a t h " encompasses the beliefs and suggestions espoused by both the N C T M Standards, and the B C M E curriculum. It is also consistent with the findings o f recent constructivist research.  A c c o r d i n g to the group o f  authors who have not only written about " W h o l e M a t h " , but also used and researched it in their own classrooms, the philosophy of "Whole M a t h " involves: • "using mathematical processes to solve practical and abstract problems" (Baker, et al., 1990, p. 3); • "the development of personal qualities such as perseverance and cooperation" (Baker, et al., 1990, p. 3); • providing "children with tools to think and to communicate w i t h " (Baker, et al., 1990, p. 3); • developing a positive attitude toward math " i n order to effectively acquire math skills and understandings" (Baker, et al., 1990, p. 6); • no breaking down of a skill into it's tiniest steps and parts for the children to somehow piece back into a whole (Baker & Baker, 1991); and  — 42 —  • fostering "creative and divergent thinking rather than the idea that maths is about finding the 'right' answer" (Edwards, 1990, p.3) using the "one method of solution" (Edwards, 1990, p. 4).  These authors' shared vision of "Whole M a t h " is that it is a method of teaching mathematics in which the children are presented with a problem o f interest to them, one that arises from a classroom situation, a home experience, or a story read to them, and then are encouraged to explore, discuss, share, demonstrate, and represent their mathematical thinking and solutions in their own ways. This thinking, and the solutions that are arrived at, are then shared and discussed with the whole class i n order to find generalized understandings, rules, and/or strategies for both problem solving and computation. These authors all agree that teaching math using this " W h o l e M a t h " method results i n children who enjoy and understand the mathematics they are expected to learn, and who become good problem solvers. I believe that Wall Math  is an example of "Whole M a t h " as described by these authors.  CHAPTER  4  METHODOLOGY OF T H E STUDY  The purpose o f this thesis is not to conduct a study that can be. generalized to a larger population using sampling logic, but rather to describe, from careful observation, how some Grade 1 children interact with the Wall Math their peers.  mathematics programme, their teacher, and  W h i l e it is recognized that a larger study that includes participants randomly  selected from the population of primary classes using Wall Math  w o u l d provide valuable  information and insights, it is beyond the scope o f this study. B y following a small number of students over a period o f time as they interacted with myself (their teacher), the other students in the class, and the particulars o f the programme itself, it is hoped that a better understanding might be attained of how students come to learn and understand mathematics when they work within a programme designed to take advantage of what is currently believed about primary mathematics education. The students' actions can provide data to strengthen or discard the beliefs that are generally accepted about the ways in which children learn by demonstrating behaviours that might be predicted by a teacher who holds those beliefs and teaches according to those beliefs. Further, the classroom anecdotes, provided through the diaries (see Chapter 5) can provide concrete examples of curriculum and teaching methods that are grounded in our current beliefs, and show effects on student learning; their successes and failures, their revelations and discoveries, their confusions and frustrations, their development and progress.  Postpositivist qualitative beliefs, rather than positivist, quantitative principles are the basis of the methodology o f this study.  Whereas positivist research aims to explain  (erklaren)  through the discovery or confirmation o f laws, postpositivist qualitative research aims to elicit personal understanding (verstehen)  (Gilbert & Watts, 1983). The methods are also  strongly influenced by the growing constructivist beliefs and research tradition. Because  — 44 —  students have experienced and thought about the world, they enter school with a complex cluster o f ideas, beliefs, values, and emotions which they use to understand the world (Snively, 1986). These ideas, beliefs, values, and emotions serve as a conceptual 'filter' by which they (along with a l l other humans) interpret sensory impressions. Authors describe learners incorporating, integrating, assimilating, and accommodating scientific and mathematical interpretations into this existing knowledge (e.g., Posner, Strike, Hewson, & Gerzog, 1982; Solomon, 1988). " A goal o f qualitative science [and mathematics] education research has been to probe this knowledge and make it e x p l i c i t " (Robertson, 1994, p. 47).  Fundamentally, qualitative methods were employed because o f my belief that: the world is not an objective thing out there, but a function o f personal interaction and perception. It is a highly subjective phenomenon i n need of interpreting rather than measuring. Beliefs rather than facts form the basis o f perception. Research is exploratory, inductive and emphasizes processes rather than ends (Merriam, 1988, p. 17).  Practitioner Researcher M a n y terms are used to describe the research that teachers do within their own classrooms (Anderson, Herr, & N i h l e n , 1994, p. 1).  O f those presented i n the literature, the term  "practitioner researcher" seemed to best describe my research.  The authors define  "practitioner research" as "'Insider' research done by practitioners (in this book, those working i n educational settings) using their own site (classroom, institution, school district, community) as the focus of their study" (Anderson et al., 1994, p. 2). This type of research turns the "informal questioning of practice to one of more systematic inquiry that lends itself to problem solving as well as possible dissemination to a larger audience" (p. 47). However, they also point out that most practitioner researchers "often enter the research process as a means o f solving their own practice dilemmas or questions" (p. 47), and not to "contribute to the field o f education" (p. 47) at large. Being an 'insider' "creates obvious advantages for the practitioner researcher, but it also makes it harder for the practitioner researcher to 'step  — 45 —  back' and take a dispassionate look at the setting" (p. 4).  H o w e v e r , the practitioner  researcher is i n a unique position to watch and document just how s/he helps a particular group of children construct the growing body o f knowledge and skills that is required o f them. O n l y an 'insider', such as the teacher, has the intimate knowledge that is necessary to accurately predict how each child w i l l react to a specific question or situation. M a n y [educational researchers] are becoming convinced that practitioners have a lot to gain through systematic observation and intervention i n our own practice sites; in addition, there is a growing sense that practitioner researchers can help inform the larger knowledge base o f education through our findings (Anderson, et al. 1994, p. 175).  The Collection of Data Data for this study were collected i n several ways. The first, and most frequently used method was to observe and tape record the children while they worked alone, i n small groups, or as a whole group. A discussion of this process follows. The second method o f data collection was to take photographs of the children at work, their completed work, and the Wall Math board itself. Copies of a few of these photographs are included in this thesis in an effort to give the reader a better understanding of the ideas presented. The third method of data collection was to have the children write about Wall Math, to tell me what they thought about it, and what they liked or disliked about it. Finally, copies o f the children's record/answer sheets were collected and examined. Descriptions o f many parts o f these sheets can be found i n the diaries i n Chapter 5. A l l of these activities, except the audio taping o f discussions, are regular occurrences i n our normal classroom, that take place as I collect data for the purposes of assessment and evaluation. Although the collection of data became more intense during the three months of this study, the children appeared to accept it as a normal part o f our classroom routine. In fact, they enjoy, and frequently ask to have their photograph taken because they usually get to take the photographs home. A problem that occurred when the tape recorder first became obvious to the children was resolved part way through the study by a combination of using a more sensitive "conference" recorder and  concealing the smaller, voice-activated recorder. A complete discussion of this problem and its resolution can be found later in this chapter.  Sharon Jeroski (1992), in her book Field-based Research: A Working Guide, says that " M o s t teachers are skillful observers who are able to work on several levels at the same time — observing, helping, answering questions, [and/or] anticipating the next activity" (p.21). I found that being an observer in my own classroom for the purpose o f doing this research was a very interesting, enlightening, and challenging experience. Although I had been quite used to " k i d watching" to gain evaluative information for the purposes of reporting to parents, to assess skills and achievement i n order to help me know where to go next with a concept, and to watch and manipulate constructive group dynamics, I had not needed to record exact conversations and actions before taking on the role of "practitioner researcher" (Anderson, Herr, & N i h l n , 1994), that is by adding the title of "researcher" to that of "teacher".  In a discussion about record keeping and language use, Spradley (1980) suggests that in order to make an ethnographic report as accurate as possible, it is important that the actual language and words o f those being observed be used in the report.  In an effort to record  students' actual language and words, I tried, on several occasions, with different small groups of children, and in a variety of ways, to use a small tape recorder to record the conversations of the children as they worked. However, I found that trying to make sense later o f what I could hear o f the conversations was impossible, unless I knew what was going on at that particular table, because the conversations tended to be a mix of talk about the math problem, what they had seen on television last night, or whatever else was on their minds at the moment. I also found that i f I hovered around the group while the recorder was obvious, the children became so conscious of the tape recorder that I had many comments such as, " O h , oh, she's recording us", "Shh, the tape recorder is on", and even, " D o n ' t talk to me; she's taping us." A n d yet, without the tape recorder, I can hover, write notes, and take part i n what  the group is doing, and the children do not seem to be concerned. However, I have been doing this latter type o f " k i d watching" since the beginning o f the school year, so they are quite used to it. In time, the children may have become as used to the tape recorder as they were to my hovering, but time constraints did not permit this 'getting used to' to occur. I was more successful using the tape recorder during whole group discussions. A t these times, I simply put the recorder i n my pocket, or under a paper on the bench i n front of the group centre chalkboard, and the children either did not know it was there, or ignored it i n their efforts to take part in the discussions. Part way through the study, I acquired a recorder with a "conference" microphone setting and a special "conference" microphone, that was very unobtrusive, but could pick up even more of the whole group discussions at a much clearer level. I was very pleased that the children ignored the tape recorder during these whole group discussions because it would have been impossible for me to record actual comments made in these intense, fast moving discussions in any other way.  I already knew that using a video recorder was out of the question since I had tried, on several previous occasions, to video tape the children as they worked on several different types of activities. Whenever I tried video taping, the children were so conscious of the camera's presence that I got nothing that was natural. This group even had problems when I tried to video them doing a puppet play. So, technology was not going to be a whole lot of help to me. I had to do much of my recording the o l d fashioned way, by hovering, jotting down what I could, when I could, and writing about the episodes as soon as possible after the lessons were completed. A s it turned out, writing up the descriptions o f the individual and small group discussions almost immediately was not difficult, and, I think, led to more precise descriptions of what was actually happening because I could mentally "see" the body language and "hear" the tone of voice used by the individuals as they made the comments that I had jotted on my paper.  In addition, part way through the study I began carrying a  small, voice-activated, audio tape recorder, that I acquired, concealed i n one o f m y  voluminous pockets that are a characteristic of many of my school clothes. A s I wrote the story o f a Wall Math session, which I always did as soon as possible after each lesson, the audio tape contents provided many useful 'memory jogs', as w e l l as the actual words o f both myself and the children, and thereby provided valuable information to the study.  In his discussion about language, Spradley (1980) states that after years o f learning to be concise about what we want to say, "In writing up field notes we must reverse this deeply ingrained habit o f generalization and expand,fillout, enlarge, and give as much specific detail as possible" (p. 68, emphasis in original).  This can best be done i f the actual  "expansion of the condensed version" (Spradley, 1980, p. 70) is completed as soon as possible after each field session (or after making a condensed account), [so that], the ethnographer should fill i n details and recall things that were not recorded on the spot. The key words and phrases jotted down can serve as useful reminders to create the expanded account (Spradley, 1980, p. 70). Sanjek (1990) comments that "field notes are meant to be read by the ethnographer and to produce meaning through interaction with the ethnographer's headnotes" (p.92). Keeping these ideas in mind, and being sure to schedule most Wall Math sessions between recess and lunch, I spent many of my lunch breaks writing expanded versions o f the written and/or "headnotes" made while I talked to individuals, or with small groups, during the previous hour. The tape recordings were usually listened to after school on the day on which they were made. Parts of them were transcribed, and notes were added so that these, along with the expanded versions of the individual and small group discussions, could be used to write a detailed diary o f each Wall Math work session when time permitted, usually the following weekend. M y knowledge of the children in my classroom helped make this an easier task than it may have been for an outside observer/researcher.  I know the differences between  Douglas's 'thinking out l o u d ' voice and his 'I know this answer' voice, I k n o w M a n d y ' s body language that says ' I ' m getting annoyed', and I know that i f I want M a r g or M i k e to share their thinking, then I need to ask them to do so since they rarely offer to share, but  — 49 —  usually have well-thought-out procedures that the other children would benefit from hearing. I also know how each child expresses him or herself, so that actual conversations can be recalled from notes that contain only a few written words. Although being the teacher, the observer, and the researcher all i n one was a challenge at times, I have learned a great deal about m y children and the way they learn, about myself as a teacher, and about how Wall Math  can be used as a teaching technique and tool to aid i n both conveying new material  and i n keeping the taught/learned concepts growing throughout the year.  Data Analyses The diaries, once written, became the data for this research. Analyses were then done by finding evidence o f links between the desired outcomes o f the programme and what was actually happening i n the classroom, as shown i n the diaries. Evidence o f links among the programme itself and the overall goals o f the I R P and the N C T M Standards  were found and  documented from the events recorded i n the diaries.  However, the act o f writing the diaries themselves was also a major form of data analyses, all be it preliminary analyses, i n that I had to mentally review events which had occurred, and then, stimulated by notes, tape recordings, student answer sheets, and my o w n memory, record the events i n such a way that both I and the reader would be able to make sense o f what was occurring in the classroom. In this way, the diaries themselves can be considered part o f the conclusions of this research study. They represent the description o f how children interacted with the programme, and thus provide an 'answer' to the main research question of this research. F o r many readers, they w i l l take more from this study in the form of anecdotes from the diaries, from which they w i l l draw their o w n conclusions, than they w i l l from my conclusions reported i n Chapter 6.  — 50 —  The School, the Teacher, and the Students of the Study This study was conducted in my own class in the school in which I have been teaching for almost 25 years. Although guarantees of anonymity prohibit disclosing names of the school and the students that form the basis of this study, a description of the school and o f the six students who were the primary participants, and thus the data source, w i l l give the reader a better understanding of the context within which the study was conducted, and provide an aid for the reader's interpretation o f the diaries presented in Chapter 5. The following section therefore describes the school used i n this study, the background o f the teacher (myself), and each of the six students on whom the study is focused.  The School The school i n which this study was conducted is an inner city, sub-urban, medium sized school with just over 200 children, a teaching staff of 15, a full time principal, and seven support staff. It is considered an inner city school with "approximately 45% o f the students living in poverty" (Oldtown Elementary School, 1997, p. 2), and a "varied, poor to affluent" (p. 2) socio-economic clientele. Although the children are predominantly English speaking, there are "80-85 E S L students speaking over 25 different languages" as w e l l as "approximately 5% [of the students of] Native ancestry" (p. 2). M a n y o f the children are from single parent families. M y class and the six children on whom I have chosen to focus this study reflect this official school profile. M y class consists o f 21 children, four o f these are E S L students, about half are from families living below the poverty line, three have moderate behavioural problems, two are diagnosed as A D H D with oppositional behaviours, one is mildly mentally handicapped, and eight are from single parent families.  The Teacher I have been teaching for a total o f 31 years, in three different school districts i n British Columbia. During this time, I have taught Grades 2 and 3 in an isolated town o f about 600  — 51 —  people, Grade 1 i n a rural suburb of a town o f about 15,000, and Grades 1 and 2 (most dual age groups) in a large suburb of Vancouver. Throughout these years I have attended many i n service sessions, workshops, and conferences, as well as upgrading my original three years of teacher training to first a four year Bachelor o f Education, and then a fifth year, concentrating on reading instruction and teaching children with learning disabilities. During these years o f teaching, my style and beliefs about teaching and the ways i n which children learn have been influenced and changed according to the current trends.  However, I have always worked  within the philosophy that my job is to assist children to learn the developmentally appropriate skills required at that time, and needed on their way to becoming educated adults, within a safe, caring environment, using whatever materials were available, and whatever methods worked for the particular group o f students I had at that time. Further, over the years I have worked extensively with students in the out of doors, both during regular school times and during outdoor summer schools that my husband and I have developed and conducted. W o r k i n g in the out of doors necessitates that one uses real life teaching/learning situations; that the teacher become adept at taking advantage o f what interests students at a particular teachable moment — one learns to use the A I C (As It Comes) programme. During a recent discussion with my principal, as part of a university course assignment about my teaching style, she commented, "I hope you know how to spell 'eclectic', because that's your style." I frequently tell new teachers who ask me about a strategy or material new to them, "Try it. If it works, then use it. If not, then change it or toss it."  The Children Choosing the children on whom to focus this study became an easy task when m y principal informed me that all but one of the Parent(s)/Guardian(s) Consent Forms were returned with positive responses, giving me permission to use any of my children (with one exception) i n my research. A s It turned out, this one child for whom permission had not been obtained was an English as a Second Language ( E S L ) student whom my principal already knew would not  — 52 —  have been chosen. Although my class has both Grades 1 and 2 children, I chose to focus on six of the Grade 1 students because they represented a group most typical o f the school population. These six children represent the variety of socio-economic groups, and, although I chose not to use any of the E S L students, two that I chose are of native ancestry, and two are from single parent families. The children were chosen on the basis o f gender and its intersection with mathematics ability, as judged by me; one boy and one girl each o f high, average, and low mathematics ability, as displayed to me during classroom discussions and by their written work so far this school year.  D o n was chosen as the high ability boy because he is a bright, verbal student who is not afraid to take a risk, but becomes quickly frustrated i f he can't come up with a 'right answer'. H e often 'thinks aloud', even during whole group discussions, and many o f the others have learned to listen to, and follow his ideas.  However, his vision o f doing "real math" is  working out sheets of standard computation questions, which he seems to enjoy doing. This vision seems to be shared by his mother, who supplies h i m with purchased mathematics workbooks and helps him learn new processes by rote. H e thinks Wall Math  is "easy and  fun", and he is usually one of the first to investigate a new board as soon as it is put up. H e is also usually one of the first to say he has finished a whole board, even i f he has left out a 'challenge' question or two. Wall Math  A s an example of how students provided their thoughts about  to me, and also to show you, the reader, what D o n thought about Wall Math,  at  the end o f the year, D o n wrote, "I liked the W a l e M a t h B o r d it was F u n doing math ad subtracting and pictures and evry thing[sic]."  Mandy was chosen as the high ability girl because she is a confident, verbal student who, like D o n , is always looking for the 'right' answers.  She has many strategies for solving  problems, but becomes frustrated i f she cannot find the 'right' solution quickly. She then usually leaves the problem unsolved, unwilling to ask for help until the problem is discussed  with the whole group. Although she is willing to share her strategies with me and her classmates, she often has difficulty explaining what she has done in a way that the others can understand because many of her skills appear to have been learned by rote. About Wall Math, she says, "I do not like Wall Math because some of it is hard!"  Mike was chosen as the average ability boy because he is a confident, quiet student who watches and listens before tackling a new problem. He usually works with Don when doing Wall Math, watching and giving suggestions as Don verbalizes about what he is doing. Mike then re-creates the solution with his own materials, often in a different way, before recording his answer. He will share his strategies with me and with his classmates, but only when asked. He thinks Wall Math is "great", commenting on his sheet which he turned in one day, "I like Wall Math. It is vey[sic] fun and I love the adds and I love estimate. Its [sic] all fun."  Marg was chosen as the average ability girl because she is a quiet student who works slowly, and who, like Mike, listens and watches before trying to solve any problem. She is very unsure of herself and her abilities, and assumes she has made a mistake just because I ask her to explain something. "I don't know," is usually her first response to any question I ask her. Although she will share her strategies with me, she is reluctant to share them with her classmates, only doing so when specifically asked, and usually only after she has explained it to me privately. She told me that she thinks Wall Math is "fun and easy, but sometimes hard". At the end of the year, she wrote to me, "I like Wall Math. 1 -> 20. I like all Wall Math. I likeed [sic] Wall Math wen [sic] September sins [sic] May. I liked + and - and between to [sic]."  Douglas was chosen as the low ability boy because of his lower than expected computation skills, his immaturity, and his numerous 'off-task' behaviours. He is a verbal student who enjoys socializing with his tablemates as he works. Although his basic mathematics and  — 54 —  problem solving skills are just adequate, he frequently accomplishes little during work sessions because he is often unfocused.  W h e n w o r k i n g with adult supervision, he is a  persistent, but slow-working problem solver. H e shows excitement and real pride when he discovers a correct solution. H e is w i l l i n g to share his strategies with his classmates, who have learned to listen to his simple, straight forward solutions and explanations. H i s thoughts about Wall Math  are that "parts of it is easy, and parts of it are hard."  Cathy was chosen as the low ability girl because she is a quiet student who spends much o f her time watching the other children or off in her own world. She accomplishes little during work sessions unless supervised by an adult. However, she shows great pleasure and pride when an activity has been completed successfully. H e r basic mathematics skills are just adequate, and she has few problem solving strategies beyond those o f simple j o i n i n g or separating actions.  During whole group discussions, she is usually unfocused, but w i l l  sometimes offer to share her "I counted three and two and got five" type of thinking with the group. She thinks Wall Math is "really hard."  Although the focus o f this study was on the work and the thinking of these six children, it must be noted that the other children in the class 'became visible' to the study because o f the types o f activities I was recording and observing. The names o f a l l the children have been changed i n order to maintain anonymity.  Ethical Review and Consent A s with a l l studies conducted at the University o f British C o l u m b i a i n v o l v i n g human subjects, the proposal for this study was reviewed and approved by the appropriate university ethical review committee.  A copy of their certificate o f approval is included i n the  Appendix. A s part o f the requirements of that review, all students involved in the study had to give their consent, through their parent(s) or guardian(s), for participation. A copy of the  — 55 —  letter that was sent home to the parents explaining the study, and a copy of the consent form that the parent(s) or guardian(s) were to sign and return to the principal are also shown in the Appendix.  This study documents how and why Wall Math was developed. The data, i n the form o f the diaries presented in Chapter 5, w i l l link the implementation of the programme with the B . C . primary mathematics curriculum, the N C M E Standards,  and principles of, and beliefs about  learning derived from constructivist research and illustrated i n C o g n i t i v e l y G u i d e d Instruction. Most importantly, the data w i l l provide examples from student learning that support these linkages. In presenting the data in this way, it is hoped that the reader w i l l develop a better understanding o f the ways children think and learn about mathematics and mathematical literacy, and ways that this teacher and her programme promoted such development.  — 56 —  CHAPTER 5 WALL  MATH  B O A R D DIARIES  S t o r y is a m o d e o f k n o w i n g that captures i n a s p e c i a l f a s h i o n the r i c h n e s s a n d the nuances o f m e a n i n g i n h u m a n a f f a i r s . W e c o m e t o u n d e r s t a n d s o r r o w o r l o v e o r j o y o r i n d e c i s i o n i n p a r t i c u l a r l y r i c h w a y s t h r o u g h the characters and incidents w e become f a m i l i a r w i t h i n [stories]. This r i c h n e s s a n d nuance cannot be e x p r e s s e d i n d e f i n i t i o n s , statements o f fact, o r abstract p r o p o s i t i o n s . It c a n o n l y be e v o k e d t h r o u g h story ( C a r t e r , 1 9 9 3 , p.70). T h i s c h a p t e r c o n t a i n s s u c h s t o r i e s ; i t i n c l u d e s the r a w d a t a o f the s t u d y i n the f o r m o f narratives about e a c h o f f i v e d i f f e r e n t Wall Math  boards. I n w r i t i n g these d i a r i e s , the actual  w o r d s o f m y s e l f a n d the students are u s e d w h e r e v e r p o s s i b l e , a n d a sincere attempt is m a d e to capture the f l a v o u r a n d tone o f the v a r i o u s a c t i v i t i e s . E a c h b o a r d w a s d i s p l a y e d i n the c l a s s r o o m f o r a p p r o x i m a t e l y one to t w o w e e k s , a n d e a c h diary/story u s u a l l y c o v e r s a p e r i o d o f f o u r to s i x s c h o o l d a y s w i t h i n that p e r i o d .  Diary #1 — January 8 -14,1998 T h i s d i a r y applies to the f o l l o w i n g b o a r d o f questions. (* = " T r y i t " c h a l l e n g e question) 1. Pattern: o oh r— use Unifix cubes. Make it as long as your leg and show a friend before showing Mrs. Bateson. 2. N u m b e r l i n e : 43,44, and 45 printed above the first three number points. 3. W o r d please: 28 4. N u m b e r please:  sixteen  5. B e f o r e a n d A f t e r : 38 printed above the middle number point. 6. B e t w e e n : 56 and 58 printed above the first and last number points  respectively.  1. D r a w a n d p r i n t as a n u m b e r : Two tens and seven 8. A d d : 5 + 3 = 9. Subtract: 10-4  =  10. E q u a t i o n please: Six balloons (two rows of three) with the forth and fifth ones crossed out.  — 57 —  11. Order please: 18, 9,12, and 15 12. Tally count:  Eight marks in tallies of five.  13. Picture please:  3+3  14. W h i c h one doesn't belong? W h y ? :  "Squareys" Weeple People (Gillespie, 1971, p.8)  Squareys have square shaped bodies while non-Squareys will have other shapes for bodies. 15. H o w many: 16. Geometry:  How many vehicles are in the school parking lot? *How many wheels? Name the shape of a traffic light. Draw and colour it.  17. Number wheel: "+  4" in the centre.  18. Estimate:  Nineteen erasers in a jar.  19. Measure:  Show today's temperature (+4 C ).  20. Stumper:  Write out your name using the numbers on a telephone dial corresponding to  the letters of your name. Add up all the numbers, (e.g., "David" would be 3 2 8 4 3, with a sum of 20).  Day 1 — Wed., Jan. 8 It has been my experience from past years that the first board after the Christmas holiday is usually a difficult one for the children to complete. This is probably because we only do one board in December — too many Christmas activities going on — and the two week holiday causes many to 'forget how'. In order to get things restarted in a positive way, we worked on much o f this board as a whole group. Because Question 1 —  Pattern tends to be time  consuming for many children and children can complete it alone, it was left for a later time. Generic answer/record sheets (see Figures 2-6 and 2-7) were given out, and Question 2 —  Number line was read aloud as a whole group. " W h o can show us these numbers on our number line?" I asked.  Several hands went up, and many others pointed i n the general  direction of this part o f the calendar number line along the side o f the classroom. M a n d y was invited to show the " 4 3 " by touching it with the pointer stick. The children then copied the  — 58 —  numbers from 43 to 49 above the dots i n this section of their record sheets. I also asked them to check their neighbour's paper, both to be sure they were correct and to help them get finished more quickly. When all were finished, we read the number on Question 3 — Word please  as a group. "Where can we find the words to print this number?" I asked.  children a l l pointed to the number word, number line above the Wall Math  The  board. D o n  showed us the card with the word for 20, and Douglas showed us the card with the word for 8. "So, what do we need to copy to print the words for the number 28?" I asked. " W e have to copy both those words," said Mandy. I instructed the children to copy the words onto their papers as I walked around checking to be sure everyone was working in the correct space. Again, I asked early finishers to help their neighbours. This process of reading the question as a whole group, then finding and copying the answer was repeated for Questions 4 — Number please, 5 — Before and After, and 6 — Between. Draw and print as a number  H o w to answer Question 7 —  became a fairly lengthy lesson since many children had  forgotten or still didn't understand how to record tens and ones as pictures — the transfer from doing this with sticks as part o f our daily calendar activity does not seem to be happening. A s a whole group, we reviewed how we do this activity on the calendar; we drew pictures o f groups of ten tally marks, balls, trees, and other shapes; then drew circles around them to show that each set was a group of ten. I then asked them to draw two groups of ten similar to the examples on the board onto their answer sheets. Most d i d this correctly. One child circled a l l twenty within one circle, but, when questioned, realized what he had done and drew a line down the centre to divide his twenty into two groups of ten. D o n , however, drew two bundles of five tally marks, circled each bundle of five, but then counted his circles as groups of ten. "Ten, twenty", he counted as he pointed to each group when I asked him how many tally marks he had drawn. When I asked him to count the individual sticks, he counted them accurately to ten, but again counted the groups within each circle as ten. H e seemed satisfied that because the group was circled, it meant it was a group o f ten. There was not sufficient time to correct his thinking right away, so, knowing there would be others  who would need extra help to understand this concept, I made a mental note to plan a small group lesson on this concept i n the near future. M i k e and Mandy carefully copied the tally marks from the board, Marge copied the groups of balls, and Cathy didn't seem to know what to do, so was sitting watching her neighbour draw her groups o f ten balls. A s I stood beside her, I asked her to draw a group of ten balls touching each other, as shown on the board. Once done, I asked her to circle them as i f they were in a basket. W h e n I asked her to draw another basket of ten balls, she started to draw, got distracted after doing nine, then instead o f checking her drawing, simply drew a circle around the nine and left it as finished. She seemed so pleased with completing the task that I didn't want to destroy her feeling o f accomplishment by asking her to correct it for the sake of one ball. Douglas's mother came to take him for a dental appointment, so his didn't get finished. The next step was to draw the seven ones.  Several children demonstrated how to draw a group o f seven on the  chalkboard, and a l l the children completed this part o f the task correctly. Although Cathy circled her group of seven, when I asked her how many were there, she told me, "Seven." W h e n I asked her to count her pictures, she counted as she pointed to each circled group, "ten, twenty, and this one only has seven." Wonderful!  However, I wonder i f she w i l l  remember that tomorrow? The last part o f this question, and print as a number, was easier for the children to complete than I had anticipated it would be. In this case, there does seem to be at least some transfer from this part of the calendar activity to the Wall Math  activity.  A l l but three children wrote the correct number, and those three who were incorrect wrote down "207" as I had anticipated they might. When I asked them, " W o u l d two tens and seven look like that on the number line?" they shook their heads ' n o ' , recounted their pictures, and printed the corrected number.  A g a i n Cathy didn't seem to know what to do, but as she  listened to me talking to the others, she watched what they were doing and wrote the correct number.  Questions 8 — Add, and 9 — Subtract  were copied, completed, and checked for accuracy  with a table mate, then with the whole group.  The children were beginning to get restless, so we moved from tables to the group centre to discuss and answer Question 10 — Equation  please.  The fact that each question is on a  separate paper held onto the Wall Math board with a magnet, or, as in a few cases, a pin on a tackboard, allows us to move each question around the room as we move to more convenient work areas. The paper with Question 10 — Equation please  was therefore easily  moved to the chalk board in the group centre to make reading and/or review o f the problem more convenient. In order to help the children understand both what was happening in the picture, and to help them see the action of subtraction, I suggested that we act out the picture. " W e need six kids up there," came from the back o f the group. " Y e a , and they a l l have to have a balloon," said Mandy. When I explained that I didn't have any balloons, she thought a moment, then handed each o f the six children a yellow pattern block saying, "Here, these are the balloons." M y question of, "What happens now?" resulted in M i k e ' s hand shooting up while several others started to call out answers. When I called M i k e ' s name for him to provide an answer, the others settled down, and he explained, "The w i n d blew two o f the k i d ' s balloons away." I asked M i k e to be the wind, and he came and took blocks away from two children: Pat and D o n . Pat pretended to cry, and D o n stamped his feet as he pretended to be angry. Both boys tend to be hams, and were chuckling as they pretended to be upset. Everyone agreed that four children still had their balloons and that the recorded number sentence should be, " S i x take away two equals four." Elaine, who has large neat printing, was invited to transcribe the equation onto the chalkboard. The children were then asked to walk to their tables, record the sentence as demonstrated on the chalkboard, and then return to the group centre to work on Question 11 — Order please. the equation exactly as demonstrated on the board.  A l l but a few children recorded  Marge drew the picture as w e l l as  recording the equation. M i k e drew the picture and added an "= 4" but d i d not record the  — 61 —  complete equation. I decided that acting out the equation was a good idea and made a mental note to use it again in the future since it seemed to help enhance the understanding of the procedures represented by the picture and/or the equation.  Once again at the group centre, we read and discussed Question 11 — Order please had been moved to this board when Question 10 — Equation please  which  had been returned.  "The 9 goes first," called out D o n . " W h y ? " I asked. "Its the smallest number there," he replied. "If that's correct, what comes next?" "The 12," responded Mandy, "cuz it's the next biggest," she added, as I waited for her to explain her thinking. " I f that's right put your hand up when you can give us the next biggest number," I said, wanting to give the quiet ones a chance to answer. A s I got the answer o f " 1 5 " from Susan, Cathy's hand shot up and her usually somber face lit up with a smile. "Eighteen," she said with a grin when I called on her, "it's the only one left," she added.  She may not understand the concepts o f 'skip  counting' or 'greater than', but she was using the logic o f elimination correcdy. A s the children gave me the numbers, I put them on the chalkboard as a number line, leaving large spaces between each number. W e talked about what numbers were missing, and I put them into the spaces I had left with a pink coloured piece o f chalk. W e also discussed what numbers came before nine and after 18 in the pattern, and what we called this k i n d o f counting. A t this point, I heard Mandy say to her neighbour, " W h y do we need to know how to count by threes?" When I questioned her about this comment, she explained, " W e need to know how to count by twos 'cause its faster to count stuff, and we need to k n o w fives and tens so we can count money and big bunches o f things, but why do we need to count by threes?" This resulted in a short but lively discussion of what comes i n groups of two, five, and ten, but no one could think o f anything that came i n groups of three. When I suggested that packages of things from Costco often come in threes, Mandy quickly reminded us, " Y e a , but that's only cuz someone did that; they don't really come in threes." The nodding heads around me showed that everyone agreed with her, and not me. I have sometimes used this  — 62 —  same comment made by M a n d y as a means of encouraging children to look for shapes, colours, and groups that appear naturally, and are not created by people; I guess my message is getting through. B y the time we were finished, we had a number line of counting by threes from zero to 60, and D o n had discovered that, "the ones start over again from here.  See?  Both o f these [he pointed to the ' 3 ' and the '33'] end with three, and these [he pointed to the ' 6 ' and '36'] end with six. and these end with nine [he pointed to the ' 9 ' and ' 3 9 ' ] . " H e continued to point out this pattern to "60". A t this point the lunch monitors arrived, a sure sign that we had again gone overtime. This tends to happen quite often when we are doing Wall Math, but the children never seem to mind; they get very involved and excited, and rarely 'clock watch' when we are doing Wall Math. The children quickly returned to their tables to record their answers for Question 11 — Order please,  then went for lunch.  Although I had intended to do so, I never did find the time to go back to this question to look for a possible second way that it could be answered (counting backwards).  Day 2 — Thurs., Jan. 9 Question 1 — Pattern for this board, ...make your pattern as long as your leg has three purposes: making a four block, three colour pattern; measuring with non-standard units; and cooperatively working with a buddy. However, because it requires the use o f so many Unifix cubes, and I have a limited supply of them, no more than six students could work on the activity at any given time. In order to keep the other children working on a constructive task while six of them worked on their pattern, I set up two other 'fun type' mathematics activities that were not related to any of our W a l l Math questions, and the children rotated through the three activities. The patterning activity itself proved to be relatively easy, and the children in each group sounded as though they were having fun while they made their patterns and compared the length of their Unifix train Tegs' to see who had the longest and shortest legs. M a n y children compared the lengths of their real legs by standing closely together side by side to be sure the relative lengths of their Unifix train legs were 'right'. D o n was surprised  — 63 —  to discover that his leg was shorter than M i k e ' s , even though both boys are the same height. "Our legs should be the same length when we're the same height," commented D o n . Several children attempted to count the number o f blocks used, but only a few were successful, probably because there were too many distractions and/or interruptions. Counting small nonmovable objects is still a difficult task for many children at this time o f year.  When I  suggested to Marge that she break her train into units of ten blocks so she could more easily count them, she said, " B u t I want to see i f I have the shortest leg." M a n d y and M i k e successfully counted theirs and told me how many cubes they used, but neither o f them recorded the numbers on their record sheets. Cathy, who was i n the first group to work on the patterns, did only a three block, three colour train, rather than the requested four block, three colour train, but the three block, three colour train was completed correctly. F o r her, this was a job well done, despite the fact that it was not the exact task that was asked o f her. When her group compared leg lengths, she was quite pleased to discover that she had the shortest leg, and asked i f she could keep her train, "to see i f I have the shortest legs i n the whole class." Because of her request, I suggested that each group keep both their longest and shortest "leg trains" so that we could compare among each group's extremes when everyone was finished to find out who had the longest and shortest legs overall. This meant that we had to borrow Unifix cubes from another classroom. When we compared all the "leg trains", Cathy was delighted to find that her leg was indeed the shortest i n the class, by five cubes. " C u z , I'm the smallest k i d , " she said with a smile. This was a rare time for her to be singled out as the best — a welcome occurrence that greatly promoted her self-esteem.  D a y 3 — F r i . , J a n . 10 Today was a cold, but dry day, so we used the opportunity to work on Question 15 — How many....vehicles  are in the school parking lot?  right after recess. Before going outside, we  talked about how to count the vehicles, decided how we could keep track of how many were there, and went over safety in the parking lot. The children all agreed that they could count  — 64 —  the vehicles in their heads because, "there's not that many," according to Douglas. W h i l e we discussed our strategies, I questioned the children abut how they could use their buddy to help them know i f they had counted correctly. " I f our numbers are both the same we d i d it right," came a quiet statement from Marge. " A n d , i f we don't get the same, one o f us d i d it wrong and we need to do it again," agreed D o n . "There's not that many, we won't count them wrong," boasted Mandy. A s we talked, I noticed that one o f my E . S . L . students was looking quite puzzled, but was not asking any questions. I considered m y question, and wondered i f he knew what a "vehicle" was. I asked i f someone could explain why I had used the word "vehicles" rather than "cars" in my question. " C u z you don't want us to count just the cars. Y o u want us to count M r s . R ' s van and M r . H ' s truck too," D o n told us. A s we talked about the variety o f types o f vehicles we would find i n the lot, the puzzled look gradually disappeared from his and several other students' faces. I was sure that at least they now understood the vocabulary used in the problem. Everyone seemed clear about their task, so we put on our coats, chose our buddies, and headed out to count vehicles. Just as we got to the front door, the van that brings the school meal programme lunches pulled into the parking lot. " D o we count the lunch van?" questioned someone. "Sure," said M i k e , "It's i n the parking lot." " B u t it's not parked properly," argued Mandy. " S o what," came another voice, "it's i n the parking lot." A quick discussion resulted i n an agreement that either counting or not counting the lunch van would be acceptable. The children then counted the 16 or 17 (depending on whether or not they included the lunch van) vehicles in the lot. A few pairs had to count twice. "I forgot what I counted," said Cathy. " W e didn't get the same number at first," said Douglas's partner. When we got inside and the answers were recorded on their record sheets, I called a l l the children to the group centre where we d i d a quick bar graph to see i f more or fewer students counted the lunch van. This is a quick and easy activity because there is always a set of magnetic photographs of the children across the top of the group centre chalkboard. T o make the graph, the children simply move their picture into the correct column, row, or circle, depending on the type of graph we are doing. " M o r e  — 65 —  kids counted the van," called out Brian. " H o w many more?" I asked. "Three," sang out several voices.  W e were now ready to tackle the challenge part of the counting activity — How many wheels? — and I wanted everyone to at least try it. D o n was first to respond to my question, " H o w many wheels did we see on these vehicles?" with, " A lot!" and, after a few seconds thought, " C a n we use the calculator?" I assured him he could, once he had shown me how to figure out the answer. " O K , " he said, and headed off toward the bin unit where the Unifix cubes are stored. A s he began his work in a far comer of the room, the rest o f us talked about how they were going to approach the problem. A s each pair of children felt sure enough to work on the problem on their own, they moved off to get a manipulative of their own choice, and started on their task. A s I wandered and asked questions, I got comments such as, "I have to put out 16 cars like this." from D o n as he pointed to a group of four U n i f i x cubes. "This is one car," concluded Cathy, as she set out a group of four blocks. B r i a n had 16 groups o f four cubes each, and one group o f five cubes that he was beginning to count. When I questioned him about the group of five, he told me with confidence, " M r s . R ' s car has five wheels." I guess my puzzled look prompted the explanation, "It's got a wheel on the back." (Mrs. R drives a Pathfinder sports utility vehicle that has the spare tire on the back door). I couldn't argue with his reasoning since it fit within the question, but I made a mental note to be sure to discuss his answer and reasoning when the whole class compared results. I watched as the children finished putting out their groups o f four and began counting to get a total. M i k e counted his pattern blocks by twos, and came up with the correct answer, and D o n again requested the calculator. " I t ' l l be faster, I just need to do 4 and 4 , 1 6 times and I ' l l have the answer." I gave him the calculator and left him to do his work, but he eventually came up with an answer that was four less than M i k e ' s . " H m m , one o f us d i d it wrong," he reasoned. I suggested that he check his answer by putting i n the 'groups o f equation. W e talked through this procedure while M i k e watched. W e had done this a few times before, so  he was familiar with the procedure, but he does not yet feel secure enough to try it on his own. A puzzled look, then the comment, "I musta missed one," showed me that his total now agreed with M i k e ' s answer, and that he understood what had most likely caused his initial error. When I checked on Cathy, she was very carefully putting her groups of four into two lines; one line had ten groups of four, and the other had six groups of four. However, she was still unable to count them. "There's too many," she said with a disappointed look when I asked her how many blocks she had. I asked her to try again while I sat and watched. She carefully pointed to each block as she counted, but she got lost after 32, then quickly stopped when she knew she was not counting correctly. Although she can rote count to 50, she continues to have some difficulty with one-to-one counting.  She was w o r k i n g beside  Douglas, who also was having difficulty counting his blocks and was watching Cathy and me. The three o f us talked about how to make the counting easier, and Douglas finally said, "I can count to 100 by 10's, see?" and he proceeded to show us. " C a n we put these in 10's?" I asked, and I agreed to put all Cathy 's blocks into groups o f 10, while she watched. When I returned, he proudly displayed his work and told me he had 64 blocks, "See?" he said as he pointed to, and counted his groups of ten by 10's, and then added the group o f four without counting it. I wondered i f he remembered what he was counting, so asked, " S o , how many wheels are on the cars?"  H e appeared puzzled for a moment, with a b i g smile, then  answered, "Sixty-four."  W e were running out of time, and we needed to gather together to discuss our answers, even though not everyone had completed the task. Three pairs o f children were completely off task, and were now playing with their materials. I asked several children to leave their work on the floor to use as demonstrations of how to do this type of activity, and asked the rest to clean up and come to the group centre.  Once gathered, I asked the children who had  completed the task to give me their answers, which I recorded on the chalkboard. The answers ranged from 60 to 68 and caused some puzzled looks. "But, shouldn't they a l l be the  same?" questioned someone. " N o , " M i k e assured them. " I did 17 cars, but D o n only did 16."  " A n d I know mine's right cuz I used the calculator." added D o n . A s the children  explained what they did, I wrote both the addition and multiplication equations for 16 cars on the chalkboard. Then, we carefully counted by fours to do the addition, and, as time was running out, I asked D o n to demonstrate how to do the multiplication equation on the calculator. "Then mine's right too,' said M i k e , " I got 68, and that's four more than D o n ' s , and I did 17 cars and he only d i d 16." I added the extra group of four, representing the wheels on the lunch truck, on to each equation, and everyone agreed that this was correct too. " A r e there any other answers that could be correct?" I asked, thinking of Brian's answer. Sure enough, " M i n e i s . " said Brian. I pointed to his answer, "65", on the board. " B u t that's an odd number." called D o n , giving Brian a look o f disdain. When I asked him to explain why this was a problem he said, "Four is an even number, so the answer has to be an even number too.  Sixty-five is an odd number, so it can't be right." I assured him that his  thinking was correct, but, "listen to Brian's reason for his answer." A s Brian explained about the spare tire, the nodding heads told me they understood, and accepted his answer as correct too. I then asked i f anyone could explain the answer of "60". "Someone missed a car," said Mandy, "cuz it's four less than the right answer," she added as we waited for her complete explanation. Again, nodding heads showed that the others agreed with her reason, so there was no need to discuss it further — a good thing since it was time to clean up for lunch.  Day 4 — Tues., J a n . 14 This was to be the final work period for this board. W e gathered around the board to talk about the next three questions. W e counted the tally marks for Question 12 — Tally count, and I asked Ronald to print the number on the chalkboard for the others to copy later. W e then discussed the equation 3+3 in Question 13 — Picture please, and drew a few examples of possible pictures for it. W e then looked at the "Weeple People" on Question 14 — Which one doesn't belong? Why? and discussed several possible answers. The children were then  — 68 —  sent back to their tables to record their answers, and called together again after about 10 minutes to remind them how to do Questions 16 to 19. They went back to their tables to complete the work alone, or with their table mates. A s I wandered, I noticed that M a n d y had printed the word circle as her answer to Question 16 — Geometry: Name the shape of a traffic light. When I questioned her answer, she pointed to an individual light in the picture of the whole traffic light and said, "It's a circle." When I silently traced the outside o f the whole set o f lights with my finger, she told me it was a rectangle, " B u t the light is a circle," she insisted. I couldn't argue with her logic, and made a mental note to be sure to discuss her answer with the whole group. Everyone else had called it a rectangle, but when M a n d y explained her reasoning, all the children agreed that her answer was also correct. About half of the class had difficulty completing the number wheel, so it was left uncorrected and a review lesson was planned for the next day. A l l o f them copied the temperature correctly from the calendar board as the answer to Question 19 — Measure: Show today's temperature. After ensuring that everyone had estimated the number o f erasers i n the 'estimating jar', I called the children to the group centre, hoping that we had enough time to count the erasers and discuss the "Stumper", which I had planned to send home as a homework challenge, before lunch.  I recorded all the children's estimates o f the number of erasers i n the jar on the chalkboard as a number line. The range was from 16 to 21. Once we had counted the first group o f ten onto the counting board (see Figure 5-1 for an example o f a "counting board"), I heard someone say, " W e guessed really close today." When I asked i f there was another group o f ten i n the jar, the reply was a pretty even m i x o f "yes" and "no". W e counted the last nine erasers, and I asked Marge to print the total number on the chalkboard. A s she d i d this, D o n called out, "Hey, there's one for each of us." " B u t there's only 18 in our class," stated Douglas. After a short silence, Marge added, " M r s . Bateson needs one too!" They were a l l sent back to their  — 69 —  tables with a new eraser to use, but we had run out of time and we hadn't discussed the "Stumper" yet.  W e took about 20 minutes in the afternoon to work on the Stumper, which, surprisingly to me, they found fairly easy to understand. When I questioned them about this, I was told that "lots o f phone numbers on T V are words." — I had not thought o f that. M a n y o f the children found their names as numbers very quickly, and several took up the challenge to use the calculators to find the value of their names — that is, to add the numbers together to find the total. This "Stumper" turned out not to be such a stumper after all! Since they had all found the task easy, as a supplementary activity I sent a list o f words home for them to work on with their parents.  — 70 —  Diary #2 — January 20 - February 2, 1998 This diary applies to the following board of questions. (* = "Try it" Challenge Question) 1. Pattern: A money pattern of penny, nickel, penny; penny, nickel, penny. 2. Number line: 32,33, and 34 printed above the first three number points. 3. W o r d please: 86 4. Number please: nineteen 5. Before and After: 43 printed above the middle number point. 6. Between: 90 and 92 printed above the first and last number points respectively. 7. Draw and print as a number: three tens and five. 8. A d d : 4+3  in columnar form; and 6 + 6 =  9. Subtract: 7 - 2 = ; and 13 - 3 = 10. Equation please: Four birds in a cage with one flying out. *Write the problem. 11. Orderplease: Four lines labeled "A", "B", "C", and "D", with the appropriate order from shortest to longest being B, D, A, and C (note that longest to shortest would be reversed) — it was not specified whether it was to be shortest to longest, or longest to shortest. Also, there is a remote possibility that children might order them by thickness Idarkness of the lines. From thinnest/faintest to thickest/darkest the appropriate order would  beC,A,  B, then D (again it could be reversed). 12. Tally count: Thirteen marks in tallies of five (five, five, and three). 13. Picture please: 4 + 5 14. W h i c h one doesn't belong? W h y ? : "Jolls" Weeple People (Gillespie,  1971, p.7)  Jolls  have completed outlines of their faces, whereas non-Jolls have a gap in the outline of the face. 15. H o w many: Cars going past our school on "Alias" Street in five minutes. 16. Geometry: Name the shape of a STOP sign. Draw a picture of it. Name something else that is this shape. 17. Number wheel: "+5" in the centre.  —71—  18. Estimate: Twenty-six candies in a jar. 19. Measure: Show the length of your foot using Unifix cubes 20. Stumper: A picture of a magician with a "cross of 5 stars (three horizontal  intersecting  three vertical, with the centre star common to both). The instructions accompanying  the  picture were: See how many solutions you can find to this puzzle. Here are the rules: 1. The sum of each line must equal 10. 2. You can only use the numbers 1, 2, 3,4, 5, 6, and 7. 3. A number can be used only once in a solution.  There are many different  solutions.  How many did you find? It is Mid-January, and it's time to do some work with money. A s a homework assignment, prior to starting this board, the children were asked to print the names of our six Canadian coins (penny, nickel, dime, quarter, loonie, and toonie).  Day 1 —  Fri., Jan. 23  A s a group, we looked at the coin chart and discussed what each coin was called, what picture was on it, its colour, relative size, and shape. M a n y children looked puzzled, and could not answer the questions. So, we went to the group centre and, after digging around in m y wallet, we experienced a penny, nickel, dime, and quarter. W e looked at the pictures on both sides. "Hey, this Queen is different from this one!" led to a short discussion and comparison of the Queen's pictures, and who she was. A graph o f the numbers o f each type o f the Queen pictures and comparisons of dates would have been appropriate, but we were running out o f time, and I wanted them to ' p l a y ' with the edge patterns. After feeling, drawing, discussing, and comparing the edge patterns we talked about w h y the coins were so different. " T o look pretty," came from Cathy; "So they can make patterns," was Douglas's contribution; " S o they are easy to tell apart," suggested Don. D o n ' s comment led to a quick discussion about rinding different denominations of coins without seeing them, and who or when one might need to be able to do this. W e played a guessing game with our buddy where one child put a coin into the cupped hands of the partner, and that person had to guess which coin it was. After several tries, most children became proficient at knowing which coin they had without looking at it.  —72—  "It's a dime — it's so little," answered M i k e ; "It's a quarter — it's so big," another said. Just before putting the coins away, I asked the children to put them i n order according to their value. A l l students knew that a quarter was worth the most — "I can buy a ring with a quarter, but I can't buy anything with a penny," and that a penny was worth the least. However, many children were confused about the nickel and dime. " B u t the nickel is bigger." Another day's lesson! W e then also took a quick look at the pictures and edge pattern on the two loonies that I had in my wallet.  Although this lesson was supposed to be just a short reminder o f the appearance o f various coins that I wanted to conduct before we started on the pattern question (Question 1—Pattern), it took about 45 minutes, and the kids were on task the whole time. Money fascinated them; apparently the majority o f this group were completely unaccustomed to seeing or handling coins. H o w d i d they do the pre-assigned homework? Today's homework was to find out how many bears were on a toonie, and to draw a toonie's edge pattern. O n l y about half the class did this homework. Most explained that " M o m didn't have a toonie."  D a y 2 — T u e s . , J a n . 27 W o r k time for this day started with the children gathered around the board for a quick, oral review o f the first six questions. The pattern was again discussed with regard to the coins and their pictures on them, as well as their different colours. " B u t you didn't start it right," said M i k e , "there's only one penny at the beginning and there should be two." This led to a discussion about just what was the pattern. W e decided there could be two patterns: p, n, p and p, p, n, and proceeded to say the pattern using each o f the two rhythms, using both the coin name and their value, as well as with 'claps and snaps'. It's nice to see at least some o f them are looking for the possibility o f more than one answer after five months o f trying to convince them to do this. Question 2 — Number line was read, and, after some discussion, the answer was found on the "days we've been i n school" number line. T h e number i n  —73—  Question 3 — Word please was read, and the place value meaning of each number discussed and demonstrated using our fingers. Eighty was shown by holding up all ten fingers with hands touching to show a group o f 10, 8 times, as we counted to 80 by tens. Six was shown by holding up six fingers. These two words were then found on the number word frieze for copying. The number word on Question 4 — Number please  was read, found on the number  frieze, and demonstrated with fingers. The number on Question 5 —Before and After was read, and a chorus of voices called out, "42, 43, 44." Forty-three was found on the number line so everyone could copy the before and after numbers. The numbers in Question 6 — Between were read as a whole group, and were found on the section o f number line which was still on the calendar board. A t this point, the children were sent back to their work spaces with small containers of paper pennies and nickels with which to make their patterns, and to do the next five questions on their own or with a buddy. Some decided to colour the coins first, and a few decided to cut each coin into its round shape. Both these activities meant that the task took longer than I had anticipated. When I asked Cathy why she was cutting rounds, she said, "...cuz they're round." W h e n asked, " B u t isn't the drawn shape good enough?" her reply was a puzzled look that said ' n o ' , as she kept cutting. B y lunch time, she had managed to cut out what she thought were enough coin shapes and was ready to glue them onto her answer sheet. However, she was out of time, so she put the pieces back in the box to be dealt with next time. Meanwhile, M i k e and D o n completed all six questions without assistance other than from each other, and had gone off to play More War with a deck of cards. Mandy, on the other hand, had completed the first six questions, and was busy working on the next questions. "What do I do on Number 11?" W e quietly discussed the two ways these lines could be put in order by length, and she went off to record her chosen answer. Once done, she brought me her paper to check. A l l were correct except her answer for Question 7 — Draw and print as a number which was only the printed numeral "35"; she had not done a drawing. Mandy tends to leave out parts of all activities that she feels unsure of completing correctly, or that she thinks are unnecessary.  When asked, "Where is the picture of this number? The  —74—  instructions say 'draw and print'," she shrugged her shoulders, picked up her paper, and returned to her table. In a few minutes she returned to me with large, messy, uncountable tally marks all over the section. When I questioned her about the number, she told me there were 35. I asked her to count them, which she started to do, but she quickly gave up with a frustrated sigh after counting 22. She decided that I should erase the space — she usually asks me to do this because she has discovered that she can't do as clean a job as I can. W e had a quick discussion about groups o f ten, a reminder o f other, similar tasks we had done, and a look at the tally system used on the calendar. Then she went off to redo hers — correctly and neatly (for her) this time. Marge quietly worked to complete the first six questions with typical care and correctness, then went off to work on some tile cards. Douglas settled down to work quickly and got the pattern and number line done correctly without help, but then got distracted by a table mate who asked him a question about the coin pictures that resulted i n a lengthy discussion about the coins and the pattern.  Once he got settled again, he quickly d i d  Question 3 — Word please, but came to me for help with Question 4 — Number please. "What does the word on Number 4 say?" I helped him sound out "nine", and as soon as I uncovered the "teen" part o f the word, he said, "nineteen, a one and a nine," with a big grin, and he went off to print the number. Question 5 — Before and After was completed alone, but he again came to me for help with Question 6 — Between. "I can't find any numbers that start with ' 9 ' on the number line," he said, as he pointed to the line above the board. I reminded him that this section of the line was still on the calendar board, and away he went to find it. Once finished, he brought his completed work to me for final approval, then took it back to his table to share with his table mate who was working much more slowly.  D a y 3 — W e d . , J a n . 28 The second day with this board of questions started with the children sitting in the group centre to do some counting o f the money that was used to make the patterns. W e also discussed the pattern, and decided that it could be considered to be correct i f there were three nickels and  —75—  between five and seven pennies in the row. " M y row is full," said Marge, "and I have three nickels and seven pennies."  "I only had room for three nickels and five pennies," said  Douglas. However, after a comparison of different uses o f space, he decided he had done the correct pattern, but had left bigger spaces between his coins than others had. Cathy was looking at hers, which she had glued together yesterday in math time, with a questioning look. When I went to see what the problem was, she said, "I have too many five cents, my pattern is wrong." I told her to leave it for now and we would work on it together later. Once the others had set to work, I went back to Cathy. W e went over the paper to discover where the error was, and she said, " I ' l l just colour this one and this one like pennies. Then it w i l l be right." She pointed to and coloured the nickels, that were now i n the wrong places, brown, so that they became pennies. She then seemed satisfied that her pattern was correct, even though none of the" other coins were coloured. I wondered i f the other pennies were brown in her head so that she was indeed seeing a correct pattern?  We then moved to sit in front of the Wall Math board for a quick review o f the next set o f questions. W e started with both a verbal and pictorial reminder o f how to do Question 7 — Draw and print as a number. This one continues to be a problem even though we use groups of ten daily on our calendar and also every time we count large numbers of anything.  Questions 8 — Addition and 9 — Subtraction had two equations, each for the first time, so we discussed and mapped how these might be done i n the space given. W e talked about the action that was happening i n the picture on Question 10 — Equation please, and what the correct equation was, and why. W e also talked about a story problem to go with the picture since this was a challenge that had been added for the first time. It is interesting to note that, despite our discussion, only Douglas tried to write a story problem. W e talked about putting the lines in Question 11 — Order please in order by length in two ways that would be correct. A l l the children seemed clear on this, but both Douglas and Cathy made mistakes when  —76—  recording their answers. A reminder that they were to print the number for Question 12 — Tally count was a l l that was required for this review of a daily calendar activity. W e then talked about the pictures they were to draw for Question 13 — Picture please. " I ' m going to draw hearts and ice cream," said Marge. " I ' m going to draw fish and hats," said Douglas. " I ' m doing circles and triangles because they're easy," said Mandy. Although M a n d y had already completed Questions 7 to 11,1 asked her to join us in the group discussions because she needs to listen to other people's thinking, and learn to look for, and accept different opinions and solutions to problems. Cathy shrugged her shoulders when I asked her what she was going to draw, but answered, " N o , " when I asked i f all the things could be the same; "I have to draw four of something and five o f something else." The children were then sent off to work with a reminder that they should do Questions 14 — Which one doesn't belong, 16 — Geometry, 17—Number wheel, 18 — Estimate , and 19 —Measure by themselves or with a buddy, that Question 15 — How many, which was turned over so they could not see it at that time, was one we had to do as a whole group at about 11:50, and to try Question 20  —  Stumper with a friend.  A s I wandered around helping various students, it again became clear that Question 7 — Draw and print as a number was causing major problems with several o f the children. I therefore pulled together a small group of the children who were having problems with it (including Douglas and Cathy) to draw the pictures on small chalkboards while we counted ten objects inside each of three circles, and five ones beside the circles. The children were then asked to transfer this drawing to their record sheet. Douglas did so successfully, but Cathy decided to finish Questions 2 — Number line, 3 — Word please, and 4 — Number please first, and by that time, her picture had been rubbed off enough that she couldn't copy it. I asked her to show me the number using the place value blocks instead. She d i d this successfully, but it required some help from me. She then told me which numbers to use to print "35"; and I sent her back to her work table to print this number on her answer sheet. M a n d y completed  —77—  Questions 12 to 19 (except 15 — How many ) on her own, then wanted to know how to do Question 20 — Stumper.  Upon inspection o f her paper, I discovered that she had done none  of the challenges, so I sent her away with the request that she do the second part (the challenge) o f Questions 10 — Equation  please,  attempted Question 20 — Stumper.  16 — Geometry,  and 19 — Measure  She did 16 —Geometry  before she  and 19 —Measure before it  was time to put things away in order to work on Question 15 — How many. M i k e completed Questions 7 to 13, then skipped to Question 17 — Number wheel because, " D o n is my buddy and he's not finished to do what doesn't belong  yet." (Question 14). B y the time D o n was  ready, and M i k e had done Question 17 — Number wheel , it was time to clean up and get ready to do Question 15 — How many. D o n had completed all the Questions 7 - 1 3 correctly by clean up time. Marge had also completed Questions 7 - 1 3 alone correctly, as well as having done Question 1 4 — Which one doesn't belong? Why? (helped by a suggestion from Mandy as to the 'correct' answer). Cathy had accomplished little during this work time. She had written the number " 3 5 " in Space 7, drawn a picture in Space 10 that sort of resembled the picture on the board, but did not attempt an equation for Question 10 — Equation please.  She  had printed the letters for the lines in Question 11 — Order please, but could not tell me why they were in this order, even though she showed me, by pointing at the correct line, when asked which was longest, next longest, next, and shortest. examining the pictures in Question 14 — Which  She had also spent some time  one doesn't belong, but just shrugged her  shoulders when I asked her which one was different.  Douglas worked alone quietly on  Questions 8 - 1 3 , including writing a story problem for Question 10 — Equation please, and spent some time looking at the pictures for Question 14 — Which one doesn't belong? Why? but had not made a decision by clean up time. A l l o f his answers were correct, except for the response to Question 11 — Order please.  However, he answered it correctly when asked to  point to the shortest, the next longest, the next, and the longest. B y the time we were cleaned up for lunch, we had no time to work on Question 15 — How many, so we had to leave it for Friday. After all this time in the classroom, one would think that I would have learned that it  usually takes children twice as long to clean up as I think it should. I guess I ' m just a slow learner!  D a y 4 — F r i . , J a n . 30 Quick reminders were given about where to copy the words for Question 16 —- Geometry, that they were to measure their foot with Unifix cubes and with something else, and to be sure to write down the name o f what they used, as well as recording the number. W e had a 30 minute work period before we d i d Question 15 — How many. During that time, Mandy finished all the questions, except for Question 20 — Stumper, and she was looking at and thinking about that one. Marge said that she had completed all the work, but she had forgotten to name her geometric shape (Question 16 — Geometry once.  ), and had only measured her foot  Cathy was away from school, so her sheet remains incomplete. M i k e completed  everything up to Question 20 — Stumper, and was looking at that one. D o n had completed everything but the second measurement for Question 19 — Measure. Douglas was working on Question 19 — Measure.  B y 11:45 everyone was ready to work on Question 15 — How  many . W e gathered in the group centre to read the question and discuss our strategy. "But we can't see Alias Street," said one child. " Y e s , we can; that's it right there," said Marge, as she pointed out the window. " O h , lots of cars go by there," was heard from the group. " H o w w i l l we know when five minutes is up?" asked D o n . " W e could set the alarm," suggested one o f my learning assistance children who uses the alarm clock as a reminder of when it is time to go to his other teacher. " W e could use the sand timer," someone said. "It's not long enough is it?" came, with an inquiring look, from M a n d y . "Someone could watch the hands o f the clock," contributed Douglas. Most seemed to think this was the best idea, but no one wanted to watch the clock, so it was a unanimous decision that watching the clock would be my job. Next, we had to decide how to keep track o f the cars seen. It was decided that either tally marks on their papers, or simply putting up a finger each time a car went past was the best way. There was no discussion of how the finger method would work i f there were more than  —79—  ten cars. A few, including D o n , Mandy, and Marge, went to get their papers and a pencil to record with tally marks. The others all used their fingers, and, as it turned out, counted out loud. W e all went to the window, and I gave the start signal. During the next five minutes we sang songs and counted the four cars that went past. "That was a long time," said Pat when we were through counting. The children were surprised that there were so few cars passing by in such a ' l o n g ' time. "I thought w e ' d have to count lots," said Douglas. " W e should do it again just before we go home. There w i l l be lots o f cars then," commented Susan, whose mother drives her to and from school. The children then returned to their papers to record the answer. W e still had to count the candies in the estimating jar and work out the "Stumper". The counting I knew was a good Friday afternoon activity, but Question 20 — Stumper would have to wait until Monday.  About 2:30 we cleaned up for the weekend and gathered in a big circle in the group centre to count the candies. The incentive o f getting the candies to eat hastened the cleanup. A s I had noticed several different answers to Question 14 — Which one doesn't belong? Why?, we took a few minutes to discuss their answers as to which doesn't belong, and their reasons for their choice not belonging with the others, before we started our counting. Question 14 — Which one doesn't belong? Why? consisted o f five pictures of "Weeple People". Although most of the children had chosen Figure D as not belonging, "becos thay hav now opin spasis" (copied letter for letter from D o n ' s paper), other choices included E : " H e is sleppy (copied from Mandy's paper); or A : "It has a krak" (copied from Douglas's paper). W e had a debate as to which answer was really right, but finally came to the conclusion that all were correct. " O h yea," said D o n when Mandy and Marge pointed out the sleepy eyes of Weeple E . "It looks like my little sister when she's ready to go to bed," explained Mandy. " Y e a , and he looks like the sleepy dwarf in Snow White," commented Marge. " B u t it has an opening," argued D o n . " I know," replied Mandy, "but none of the other ones look sleepy." " O h , yea," was all D o n could comment. When Douglas gave his reason for choosing Figure A , someone suggested  —80—  that the "krak" looked like it was in the picture and was not really supposed to be there. Douglas agreed, but insisted " B u t it still makes it not belong." N o child challenged h i m further.  N o w we were ready to count our candies. Each child gave me their estimate, and I recorded them as a partial number line across the board. The highest estimate was 36, and the lowest was 13 (from Cathy). W e counted the candies by 2's as I pulled them out of the jar and placed them on the "ones" section o f our counting tray. A s we got a group of 10, someone would call out "Bundle them," and the group of 10 was placed in a small paper cup and transferred to the "tens" section of the tray. W e then looked at what was left in the jar, and estimated whether there were more or less that 10 left. Everyone agreed that there were more than 10 candies left, so we continued counting by 2's  to make another group of 10. I asked Cathy to do the  bundling this time since she and a few others whose estimates were now obviously too l o w were beginning to loose interest, as were the few whose estimates were over 30 — they could now see that there w o u l d not be another group o f 10. This was happening despite the discussions that an estimate was just a 'guess', and that just being close was as important, and more easily attained than choosing the exact correct number. Some children, including Mandy and Marge, often do not record an estimate until after we have counted, then they put the counted number in both spaces. Or, i f specifically told to record an estimate, they w i l l erase the estimate and print the counted number i n both spaces after they know what the 'correct' number is. When we first started doing this type o f activity, it was often difficult to get these children to verbalize an estimate that could be recorded on the board. Risk taking is not one of their strengths, but it is slowly improving. A s Cathy transferred this second group o f 10 to the "tens" section, D o n announced to the class that there is, "at least one for everyone." I asked him i f he thought there might be enough for two each. A s he pondered this question, M a n d y called out, " N o , cause we only have twenty-something and there are 18 o f us." " Y e a , w e ' d need 2 times 18 to give us all two," decided D o n . " A n d how many would that be?" I asked.  Puzzled or blank looks appeared on most faces around the group as I put the equation 18+18=  on the board.  D o n was the only one who tried to talk his way through this  equation, but as he talked "1+1=2 and 8+8=16," his body language told me that he knew there was something wrong, but he could not explain to me exactly what was wrong. A t this same time, I noticed Marge's and several other children's heads nodding as they counted the children around the circle. Almost in unison, they called out "36". " W e ' d need 36 to give everybody two candies," confirmed Marge, and the other counters nodded their heads i n agreement. A t this point, I talked my way through the algorithm for adding with regrouping, pointing out that this is the way an experienced mathematician would do it. "Nope," said D o n , getting back to the question of whether or not there were enough for two candies each, "we only get one. W e ' d need another group of ten, and there isn't another group of ten in there." M a n y heads around the group bobbed in agreement. W e quickly counted the remaining six candies, and discussed how to record the number. The children went to their tables to record the number, and quickly returned to the group centre for the sharing which, because o f our earlier discussion, simply became a task of giving one candy to each child. "What are you going to do with the rest?" asked Mandy. Several children immediately suggested they go into the bingo prize jar. Everyone agreed, so that's where they went. Everyone left happily munching on their candy and reassured that there were new candies in the almost depleted bingo prize container. Figure 5-1 displays a counting tray/board with the candies set out on it.  Figure 5-1. Counting the candies from the estimation jar on a counting tray.  Day 5 — Mon., Feb. 2 Question 20 — Stumper was an entirely new type o f puzzle, and one which I knew would be beyond the understanding of about half of the class. However, I wanted everyone to have a try at this type of playing with numbers. W e gathered around the large version of the problem on the wall and discussed the rules for solving it. "The number in the middle has to be used for both sentences," reasoned Marge, "but I thought it said you could only use the number one time." " Y e a , " said D o n in a puzzled tone of voice, then almost immediately, " O h , I know, you can only w r i t e the number once!" I assured them that this was correct, and sent them off to work, reminding them that they should finish the rest of the questions before they attempted Question 20 — Stumper, knowing that some of them would try it first anyway, then finish the rest. D o n and M i k e immediately went to work on the "Stumper", trying combinations o f numbers they knew equaled ten. After several tries at recording one that worked while the other didn't, they came for more help. Mandy decided she'd work with Marge, but basically she works on her own while Marge sits, watches, and makes comments such as, " Y o u only need one more to make ten, so that won't work", or "That one works that way, w i l l it work the other way?" Despite the fact that she is not 'doing' anything, her comments to Mandy let me know that Marge was actually actively involved, even though she was not writing anything down. Marge is not a risk taker! This commentary from Marge causes Mandy to give her frequent inquiring looks before she continues her work, sometimes adjusting her work to include Marge's suggestion, and sometimes not. M a n d y became frustrated quite quickly, and both girls came for help about the same time as did the boys. Douglas had been watching M i k e and Don, and decided that i f they couldn't get it, then he too would need help. I invited all the children who were ready for this "Stumper" to come to the group centre for some help with a strategy; eleven children came up. "What are the numbers we have to work with?" I asked. "One, two, three, four, and five," they replied. "Show me these groups with Unifix cubes," I  —83—  asked. The children made Unifix trains of one, two, three, four, and five cubes. " H o w many of these groups does your question want you to add together to make ten?" I asked. "Three" came the answer. "Put three groups together to make a group o f ten, but don't j o i n them," I instructed.  M i k e made 1+4+5, D o n made 2+3+5, Douglas copied D o n ' s , M a n d y made  1+5+4, and Marge made 4+5+1. Others in the group had these or a combination with the " 2 " in the middle, but no one made a combination with the " 1 " in the middle. " H e y , this equals 10 but when I put the 1 and 4 on the other way it doesn't work." said a disappointed D o n . "But, 1+4+5 equals 10," he said with a confused tone of voice and puzzled expression as he looked at me. A s he was saying this, M i k e began changing the order of his blocks so that he had the " 5 " in the middle. A smile appeared on his face as he announced, "There, that way works. Y o u have to put the five blocks i n the middle. See, 1 and 5 and 4 is 10, and 2 and 5 and 3 are 10." H e pointed to each group as he spoke so that everyone could follow his thinking. " H e y , that's what I've got!" called M a n d y , just as M i k e completed his explanation. " O h , yea!" declared an excited group as the lights of understanding went on. Although both Marge and Mandy had put the group o f five blocks in the centre, they were so busy watching and listening to the others that they didn't realize they already had the correct combination until M i k e pointed it out. A l l the children except Douglas quickly threw their cubes back into the bin and went to record the answers on their sheet. Douglas continued to look at his blocks with a puzzled look. When I questioned him he said, " B u t this (pointing to his 1+4+5) equals 10." H e seemed to have missed the need to use the middle number i n both combinations. W e drew the diagram on a small chalkboard, and then put the blocks directly onto the diagram. A s he moved the blocks around, a smile spread across his face. " O h , I get it now," he said, as he took the board with the blocks still on it and carefully returned to his table to record his answer. Figure 5-2 is a picture of Douglas' chalkboard diagram with his Unifix cubes  —84—  Figure 5-2. The chalkboard and Unifix cubes that Douglas used to solve the "Stumper". A s I walked past him to check on another student, he was trying to explain to Cathy what he had done, and why it didn't work, "with the ' 4 ' in the middle," but it did work, "with the ' 5 ' in the middle." Cathy still did not understand, so Douglas carefully instructed her how to do it. "Put the ' 5 ' in the middle, just like this," he instructed as he pointed to the five blocks on the chalkboard diagram. She wrote the " 5 " . " N o w , put the ' 2 ' in this star." A s he pointed to the group of two blocks on the diagram, Cathy printed a " 2 " in the corresponding star. " A n d the ' 3 ' goes down here," and he pointed to the group o f three blocks on the diagram. A s Cathy printed the " 3 " , Douglas said, "See, that equals 10. N o w put the ' 4 ' here and the ' 1 ' there," he instructed as he pointed directly to the stars on Cathy's paper. "That equals 10 too!" he declared with a proud smile of accomplishment. Cathy smiled at him as he got up to return his chalkboard and cubes to their storage places, then she went back to watching the child beside her measure her foot to complete Question 19 — Measure.  B y lunch time, the group that had  been working with me on the floor had shared their answers with everyone who would listen — so much for this board's "Stumper". However, the next "Stamper" w i l l be of the same type to see i f they really understood the strategy.  —85—  Transfer of knowledge is one of the things that I really hope to accomplish by having many of my questions related to other curricular areas and to aspects of students' out-of-school lives.  —86—  Diary #3 — February 17-23, 1998 This diary applies to the following board of questions. (* = " T r y i t " Challenge Question) 1. Pattern: No pattern was posted on this board. for creating their own patterns using pictures beaver, chipmunk,  The children were given oral of Canadian  animals:  instructions  e.g.,  chipmunk,  raccoon.  2. Number line: 16,17, and 18 printed above the first three number points. 3. W o r d please: 25 4. Number please: nineteen 5. Before and After: 24 printed above the middle number point. 6. Between: 79 and 81 printed above the first and last number points  respectively.  7. Draw and print as a number: Two tens and four 8. A d d : 7 + 6=,and2  +7=  9. Subtract: 12 - 5 =, and 3 - 0 = 10. Equation please: A picture of eight bugs with three in a cloud and five flying away 11. Order please: 45, 25, 35, and 55 12. Tally count: Twenty marks in tallies offive (five, five, five, and five) 13. Picture please: 8 - 4 14. W h i c h one doesn't belong? W h y ? : Three Canadian animals (goose, bear, and beaver) along with one African animal — giraffe.  Note that some students might pick the goose  as not belonging since it is a bird and the others are mammals. 15. H o w many: How many provinces in Canada (note the distinction between provinces and territories). 16. Geometry: Name the shape of the Canadian flag. Draw and colour it. 17. Number wheel: "- 3" in the centre. 18. Estimate: Forty-two white Unifix cubes (I called them 'ice chunks') in a jar.  — 87 —  19. Measure: Show what time your family eats dinner. Show it on both clocks (both digital and analog clock faces were provided). Note that this was a homework assignment the data from which was made into a graph at school. 20.  Stumper:  Two pictures of clowns, each with a "cross of 5 balls (three horizontal  intersecting three vertical, with the centre star common to both). The instructions accompanying the picture were: Solve this puzzle. Here are the rules: 1. The sum of each line must equal 10. 2. You can only use the numbers 1, 2, 3,4, and 5. 3. You can only use each number once.  D a y 1 — Tues., F e b . 17 It's Canadathon time, a fund raising quiz similar to a spellathon, organized by the teachers to earn extra money for our school, and a time for a school-wide learning theme. Math  The Wall  board reflected this theme with questions about the Canadian flag, provinces, and  Canadian w i l d animals. It was also a fairly easy board that most o f the children could complete with little or no instruction, except for the pattern i n Question 1 — Pattern.  Question 1 — Pattern was quite different from what the children have been used to, i n that they actually had to make their own pattern following the rules I gave them; previously they had been given a pattern and had to copy, use, and/or manipulate it. I knew this problem and the activity around it would take more time than normal, so I planned to spend our whole work session between recess and lunch doing the calendar and this one activity. Once they had a chance to look at, and talk about the animal pictures, they were to use them to respond to Question 1 — Pattern.  I gave them these oral instructions. Choose any three of these  animals. Make a pattern that is four animals long. I received many puzzled or blank looks, and a few, "Huh?"s, and several, " H o w can we do that?" But, as I watched them looking at the animal pictures, facial expressions told me that a lot o f thinking was going on. Then  — 88 —  Mandy yelled, "I know, you have to use two of one animal!" and she quickly got up to take a fox, a chipmunk, and two geese out of the containers, and then put them on the board for the others to see what she was thinking. W e made M a n d y ' s pattern and several others with different sets o f animals, then clapped the rhythms, and said the pattern using the animal names, colour words (red, brown, gray, gray), and even animal sounds (yip, chatter, honk, honk). This almost turned into a 'science' lesson since many of the children did not know the types of sounds made by some of the animals. B y this time, everyone said they knew what to do and were ready to go off to get scissors, glue, and the long paper strips needed to complete the activity. However, at this point I added one more rule to the problem. Your finished answer has to be five patterns long. This brought more questioning looks, since previously their completed patterns could be as long as they wanted, as long as the space provided was filled, or it was a dictated measurement long (as long as your arm). "I k n o w , " said D o n i n his ' I ' m thinking out loud voice', "each pattern is four animals, right?" Everyone agreed that this was correct. "Then the pattern has to be 4 + 4 + 4 + 4 + 4 animals long." A s he said this number sentence, I wrote it on the board, and I could see several children counting to find the answer.  "That makes 20," came M i k e ' s quiet voice from the back o f the group.  After  several others came up with the same answer, I assured them it was correct, and asked i f there was anyone who could remember how we could write this equation so that it only used two numbers. O n previous occasions, we had several discussions about the mathematician's short way o f printing long adding questions, and I hoped that someone would remember. Mandy did. "I know! I know! Y o u have to use the times sign i n the middle." "Four times five", called out Douglas, and I wrote his equation on the board as M a n d y glared at h i m . " W e l l , almost.  If you say groups o f instead o f times, does it tell you about the adding  sentence?" I inquired. " N o , " said D o n , sure o f himself this time. "There are five groups o f four in my sentence, so that means it must be five times four," and I wrote this equation on the board under the addition equation. The others that were following this conversation nodded in agreement.  About a third of the class were totally lost, but were at least sitting  — 89 —  quietly and watching what was going on. I believe that exposure is the first step to learning a new concept. W e then had a quick discussion about the difference between the two 'groups o f sentences before they were sent off to complete the activity. " M i n e was almost right," said Douglas, after this discussion. "The answer was the same, I just got the numbers mixed up." I assured him that his sentence did indeed give the correct answer; it was just a slightly different picture, and, i f he could remember to use the words 'groups o f instead o f 'times', I was sure his numbers would go with the picture next time. N o w they were ready to go to work actually creating their own patterns.  Once materials were collected, each table group was called up to get sufficient pictures to make one block o f their chosen patterns. Once this was glued onto the long strip, they were welcome to come back when they were ready to take the pictures needed for another block of their pattern. I have learned that children make too many careless errors i f more that one set of pictures is taken at a time. Although doing it this way causes more movement around the classroom, and sounds like it should take longer, it actually causes fewer frustrations, more thinking, and more time on task, and is more economical i n terms o f paper used and time spent finding and trying to correct a single error that has caused many errors. The children seemed to enjoy this activity, and several table groups made a point o f everyone having a different pattern. " C a n I colour mine?" asked Cathy as she was colouring her first animal picture. M y answer was, "After you finish making your pattern!", remembering what had happened when she had started colouring her coin pattern pieces before making the actual pattern two boards ago.  She gave me a frown, but put down her crayon, and set about  making her beaver, beaver, bear, chipmunk  pattern without any mistakes. When I wandered  by her a short time later, I heard her quietly singing to herself (a common occurrence): "smack, smack, grrr, chatter") as she cut and glued her animals onto the paper strip. The fact that she never did go back to colour any of her pictures, even though she had some time to do so, was a strong indication to me that Cathy did not really want to colour the animals, but  — 90 —  was using this relatively mindless activity to avoid, doing the required work. This avoidance may be partly a result of not wanting to make mistakes, but such avoidance behaviour is a deeply ingrained part of Cathy's self; she works at avoiding ' w o r k ' .  A s I looked at Marge's work, I noticed that her original pattern was bear, raccoon, beaver, bear. I chose a spot several patterns along the strip, and asked, " W h y are there two bears together here, when there are not two bears together here?" and I pointed to the beginning of these two blocks. Her initial reply was her typical, " O h , I guess I made a mistake," but, very quickly a smile spread across her face, and she said, " N o , I didn't. There are two there because this one is the end of that pattern, and this one is the start of this pattern. Right?" as she pointed to the last and first bears of the two patterns that were side by side. "Right," I said, and I moved on to check what M i k e and D o n were doing. They had chosen to do similar patterns using two bears, "because we like bears," they said. A s I listened to their chatter while they worked, their whole table was telling bear stories. I wondered i f any of them had actually seen a bear in the w i l d , and where they had obtained some o f their strange ideas and stories. I guess we should do a short research study about bears to try to correct some of these fanciful ideas. Douglas had almost finished his goose, beaver, fox, fox  pattern  strip without any errors, and was happy to read it back to me using animal names, colour words, and the beginning letter of each animal name (g, b, f, f). T o m and Marge were the only children i n the class who d i d not put the two animals which were the same together in their pattern. When I asked them individually i f there was a reason for doing this, the replies were a shrug of Marge's shoulders, and, " C u z I wanted to," from T o m . Brian did his pattern vertically, even though everyone else d i d it horizontally. W h e n I asked him about this orientation of his pattern, he said, "It's easier to work on my table, I just push it down here (he indicated the space between the tables), and you didn't say which way it had to go." Everyone was finished his/her pattern by lunch time, so at noon I put them on display i n the hallway where they were the centre of interest for many days.  — 91 —  D a y 2 — W e d . , Feb. 18 Today's work session was a fairly lengthy period of time when the children were working on their o w n (i.e., no group instruction was done), but most children got a lot o f questions completed. I hovered around Cathy to catch her talking to her tablemates about her work, and to help when needed. She set right to work on the number line with a comment to no one in particular that, "This one is easy." She said the numbers aloud as she copied, "sixteen, seventeen, eighteen. There's eighteen," she continued; as she found it on the number word number line above the board and copied the numbers " 1 9 " and "20", again saying them aloud as she did so. "Nineteen, twenty. What comes next?" she inquired quietly of no one. "What comes after twenty?" she asked her tablemates. "Twenty-one," replied Brian. " H o w do I make twenty-one?" she asked. " A two and a one," said Brian with a frown that both of us read as 'stop bugging me'. She looked at me and gave a shrug, and I knew that it was time to step in before she became frustrated and went off task, and before her tablemate became more annoyed with her. She and I went off to find the twenties on the calendar number line so that she could find and copy the twenty-two on her o w n . She knew that twenty was close to the beginning of the line so she started counting and pointing from the beginning. "That's the one I have to copy, a two and a two," she said proudly after she had read, "twenty-two." I assured her she was correct, and we went back to her table to copy the numbers and to work on the next couple of questions together. She looked at the number written for the next question and said, "I need to find this number up there (she pointed to the number words number line) and copy its word, right?" "That's right. What does the number say?" I asked. She looked at the number for a second, then looked at the number line she had just completed above it before she answered correctly, "Nineteen." She quickly found and copied the word "nineteen", but looked to me for reassurance immediately after printing the first letter. The next question was more difficult for her because she had to actually read the two words, and then find the number and copy it. I helped her sound out the words, and we again found the  — 92 —  number on the number line, this time by reading the numbers starting from "ten".  She  wanted to start from the beginning, but she could tell me that twenty-five was more than ten so we started at "ten" and read all the numbers to "twenty-five". "That one!" she said as soon as she read it, and she ran back to her table to copy it out. Question 5 — Before and After  was done quite quickly after this because she remembered reading these numbers  already. "I need to put the ' 2 5 ' here too," she said after she read the "24" I had printed above the centre dot of this partial number line. Then she got up to check the calendar number line to find what to copy on the first dot. "Twenty- three, twenty-four, twenty-five," she read proudly after printing the " 2 3 " . I knew the next question w o u l d take a while for her to complete, so I asked her to do the addition questions i n Question 8 — Addition  while I  checked on what the rest of the class was doing. Everyone seemed to be working without trouble, either with their table groups or alone, so I went back to Cathy within about five minutes. She had copied and completed both addition equations correctly, and was very pleased with herself for having done so. I praised her lavishly, then I copied the two numbers for Question 6 — Between  onto her paper, and asked her to read them to me, knowing that  she couldn't. " W h e n we look for this one ["79"] on the number line, are you going to start looking before or after the ten?" I asked. " W a y after," she replied. "I need to find the numbers with the ' 7 ' at the beginning," and she started looking, without counting, along the number line. "There," she said as she pointed to the " 7 0 " and " 7 1 " . " N o w I need to find the one that is a ' 7 ' and a '9'...there," and she pointed at the "79". "What comes next?" I asked. "It's an ' 8 ' and an ' 0 ' ['oh']," she said. "Is it an ' o h ' or a 'zero'?" I inquired. " O h yea, it's a 'zero' cause it's a number, right?" She went happily back to her table to record the answer. I then asked her, " C a n you read this number for me?" She turned to me with a look that said, 'not without help'. "Let's find it on the number word number line," I suggested. Once she found it on this line, she walked back to the 'ten' card and read the numbers, "twenty, thirty, forty, fifty, sixty, seventy, eighty," stepping sideways as she said each word. "Eighty!" she said proudly. I had spent quite a bit of time on a one-to-one basis with Cathy, and I needed  — 93 —  to spend some time with the other children, so I suggested that she do the subtraction question (Question 9 — Subtraction  ) and the number wheel (Question 17) next, knowing  that she might get the subtraction question done, but would then get distracted until I got back to her. I was right; she did one of the subtraction equations, but then lost focus. B y this time, most of the others had completed at least the first eleven questions, and many were beginning to work on the problem solving parts o f the board.  F o r most of the children i n the class, the only question on the first page that continues to cause problems is Question 7 — Draw and print as a number. However, most children can now draw the picture or print the correct number rather than either leaving the question out or coming to me for help as soon as they get to it. D o n , Marge, and four others were the only ones to consistently record both the picture and the number. W h e n I ask, "What does this look like on the number line?" almost all can write the number and then read it, even though they do not record it.  I overheard several children discussing Question 11 — Order please.  There seemed to be a  question about whether or not this counting was by tens. "There has to be a zero after the number," said T o m . " N o , there doesn't," responded D o n . "See, i f you count from ' 2 5 ' to ' 3 5 ' , there's ten," and as he said each number out loud, he put up a finger. " A n d when you count from ' 3 5 ' to ' 4 5 ' , there's ten too." " O h , " said T o m , "then it goes ' 2 5 ' , ' 3 5 ' , ' 4 5 ' , ' 5 5 ' . Right?" " Y u p , " D o n assured him, as they went back to their tables to record their answers. M a n d y decided to record her numbers from largest to smallest, which caused a bit o f a discussion at her table. "That's not right," said Ashley. " Y e s it is. I just did it backwards," replied Mandy. "But, M r s . Bateson said it had to be in order." "I know. It is i n order. I just did it backwards." "But, the ' 2 5 ' has to come first," insisted Ashley. "She didn't say it had to be forwards," persisted Mandy, who was beginning to get annoyed. I stepped i n to assure them that both ways were correct. "I didn't say which way the counting should be recorded  this time. It just has to be in order, biggest to smallest, or smallest to biggest, both ways are right." I think Ashley still thought M a n d y ' s was incorrect, but she wasn't about to argue the point any further.  I made a mental note that I w i l l have to do more backwards counting i n  whole group activities, and specify some largest to smallest counting in future Order please (Question 11) questions.  D a y 3 — T h u r s . , F e b . 19 W e had about 20 minutes between doing the calendar and going to music, and I had decided this would be a good time to look at Question 15 — How many This question was tied to our Canada theme, and asked the children, How many provinces are in Canada?  In the hallway  between our classroom and the next room, there was a large, simple map of Canada at which the children had been looking for several days. So, when I asked how we were going to find the answer to this question, there was a chorus of, "Use the Canada map in the hall!" W e had a quick discussion about the need for being quiet while we worked i n the hall so that we wouldn't disturb the other class too much, and then went to sit i n front o f the map. I had already talked to the teacher i n the classroom outside of which we would be working, and she had assured me that our work would not disturb what she had planned for that period o f time. M o s t o f the children had no idea how to even start to find this answer, so it was a multipurpose lesson. After showing them again where we live, I asked i f someone could remember the name o f our province. "That's easy," said M a n d y , " B . C .  B u t I can't see  anywhere that it says B . C . " "That's cause it stands for British Columbia," answered M i k e . " O h , I see that, right there," replied Mandy, as she pointed to "British C o l u m b i a " on the map. A s I passed my hand over the province, I said, " Y e s , this is the province of British Columbia. W h o can tell me what the name o f our neighbour province i s ? " Several children became busy sounding out the word "Alberta" while I pointed to it, and there was soon a chorus o f voices announcing it's name.  " S o , i f each colour is a province, then there are nine  provinces," said T o m in his 'thinking aloud' voice. " N o , there's 10," said Ashley. I began to  — 95 —  hear a chorus of answers, some correct and others not, so I suggested that we count them together. W h e n we got to the Maritime provinces, I heard comments such as " O h , I missed that one," and "Some of those ones are so little." W e had finally agreed that there were ten provinces, when Douglas asked, " B u t what about those ones at the top? A r e n ' t they part o f Canada too?" A s a result of Douglas's question, we got into a short discussion about the territories. When we went back to record our answers, some of the children simply recorded 10 provinces, while others recorded the territories as w e l l , but i n a variety of ways: D o n printed, "10 provosinsis  2 mor provosinsis"; Marge wrote, "10 and 2"; T o m wrote, "10  provinces and 2 left"; and Pat wrote, "10 probints  2 difrint probints."  Day 4 —Mon., Feb. 23 Today was planned to be the final work session, as well as a sharing session to finish off this board, while I did some non-participant observations.  Everyone seemed to be working well on today's activities, and the actual work session was uneventful, with most children ready for the sharing session more quickly than I had foreseen.  Even the strategies for doing the "Stumper" were remembered from last time! A s  soon as D o n got to it, he commented, "I remember how to do this," and immediately went to get the U n i f i x cubes he needed.  H e had two solutions to the problem within minutes.  I  challenged him to see i f he could find a third solution, but, after several unsuccessful tries, he gave up and asked i f he could be a helper. Since several of the other children were beginning to get off task, I said, " Y e s . " However, M i k e , Marge, Mandy, and Pat persisted, and each found three solutions to the "Stumper".  About 30 minutes before lunch, we gathered into the group centre to share our solutions for questions that could have more than one correct answer. W e started with Question 14 — Which one doesn't belong? Why? The pictures on the board consisted o f three o f the  — 96 —  animals used for the pattern: the polar bear, the goose, and the beaver, and there was also a giraffe. T o my way of thinking, the giraffe 'didn't belong' because it was not a Canadian animal. However, most of the children decided that the goose didn't belong, because it was a bird, because it had wings, or because it could fly. Several children said that the polar bear did not belong because it lived i n snow. None o f the children saw it the way that I had anticipated they would (the giraffe being a non-Canadian animal) until after we had discussed it! I ' l l try a similar problem another time, but make sure that a l l the animals are mammals.  W e then went on to count our 'ice chunks' (white Unifix cubes) in the estimating jar in our usual manner.  A s I was printing the children's estimates as a partial number line on the  board, I heard D o n say to Douglas, " Y o u r number is way too low. 'Member last time she put U n i f i x i n there, there were thirty-something, and the jar was only half full. This time it's almost full so there has to be more than thirty." Douglas just shrugged his shoulders, but changed his 'too l o w ' estimate to " 3 5 " at the first opportunity. Even that was too low though since we ended up counting 51 'ice chunks'.  Question 19  — Measure  was a homework assignment for this board. A picture o f an  analog clock face and a digital clock face were taped into each child's planner (a book that goes home each night and serves as a regular c o m m u n i c a t i o n device between the school/teacher and the parents) on Thursday last week. The parents were to help their child record their family's dinner time. The clock pictures were then removed from the planner and glued onto the response sheets. W e now used this information to make a 'dinner time' graph of the whole class by putting each student's magnetic picture beside the time that s/he eats dinner. W e discovered that the earliest anyone had dinner was five p.m., and the latest was seven p.m., but most families ate at six p.m.. Douglas wondered i f this was true for the whole school, but when I suggested that we could ask everybody and make a whole school graph, he said, " N o , no!" I don't think he was thinking about the task itself, but d i d not like  — 97 —  the fact that he would have to talk to a lot of people that he didn't know. Since no one else seemed interested i n taking on this task, I let it drop.  Our last activity was to share how the "Stumper" worked. "I can do that," said D o n , as he went to get the blocks to show his combinations. H e showed and explained the two that he had found, and I then invited Marge to show and explain a third one. I told the children that didn't have the "Stumper" done that they could leave it out, but that they d i d need to complete everything else. Since it was now lunch time, I told them that we would take some time in the afternoon for those who were not finished to work with a helper who was finished to get the required questions completed. This worked w e l l , and a l l but a few (including Cathy) completed the whole board, with the exception o f the "Stumper".  — 98 —  Diary #4 — March 3 -11,1998 This diary applies to the following board of questions. (* = "Try i t " Challenge Question) 1. Pattern: No pattern was posted on this board. The children were given oral instructions for creating their own pattern in the form of a St. Patrick's Day necklace of macaroni — green coloured tubes and yellow coloured wheels threaded onto green string with a piece of tape. Pattern must be at least three beads long. Draw the pattern onto paper and then make it with "beads". 2. Number line: 27, 28, and 29 printed above the first three number points. 3. W o r d please: 58 4. Number please: fourteen 5. Before and After: 89 printed above the middle number point. 6. Between: 95 and 97 printed above the first and last number points respectively. I. Draw and print as a number: One ten and eight (done using place value stamps  —first  time for using the stamps). 8. A d d : 6 + 2 =, 7 + 7 = , and * (24 + 25 (in columnar form) 9. Subtract: 7 - 5 =, and 14 - 4 = 10. Equation please: A picture of four branches of a tree, having a total of 12 pink blossoms (three flowers on each of the four branches). *Write a story problem. II.  Order please: by value —pictures  of a quarter, nickel, dime, and penny (since it was  not specified, children could order the set from largest to smallest, or smallest to largest). 12. Tally count: Thirty-four marks in tallies of five. 13. Picture please:  9-6  14. W h i c h one doesn't belong? W h y ? : Pictures of four snakes, one of which had part of its tail missing (Some children also noticed that one of them had a different tongue from the others). 15. H o w many: How many teeth has our class lost so far this year? In which month were the most teeth lost? How many children have lost teeth? (Note that this was done using  a 'graph' that was started in September, permanently  displayed in the hallway  outside  the classroom, and updated whenever a child loses a tooth — see Figure 2-9 in Chapter 2). 16. Geometry: Name the shape of a ball. Name three kinds of balls. 17. Number wheel: "+ 5" in the centre. 18. Estimate: Forty-two animal crackers in a jar. 19. Measure: Measure the weight of a potato using Multilink  cubes. Do it again using  something else to weigh the potato. 20. Stumper: Children were given an undersea picture with a central circle emanating eight rays out to eight outside circles which are regularly arranged around the central circle (which means each outside circle is directly across the centre of the figure from another outside circle). This leads to four intersecting lines of three circles each, with the centre one being common to each line. Here are the rules for the puzzle: 1. The sum of each line must equal 12. 2. You may only use the numbers 0,1,2,3,  4,5, 6, 7, and 8.  3. Any number can only be used once.  Hint: Start by listing all the combinations of three numbers that can make 12.  D a y 1 — Tues., M a r c h 3 Spring time is a busy time at our school with lots o f school-wide activities. W i t h many students being away for a few days on holidays close to Spring Break, this board was set up as a review board that was to be completed during the first two weeks o f the month which ended with the spring break holiday.  D u r i n g this time, we were also working on an  integrated unit investigating many ideas related to St. Patrick's D a y and other things considered 'Irish'. A s with most other Wall Math our overall unit/theme of study.  boards, many o f the questions reflected  Question 1 — Pattern  was not posted on the board because it required a series of verbal  instructions and a discussion about the creation of a pattern that involved the stringing of 'macaroni' to make a necklace or bracelet.  I estimated that the Question 1 —  Pattern  activity would require about an hour to complete. Right after recess, I called the children to the group centre i n order to give instructions and discuss the project. Our Wall Math  pattern  this time is another one that you will create. You are to design a pattern for a necklace or a bracelet that uses these two shapes [I held up a sample o f the two different shapes of pasta] and colours of 'beads'.  The 'beads' were natural coloured, wheel-shaped macaroni, which  the kids said were yellow, and barrel-shaped macaroni that I had dyed green with food colouring. Your pattern design must be at least three 'beads' long. " C a n it be four 'beads' long?" interrupted M a n d y .  Sure, it can be as long as you want. But, before you start  stringing the 'beads' you must decide what your pattern will look like and draw the plan on this piece of paper, and I showed them a paper about 3 cm wide and 16 cm long. Each child then went off to create his/her pattern on the paper. The children planned their patterns using a variety o f methods: some used dots of colour, some printed the letters " Y " and " G " using a regular printing pencil to indicate the colour of 'bead' they would use, a few drew the shapes of the 'beads', and a few others used combinations of these methods. D o n and Douglas used a combination of yellow coloured " Y " s and green coloured " G " s ; Cathy used coloured dots; and Mandy, Marge, and M i k e simply recorded " Y " s and " G " s , with a regular pencil. A s they did this, I distributed a length of green string and two small containers of 'beads' to each group o f children.  Once each c h i l d had quickly shared his/her pattern with me, s/he  proceeded to string the 'beads' to make his/her own jewelry. M o s t followed the pattern that they had planned with such care that, i f an error was made, they found and corrected it themselves, leaving me with very little to do except assure them they were doing a great job, encourage them in other ways, and observe individuals or groups as they worked. B y lunch time, everyone had created a piece of jewelry that s/he appeared very proud of, several asking  — 101 —  to show them to our principal, and some of them wore his/hers for several days before it broke or disappeared.  Most patterns were simple three- or four-'beads' designs, such as: y, g, y, g or y, g, g, y, and only a few minor errors had to be found and corrected. D o n , however, created a complicated pattern that was five 'beads' long y, g, g,y, g). When I suggested that this was going to be a hard pattern to follow, his reply was, "I know, but I like it," and quickly resumed his work. A s I watched him stringing the 'beads', he carefully put a pattern length of 'beads' in front of him, then put them on the string. The pattern that I thought would be hard, because of its length and the fact that it appeared to be two patterns in one, was completed without any mistakes. A s shown in Figure 5-3, he then repeated this process until his length of string was full.  Figure 5-3. D o n stringing his St. Patrick's Day 'bead' jewelry pattern. Day 2 — Wed., M a r c h 4 Today's work session was short and relatively uneventful.  I asked the children to start  working on Question 2 — Number Une, and complete as many of the questions on the front side o f the response sheet as possible before 11:45 a.m., at which time we would gather to compare and discuss their answers. I also informed them that while they worked, I was going to ask small groups to come to work with me in the group centre, so that rather than coming to me for any help they required, they would have to help each other i f they got stuck. I  — 102 —  planned to show these small groups how to use some 'dominos' to create number sentences, and then use the answers to play a 'race to the finish' game on a grid o f one hundred numbers.  Between working with each small group, I wandered around the classroom to  answer questions, and to be sure that everyone was working and that no major errors were being made. W h i l e I was circulating after working with the first small group, I overheard Marge say to Mandy, " W h e n you get to here [she pointed at Question 8 — Addition.],  tell  me, cuz we need to do it together. W e need help, big help on i t ! " O n my next round of circulating, both girls had completed the question correctly, but each had recorded it differently.  M o s t children had completed the whole first side o f the response sheet by the time I called the whole group into the group centre to discuss the " T r y It" challenges on Question 8 — Addition,  on Question 10 — Equation  please, and on Question 11 — Order please.  The  other questions a l l required only straight forward, correct or not correct numeric responses, which I had anticipated that all but a very few children could now complete independently, correctly, and quite easily. These have become basically ' d r i l l and practice' components o f each Wall Math board. The Question 8 — Addition.  " T r y It" challenge was not as much of  a challenge as I had anticipated it might be since many o f the children not only had the correct answer, but also could explain to me how they had found their answers. M a n d y said, "I know that ' 2 5 ' plus ' 2 5 ' equals ' 5 0 ' , and ' 2 4 ' is one less than ' 2 5 ' , so the answer is one less than ' 5 0 ' . " She tried to leave her explanation at that, but the other children kept looking at her with such expectant looks that she finally completed her answer, " S o the answer is ' 4 9 ' , " M i k e explained, " ' 2 0 ' plus ' 2 0 ' is ' 4 0 ' , and ' 4 ' plus ' 5 ' is ' 9 ' , so the answer is ' 4 9 ' . " D o n had listened to these explanations very quietly, which, for him, was unusual, then said, "But you're s'posed to add the ' 4 ' and ' 5 ' first, then you add the two 2s." " Y e s , that's the way a mathematician would do it. Can you tell us why we should start with the ones?" I asked, as I pointed to the " 4 " and the " 5 " . " C u z that's the way it's s'posed to be done," was  — 103 —  the only explanation he could give us, even though he agreed that the others had come up with the correct answer.  I made a mental note to have h i m try a question that required  regrouping to see whether or not he could complete it correctly, and then appropriately explain what he had done. Although I knew Marge had the correct answer, at this time I did not ask her to explain how she had found it because I had noticed that she had recorded the equation using a combination of formats and suspected that she had simply copied the answer from Mandy. Having her try to explain her solution method would probably only embarrass her. When I asked her privately to explain how she had arrived at her answer, she shrugged her shoulders and said, "That's what Mandy told me," confirming my suspicion.  The answers recorded for Question 10 — Equation please were quite varied. A s a group, we have done many story problems that require repeated addition, and we have discussed and recorded these along with the multiplication equations that a mathematician would use, but this was the first time this type o f picture had been used on the Wall Math  board. M o s t o f  the children saw it simply as a repeated addition equation, and agreed w i t h M a n d y ' s explanation that, "It's 3 + 3 + 3 + 3 cuz there's three flowers on all the branches." Marge recorded 6 + 6 = 1 2 , and explained, "there's six flowers on both sides of the branch." A few children, including D o n and M i k e , recorded the multiplication equation. "The picture has four groups o f three flowers... see... one [group] on each o f the branches.  So the number  sentence can be a times one," explained D o n with a big smile, as he wrote " 4 X 3 = 12" on the board. " C o u l d you write it the other way around?" I inquired. H e looked at it again, and said " N o ! " " W h y not?" I asked, "cuz it's four groups of three, not three groups o f four." A s he said this, I saw many heads nodding i n agreement, so I guess the informal lessons are getting through to at least some of the children.  Question 11 — Order please  was fairly straight forward, but it could be answered in two  different ways. M o s t children chose to record the largest valued coin first, and correctly  — 104 —  ordered the other three. Douglas, however, recorded the smallest valued coin first, reasoning that, "I always start counting at 'one'." "So why d i d you chose to put the penny first? Y o u start counting the others with 'one' too," I questioned, "cuz the penny is only 'one', and I have to keep counting for the others." " H o w many do you have to count for the nickel?" I asked. " T o five," he replied, "...and for the dime and quarter?" I inquired. " T o 10 for the dime, and 25 for the quarter," he said, then added, " S o this [he pointed to the quarter] is the biggest cuz I had to count the most." I would have liked to carry on this conversation, and brought out a bag o f coins to show everyone that we would indeed have the largest pile o f pennies for the quarter i f we changed each of the coins for pennies, but we were out of time. It doesn't seem to matter how much time is set aside for these discussions, it never seems to be enough.  Day 3 — M o n . , M a r c h 9 The record sheets were given out as soon as we had completed our calendar, and the children were asked to work on the rest o f the board until about 11:40 when we would clean up for lunch and count the cookies in the estimating jar. They were reminded to make and record their estimates as soon as they could. A s I wandered around helping where needed, I noticed that counting tally marks was still causing problems for many o f the children. Although we do this activity daily on the calendar, most children do not circle groups o f ten and then count by tens, but rather they count the tally marks by fives or ones. Although most children do this correctly, Mandy frequently counts the bundles of five marks as i f they were groups o f tens. When I asked her to count the tallies for me, she confldendy pointed to, and counted, each bundle o f five as ten, so that her answer turned out to be 64, rather than the appropriate 34. I asked her to count the tallies in each bundle, and she counted five i n each. " I f you count these tallies by ones, how many would you have?" I asked. She gave me a questioning look and replied, "64." "Count them for me," I said. W i t h a look that asked, ' W h y ? ' she proceeded to count the marks one at a time. B y the time she had counted about half the  — 105 —  marks and had only counted to 15, her facial expression began to show puzzlement and confusion. B y the time she had counted to 20, she looked at me and said, "I guess I did it wrong." "Count the rest," I suggested, "then w e ' l l talk about what you d i d . " M a n d y does not deal well with failure, and by the time she was finished, she appeared quiet annoyed with both herself and the question. "So what did I do wrong?" she demanded. I started by posing the question, " D o you remember how you counted each bundle of tallies?" I received a blank look. " H o w many sticks are in each bundle?" I asked. " F i v e , " she replied. " Y e s , five. Good! So you have to count by...?" "...fives," she finished, and started to count the bundles by fives. "Thirty-four?" she said slowly, with a questioning look. " C a n you tell me how you got 64 the first time?" I asked her. She shrugged her shoulders and shook her head, ' n o ' . " H o w many would you get i f you counted each bundle as a group o f ten?" I questioned. She gave me a look that seemed to say ' W h y would I do that when there are only five in each bundle?' "Count them that way," I requested. She had only counted the first three bundles before she said, "I must have counted this way before." I nodded, 'yes' and left her to correct her paper, which she did not do. However, I didn't worry about this since I knew that she was well aware that it was incorrect, and that she would count it correctly i f requested. I was also sure that she would do this question with more care on the next board. Question 13 — Picture please  now seems to be an easy one for almost everyone, but none o f the children  are making up, or at least not writing down story problems for this as yet. I w i l l have to stress this part of the activity more during the rest of the school year.  There were several small group discussions, in which everyone was eventually involved, about which snake was different i n Question 14 — Which one doesn't belong? Why? M o s t children decided that snake " C " was missing part o f its tail, while others decided that snake " D " had no tongue. A s I listened to these discussions, I had to agree that both answers could be considered correct, and since the children seemed to accept both these answers and their  — 106 —  accompanying explanations, we d i d not need to spend time discussing this question as a whole group.  About 11:40 a.m., I asked the children to finish up what they were working on, check that they had recorded their estimate of how many cookies were i n the jar, and gather i n the group centre to count the animal cookies i n the estimating jar and discuss Question 15 — How many.  Question 15 — How many was one that I thought would be fairly easy, but proved to be confusing to most of the children until we actually talked about it as we looked at the tooth graph which is on a bulletin board in the hallway outside our classroom. Everyone had the first part of the question (How many teeth has our class lost so far this year?)  recorded  correcdy; "I counted the teeth on the graph," said M i k e , when I asked him how he got his answer. However, only a few, and none of the children observed for this study, had recorded the answers to the other two parts of the question (In which month were the most teeth lost? How many children have lost teeth?). However, when we gathered as a whole group i n front of the graph o f the lost teeth, everyone was able to tell me that the most teeth were lost i n the months o f October and A p r i l . Finding how many children had actually lost teeth was a little more difficult, but, when Brian reminded them that, "Some kids have more than one tooth up there. I have two, cuz I've lost two teeth, see." There now seemed to be an understanding o f how to answer this third part o f the question, and I could see heads nodding and hear whispered numbers as the children quietly counted the names on the teeth. "So, I don't count Brian's name twice, right?" questioned Marge, i n her quiet, questioning voice that tells me she is thinking, T think I ' m doing it right, but I need reassurance.' and anyone else who had doubts.  "Right," I assured her,  Some of the names were difficult to read because they  were printed so lightly, so small, and/or, as was the case for a few that had been printed early in the school year when the children were just learning to print, were just plain difficult to  decipher. So, we ended up doing this part of the question as a whole group activity. I asked Trudy to read each name (some with my help) while the group counted, remembering not to count a repeated name. When she read the eleventh name, T o m called out, "I don't have any more fingers, I'm gonna lose track!" Pat, who was sitting beside T o m , and I could see was also wondering what to do next, said with a big 'I k n o w ' smile, " L e t ' s do it together. I ' l l keep up my fingers and you start over for the rest." Both boys seemed satisfied with this solution, and, as Trudy carried on reading the names, I noticed several other pairs using Pat's improvised method. W e completed this task, and went back into the classroom to count cookies.  W e still had lots o f time to complete our cookie estimating activity, so I asked each child to give me their estimate and I recorded them on the chalkboard as part o f a number line. The lowest estimate was 23 from Cathy and her two tablemates, and the highest estimate was 50, with most estimates between 35 and 45. W e then started to count the cookies by twos, placing the pairs on a paper towel in the "ones" section of a counting board. A s we amassed a group o f 10, someone would remind me to move the group to the "tens" section and start over again. Once we had two groups of ten, we stopped to look at our estimates and I asked, "Does anyone wish to change their estimate n o w ? " A l l the children who had estimated below 30 decided to change theirs. "There's way more than ten left," said Susan, who had been working with Cathy. One child who had estimated 38, changed her estimate to 48, but the others chose to leave theirs. W e counted out another group o f ten, and again went through the process o f changing estimates. This time many children chose to either raise or lower their estimate because, as D o n pointed out, "There's about one more group o f ten there." The estimates were now all between 38 and 44. W e counted out one more group o f ten and had two left. "Forty-two," came a chorus of voices. " S o how many do we get?" asked Gerri. I could see heads nodding and fingers wiggling as many began counting the children in the circle. "There are 18 of us," called out Pat. "But, that doesn't tell how many  — 108 —  cookies we get," complained Gerri, who rarely gets any treats. "We get two each and there's some left," said Marge. As she said this, I saw Mike, Don, and several others nodding thenheads in agreement. "There's six left," called Don. "Tell us how you know that we can each have two cookies," I requested of Marge. "I counted the kids by twos, and got to 36," she replied. "But, that's not all the cookies," I said. "I know, but there aren't enough left to give everyone another one." "What will you do with the rest?" asked Gerri. "What would you suggest?" I asked the group. After a moment of thinking, Douglas reminded us that, "There are two kids away today. I think we should save two for each of them." "How many is that?" I asked. "Four," called out the group. "But there are six left. Does that use all the leftovers?" "No," they said. "You can have the other two," said Mandy, and the others all agreed. I distributed the cookies as I dismissed them for lunch, being sure to wrap the ones for the two children who were away in paper towels and put them into their mailboxes so they would get them when they returned to school.  Day 4 — Wed., March 11 Today's work session included an introduction to the use of a balance scale. Some of the children had been 'playing' with it during the last several days, but no one had figured out exactly what to do with it yet. I called the children to gather around the balance so we could discuss what it was and what it could be used for. I then put an eraser into one bucket and asked how many Multilink cubes they thought I needed to put into the other side to make it balance. A chorus of voices called out, "two," so I put two cubes in the bucket on the opposite side. It almost balanced, so several children called out to add one more. This time it balanced. "So, how many Multilink cubes does this eraser weigh?" I asked. "Three," came a chorus. "Let's weigh something heavier," suggested Douglas. "Such as...?" I asked. "Scissors, a pair of scissors," called Don. He got a pair of scissors from the storage bin and put them into a bucket on the balance. "How many Multilink cubes will it take to balance them?" I asked. "Lots!" replied several children. "Then, how many shall we start with?" I  — 109 —  asked.  " T e n , " suggested Marge.  I put ten cubes into the other bucket, i n pairs, as the  children counted by twos. "It didn't even move it!" commented Elaine. "Put i n another ten," suggested D o n . A s I put i n the sixteenth cube, the balance began to sway, and D o n called, " Y o u ' d better put them i n one at a time now." When we were finished, the scissors were balanced with 19 cubes. W e tried a twentieth, but it oyerbalanced the scissors, and everyone agreed that it balanced best with 19. " N o w that you k n o w how to use this for weighing things, lets look at our Wall Math  question. It asks you to weigh this potato with the  M u l t i l i n k cubes, and then with something else, i f you have time." Everyone seemed to understand what was expected, so we separated into working groups and set to work. I hovered around the balance to be sure the first pair of children knew what they were doing, which they did, then went to help where needed.  Mandy and Marge were working on Question 20 — Stumper, which was similar to those on the last two Wall Math  boards, and they seemed to know what they were doing. D o n and  M i k e were also working on Question 20 — Stumper.  However, each boy was working  individually, using a trial and error method of finding the answers, while quietly telling their partner what they were doing. D o n said, " I ' m starting with a five over here, and a three in the middle and a four at the bottom. Y u p , that makes 12. N o w , i f I put a five here, five plus three is eight, that means I need a four here, but I've already used the four, so that won't work," and he rubbed out what he had already done. M i k e was working on another piece of paper that he had drawn the diagram onto, " S o I can keep track o f what I've already tried," he explained. I suggested that they might get the blocks like they had done last time, but they both shook their heads, 'no,' and kept on working. I walked over to see how M a n d y and Marge were doing. Marge was looking at the problem with a very puzzled look, then said to both me and Mandy, "There's something funny here. There are only eight numbers and there are nine circles. It says you can only use each number once. There isn't enough numbers!" Mandy looked at me with a look that seemed to say, ' Y o u tricked us.' " W e l l , i f you can't use  — 110 —  any other numbers what do you think you could use?" I asked. I got a blank stare from both girls. "Nothing," said Mandy. "That's right," I replied, as I continued to receive blank looks. Then I saw a smile start on Marge's face as she said, "Nothing, like in a zero?" " T r y it," I suggested, and walked away.  Several children were looking at the geometric shapes chart and models as they were discussing which of the words they should copy to answer Question 16 — Geometry.  As I  listened, I heard Pat say, "It can't be a circle 'cause it's not flat." " S o it's the 'sufere' [sufear] one." replied T o m , and the three others nodded in agreement.  I stepped in to assure  them they were correct, but also to help correct the pronunciation of the word. Although we have talked about this word many times, some o f the children continue to have difficulty putting the V and ' p h ' sounds together. W e then got into a short discussion on the types of balls they knew about and could record. "I know a ball that isn't a sphere," commented Pat, who, with a struggle, had pronounced the word correctly this time, and was quite proud of himself. The other's looked on with expectation as he explained that a football wasn't round like a sphere, but that it looked more like two cones stuck together. The others looked to me for confirmation as I asked Pat to draw a football on the chalkboard. "What do you think?" I asked. "I think he's right," said Brian. " S o we can't put football as one o f our answers then," he added as the group broke up to return to their desks and record their answers.  I then went back to see how Mandy and Marge were doing with the "Stumper". Marge was almost finished, having discovered that the four needed to go i n the middle, while M a n d y was still struggling to find something else that would work as the centre number. B y the time Marge had finished hers, and checked to be sure they were a l l correct, M a n d y finally admitted that nothing else worked in the middle, and wrote Marge's answers on her sheet as Marge read them to her.  D o n and M i k e were still working on their answers, but were now working only on the diagrams M i k e had drawn on the scrap o f paper.  D o n ' s body language was beginning to  show frustration as he said to me, " W e get almost a l l the numbers done then the rest won't work." Neither o f the boys had yet discovered that they had to use a zero to complete this task, so I decided that I had better give them a hint.  " T e l l me how many numbers the  instructions told you to use? I questioned. " E i g h t , " they answered. " A n d , how many circles are in the problem?" They counted the circles and looked at me with questioning looks. "It can't work. There aren't enough numbers to fill i n a l l the circles and only use them only once. Can we use one of the numbers twice?" asked Don. "But it says we can only use them once," returned M i k e . Both boys continued to ponder the problem, and think out loud. " I f we put the four i n here [Mike put his finger in the centre circle], they all work except the eight, and 8 and 4 is already 12." A s he said this, I saw a smile begin to form on D o n ' s face. "So the other number must be a 0!" he told both M i k e and me. I confirmed his response, and asked why. "cuz zero is nothing," replied D o n . "But, the instructions are wrong then. They say to only use the numbers 1 to 8, and 0 is a number even i f it is nothing." I agreed, but asked, " H o w can you use each number only once when you have nine circles, i f you don't use the zero?" " Y o u can't", they both replied. "That was a tricky question," commented D o n as I moved off to check on a group at the measuring equipment who seemed to be having a problem. A s it turned out, the problem was one of sharing, not measuring, which I helped them solve just in time for us to get cleaned up for lunch. When I made the final check of their record sheets, I discovered that M i k e had not transferred his final "Stumper" answers from the scrap paper, on which he had been working, to his record sheet. Maybe his parents w i l l get him to do it again at home. They should be thoroughly impressed at how quickly he is now able to do it. W e had again run into lunch time, so we did not get a chance to discuss the last questions as a whole group. Although some of the children did not get all the questions completed, we d i d not have time to get back to our Wall Math  before the  holiday. Both Cathy and Douglas were away several days during these two weeks, so both their sheets remain mosdy incomplete.  — 113 —  Diary #5 — March 24 - April 1,1998 This diary applies to the following board of questions. (* = " T r y i t " Challenge Question) 1. Pattern: The pattern consisted of series of four pattern blocks each made up of two blue rhombuses and two green equilateral triangles. Instructions were to make this pattern as long as your arm. Record the number of triangles and rhombuses you used, (the actual shapes rather than the words were used in the instructions). 2. Number line: 47, 48, and 49 printed above the first three number points. 3. W o r d please: 17 4. Number please: forty-six 5. Before and After: 70 printed above the middle number point. 6. Between: 32 and 34 printed above the first and last number points respectively. 7. Draw and print as a number: One ten and five 8. A d d : 5 + 4 = , 6 + 6 = , and * 32 + 46 = (in columnar form). 9. Subtract: 4-0=,  12 - 2 = , and * 25 -13 = (in columnar form).  10. Equation please: A picture of "rocks" rolling  down a hill.  There are a total of 12  "rocks" 4 on the top of the slope, two rolling down, and six sitting at the bottom. 11. Order please: children  There were four rock samples labeled "A", "B", "C", and "D", and  were instructed to order them from heaviest to lightest.  The  appropriate  response would be: C, A, D, B. 12. Tally count: Twenty-seven marks in tallies of five. 13. Picture please: 7 + 2 14. W h i c h one doesn't belong? W h y ? : Samples of four different rocks were provided to students:  granite, obsidian,  sandstone,  and pumice.  There would be a variety of  appropriate responses to this problem: obsidian is "shiny Iglossy, whereas the other three have dull/matte surfaces; pumice is light enough to float, whereas the other three are all dense enough to sink; granite has "speckles", whereas the other three are much more "solid" in colouration;  and sandstone is "weak" enough that pieces of it will come off  — 114 —  when rubbed, whereas the other three are relatively much "harder". Other arguments might also be made. 15. H o w many: 16. Geometry:  How many pebbles can you hold in one hand? Name the shape of a volcano. Name the liquid that comes out of an erupting  volcano. 17. Number wheel: "+  9" in the centre.  18. Estimate:  How many pebbles are in the jar?  19. Measure:  Measure the weight, length, and width or your rockfriend(in science 'class'  the children had each taken on a different rock 'friend' to study). Draw and label your friend. 20. Stumper:  Tara, Michael, Timmy, Nicky and Terri have rock collections. Read the clues  to find out who owns which collection. Children were then given a chart with space for the five names in a "Names" column and the numbers 20, 12, 8, 7, and 5 in a corresponding column labeled "Rocks". They were then given the 'clues' 1. Nicky has 3 quartz crystals and 2 amethyst crystals. 2. Terri has 3 more crystals than Nicky. 3. Timmy has the most rocks. 4. Michael and Terri together have the same number of rocks as Timmy. 5. How big is Tara's collection?  Day 1 — Tues., March 24 Spring break is over and we are just beginning our last major integrated unit of study for this school year: " R o c k s " .  Besides using rocks as a theme, I also needed to include a lot of  review work on this board, as well as set some challenges. A l l the questions were on the wall, and the children were invited to work on completing them in any order they wished, without having a whole group discussion at the beginning. M o s t thought this was a good idea, but I knew that several would need help right from the start. I asked this small group,  that included Cathy, to sit together at one table so it would be easier for me to help them, and so they could listen to each other and me talk through, and about the questions. I told them that they could leave Question 1 — Pattern  until later, when they could complete it with the  help o f a classmate rather than me. O n each o f these children's papers, I wrote the first three numbers of the number line that they were to complete. W e then found the fortys on the number line by reading the red-circled numbers (which count by tens), and counted all the "forty" numbers until we got to 47, which Cathy recognized as the first one printed on her paper, "That's the first number!" she called. "That's right, but where do we start copying?" I asked. " Y o u wrote ' 4 7 ' , ' 4 8 ' , and ' 4 9 ' on our papers, so what comes next?" asked N a n . "The five and o h " replied Cathy, as she wrote it on her paper.  " H o w do we read that  number?" I asked. Cathy shrugged her shoulders as she continued to copy the next three numbers.  " F i f t y , " replied D o n , who was not part of this group, but could hear our  conversation from where he was working at the table beside us. Once this group had copied the numbers, we read them again as a group. I later asked Cathy to read them to me on her own, and, to my delight, she was able to do it. I ' l l have to ask her again tomorrow to see i f she still remembers. W e then talked about where to find the number words, and I wrote the number " 1 7 " on each of their papers beside Question 3 — Word please.  Everyone was able  to read this number, and point to the card with the number and word on it on the picture number line. However, at this point, Cathy decided she needed to go to the bathroom, and when she came back, she had forgotten which word she was to copy and copied the title for Question 17 — Number wheel rather than the word "seventeen". She d i d find the correct number, just not the correct word! W h i l e she was out of the classroom, I had printed the words for Question 4 — Number please, on her paper, but asked her to continue with the group rather than doing this one at this time. I then proceeded to forget that she hadn't done this question until after we had totally finished the board and I had collected their answer sheets!  I wrote the number for Question 5 — Before and After on the three-dot number line for each child in this small group.  "Seventy w i l l be easy to find cuz i t ' l l have a red circle,"  commented Joe, who was in this group because he was a new member o f our class, and this was his first time doing Wall Math . I already suspected that he would be able to do much o f the work on his own once he knew what was expected, but this was a good time for me to assess his numeracy skills, problem solving skills, and work habits by having him participate in this group. The others agreed with h i m , so we read the red circled numbers together. "That's it! That's the one we need, with the seven and the ' o h ' , " said Sam as soon as we had read it. I wish I could train them to say "zero" when they are working with numbers, but the "oh" is used on so many television shows and at other times i n their out-of-school life that this wish is probably a losing proposition. " S o , what comes next?" I asked. "Seventy-one," said Cathy with confidence, as the others all nodded in agreement. "Super," I commented, as they copied it beside the "70". " N o w , what comes before 70?" I asked. They a l l looked at the number line, and Joe responded with the correct answer before the others had time to think about it. They gave him a disgusted look as they copied the " 6 9 " onto their papers, and then we read the section of number line together.  Having Joe as part of this group was not working, so I suggested that he go to work beside Tom or D o n and ask them for help i f he needed it. H e seemed more than happy to comply. I kept my eye on him as he worked and fidgeted beside T o m , asking very few questions, but watching and commenting on what T o m was doing. When I checked his paper later, the work he had done was correct, but it was very messy. Some of it so messy that I had to have him tell me what some o f his numbers and words were. H e seemed more disturbed with the fact that I couldn't read his work than with the fact that his work was messy. In striving to understand his thinking on this, I rationalized that the purpose of writing something down is so that others can read it, not to create a work o f art; writing words and numbers is communication, and i f someone else cannot read or understand what another has written,  communication has not been accomplished.  In his case, his purpose had been  communication, not to make something "pretty". What he was trying to do (communicate), was being problematic, so he was upset about that fact that he was not achieving his goal (me being able to read and understand his writing). Since his goal was not to create a work o f art, there would be no reason, in his mind, to be upset about the fact that I considered his work to be "messy"; to him, this was not problematic.  Having sent Joe off to work somewhere else, my little group became more cooperative and verbal. What a difference it makes putting someone new with a group, the rest o f whom are unsure of their individual skills, but who are comfortable working together. A s we reviewed what was expected in Question 6 — Between, I wrote the two numbers on the section of number line. " H o w do we read the first number?" I asked. After a brief pause, Gerri replied in a very quiet voice, "Thirty-two?" " G o o d , three groups of ten is thirty," I said, as I pointed to, and read the red-circled numbers to thirty on the number line, " T O ' , ' 2 0 ' , '30'...and, how do we read the last number?" I asked, as I pointed to it on N a n ' s paper. "Thirty-four," responded Cathy with confidence. " G o o d , so i f we have ' 3 2 ' and ' 3 4 ' , what comes between them?"  "Three...I mean thirty-three," answered N a n . Cathy was nodding her head i n  agreement, and added, "cuz, we count 'one', 'two', 'three', 'four'." " A n d what does that tell you?" I asked. She looked puzzled for a moment, then said, " W e have to count ' 3 1 ' , ' 3 2 ' , ' 3 3 ' , ' 3 4 ' , " but she could explain no further. The children copied the " 3 3 " onto the section o f number line on their individual response sheets. A t this point, I knew the group could carry on with Question 8 — Add and Question 9 — Subtract  on their own, and I needed to have a  look at how the rest of the class was doing. I asked my little group to leave out Question 7 — Draw and print as a number, and do Questions 8 — A d d i t i o n , and 9 —  Subtraction.  without my help. When I returned after walking around the class and finding nobody else in need of assistance, all of them had finished the requested work correctly, except Cathy. When I looked at her sheet, I noticed that she had printed "42", rather than " 3 3 " as her  — 118 —  answer for Question 6 — Between, and that, although she had completed Question 8 — correctly, she had also done the equations in Question 9 — Subtract  Add  as addition questions.  However, with the exception of a backwards fourteen, which for her, was still typical, they were correct! However, corrections would have to wait for another day since it was now lunch time.  A s I had wandered, checking on the rest of the group, I noticed that most o f the children had already completed the first side of the record sheet, with the exception o f Question 11 — Order please,  and were beginning to work on the second side. Once we were cleaned up for  lunch, I asked why so many of them had left out Question 11 — Order please,  " c u z we  didn't know how to do it, and you were working with those other kids," came D o n ' s reply. I made a mental note to start the next day's work with a discussion o f this question.  D a y 2 — W e d . , M a r c h 25 Today's work time began with a whole group discussion about Question 11 — Order please. W e read the problem together, and I asked someone to explain, i n their o w n words, what they were to do. " W e need to tell which rock is the heaviest, and which is the lightest," said D o n , "That's easy, cuz the pumice is the lightest and the black one is the heaviest. But I can't tell about the other two." I asked him to explain how he knew about the heaviest and the lightest. " W e just held them i n our hands," he replied. "What did you do with them i n your hands?" I asked.  I got a puzzled look, then a tentative answer/question from M i k e , " W e weighed  them?" " Y o u sure d i d , " I assured him. "But, how do we do the middle ones? They feel the same," came the frustrated voice of Mandy. After a few moments o f thinking time, and still receiving no answer, I asked " H o w did you weigh the potato before Spring B r e a k ? " " W e used that balance thing," answered Douglas, "but, we don't want to know how many cubes it weighs, we need to know which one weighs the most." A g a i n I waited for an answer. " W e could find out how many cubes each rock weighs," said Marge. "Then what would you need  — 119 —  to do?" I asked. "See which rock used the most cubes. That one would be the heaviest," she replied, i n a quiet voice. "I know," called an excited D o n , "then we see which rock had the next most, and the next most cubes, and that's the order. W e don't need to do the pumice cuz we already know it's the lightest." "Great thinking!" I said, " That's one way to do it. Can anyone think o f a way o f comparing their weights without using the cubes?"  Again I  received silence and blank looks. " C a n we just put the rocks i n the buckets?" asked Mandy. A s I nodded my head, ' Y e s . ' , she came over to the table, picked up Rocks A and B and put one into each o f the buckets on the balance. "So, this one (she pointed to the bucket that was touching the table) is heavier than that one (she pointed to the other bucket)," she explained. She seemed unsure o f what to do next, so I prompted, " A r e any of the other rocks lighter than R o c k B ? " which was the rock i n the bucket i n the air. " N o , " answered Pat, "That's the pumice, and it's the lightest one." Mandy gave him a nasty look for answering a question she felt was directed only to her. " S o , let's put Rock B here on the chalk ledge and remember that it is the lightest," I said, as I moved the rock. " N o w , you told me that Rock C was the heaviest one. (I held up the black rock that D o n had earlier said was the heaviest) H o w can we prove i t ? " "Put it in one o f the buckets and see i f it stays down when you put the other two i n , " answered M i k e . I invited him to come and show us. H e put Rocks A and D into the other bin, one at a time, and everyone agreed that Rock C was indeed the heaviest. I then put Rock C on the chalk ledge to the left of Rock B , and left a large space for the other two rocks to be placed. " S o now all we have to do is find out which o f these two rocks is the heaviest and w e ' l l be done," commented D o n . I assured him he was correct, told them that I thought they could finish the rest on their own, and sent them off to work.  M a n y of the children were beginning to work on Question 15 — How many, and were busily counting their "handful of pebbles", which they had brought i n from the playground. "I can't count a l l these," came a plaintive cry from Douglas, as he put his pebbles down on his table. When I went to see what his problem was, I discovered that he had brought in all the pebbles  he could carry in his two hands cupped together!  After a quiet discussion about what a  "handful of pebbles" meant, he took the extras back to the playground, and began counting the ones left on his table. H e started counting them by ones, but got lost after counting about 40 of them. However, by this time several others had already discovered that putting the pebbles into groups o f ten was making the counting much easier. "See, then a l l we need to do is count them by tens," said M i k e , who was working side by side with D o n , but each was working on his own sets of pebbles. I suggested to Douglas that he go and watch how M i k e and D o n were counting theirs. H e watched for a few minutes, then went back to his table and began putting his pebbles into groups of ten. W h e n he was finished, I stood behind him while he counted his groups o f ten. "I've got seventeen groups, and one left over," he said proudly. "Great! N o w how do you record that number?" I asked, knowing that he can do this type o f activity at the calendar, and hoping that he would see the connection. H e did! "I can't write seventeen tens cuz that's more than ten tens," he said, and looked at me for some help. " G o o d for you," I encouraged him, "So, how many of these groups do you need to put together to make a hundred?"  " T e n , " he answered quickly, and he pushed ten groups  together onto the left side of his paper. " C a n you print a number n o w ? " I asked. H e looked at his paper, and, although he nodded in the affirmative, he seemed unsure. " H o w many groups o f one hundred do you have?" I prompted. " O n e , " he answered. " A n d how many groups of ten do you have?" "Seven..." he replied with more confidence and the beginnings of a smile, "...and one left over," he added, as he picked up his pencil and wrote the numbers " 1 " , " 7 " , and " 1 " on his table top. The children are encouraged to use their arborite table tops as 'thinking boards' because they are easily cleaned off, and I have found that they are more willing to put answers that may be incorrect here, where they know they w i l l be erased, than on a piece o f paper. A s he printed these numbers, he was mumbling to himself, "One group o f a hundred, and seven groups of ten, and one more makes one hundred and seventy one. I picked up one hundred and seventy one rocks," he announced, with a proud smile.  "Great job," I assured him, and reminded him that he now needed to record his answer on his record sheet.  Marge seemed to find Question 15 — How many a fairly easy task to complete. When I looked at her working paper, she had put all her groups o f ten pebbles into neat rows, circled each group, and had printed the counting by tens below each circle. When I asked her to read the counting for me, she confidently counted her one hundred thirty-two pebbles, with no mistakes.  Mandy and M i k e had both picked up double handfuls o f pebbles, but after a short private discussion with each of them, they decided they still wanted to count what they had brought in rather than single handfuls. I watched with interest to see how they would accomplish their task. Both, although sitting at different tables, used similar strategies. Each counted ten groups o f ten (100), then pushed the group o f 100 together i n one corner o f their work paper. Each of them then repeated this until they had as many groups o f one hundred as they could make, with several groups o f ten and some single pebbles left. W h e n they were finished, they were both able to read their numbers to me, and show me the pebbles that they had counted. "I have three groups of a hundred, eight groups o f ten, and nine more," said Mandy, as she pointed to the appropriate groupings o f pebbles.  "That means I picked up three  hundred and eighty-nine rocks," she informed me proudly, " D i d anyone else pick up that many?" "I don't know, w e ' l l have to see," I told her, and decided that we could do a comparison o f large numbers as part o f tomorrow's mathematics lesson. H e r "handful" turned out to be the largest group counted by an individual, but it was obvious that hers was still much smaller than the group of rocks i n the estimating jar.  Cathy started counting her small handful o f pebbles, but very quickly tired o f the activity, and simply put what she thought looked like ten pebbles into each of ten groups on her paper.  — 122 —  When I asked her i f there were ten pebbles in each group, she shrugged her shoulders and said, "I just guessed," then added with a smile, "but I have ten groups. That means I have a hundred rocks." " A r e there really a hundred pebbles there?" I asked with a frown. "I don't know," she admitted. " I f there were ten pebbles in each group would you have a hundred?" I asked. She nodded her head, 'yes', and started to count the pebbles in each group, adding or removing, as needed, so that each group had ten pebbles. When she was finished, I helped her count her one hundred and fourteen pebbles, by tens.  W e had about five minutes left before lunch time, and I hoped this would be enough time to have a quick discussion about Question 14 — Which one doesn't belong? Why?, so I asked the children to gather in the group centre, and moved the question sheet to the group centre chalkboard. "The pumice doesn't belong, cuz it floats," called Douglas. "I did that too, but because it has holes," said D o n . "I said the sandstone didn't belong, cuz it's the biggest one," said M i k e . " M e too," added T o m , "cuz, it's not an igneous rock." W o w , our science lessons are showing up! Mandy called out, "I said the obsidian didn't belong cuz it looks like glass and it's a funny shape."  B y this time, there was so much discussion going on about the  various reasons for the different decisions that the children had made that no one was listening to anyone else, so I called the children to order, then dismissed them for lunch since everyone seemed satisfied that there were many acceptable answers to this question.  A s I checked through the children's papers at noon, I noticed that most o f them had printed the equation "12 - 8 = 4 " as an answer for Question 10 — Equation please,  and those that  had printed a story problem had written that eight rocks had rolled down the h i l l . However, there was no group of eight rocks in the picture. I made a note to discuss this question with the children during our next session to discover where the ' 8 ' came from.  — 123 —  Day 3 — Tues., M a r c h 31 I had hoped that we would complete this board today, but because of an interruption and two lengthy, but productive discussions, we w i l l need one more short session to complete a l l the questions.  A s the record sheets were handed out, I talked to the children about my problem with their answer for Question 10 — Equation please, and challenged them to write an equation that showed where the "eight" in the equation "12 - 8 = 4 " came from.  D o n and M i k e  immediately went to look at the picture, and they decided that they had got the "eight" by adding together the two groups that had rolled off the hilltop. They went back to their papers and wrote "12 - 2 + 6 = 4". When I asked them to explain this equation to me D o n said, "First we added the two and the six to get eight, then we took the eight away from the twelve and there are four left. See? There are four rocks left on top o f the h i l l . " When I asked how their number sentence told me this, they looked a little puzzled. I read the sentence, and did the arithmetic in the order presented. "When I read your equation it says, T 2 - 2 ' which equals 10, then 10 + 6 = 16." "But that's not right," said Don, as M i k e shook his head, 'no', in agreement with D o n ' s statement, "The answer is four...there's only four rocks left." After a moment o f looking at the equation, D o n began to think out loud. "What would happen i f we add the two and the six first, (he wrote '2 + 6' on his paper) and then d i d the take away twelve? (he wrote '-12 =')" Both boys puzzled over this equation for a second or so, then shook their heads as M i k e stated, "That doesn't work, two plus six is only eight, and you can't take twelve away from eight." "The other way has to be right then," said Don, as both looked at me for assurance and help. I assured them that their first equation was the correct one, but they needed to add some brackets. I had demonstrated the use o f brackets on several previous occasions when we had been working on multi-step story problems, and wondered i f they would remember. "Think about where a mathematician might put a set of brackets to show what part of the equation was to be done first," I prompted. A n 'ah-ha' look soon  — 124 —  replaced the stumped look from D o n , as he picked up his pencil and put a bracket in front of the " 2 " i n his equation. M i k e still looked doubtful, and said, "I thought there was s'posed to be two of those." I nodded, 'yes' in encouragement  Immediately, D o n exclaimed, " O h yea,  there's s'posed to be one at the end too!" and he added the second bracket behind the " 6 " in the equation. Both boys now had an appropriate equation on their papers, and could explain how it worked well enough to share it with the rest of the class. A s I wandered around, I noticed that many children now had the correct equation, but without the brackets. Elaine however, had written "12 - 2 = 10". She explained, "There's twelve rocks on the hill, two are rolling, so that means that there's ten not moving." When I asked where the ten came from, she replied "Four on top, and six at the bottom; that makes ten." I asked her i f she could write this as part o f her equation so that the others would know how she got the ten. I also asked her to write the story problem for her equation.  She shrugged her shoulders, and  picked up her pencil to go back to work just as the Principal called for an earthquake drill!  When we came back inside, I gave them about ten more minutes to work before we gathered in the group centre to discuss our solutions to the problem about the rocks rolling on the hill. A s I wandered around the classroom during this time, I listened to the table group that D o n and M i k e were working with discussing Question 16 — Geometry. children to Draw and name the shape of volcano.  The question asked the  A s the boys talked, they decided that the  answer was "triangle". Although we had not talked about a cone previous to this, we have discussed the differences between other three-dimensional shapes and two-dimensional shapes, such as a circle versus a sphere, and a square versus a cube, and have worked with both the three-dimensional geometric solids and the two-dimensional logic blocks on several occasions. I wondered i f the concept of a three-dimensional object would transfer to this problem.  A s I listened to the discussion, I heard M i k e say, " B u t a triangle's flat, and a  mountain's round. I think it looks more like that 'cone' (he pointed to the geometric shapes chart), but with the pointy end cut off." A puzzled look on the faces of his table mates  — 125 —  indicated to me that the others were unsure of what a "cone" was, so I went over to take part in the discussion. After a brief discussion, a look at the model of a cone, and another look at the geometric shapes chart, the boys seemed satisfied that the shape of a volcanic mountain most closely resembled that of a "cone". Several other children listened as we discussed this problem, and they went back to their papers to change or add this newly-discovered response. A s I continued to watch, and to help where needed, I noticed that T o m printed the word "circle" for his answer to this question. H i s response to m y request for an explanation was, " W h e n you look into a volcano it looks like a circle." Further questioning resulted i n him telling me that, "That's what it looks like when you fly over top o f it; I saw it on T V . The people were in a helicopter, and they were looking into the volcano and it looked like a circle full of red, bubbly stuff." W h o would I be to argue with such experience?  A t the end o f the work session, we gathered into the group centre to talk about Questions 10 — Equation please and 16 — Geometry, and, i f time permitted, the "Stumper". Most o f the children had not heard the discussion about the geometry question because they had been busy working on some other section, so the whole process happened again. A s a whole group, we now looked at the geometric solids and talked about how the "cone" was "like a sphere, it's not flat," according to D o n . "But,..." came an unsure voice from the middle o f the group, "when you look at a mountain, it looks like a triangle." " Y e a , " said Pat, who had been part of the earlier discussion, "but, you know it's not flat, so it has to be a cone." I continued to see some unconvinced faces i n the group, so I put models o f a cone and a triangle side by side on the chalkboard ledge for the children to look at. A t this point, I decided that we needed to take an imaginary flight over the mountains. I put the cone and the triangle (standing upright) in the centre of the circle we had formed, and asked the children to stand and look down on them. Then, I said my magic words, cast my magic spell, and we were ' f l y i n g ' above the mountains on the other side of the water! The children began talking about what they were 'seeing' (very few had ever been on an airplane but most have good  — 126 —  imaginations and have watched a lot of television), and, as I watched and listened, I saw most of the puzzled looks disappear.  I took the ' s p e l l ' off, and we continued our discussion.  Everyone now seemed to understand that a volcano was a cone-shaped mountain "with the top cut o f f . When I asked T o m to share his circle answer and explanation with the class, everyone seemed to understand and accept it as an alternate, but correct response. However, no one else recorded that answer. I then asked i f they could name anything else that was a cone shape. " I f you turn it over, like this (the child took the cone out of my hand and gave it back to me with the point down), it looks like an ice-cream cone." " O h ! . . . " came another voice, "and i f you take the half ball and put it on top you've even got ice cream i n the icecream cone!" A s this comment was made, the child did what she was saying. The result o f this comment and demonstration was a seemingly ' o f f task', short discussion that led to a homework assignment to make a graph about the favorite flavours o f ice-cream of both themselves and their family members. The children could think of only one more example o f a cone shaped item. D o n said that "a Christmas tree looks like a cone with the pointed end up."  L o o k i n g around to find other examples o f cone shaped items became another  homework assignment. W e gathered in the group centre the next morning to share the results of our homework search. The suggestions that came back were mostly what many children called "sort-of cones". " W e couldn't find anything except an ice-cream cone that had a pointed end," commented M a n d y , as many nodding heads indicated that they too had suffered with this problem. Examples included "flowers like the ones on my Nana's l i l y " , "a bunch o f grapes looks like a bumpy cone laying down", "a candle and a carrot". " M y bird's beak...and it even has a pointy end!" shared Cathy. " A witch's hat has a pointy cone too", called out an excited Gerri, as Cathy's suggestion brought this idea to her. "The funnel my dad uses to put o i l in his car with"; " M y M u m ' s pretty bell"; and Marge brought a sample of, "these b i g thread cones that my M u m uses on her sewing machine." D u r i n g the short discussion that followed, everyone agreed that these ' s o r t - o f cones c o u l d indeed be considered cones because, as Mandy pointed out and demonstrated by drawing a picture on  — 127 —  the board, " I f you draw the lines of the bell longer they come together i n a point just like that one," and she pointed to the model.  H a v i n g completed our discussion about the geometry question, we now needed to talk about the equations recorded for Question 10 — Equation please.  I wanted them to have more  exposure to multi-step problems, and I also wanted M i k e and D o n to explain their thinking to the whole group. T o start off this discussion, I asked M i k e to write their equation on the chalkboard. Then I asked both boys to share their story problem and explain their equation to the group .  They explained their work and answered their classmates' questions with  confidence. The explanation o f the brackets was very simply stated, " Y o u put the brackets around what you do first." I then asked Elaine to share her equation. She wrote, 12 - 2 = 10, on the board and explained her thinking to the group. Although she had not gone back after the earthquake drill to change her equation as requested, I wanted her way o f thinking about this picture shared with the others. Some o f the children looked skeptical, but most seemed to understand and accept what she had done as another way o f looking at the problem. She had not written a story problem for her equation and because she does not think well under pressure I asked the group to suggest a story for the equation.  M a n d y ' s hand shot up.  "That's easy. There were twelve rocks on the hill. T w o o f the top rocks are rolling off. H o w many rocks are not rolling?" "I've got another one," said K y l e . "There were six rocks on top of a hill and six rocks at the bottom. There was an earthquake and two o f the top rocks rolled off. H o w many rocks didn't move?" Several children chuckled as they realized that our earthquake drill had probably sparked K y l e ' s story.  " B u t that's not the same number  sentence," commented D o n . "The sentence for that story is six plus six take away two," he said, and he wrote it on the board complete with brackets around the '6 + 6'.  Everyone  agreed that he was right, and K y l e ' s face lit up with delight as he realized that he had found yet a different equation and story for the picture. Marge then said, in a tentative voice, that she thought she had still another equation. She wrote "6 - 2 + 6 = 10" on the board. A s she  wrote she explained, "There were six rocks on top of the hill. T w o are rolling off. A n d there are six at the bottom." " A n d what is your question?" I asked, uncertain o f what she was trying to do. She thought a moment, shrugged her shoulders, and said with a question, " H o w many didn't move?" " O K , " I said, then asked i f this two-step equation needed brackets. She looked toward D o n and M i k e who were nodding i n the affirmative and had wide grins on their faces. Marge handed the chalk to M i k e with a shrug o f her shoulders and he put the brackets around the '6 - 2 ' . W e now had four equations and story problems for this picture, so we left it, went for a run to the far end of the playground and back, then began to work on Question 20 — Stumper. This logic problem was fairly straight forward and d i d not take long to discuss because the children who are interested i n these types o f problems had already solved it, and they just needed to explain their reasoning to the group. "That's an easy one," commented M a n d y when I put the problem on the chalkboard i n front o f the group. " Y e a , it tells you the answers," D o n agreed.  However, many of the others were  looking at it with blank faces. W e quickly discussed what had to be done, then read each clue, d i d the little bits of arithmetic required, and wrote the names beside the numbers that tell how many rocks each child has. " Y o u ' r e right," said T o m , "that one was easy." I asked the children to return to their seats, copy the names on the correct lines, and then go for lunch. W e would have to count the pebbles in the estimating jar tomorrow.  D a y 4 — W e d . , April 1 Today we only had one question left to work on, Question 18 — Estimate and Count. I had filled the jar half full of pebbles so that the children would be forced to count a large number of objects. "That's a lot o f rocks!" said Douglas, as I dumped the pebbles onto a towel on the floor i n the middle of the group centre. "It's way more than I counted for my handful, and I counted 389," commented Mandy. I had noticed that the estimates the children had made were no where near the number in the jar, so I decided not to ask them to share these numbers today, but to simply count the pebbles. " H o w can we count so many?" asked Marge. I asked  for suggestions. "There's way more than a hundred. So can we make groups of a hundred?" asked D o n . I assured him this would be a good way to start. The children quickly put themselves into groups o f two or three and began counting out groups o f ten pebbles, and then putting them into groups of one hundred. Within about five minutes there were calls of, " W e can't make any more groups of a hundred," from several o f the groups. "Sure we can," said D o n as he looked around him. " B u t we need to put some o f our groups of ten together. W e need three more tens to make another group o f a hundred." A s he said this he leaned over to the group beside him and helped himself to three groups o f ten.  Although his  neighbours appeared somewhat distressed by his action, when they realized what he was doing they d i d not protest. The rest of the children watched and began joining their groups together. Soon we had seven groups o f one hundred pebbles, and two groups of ten pebbles spread on the floor in the centre of our circle. " S o , how many pebbles are here?" I asked as the children looked at their work.  "Seven groups of a hundred, and twenty," answered  several children i n chorus. " W h o would like to try to print this number on the board?" I asked. Although several hands went up, I asked D o n because he was the only one I was sure could do it correctly. Once it was written on the chalkboard, and we had read it together, I sent the children back to their tables to copy it onto their papers, or to finish any questions that were not completed, and then to return the answer sheet to me to be three hole punched so they could put it into their binder.  — 130 —  CHAPTER 6 SUMMARY AND CONCLUSIONS  The major purpose o f any thesis is to contribute to a body of knowledge and/or pedagogy by allowing the reader to learn something new or develop/expand their own knowledge and/or beliefs.  This being the case, for many readers o f this thesis, this chapter may be either  redundant or totally unnecessary, since reading " m y experiences i n narrative form provides an opportunity for the reader to experience vicariously and deliberate with me about the mathematical, pedagogical, and educational issues and problems" (Nichol, 1997, p. 70) of teaching and learning through Wall Math. M a n y researchers/authors have argued the case that the narratives or stories, written such as those i n Chapter 5 of this thesis have been written, stand alone i n this style of research; the stories become the normal "conclusions" of other forms o f research. Statements such as: "stories are a powerful way of depicting the complex, unpredictable, and dilemma oriented nature o f learning" ( N i c h o l , 1997, p. 70); stories or narratives are "concerned with the explication of human intentions in the context of action" (Bruner, 1985, p. 100); "narrative ... is an appropriate form for reconstructing pedagogical experiences and for making them accessible to reflection" (Nichol, 1997, p. 70); stories are "a distinctive mode o f explanation characterized by an intrinsic multiplicity o f meaning" (Carter, 1993, p. 6), "stories function as arguments i n which we learn something essentially human" (Connelly & Clandinin, 1990, p.8), and "story represents a way of knowing and thinking that is particularly suited to explicating the issues with which we deal" (Carter, 1993, p. 6), all direct one toward a conclusion that the stories themselves become the thesis. Indeed, even renowned and respected researchers such as E l l i o t Eisner (1996) are now arguing the case that stories or narratives themselves can legitimately become the sole 'report' o f educational research.  If one is to conclude that the stories are essentially the  conclusions, or that the reader comes to his or her o w n conclusions by vicariously  — 131 —  participating in the teaching/learning process through the stories told, then the conclusions have already been presented.  However, some conclusions that I have drawn from the totality of the five stories may assist some readers in beginning their own interpretations of the stories. Some interpretations have to do with how children develop and learn, some may have to do with how teachers can be facilitating guides for students, and others may have to do with the programme itself.  This study set out to show how the Wall Math  programme, as used i n my classroom, assists  my students to attain the individual ILOs o f the provincial Grade 1 mathematics curriculum. Although I have not discussed these individual I L O s because there are so many of them, anyone reading the diaries in Chapter 5, and familiar with the Grade 1 curriculum, w i l l realize that many of the I L O s have been addressed through the Wall Math activities, and w i l l also see the potential for including others i n other Wall Math  questions. In addition, as  discussed i n Chapter 2, an unintended but important result o f this study was the finding that by matching Wall Math to the goals of the B C M E ' s I R P , and applying it the way I do i n the classroom, the programme also met the N C T M  Standards.  W h i l e listening to the tape recordings, looking at the children's response sheets, and reflecting and writing about the individual episodes that occurred during each Wall Math session, I began to realize that the use o f the structured framework o f the Wall Math  system  was helping me, the teacher, help the children i n my class attain a better understanding o f more o f the I L O s and overall goals of the provincial curriculum than they would i f I were using a traditional type of programme. The use o f a traditional programme usually means chunking the mathematics into separate units o f study, and, once that unit o f study is completed, unless a conscious effort is made to review these units occasionally, the children probably do not use much of the specific knowledge or skills of that unit until the following  — 132 —  year. This often results in the children not making the connections between their schoollearned mathematics skills and the mathematics they need for real life. However, because the set categories on the Wall Math  board, as reflected i n the specific questions, cover a l l the  major, prescribed outcomes o f the curriculum as presented i n the I R P , (number, number operations, patterns, measurement, 3-D objects and 2 - D shapes, transformations, data analysis, and chance and uncertainty) almost every one is reviewed in some form each time a new set o f questions is posted. Because many of the questions are set as problems, because I give the children time and encouragement to discover their own solutions, and because I encourage them to talk to each other, and me, about what they are doing, they are learning not only how to problem solve, but also to communicate, to make connections between mathematics strands and real life, to reason mathematically, and to develop a positive, T can do it' attitude toward mathematics. Thus the overall goals o f both the I R P and the N C T M are being taught and learned. The diaries show this growth as they recreate the small and whole group discussions about the variety of problems set for the children to solve.  A s I read the research, wrote the diaries, and reflected on the way that I use Wall Math in my classroom, I began to realize that, although I had no previous knowledge of Cognitively G u i d e d Instruction ( C G I ) , m y beliefs and classroom practice reflected many o f the characteristics presented by Peterson, Fennema, Carpenter, and their colleagues as typical o f a C G I classroom. I know that children come to school with many mathematical skills. I know that it is my job to create a learning environment i n which these children feel secure and comfortable enough to take the risks needed to construct the conventional mathematics processes and algorithms within their personal understandings and knowledge. In order to create this environment and assist the children's learning, I am constantly on the watch for situations that can be posed as problems for the children to solve. I encourage my children to talk about their thinking, I listen to this talk, and I use it as a guide for determining what to do next. I use word problems as a means o f teaching computation skills, allowing, and even  — 133 —  encouraging lots of talk, use of manipulatives, and demonstrations of a variety of methods for finding solutions before the children are asked to record their solution(s) on paper. I also use word problems as a means of encouraging the children to improve their creative problem solving skills, the problems i n these instances being o f Schoenfeld's (1992) "problematic" (p. 338) type.  The graphic model (see Figure 3-1 on p. 33), created to show how C G I works, equally demonstrates how Wall Math is used in my classroom. However, for the application o f this model to my classroom, I would add to the model the very important processes o f carefully observing and, most importantly,  listening to the students i n order that appropriate  instructional decisions can be made; listening to students is critical. M y knowledge and beliefs about teaching and learning, and the needs and interests of my particular children, as gleaned from watching them and listening to them, are combined with the requirements o f the curriculum to decide what to teach, when to teach it, and how to teach it.  Once  instruction has begun, the talk and behaviour demonstrated by the children assists me in deciding what to do next in order to ensure that the skill or knowledge is being added i n some way to each child's learning. I have found that Wall Math  provides not only a means o f  frequent practice of the skills and use o f the knowledge learned i n regular mathematics lessons, but also it provides a means of continuing the dialogue of problem solving at a variety of levels that the originators of C G I suggest are important to children's learning.  The fact that the C G I model of teaching and learning has been shown to be successful with even a group of "disadvantaged minorities" (Villasenor & Kepner, 1993, p. 62) suggests to me that it can be used to advance the mathematical understanding and mathematics learning of all children.  The study set out to explore the ways i n which the programme facilitates the understanding and retention of learned mathematics skills by Grade 1 children. The diaries i n Chapter 5 show a variety o f ways the programme facilitates the understanding o f mathematics concepts by assisting the children to construct their own connections between classroom mathematics and other areas o f their lives. Question 19 —Measure, i n Diary #3 (See Chapter 5), is an example of a home-related task. Question, 16 — Geometry  frequently asks the children to  name the shape of something that they might see outside. Diary #1 contains a description of the results o f posing the seemingly straight foreword challenge of Name the shape of a traffic light. Diaries #3 and #5 present several examples of questions that relate to the current units of study i n Social Studies and Science respectively. Diary #4 has examples of questions that relate to a special day (St. Patrick's Day). Although a typical Wall Math  board contains no  explicit descriptions of literature or holiday related activities, the questions frequently reflect them. A t the time that we read and studied the story of The Three Little Pigs, the pattern used i n Question 1 of the Wall Math board used pictures of the three types o f houses built by the pigs, and the geometry question for the Christmas board asked the children to Name the shape of a Christmas tree.  The diaries also show that by providing frequent, practical use of mathematics over an extended time, the children begin to use their classroom learned knowledge and skills more easily in every-day situations. While listening to the children's conversations with each other and me, I often heard examples of this transfer of knowledge. W h i l e admiring the new street banners as we walked to the park one day, I heard someone comment, " H e y , look, it's a pattern! Banner, no banner, banner, no banner..." The banners had been put up on every other lamp post and this student was viewing the situation as a 'pattern'. Another example occurred on Sports D a y when, at their first station, one of my children counted the members of his obviously smaller team and those o f the other teams, and then informed me that, "It's not fair! W e have two less kids on our team." When the traditional solution to this problem  — 135 —  was described to him (have two children on his team go twice each), his smile and, " O K ! " o f acceptance let me know he understood.  The study set out to submit Wall Math  to a systematic inquiry into its outcomes with  students when used with Grade 1 children, and, again to stimulate the reader to draw his or her own interpretations from the diaries, the following provide some examples o f how children have or have not at least partly achieved some of the intended curricular outcomes.  The diaries show the growth in persistence of the children when solving the problems. O f the six children studied, two were very persistent right from the beginning; two are beginning to be more persistent; one, who chose not to do the problems without help at the start of the study period, is beginning to participate in these activities as part o f a small group; and one continues to be a non-participant.  The diaries and the children's own writing show that the children are developing a positive attitude toward mathematics. Frequently our Wall Math  discussions ran overtime into the  lunch break, but the children never once complained or reminded me that our time was up. Most of the children were totally unafraid to try to solve the problems once they understood the question. Evidence o f this can be found in Diary #1 as Douglas struggled to discover how many wheels were on the cars i n the parking lot. The quotations from the children's own writing about their feelings toward Wall Math  that I had them do at the end o f the year  and are found i n the descriptions of the students in Chapter 4 show that all but one o f the six students used i n this study, and many of the other students i n the class, indicated that they considered at least most of Wall Math to be easy and fun.  The diaries show the growth i n the children's number operation skills. A t the beginning o f the study, most o f the children were confident working with one-step story problems, and  addition and subtraction equations that used numbers to ten. B y the end of the study, many of the children were able to complete multi-step story problems, using addition and subtraction equations with large numbers, and multiplication equations.  Frequently the  children asked to use a calculator to do the actual computation, after the equation had been recorded and explained.  The diaries show the growth i n the children's confidence at explaining their thinking and at sharing their ideas with their classmates and myself. A t the beginning o f the study, although the children had been used to talking about their thinking, they were very reluctant to talk, particularly when they knew that their ideas were being recorded. B y the end of the study, most of the children were much more w i l l i n g to voice their ideas and thinking, and did not seem to be concerned that their explanations were being taped. Evidence o f this is i n the length of the discussions for individual children and the numbers of ideas that were presented by an increasing number o f students.  A n excellent example o f this can be seen i n the  problem involving the rolling rocks in Diary #5.  The diaries and other classroom discussions show the beginnings o f change of attitude about "real math" in at least some o f the children. A t the beginning of the study, D o n , Mandy, and several other children who are very good at computation, often question the strategies and explanations o f others, but frequently have difficulty explaining their own strategies. "I just added them together," came the explanation from Ronald, but, when asked to talk us through his thinking about adding double digit numbers that required regrouping, he could not do it. However, M i k e could! "First I added the one and the two to get 30, then I added the six and the six to get twelve. So the answer is 42." When I asked him where the " 4 0 " came from, he gave me a look that seemed to say, ' C a n ' t you see?', and said, " F r o m the 12. Twelve is one 10 and two so you have to put the one 10 with the 30 and that makes 40; so the answer is 42." A s he commented, he pointed to the numbers on the chalkboard. Although this concept has  — 137 —  not been officially taught to this group of children, we have talked our way through this type of equation several times. However, I have always demonstrated the standard method o f adding with regrouping. This was evidence, to me, of how one child is constructing his own knowledge and meaning from what was actually presented to him. Ronald can complete a sheet of computation quickly and accurately, while M i k e continues to use counting and some direct modeling strategies to find the answers for equations with answers between 10 and 20. These same children often had difficulty accepting more than one solution to a problem. "That's not right," declared D o n , when M a n d y explained her reason for choosing the "sleepy" Weeple People (Gillespie, 1971) rather than the one that had "no opening", which most of the other children chose as their answer and explanation for Question 14 — Which doesn't belong? Why?. B y the end of the study, although I continued to hear the words "real math" being used i n connection with computation and conventional word problems, these children were more able to explain their thinking and more accepting o f the variety o f answers and strategies presented by their classmates.  If one were to follow the episodes o f the diaries involving Cathy, they w o u l d see that, although she continues to be basically a non-participant i n group discussions, and is reluctant to actually put answers onto paper, she has shown some growth in these areas, as well as i n both her basic mathematics skills and her ability to solve some types o f problems. Diary #1 shows that at the beginning of this study she had difficulty creating and following a pattern, while Diaries #3 and #4 show that she is beginning to be more confident and successful with both these skills. Diary #5 shows growth i n her ability to find numbers on the number line, and to do addition equations. Cathy's immaturity and low self-esteem have proven to be a major hindrance to academic growth in all areas of the curriculum. However, with continued successes, and with praise for those successes, her mathematics skills are gradually improving.  — 138 —  A t the beginning of this study, I chose six students to study on the basis o f mathematics skills as displayed in their work and participation i n group discussions. However, as I focused on this group of children and read the literature, I began to realize that, although I had chosen a cross section of children, I had probably placed four of them i n incorrect positions. I had chosen D o n and Mandy as the two top achievers, and M a r g and M i k e as the average students. Part way through the study, I realized that these positions could be reversed. A s I observed and listened to these four children work on their activities, I began to see that, although D o n and M a n d y ' s ability to find the right answers and their participation i n discussions was superior to those of M a r g and M i k e , their skills had been learned by rote, and they were frequently unable to give a detailed explanation of their thinking. M a r g and M i k e , on the other hand, with a little coaxing, could readily and easily explain their reasoning and thinking as they solved problems and d i d number operations. These two children were also more w i l l i n g to continue looking for a solution to a problem after the other two had given up. They were also more willing to take a risk to try finding a different solution or method o f solving a problem for which one solution or method of solving had already been found. M a r g and M i k e ' s persistence at solving Question 20 — Stumper in Diary #3, and M a r g ' s last equation for Question 10 — Equation please, are examples o f their willingness to take risks, their understanding of mathematics, and their ability to communicate their thinking. D o n and M a n d y proved to me that one should not judge mathematical ability by verbal skills and correct answers alone. M a r g and M i k e were the two who were making those important connections and constructing their own knowledge, and therefore understanding mathematics in a more complete way than were D o n and Mandy.  One could go on almost indefinitely at making inferences and drawing conclusions from the stories i n Chapter 5. However, such analyses would be redundant to the stories themselves, and the above examples provide the reader with a flavour o f the vast number o f  interpretations that can be made. The reader is left to take whatever he or she wants or needs from those stories.  W h i l e writing and reflecting on the diaries i n Chapter 5, I found several places i n which I could make changes that would probably improve my Wall Math programme for m y future classes. One such change came to mind as I wrote the observations about Question 7 — Draw and print as a number in Diary #3. Although the skill demanded by this question has always been problematic for my students, it is a required learning outcome according to the B C M E ' s I R P . After watching and listening to the children work on this question, I now suspect that the problem lies not so much in their lack o f knowledge, but i n the format o f the question and how I ask them to record their answer on the record sheet. The question requires a two-part answer, but the recording sheet only provides a single, large space. A change to the record sheet so that there are distinct spaces for both parts o f the response may be a simple solution to this problem.  Another such change presented itself as I reflected on the Order please  questions and  answers. The children's usual response is to record the smallest, shortest, thinnest, etc. first (the 'least'), and then end with the biggest, longest, widest, etc. (the 'most'). However, the reverse order is frequently discovered during discussions. Since one o f the goals o f my mathematics programme is to encourage students to look for more than a single way to solve any problem, I should occasionally specify that the order be recorded in more than one way (i.e., both 'least' to 'most' A N D 'most' to 'least').  Wall Math  is a system used by myself as a means of introducing and teaching new material,  keeping previously taught knowledge current, and providing practice for students who need it, and challenges for others.  It is a means o f helping students make connections among  mathematics strands, other curricular areas, and real life.  It can be used as a means o f  gradually introducing new concepts, of encouraging children to find solutions to a variety of standard and nonstandard problems i n their own ways , and as a vehicle for allowing students to learn from each other as they talk about their strategies and solutions. It is not a total mathematics programme, but, as one knows by reading the diaries, i n my classroom it has become an important part of my mathematics programme.  P r o b l e m s A r i s i n g i n the Study Thankfully, this study was conducted relatively 'trouble free'; almost everything happened as it was planned. However, one of the annoying problems o f this research had to do with the obtrusiveness o f "technology". I tried, on several occasions, to video-tape m y class ' i n action'.  H o w e v e r , despite mounting the camera i n a corner where it w o u l d be as  unnoticeable as possible, and using it several times, the students never d i d 'get used' to it, and their behaviours were always influenced by the camera; they just were not 'natural'. Similarly, trying to audio tape small groups of students led to unnatural behaviours and speech.  E v e n after taping numerous groups o f students numerous times, I still heard  comments such as, "Watch it; she's taping us!" or, "Quiet...tape recorder!" i f I came near. It was only by finally concealing a small, voice-activated tape recorder in m y pocket that I could get usable audio tapes of individual students or small groups of students going about their Wall Math  work i n an 'honest' way.  Part way through the study I acquired a  'conference' microphone and a Dictaphone/transcriber that had a 'conference' setting for recording. B y leaving this recorder on my desk and leaving it on throughout each Wall Math class, I could get excellent recordings o f 'whole-class' discussions, but that technology still did not solve the problem of getting usable audio tapes for small groups when several other groups were discussing various aspects o f their work at the same time. The best I could get for individuals and/or small groups was 'muffled' tapes, recorded v i a my 'concealed' tape. It requires a lot of time, much 'trial and error', and a lot of practice for a practitioner/teacher-  — 141 —  researcher to acquire the appropriate equipment and to become proficient with the technology necessary to collect quality 'data' for qualitative research with primary students.  Suggestions for F u r t h e r Research W h i l e working on this study I began to realize that, although I was seeing growth i n my students i n many areas, a study o f longer duration would be needed to provide stronger indications that the long term retention o f skills was indeed taking place, and that the children thoroughly understood, and could use, i n real-life situations, the concepts that were taught. It would also have been very interesting, and would have contributed immensely to the study, i f I had been able to have the time and/or opportunity to act as a helping teacher/observer i n another Grade 1 classroom using Wall Math, or at least i n another classroom at some grade level using Wall Math.  However, no other teacher in my school was using Wall Math  this  year; m y 'buddy' teacher, who worked with me to develop the initial programme, moved from our school to a middle school this year, and so no practical opportunity arose for me to spend time watching other teachers and classes using Wall Math.  A follow-up study to this  research might include collecting data over a whole year o f study from a variety o f Wall Math classrooms.  This study by no means "proves" that Wall Math "works", that it is a beneficial programme for students, or that it works any better than any other programme. Such conclusions would require a large scale study, at a minimum including: a detailed documentation o f a particular programme of Wall Math; extensive in-service of participating teachers on that programme; utilization of pre- and post- (and perhaps intermediate) interviews o f students and teachers, pre- and post-tests of attitude scales and cognitive mathematics items; and employment o f randomly assigned students i n classes, some o f which would be randomly assigned to a 'treatment' of Wall Math, and others to a 'control' treatment. Such a study is far beyond the scope o f this modest piece o f research.  However, i f major programme decisions about  curriculum , requiring a major outlay of resources, are contemplated, perhaps such a study would be desirable.  The method which I used for this study, that of writing narrative 'diaries' from notes, audio tapes, photographs, children's 'writings', and short term memory, proved to be very valuable for me to try to make sense o f Wall Math  and the way i n which students were interacting  with the programme, each other, and me. Despite having taught for over 30 years, I learned a tremendous amount about the programme, my students, and myself by conducting the study in the way that I did. The method of teacher enquirer or practitioner researcher is a method that I would recommend to other researchers or educators who wish to learn more about their own classes, their programmes, and themselves.  Conclusion After careful observation and documentation o f the Wall Math used in my classroom, I believe I have shown that Wall Math  programme and how it is  does what it was designed to  do for me and my class; it facilitates the understanding and retention of learned math skills for my Grade 1 children.  — 143 —  REFERENCES  Anderson, G . L . , Herr, K . , & N i h l e n , A . S. educator's guide to qualitative practitioner Press.  (1994). Studying your own school: An research. Thousand Oaks, C A : C o r w i n  Baker, A . , & Baker, J. (1991). Maths in the Mind: A Process Strategies. Portsmouth, N J : Heinemann.  Approach  to  Mental  Baker, D . , Semple, C , & Stead, T. (1990). How Big is the Moon: Whole Maths in Action . Portsmouth, N H : Heinemann. Bebout, H . C . (1993). Investigating the role of basic facts in early mathematics word problem solving. 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Greenwich, C T : J A I Press. Folkson, S. (1995). W h o ' s behind the fence? Creating a rich learning environment with a nontraditional problem. Teaching Children Mathematics, 1 (6), 382-385. Franke, M . L . , & Carey, D . A . (1997). Y o u n g children's perceptions o f mathematics i n problem-solving environments. Journal for Research in Mathematics Education, 28 (1), 8-25. Garofalo, J. (1989). Beliefs, responses, and mathematics education: Observations from the back of the classroom. School Science and Mathematics, 89, 451-455.  — 145 —  Gilbert, J . K . , & Watts, D . M . (1983). Concepts, misconceptions, and alternative conceptions: Changing perspectives in science education. Studies in Science Education, 10, 61-98. Gillespie, D . C . (1971). Weeple people. Toronto: M c G r a w - H i l l . Harper, G . F . , Mallette, B . , Maheady, L . , & Brennan, G . (1993). Classwide student tutoring teams and direct instruction as a combined instructional program to teach generalizable strategies for mathematics word problems. Education and the Treatment of Children, 16 (2), 115-134. Hankes, J . E . (1996). A n alternative to basic skills remediation. Mathematics, 2 (8), 452-458.  Teaching  Children  Hiebert, J . , & Lindquist, M . M . (1990). Developing mathematical knowledge i n the young child. In Payne, J . N . (Ed.), Mathematics for the young child (pp. 17-38). Reston: V A : National Council of Teachers of Mathematics. Jeroski, S. (1992). Field-based research: A working guide. V i c t o r i a , B C : M i n i s t r y of Education and Ministry Responsible for Multiculturalism and Human Rights. Jitendra, A . K . , & Kameenui, E . J. (1993). A n exploratory study o f dynamic assessment involving two instructional strategies on experts and novices' performance i n solving part-whole mathematical word problems. Diagnostique, 18 (4), 305-324. Jordan, N . C , Levine, S. C , & Huttenlocker, J . (1995). Calculation abilities in young children with different patterns of cognitive functioning. Journal of Learning Disabilities, 28 (1), 53-64. Knapp, N . F . , & Peterson, P. L . (1995). Teachers' interpretations of " C G I " after four years: Meanings and practices. Journal for Research in Mathematics Education, 26 (1), 40-65. Koehler, M . S., & Grouws, D . A . (1992). Mathematics teaching practices and their effects. In D . A . Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 115-126). N e w Y o r k : Macmillan. Lampert, M . (1988). Connecting mathematical teaching and learning. In E . Fennema, T. P. Carpenter, & S. J. L a m o i n (Eds..) Integrating research on teaching and learning mathematics (pp. 132-167). Madison, W I : University o f Wisconsin, Wisconsin Center for Education Research. Lampert, M . (1989). Choosing and using mathematical tools in classroom discourse In J. Brophy (Ed.), Advances in research on teaching , V o l . 1 (pp. 223-264). Greenwich, C T : J A I Press. Lampert, M . (1990). W h e n the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63. Ministry o f Education. (1989). Primary Program Foundation Author. M i n i s t r y o f Education. (1995). Victoria, B . C : Author.  Mathematics  — 146 —  Document.  K to 7: Integrated  Victoria, B . C :  Resource  Package.  r  Merriam, S. B . (1988). Case study research in education: A qualitative Francisco, C A : Jossey-Bass. National C o u n c i l o f Teachers of Mathematics. (1989). Curriculum Standards for School Mathematics. Reston, V A : Author.  approach. and  Evaluation  National C o u n c i l o f Teachers of Mathematics. (1991). Curriculum and Standards for School Mathematics.: First grade book. Reston, V A : Author. N i c h o l , C . (1997). 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(1990) Alternative perspectives on knowing mathematics i n elementary schools. In C . B . Cazden (Ed.), Review of research in education (pp. 57-150). Washington, D C : American Educational Research Association. Robertson, A . S. (1994). Student-teacher conceptualizations of environment and humannature relationships. U n p u b l i s h e d doctoral dissertation, . U n i v e r s i t y o f B r i t i s h Columbia. Sanjek, R . (1990). A vocabulary for fieldnotes. In R . Sanjek (Ed.), Fieldnotes 121). Ithaca: Cornell University Press. Schifter, D . , & Fosnot, C . T . (1993). Reconstructing Teachers College Press.  mathematics education.  (pp. 92-  New York:  Schoenfeld, A . (1983). Beyond the purely cognitive: Belief systems, social cognitions, and metacognitions as driving forces in intellectual performance. Cognitive Science, 7, 329363. Schoenfeld, A . (1992). Learning to think mathematically: Problem solving, metacognition, and sense making i n mathematics. In D . A . Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). N e w Y o r k : Macmillan. Simon, M . (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.  — 147 —  Simon, M . , & Schifter, D . (1991). Towards a constructivist perspective: A n intervention study of mathematics teacher development. Educational Studies in Mathematics, 22, 309-331. Snively, G . J . (1986). Sea of images: A study of the relationships amongst students' orientations, beliefs, and science education. U n p u b l i s h e d doctoral dissertation. University of British Columbia. Solomon, J . (1988). Learning through experiment. Studies in Science Education, 108. Spratley, J. P. (1980). Participant Observation.  5, 103-  N e w Y o r k : Holt, Rinehart, and Winston.  Steinberg, R . M . , Carpenter, T . P . , & Fennema, E . (1994, A p r i l ) . Toward instructional reform in the math classroom. Paper presented at the annual meetings o f the American Educational Research Association, N e w Orleans, L A . (ED375000) Suydam, M . N . (1990). Planning for mathematics instruction. In Payne, J . N . (Ed.), Mathematics for the young child (pp. 285-304). Reston: V A : National C o u n c i l o f Teachers of Mathematics. U n d e r h i l l , R . (1988). Focus on research into practice i n diagnostic and prescriptive mathematics. Mathematics learners' beliefs: A review. Focus on Leaning Problems in Mathematics, 10, 55-69. Villasenor, A . , & Kepner, H . S. (1993). Arithmetic from a problem solving perspective: A n urban implementation. Journal for Research in Mathematics Education, 24 (1), 62-69. W i l l i a m s , M , M c P h e r s o n , T., M a c k i n t o s h , M . , & W i l l i a m s , M . (1991). Active maths: Primary maths form the environment. Surrey, Great Britain: W o r l d W i d e F u n d for Nature. W i l s o n , K . J . (1991). Calculators and w o r d problems i n the primary grades. Teacher 38 (9), 12-14.  Arithmetic  Worth, J. (1990). Developing problem-solving abilities and attitudes. In Payne, J. N . (Ed.), Mathematics for the young child (pp. 39-62). Reston: V A : National Council of Teachers of Mathematics. Y a c k e l , E . , Cobb, P., W o o d , T., Wheatley, G . , & M e r k e l , G . (1990). The importance o f social interaction i n children's construction of mathematical knowledge. In T. J. Cooney & C . R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s (pp. 12-21). Reston: V A : National Council of Teachers of Mathematics.  — 148 —  Parent(s)/Guardian(s) Consent F o r m  Top Portion is a Copy Only consent / do not consent for my child. •  to participate in the research project entided Wall  Math: A Study of Children's Understanding of Mathematics, as proposed by Mrs. D. Lynne Bateson and Dr. Ann Anderson in the attached letter. If parental consent is given, please ensure verbal consent from your child.  Parent's Signature:. Date:  Telephone Numben_  Please detach here and return the bottom portion of this form  CONSENT FORM  This is to acknowledge that I have received a copy of the consent form and all attachments for my own records. Signature:  ]  Date:  .  .consent/do not consent for my child, to participate in the research project entided Wall  Math: A Study of Children's Understanding of Mathematics, as proposed by Mrs. D . Lynne Bateson and Dr. Ann Anderson in the attached letter. If parental'consent is given, please ensure verbal consent from your child.  Parent's Signature: Date:  Telephone Number:_  151 —  


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