THE EFFECT OF FRONT-LOADING MATHEMATICAL VOCABULARYiPRIOR TO TEACHING A UNIT ON TRANSFORMATION GEOMETRYbyHallie RockelB.Sc. Carleton University, 1978 B.Ed. The University o f British Columbia 1998 A GRADUATING PAPER SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF EDUCATION inTHE FACULTY OF GRADUATE STUDIES Department of Language and Literacy Education sDr. Theresa Rogers (Graduate Advisor)Dr. Marlene AsselinTHE UNIVERSITY OF BRITISH COLUMBIANovember 2010 © H . M. ROCKEL, 2010ABSTRACTIt has been my personal observation that students are more comfortable with solving mathematical problems when only numbers are involved. However, to be mathematically literate, a student must also understand the unique and sometimes confusing vocabulary of mathematics. This paper addresses the definition o f mathematical literacy, the importance of being mathematically literate, and how students can become mathematically literate. After reviewing the research, I decided to implement the strategy of front-loading vocabulary prior to teaching the concept of transformation geometry in an effort to find out, to what extent does pre-teaching o f math vocabulary impact students' ability to grasp an understanding o f a math concept. The end of the unit assessment indicated that many of the students met the prescribed learning outcomes for transformation geometry as they were able to communicate their understanding using appropriate mathematical vocabulary. It is my belief that the strategy o f front-loading vocabulary is an effective strategy that can help students acquire mathematical literacy.TABLE OF CONTENTSABSTRACT.............................................. . ......................iiTABLE OF CONTENTS .............................. ................................................ iiiACKNOWLEDGEMENTS................................................................................. ivINTRODUCTION...............................................................................................1LITERATURE REVIEW ...................................................................................2What is mathematical literacy?....................................... :.................... 2Why is it important for students to be mathematically literate? 6Why do some students find it challenging to become mathematically literate?........................................................................ 7What does the research say about the best practices for teaching mathematical literacy?........................................................ 11What is front-loading of vocabulary strategy?.............................. 12CONNECTIONS TO PRACTICE............................................ 15Context................................................................................................. 15Unit P lan ................................................................................................16Assessment...........................................................................................22REFLECTIONS............................................................................................... 23REFERENCES................................................................................................ 26Appendix A: Prescribed Learning Outcomes..............................................29Appendix B: Transformation Geometry Vocabulary................................. 32Appendix C: Transformation Geometry Unit T es t.................................... 37ACKNOWLEDGEMENTSI would like to acknowledge the amazing twenty-one Grade 6 students of Division 3 who were willing subjects in this study. Their desire to learn not only helped me to formulate my research question, but also allowed me to implement theory into practice.I began this program secure in my ability as a teacher but unsure o f my ability as a graduate student. However, the encouragement from my fellow students in the Delta Language and Literacy M.Ed. Cohort, helped me to believe in myself as a student. Thank you for your gentle bullying when I felt I could not complete the program. You would not let me give up!Much of the curriculum materials used in schools are deemed to follow best teaching practices backed by research. I would like to thank the many professors I have met these past two years, who have defined what is meant by best teaching practice research but more importantly, have taught me how to critically evaluate the research behind the curriculum so that I could adapt the program to best suit the needs o f my students.It has been said, that behind any great man is a woman. In my case, the saying becomes, behind any successful student is a great husband. Without the support o f my husband Ed, I would not have acquired a B.Ed. let alone a M.Ed. Thank you for all that you have done, from having a hot meal waiting for me upon my return from a night class, to giving me a quiet house to complete my studies but most importantly, believing that I could and would complete these two years of hard work.INTRODUCTIONOn a number o f occasions during parent-teacher interviews, parents have expressed concern over their child’s struggle with mathematics. They know their child can count, add and subtract, and recall multiplication facts quickly and correctly. And yet their child is not doing well in Grade 6 Mathematics. I patiently explain that mathematics is not just about numbers, and that their child also needs to become mathematically literate. Their child needs to know and understand the unique vocabulary o f mathematics. They need to question, reason and solve problems that are presented not just as numbers, but also in words. They need to be able to communicate their understanding of mathematical concepts. They need to be able to apply their knowledge of mathematics taught in the classroom to real-life situations. If a child struggles with mathematical literacy, the concepts of mathematics are difficult to grasp and then, to apply to the real world.The purpose o f this paper is to define, explore, and discuss research on mathematical literacy, to explain the importance of mathematical literacy, and to suggest how a student can become mathematically literate. Upon review of the literature, I decided to implement the strategy of front-loading vocabulary prior to teaching a unit on transformation geometry. The paper describes how the front-loading vocabulary strategy was taught, how the vocabulary was reviewed using games and manipulatives, and how this strategy helped students understand the concept of transformation geometry.The strategy was very successful in helping students understand transformation geometry as indicated by the end-of-unit assessment. Since undertaking this research, I have used front-loading o f vocabulary prior to teaching other mathematical concepts and I plan on implementing the strategy in other content-area subjects.1LITERATURE REVIEW What is Mathematical Literacy?Two of the reports I read (ICME 11, 2008, and Ontario Ministry of Education, 2004a) quoted the OECD Programme for International Student Assessment’s (PISA) (OECD, 1999) definition of mathematical literacy as: an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded mathematical judgments and to engage in mathematics, in ways that meet the needs o f that individual’s current and future life as a constructive, concerned and reflective citizen.The authors of the Ontario Ministry of Education document on mathematical literacy include the following as necessary components of mathematical literacy. Students should be able to:• estimate in numerical or geometric situations• know and understand mathematical concepts and procedures• question, reason, and solve problems• make connections within mathematics and between mathematics and life• generate, interpret, and compare data• communicate mathematical reasoning(Ontario Ministry o f Education, 2004b, p. 24).A study undertaken by the National Research Council, states that students who are mathematically literate “understand basic concepts, are fluent in performing basic operations, exercise a repertoire of strategic knowledge, reason clearly and flexibly, and maintain a positive outlook toward mathematics. Moreover, they possess and use thesestrands of mathematical literacy in an integrated manner, so that each reinforces the others” (National Research Council, 2001, p. 409).The British Columbia Performance Standards (BC Ministry o f Education, 2002) equates mathematical literacy with the term “numeracy.” As they state: “Just as there is more to literacy than teaching the rules and procedures o f language, there is more to numeracy than teaching the rules and procedures o f mathematics. Numerate individuals not only “know” mathematics, but understand it in personally meaningful terms. ' They feel competent and confident about their ability to draw on the necessary knowledge and apply it in new and relevant ways”(BC Ministry of Education, 2002).Mathematical literacy involves the student learning and using many different skills, as the following graphic organizer shown in Figure 1 demonstrates.THEFCO RN The components of Mathematical Literacy (from a PRIME workshop attended in the Fall of 2007 (Small, 2005)).As students read and answer questions found in math textbooks, they are required to represent mathematical concepts in many different ways. They must be able to solve3Iproblems and then communicate their knowledge in a manner that involves different forms of representation.THEFCO 2. Isosceles TrianglesFor example, a mathematically literate student knows that the picture in Figure 2 represents four isosceles triangles. The student is required to know that not only are there four geometric figures but that they are four specific geometric figures - congruent isosceles triangles. The student needs to convert a picture representation into a word representation involving correct vocabulary and the word representation of a specific number.Star, Stickland, & Hawkins (2008) see the connection between literacy and mathematics in a different way. Their research focuses on the relationship between content-area literacy and content-area literacy. Content-area literacy might involve an emphasis on developing students’ reading and writing skills by reading mathematics texts or writing papers in mathematics classes. It is their opinion that “content-area literacy activities implemented in a mathematics class typically do not help students learn mathematics” (Star et al., 2008, p. 106). Such activities tend to focus on literacy goals such as how to read the text found in content material and not on mathematics learning goals. Content-area literacy in mathematics is concerned with what it means to be literate in mathematics. Star et al (2008) assert that to be mathematically literate includes4understanding mathematical concepts and the ability to analyze mathematical problems logically, analytically and thoughtfully. A person who is mathematically literate has the ability to recognize and understand when and how mathematics is used in the real world (e.g. statistics reported in the media). They are able to apply their mathematical knowledge to solve real world mathematical problems (e.g. creating a budget). To be mathematically literate, one is able to reason mathematically and to communicate mathematically.I would argue that content-area literacy and content-area literacy should not be treated as separate entities. Teachers need to connect the learning of mathematical concepts with reading and writing strategies. As an elementary school teacher, I try to incorporate reading comprehension strategies at the same time as teaching math concepts.Mathematical literacy in my classroom is the ability to communicate the strategy used to solve a problem and to summarize the answer in a sentence. By analyzing students’ written answers to a word problem, I can assess their reasoning process and their level of understanding. All too often, I find students can solve the algorithm component of a word problem, but when asked what the number answer means, they are unsure or do not know even the unit attached to the answer. This demonstrates a lack of true understanding of the mathematical concept. It is my observation that students are very weak in their understanding of mathematical vocabulary and this inhibits their ability to communicate their understanding of a math concept. They are not fulfilling the definition of a mathematically literate person and this is troubling to me as a mathematics teacher. This problem is worth pursuing because in the “real world”, mathematics exists in word problems and not as sheets of algorithms requiring a calculator to solve them.5Students need to understand the specific meaning of words to help them solve a question involving mathematics.Why is it important for students to be mathematically literate?Mathematical literacy does not exist only in the classroom. It is necessary to be mathematically literate outside of the school setting. “Mathematical literacy involves more than executing procedures. It implies a knowledge base and the competence and confidence to apply this knowledge in the practical world. A mathematically literate person can: estimate interpret data; solve day-to-day problems; reason in numerical, graphical, and geometric situations; and communicate using mathematics” (Ontario Ministry of Education, 2004, p. 10). To be mathematically literate, one is able to make informed opinions on changes in the economy, the rise and fall o f political parties and the true value of an item on sale.Mathematical literacy is as important as proficiency in reading and writing. Competence in both domains is important. Not only do students need to be able to pronounce and decode words, but they need to be able to comprehend what has been read. “Learning mathematics with understanding is essentially the same thing. No matter how well your child can perform calculations, this ability is not very useful if he or she does not know how to apply these skills. The goal is for your child to develop a lifelong understanding of mathematics that is useful both at home and in the workplace” (Ontario Ministry of Education, 2004, p. 10).6Why do some students And it challenging to be mathematically literate?There are several reasons why students Tind mathematics to be challenging. Some of the reasons are: the different types o f ^ ^ ^ ^ ^ a t i o n ^ m d in mathematics, parental attitude,a teacher’s knowledge of m a th e n l || | |^ j |^ k ol^§j|yy£||gjvledge, and the vocabulary that is unique to mathematics.Types o f representations found in mathematics: Whereas the written Englishlanguage uses the twenty-six letters of the alphabet to create words and ultimatelyjmeaning, mathematics uses several different types of representation. This includes symbols for numbers (1, 2, 3 etc.), words for numbers (one, two, three, etc.), letters used to indicate variables (n, x, y), symbols used to indicate constants (TC for pi), symbols used to indicate a mathematical operation (+ for addition), and symbols to indicate the relationship between numbers (< means a number less than or equal to another number). Mathematical information can also be represented in many different types of graphs and tables and students need to learn how to read this representation and to determine what it means. Geometric shapes are pictorial representations that have a definition imbedded in the drawing. All of these representations require the student to learn and to build on, as mathematical Concepts become more abstract when meaning can be found in solving equations.Parental attitude: Sometimes a parent’s attitude towards mathematics can negatively influence a child’s ability to become mathematically literate. I have heard parents tell their children that they also struggled with mathematics when they were in school. This can create the impression that being mathematically literate is connected to their parent’s genes and that if their mom or dad failed at mathematics, they will fail too.7Based on a review of 66 studies, Henderson and Berla concluded that “the most accurate home predictors of student success in school are the ability o f the family, with the help and support o f school personnel, to create a positive home learning environment, communicate high and realistic expectations for their children’s school performance and future careers, and to become involved in their children’s schooling. All emphasize a positive parental attitude about the value o f education” (as quoted in a paper by Christenson, 2004, p. 90).Based on my own past experience as an elementary student and that of my students’ parents, math classes o f the past involved solving algorithms. The students of today are required to read and understand a word based problem before solving the algorithmic component. If parent’s struggle with English and the vocabulary used in word problems, they will have difficulty helping their child complete math homework.A teacher’s knowledge o f mathematics: A teacher’s understanding of mathematics is important to a student’s understanding o f mathematics. Teachers need to be confident in their knowledge and they need to be current in the best ways to teach mathematics. “Teacher knowledge about the teaching and learning process is the most powerful predictor of student success” (Ontario Ministry of Education, 2004b, p. 34). Unfortunately, I am not sure how to solve that problem as a teacher’s reason for lack of mathematical understanding may stem from the way that they themselves were taught in elementary school. I hope that by offering workshops on understanding mathematical literacy, I can help teachers find a way to improve their knowledge of mathematical vocabulary and ultimately pass this knowledge on to their students.8Lack o f prior knowledge: Understanding mathematical concepts requires scaffolding of prior knowledge (Altieri, 2009, Bay-Williams & Livers, 2009). Learning to read and developing mathematical proficiency both rest on a foundation o f concepts and skills that are acquired by many children before they leave kindergarten. In the primary years, students are taught simple equations to develop number sense. In later years, the equations are presented in word problems and the student is introduced to vocabulary related to a mathematical operation. For example, the word “difference” means “to subtract”. When a word problem involves more than one step in solving the problem, the ability o f the student to understand mathematical vocabulary is tested. If the student lacks the required prior knowledge, learning a new mathematical concept can be very challenging. For some of my students, gaps in knowledge are the result o f extended family vacations out o f the country during the school year.Another reason reading mathematics can be difficult is that it requires the reader to recall or find all information, definitions, theorems or notation pertinent to the concept currently being read (Hyde, 2006). For example, if the definition of a square is given as a quadrilateral with four right angles and four congruent sides, then the student must know the meanings o f the words quadrilateral, right angle, and congruent. Finding this information may require looking at previous sections in the book, looking at the mathematical glossary (if it is included in the textbook), or looking in the texts of disciplines other than mathematics. It may require the reader to scan the page to find any pictures, graphs, tables or charts that accompany the definition and then be able to understand how they are related to the meaning. Thus, mathematical reading is not linear, and students should not try to read mathematical text as they would a fiction novel;9straight through, cover to cover. Reading mathematical text may require the reader to reread passages several times in order to gain understanding. It is my observation that students are reluctant to use a dictionary when they do not know the meaning of a word let alone re-read a passage in an effort to comprehend text.Mathematical Vocabulary: To be mathematically literate includes understanding the vocabulary unique to mathematics (Wilson, 2010). Some o f the terminology used in mathematics can cause confusion because words have a different meaning in everyday language (e.g. right, table, product or odd). Confusion occurs when the meaning of some mathematical terms are not related with the meaning in the real world. For example “mean” in math means “the average of a set of numbers” whereas in the real world, “mean” can be defined as “to be unfriendly”. Another problem is misleading terminology that is used when teaching some math concepts. For example, if a student is asked “Which number is bigger?” when the question should be “Which number has the greater value?”, a student may think that the number printed in a large font size is the correct answer. Words can have multiple meanings that are dependent on the context (e.g. similar, area, multiply, reflect). Some words used in mathematics are homonyms and have a sound similar to everyday words (e.g. sum and some, plain and plane, ad and add) which can be confusing, especially to ELL students (Bay-Williams & Livers, 2009).Thus the reasons for why some students find it challenging to become mathematically literate are varied. Teachers should be aware o f these reasons and try to find strategies to help overcome the difficulties faced by their students and adapt their lessons as necessary.10What does the research say about the best practices for teaching mathematical literacy?As has been indicated earlier, mathematical literacy involves understanding the multiple representations used to create mathematical concepts. These concepts rely on the ability of the student to access prior knowledge and make connections with current knowledge. “Translating and moving flexibly between representations is a key aspect o f students’ mathematical understanding. Presenting students with opportunities to make connections between multiple representations makes math meaningful and can help students see the subject as a web of connected ideas as opposed to a collection o f arbitrary, disconnected rules and procedures” (Marshall, Superfine, & Canty, 2010, p. 39).According to the National Council of Teachers of Mathematics, students must apply their prior knowledge to explore and understand new concepts (National Council of Teachers of Mathematics, 2000). Often this prior knowledge refers to knowing and understanding the vocabulary related to a mathematical concept. If the students’ understanding of the vocabulary is weak or non-existent, then explicit teaching of the vocabulary will be required. Launching into a new lesson without addressing the vocabulary can hinder development o f mathematical understanding (Bay-Williams & Livers, 2009). Understanding mathematical vocabulary is an important part of mathematical literacy and o f understanding of mathematical concepts. I decided to focus on pre-teaching vocabulary prior to teaching a new mathematical concept.It is interesting that much o f the research on improving mathematical literacy and suggested strategies that I found, is directed towards the English Language Learner in a mathematics class. I believe the strategies based on this research can be applied to all11learners be they an English Language Learner, a student with learning disabilities or a student who has forgotten what was taught in previous years.What is front-loading of vocabulary strategy?Some of the reasons that hinder a student’s ability to become mathematically literate are not easy to rectify (e.g. parental attitude and teacher’s knowledge) so I decided to focus on improving students’ mathematical vocabulary. There are many strategies available to help students learn content-area vocabulary but I decided to use the strategy of front- loading prior to teaching transformation geometry.Children readily learn vocabulary and ways of speaking that allow them to negotiate the tangible, everyday world. However, students struggle when they are expected to use language in an abstract manner, such as in math. Math vocabulary should not be taught the same way as vocabulary in a language arts class as more than words are needed in the definition. As indicated earlier, math vocabulary is represented with numbers, symbols, and drawings as well as words. The strategy known as front-loading vocabulary, which has been used to help English Language Learners learn English, can be adapted to teach mathematical vocabulary and hence improve mathematical literacy.Front-loading vocabulary entails focusing on pre-teaching vocabulary needed to understand content or concepts. This is an especially effective tool for acquiring academic-language proficiency. “Front-loading of English Language Development refers to focusing on language prior to a content lesson. The linguistic demands of a content task are analyzed and taught in an up-front investment o f time devoted to rendering the content understandable to the student - which takes in not only vocabulary, but also the forms or structures o f language needed to discuss the content” (Garcia, p. 230).12Front-loading vocabulary is a logical process. “If one front-loads the discussion to gain clarity on the task, it is time well spent because there is always danger in assuming one knows what is the problem and charging ahead to solve what was not the problem” (Hyde, 2006, p. 29). Although this quote is referring to solving word problems, it does raise the issue o f assuming that students know the meaning of the vocabulary. As was mentioned earlier, one of the difficulties with mathematical vocabulary is that some terms can have different meanings in the world outside o f the math classroom. I tried to anticipate if the transformation geometry vocabulary and its definition could have a different meaning in everyday language and therefore try to be prepared to address what would be potentially confusing to the student. For example, when the transformational geometry term, “translate” was introduced, I did not assume that the student knew to use the mathematical definition. Since a student may think the definition o f “translate” means to decode from one language to another when in mathematics it means, “to move a 2-D shape across a co-ordinate grid”, I made sure to point out the different meanings and which one was the correct mathematical definition.For any strategy to be effective, one must teach the strategy using different methods. Marzano (2004), states that effective direct vocabulary instruction is characterized by the following:1. does not rely only on definitions of the vocabulary2. both linguistic and non-linguistic representations are given3. students receive multiple exposure to the vocabulary4. word parts and derivations are included in understanding the definition5. specific instruction is given for different content vocabulary136. students are encouraged to discuss the vocabulary7. games are used to teach and re-enforce understanding of the vocabularyTherefore, when using the strategy of front-loading vocabulary prior to teachingtransformation geometry, I utilized some of Marzano’s recommendations. My school district is currently using math textbooks published by Nelson under the series called MathFocus. I am using the edition MathFocus 6 (Nelson Education, 2002) in my classroom. The publisher conveniently included a list o f the vocabulary students would encounter in this unit. Each word was printed in a font large enough to be seen from the back of the classroom. This provided me with a manipulative that I could readily incorporate into my lesson plans. Although I did not know this at the beginning of this research, front-loading of vocabulary would also help to activate (somewhat) my students’ prior knowledge o f the mathematical concept being taught.When a term was introduced, the definition was given orally, followed by a written definition of the word on the board and if required, accompanied with a drawing thus providing both linguistic and non-linguistic representations. Through oral reviews and games prior to a new lesson in the unit on transformation geometry, and with the assistance of the students, multiple exposure of the vocabulary was attained. Students were encouraged to talk with one another about the questions assigned after a lesson as the ability to communicate is an important component o f mathematical literacy.Transformation geometry is a mathematical concept that I struggled with as a student and as a teacher. The terms used to describe transformations in older textbooks that I used as a student and as a beginning teacher were vague and confusing. Instead of the more mathematically correct terms of: translate, rotate and reflect, the terms slide,turn and flip were used. This variation in vocabulary was also discussed with the students should their parent use these terms instead of the terms used in the textbook. This provided some consistency in definitions for both the student and the helping parent.It is my belief that pre-teaching vocabulary through front-loading is a strategy that promotes both content-area literacy and content-area literacy. When teaching content- area literacy, a teacher will draw the students’ attention to text features as a tool for comprehension. The MathFocus 6 (Nelson Education, 2002) textbook uses many text features that aid in the implementation of the front-loading vocabulary strategy. This includes: bullets to summarize key ideas, highlighting o f vocabulary, text boxes with definitions and other text boxes with communication tips. A glossary of instructional words and mathematical words is also included at the end o f the book. Because the vocabulary for transformation geometry is very specific in its definition and representation, the front-loading of vocabulary strategy also addresses content-area literacy. Once students have acquired the vocabulary for transformation geometry, they now have the tools to analyze what happened and to communicate a transformation when a 2-D shape is reflected, rotated, or translated across a co-ordinate grid.CONNECTIONS TO PRACTICE ContextI teach Grade 6 students at a school located in a suburb o f a large city in Western Canada. The neighbourhood is described as low on the socio-economic scale. In many o f the families, English is not the language spoken most often at home. There is a wide range in the needs and abilities o f my students which is typical of all classes in this school. ELL students’ ability to communicate in English range from poor to fair and very poor to poor15in their writing skills. Many students are not meeting expectations with respect to Math which can be the result o f poor English skills, poor computation skills and difficulty with learning academic concepts. Typically math is taught in the morning for ninety minutes. Even though many of the students have learning deficits, a Learning Support Teacher is available to help students only two times a week for forty-five minutes. This lack of extra support is also typical for many of the classrooms in my school. Therefore, it is imperative for the teachers in my school to be equipped with as many effective strategies as possible to help our students achieve their best.The textbook, MathFocus 6 was designed to meet the Prescribed Learning Outcomes for Mathematics by the Ministry of Education. Appendix A lists the Prescribed Learning Outcomes for Transformational Geometry at the Grade 6 level. It is expected that the students will be able to perform the three types of transformations on a 2-D shape - translation, rotation and reflection - using the first quadrant o f a co-ordinate grid. The suggested achievement indicators state that the student can demonstrate, model, draw and describe, perform and record, analyze, create, label and plot various components of transformation geometry. In all tasks, the student is required to use correct transformation geometry vocabulary.Unit PlanLesson One: I began this unit on transformation geometry by having the students do a “chapter walk” of their textbook, MathFocus 6. The students flipped through the chapter looking at the diagrams, photos, and text features. They were asked to share what they thought was interesting to them. How artwork can be created using transformation16geometry caught the interest o f most o f the students. We also read the goals for the chapter which stated; “You will be able to:• translate, reflect, and rotate 2-D shapes on a coordinate grid• identify transformations in a design• combine transformations to produce a design, and• communicate about transformations. (Nelson Education, 2002, p. 145)We then focused on the text features such as bold and different coloured print, highlighted words, and the different lesson goals listed at the beginning of each lesson. Prior to using the lessons in the textbook, I decided to front-load the vocabulary associated with transformation geometry. Appendix B lists the vocabulary and definitions used in this unit. The vocabulary words describing transformation geometry were introduced systematically from the overall description of transformations to the three types of transformations (rotation, reflection, and translation) each with their own specific terms. Each vocabulary word was previously printed on yellow stock card, laminated, and a magnet was attached to the back to allow placement of the words on the chalkboard. The students wrote the definitions o f the words they would encounter in transformation geometry in their glossary (a three holed exercise book that fits in the front o f their math binder). Some o f the definitions were copied directly from the textbook while others were rephrased in simpler language by the teacher. Some o f the words used in transformation geometry vocabulary required a drawing to aid understanding. “When the student create the images, they know what they are showing and can more easily remember the word’s definition” (Altieri, 2009, p. 15). This was a tedious lesson for the students as it took the entire ninety minutes o f class. The 15 minute17Daily Physical Activity break plus breaks for student questions and comments, helped to alleviate restlessness. This front-loading step is necessary not only as a method for recording the necessary vocabulary but as a possible tool for activating prior knowledge.Lesson Two: Research has shown that, front-loading, where students learn the specific meanings of mathematical terms on their own, is not useful. “Math lessons take two fundamental kinds o f telling. Some telling is finite, describing a procedure or giving an answer to complete students’ thinking. The other kind o f telling is generative; it promotes conceptual understanding, encourages the search for greater understanding, and furthers students’ thinking. “Generative telling must be initiated by students’ talk” (Wilson, 2010, p. 498). To promote this generative talk as the unit progressed, the students were encouraged to ask for a classmate’s help during whole class discussions and to work with their “elbow partner” (the student who sits beside them) when answering questions from the textbook. Student talk was an important component of this unit.The following day, the vocabulary words were posted on the board and students were invited to read the definition from their glossary when asked what the word meant. This provided a quick review o f the vocabulary introduced the previous day. The students began the unit with the introductory lesson for transformation geometry found in the textbook. The activity was called “Roving Around Mars,” and the goal was to use translations and rotations to move a Mars Rover through a maze in its mission to examine rocks. The challenge was to move the Mars Rover in the direction it faced. An overhead of the maze the Mars Rover had to navigate was used so that all students could contribute in finding the route. The activity referred to the vocabulary words such as translations,18rotations, clockwise, and counterclockwise. The students soon realized that precise language was needed so that others could understand how the Mars Rover reached its destination.Lesson Three: On day three, the vocabulary words were again placed on the board and students were invited to draw a representation o f what the word meant. Sometimes the class took on a game show atmosphere as their understanding of the meaning o f the transformation geometry vocabulary was challenged. As in the game, “Who Wants to Be a Millionaire” whereby a contestant can call on another for help, the student was given the opportunity to ask a classmate for help if their representation was incomplete or contained any errors. The students then began Lesson 1 in the textbook - Translating Shapes. This lesson required the students to translate shapes on a coordinate grid using a rule. The rule the student created had to describe the movement as, up or down a certain number of units, and left or right a certain number of units as shown on a coordinate grid. Upon completion of the translation, the students were to correctly name the transformed shape using the prime symbol ('). For example, the image of AABC becomes AA'B'C' after the translation. (Unfortunately I was not at school the next day to ensure the students understood how to read and how to create, the rule for a translation. This gap in understanding was evident in their final test as many o f the students did not answer this question correctly).Lesson Four: Prior to beginning the lesson on the fourth day, the vocabulary words were randomly distributed to the students who were asked to sort the words according to the three types o f transformations. This helped students realize that the vocabulary words fall into one of three categories and that certain transformation19geometry expressions have unique terms attached to their definition (e.g. reflections require a line of reflection and rotations are described as clockwise or counterclockwise). This activity occurred on one o f the days when the math class lasted only forty-five minutes and the lesson took longer than expected. We ran out o f time and the proposed lesson on Reflecting Shapes could not be taught. Time is always a challenge when teaching and a teacher must decide what is most important - teaching the concept or teaching the curriculum. In this case, teaching the concept had priority.Lesson Five: The lesson on the fifth day began with a quick review of terms where I held up a term and asked a student to either give a definition or draw a diagram on the board. The students then worked on a revised lesson on Reflecting Shapes found in Lesson 2 of the textbook. Students completed a mirror image drawing using two different methods. Each student was given two simple cartoon drawings. The first drawing was the left half o f a face. The student lined up a reflective manipulative called a Mira along the line of symmetry and completed the right side of the face. The Mira “reflects” the image along the line of reflection onto the blank paper. The student then traces over the image thus completing the right side of the face resulting in a complete drawing of the face. The second drawing was also a face with two lines of symmetry at right angles to one another. The face was superimposed on a co-ordinate grid with the upper right and lower left portions omitted. The student was required to complete the drawing using the lines of symmetry and the placement o f the drawing on the coordinate grid as guides. The silence in the classroom was eerie as the students were incredibly focused on their tasks. Both methods for creating a reflection were easy to use but did require fine motor skills for20accurate drawings. The students were asked to draw and label the line(s) o f reflection in their drawings to show they understood the term and how it is represented on a drawing.Lesson Six: The lesson on the sixth day involved rotating shapes. Because the vocabulary for this transformation involved many terms (centre of rotation, clockwise, counterclockwise, coordinate grid, plot points, ordered pair, origin, x-axis, y-axis), a review o f just those vocabulary words was undertaken prior to the lesson. The method for drawing a rotation, as described in the textbook, required the students to use flimsy tracing paper (something they had not used before), that had a tendency to rip from repeated erasing or slip when rotating around the centre of rotation. This resulted in rotated images that were inaccurate. The rotated images did not line up correctly on the coordinate grid and therefore the ordered pairs describing the location o f the new plot point, were wrong. This method for creating rotations was not successful with my students therefore f need to find a different method for drawing a rotation using materials that students are familiar with and that will stand up to repeated erasing. It is interesting that the textbook included a strategy that required abilities and an unfamiliar manipulative that hindered the understanding o f a math concept. Also in future lessons, I plan on spending more time explaining that although the amount o f rotation can be any amount, it is often described as a lA turn, 'A turn, or a 3A turn. The definition for these rotations should be included as part o f the front-loading vocabulary activity.Lesson Seven: The seventh lesson was a mid-chapter review of the three transformations - translations, reflections, and rotations. The students were told that they would be having a test the next day and they were encouraged to use their glossary when studying. A final review of the vocabulary words was done by randomly distributing the21vocabulary words and asking the students to categorize the transformations and define each term. By this time the students were comfortable with both the word definition and a visual representation of the vocabulary. The atmosphere was positive as they helped one another complete the task.AssessmentThe students’ knowledge was assessed using the practice questions found in the textbook after the chapter review (see Appendix C). This test indicated that many students understood the majority of the vocabulary words unique to transformation geometry.They were able to follow instructions such as draw a shape on a coordinate grid and then translate the shape according to a rule creating a second image. They were able to describe the second image as congruent to the first image. When shown seven congruent shapes on a coordinate grid, they were able to determine which shapes were rotated, reflected or translated. The test required the student to answer each question using a complete sentence. The majority o f students used transformation geometry vocabulary and used the terms correctly.One common error made by the students that I did not anticipate, however, was inaccurate plotting o f coordinate plot points and inaccurate labeling o f ordered pairs. Plotting on a coordinate grid was taught in an earlier math unit on creating line graphs. Even though the students could have accessed this prior knowledge, I incorrectly assumed that this skill had transferred to this new lesson. I now realize that I should include a review o f plotting on a line graph as part o f the front-loading activity.As mentioned, the rule used to describe a translation was not adequately explained. This was evident because many of the students were not able to correctly22describe the rule that translated AABC to AA'B'C'. In future, I need to read the questions asked on the final test and to ensure that all learning outcomes assessed on the final test have been adequately taught prior to administering the test.REFLECTIONSThe math unit on transformation geometry requires a student to understand many unique vocabulary terms as well as to be able to communicate to others through words, diagrams, and symbols. It is my opinion that my students had greater success meeting the prescribed learning outcomes for transformation geometry through front-loading and through re-enforcing the vocabulary of transformation geometry often, and in ways that resulted in engagement by the students. I must admit that the front-loading vocabulary strategy not only helped my students but also helped me to better understand transformation geometry. I believe my students benefited from a teacher who had a better understanding of transformation geometry because I was more confident and I actually enjoyed teaching the unit.Learning support teachers and education assistants assigned to my students have commented in positive terms on the implementation and success of the front-loading vocabulary strategy. Since the front-loading vocabulary strategy required review o f the vocabulary prior to any lesson, they were introduced to the correct terms to use when communicating this mathematical concept. When these teachers were taught transformation geometry, the inadequate terms of slip, slide and flip were used which are not the correct vocabulary to use when describing a transformation. It is essential that consistent vocabulary is used by both students and teachers.23It is my belief that peer support is essential when teaching students that struggle with mathematics. Allowing the students to talk with their elbow partners and other classmates when the vocabulary was reviewed, helped to clear up misconceptions and helped to re-enforce definitions through words and pictures. These exchanges were often accompanied with laughter and cheers as students successfully completed the task.Because of the success in using this strategy, I plan on continuing using front- loading o f vocabulary in future lessons. The students would benefit from learning the unique vocabulary associated with other mathematical concepts such as 2-D and 3-D Geometry, multiples and factors, and data analysis.Students stumble when asked to solve a problem using terms such as estimate, justify, and predict. Phrases used to describe mathematical operations (e.g. What is the difference? If John had twice as many candies as Bill, how many candies does he have?) often confuse students. Because instructional vocabulary is found throughout the math textbook, front-loading the meaning of these words at the beginning o f the school year would not work. Unless the terms are reviewed regularly, their meanings would be forgotten. I would need to adapt this strategy to help students understand the instructional vocabulary found in mathematical word problems without boring them to death. It may be possible to combine some of the methods used to re-enforce the definitions of words included in the front-loading vocabulary strategy, with a highly visible mathematical operations word-wall that can be created and referred to during mathematics lessons throughout the school year.I can also envision using the strategy of front-loading vocabulary in other content area subjects. The Grade 6 Science unit on the Diversity o f Life, requires the student toknow and understand terms such as phylum, adaptation, and mimicry. The Social Studies curriculum introduces students to terms such as global citizens, famine, and genocide. Could the front-loading vocabulary be a useful strategy? Could this strategy lose its effectiveness if used to often? Because the initial lesson can be very tedious to the student, as it involves copying terms into their notebooks, I need to be very careful to not use the front-loading of vocabulary strategy too often and to not teach it the same way each time. As previously noted, student participation is an essential component o f their learning. Instead of the teacher providing the definitions o f targeted vocabulary, this strategy of front-loading could be flipped whereby the students are asked to find the definitions of vocabulary and present their knowledge in a unique way.Once I have tested and revised the front-loading of vocabulary strategy and become more aware of its successes and failures, it is my plan to share my findings with other teachers through workshops both in my school and other schools in my district.Calhoun (2002) states that educators who engage in action research are able to create instructional opportunities that are more effective and more intentional for student learning (as cited in Hendricks, 2009, p. 12). I definitely agree with this statement. The last two years as a graduate student and as an action researcher have allowed me to think critically about the textbooks and teacher guides that I use to teach my students. I am now able to understand the research that was used to create these materials and more importantly, how I can adapt these instructional aides to better fit the needs and abilities o f my students.25ReferencesAltieri, J. (2009). Strengthening connections between elementary classroom mathematics and literacy. Teaching Children Mathematics, 15(6), 346-351. Retrieved from http://www.nctm.org/eresources/ article_summary.asp?URI=TCM2009-02-346a&from=:BBC Ministry o f Education (2002). BC Performance Standards: Numeracy. Retrieved from http://www.bced.gov.bc.ca/perf_stands/numerg6.pdfBay-Williams, J., & Livers, S. (2009). Supporting math vocabulary acquisition. Teaching Children Mathematics, 16(4), 238-245. Retrieved from http://www.nctm.org/eresources/article_summary.asp?URI=TCM2009-11- 238ahttp://www.nctm.org/eresources/article_summary.asp?URI=TCM2009-l 1- 238aChristenson, S. L. (2004). The family-school partnership: An opportunity to promote the learning competence o f all students. School Psychology Review, 33(1), 83-104.Garcia, G. (Ed.). (2003). English learners: Reaching the highest level o f English literacy ( ed.). Newark, DL: International Reading Association.Hendricks, C. (2009). Improving schools through action research: A comprehensive guide fo r educators (2nd ed.). Upper Saddle River, New Jersey: Pearson Education Inc.Hyde, A. (2006). Comprehending math: Adapting reading strategies to teach mathematics, K - 6. Portsmouth, NH: Heinemann.ICME 11 (2008). What is mathematical literacy? Retrieved November 16, 2009, from dg.icmel 1 .org/document/get/46926Lobato, J., Clarke, D., & Bums Ellis, A. (2010). Initiating and eliciting in teaching: A reformulation o f telling. Journal fo r Research in Mathematics, 36(2), 101-136.Marshall, A. M., Superfine, A. C., & Canty, R. S. (2010). Star students makeconnections: Discover strategies to engage young math students in competently using multiple representations. Teaching Children Mathematics, 77(1), 38-47.Marzano, R. (2004). Building background knowledge fo r academic achievement: Research on what works in schools ( ed.). Alexandria, VA: Association for Supervision and Curriculum Development.National Council o f Teachers of Mathematics. (2000). Principles and Standards fo r School Mathematics. Reston, VA: NCTM.National Research Council (2001). Adding it up: Helping children learn mathematics (Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education). Washington, DC: National Academy Press.Nelson Education. (2002). Nelson MathFocus 6. Scarborough, Ontario: Small, M. ed.Ontario Ministry o f Education (2004a). Leading math success: Mathematical literacy Grades 7 - 12. Retrieved from http://www.edu.gov.on.ca/eng/ document/reports/numeracy/numeracyreport.pdfOntario Ministry o f Education. (2004b). Leading math success: Mathematical literacy grades 7 to 12 The report o f the expert panel on student success in Ontario. Toronto, ON: Queen’s Printer.Star, J. R., Strickland, S., & Hawkins, A. (2008). What is mathematical literacy?. In M. W. Conley, J. R. Freidhoff, M. B. Sherry, & S. F. Tuckey (Eds.), Meeting the27challenge o f adolescent literacy: Research we have, research we need (pp. 104- 112). New York, NY: The Guildford Press.Wilson, J. B. (2010). A foxyloxy and a lallapalagram. Teaching Children Mathematics, 16, 492-499.28Appendix A Prescribed Learning Outcomes (BC Ministry o f Education, 2002) for Transformation Geometry Grade 6Ha l i e l R o H i l c e kMA l R S T E A O l F DE R S UGeneral Outcome: Describe and analyze position and motion of objects and shapes.Prescribed Learning Outcomes Suggested Achievement IndicatorsIt is expected that students will:The following set o f indicators may be used to assess student achievement fo r each corresponding prescribed learning outcome.Students who have fu lly met the prescribed learning outcome are able to:0 6 perfo rm a com bination of transla tion(s), rotation(s) a n d /o r reflection(s) o n a single 2-D shape, w ith and w ithou t technology, and d raw and describe the im age (C, CN , PS, % V]J dem onstra te th a t a 2-D shape an d its tran sfo rm ation im age are cong ruen tj m odel a given se t of successive tran sla tions, successive ro ta tions o r successive reflections of a 2-D shapeJ m odel a g iven com bination o f tw o d iffe ren t ty p e s of tran sfo rm ations o f a 2-D sh apeJ d raw and describe a 2-D sh ap e an d its im age, g iven a com bination o f tran sfo rm ations□ describe the tran sfo rm ation s perfo rm ed on a 2-D shape to p roduce a given im age□ m odel a g iven se t of successive tran sfo rm ation s (translation , ro ta tion , a n d /o r reflection) o f a 2-D shape□ perfo rm an d record on e or m ore tran sfo rm ation s o f a 2-D shape th a t w ill re su lt in a given im ageG7 perfo rm a com bination o f successive tran sfo rm ation s of 2-D shapes to crea te a design , a n d identify a n d describe the tran sfo rm ations fC, CN , T, VI□ ana ly ze a g iven design c rea ted by tra n sfo rm in g one or m ore 2-D shapes, an d iden tify th e o r ig in a l shape a n d the tran sfo rm ations used to crea te th e design□ crea te a des ign u sing one o r m o re 2-D shapes an d describe the tran sfo rm ation s u sedC8 iden tify an d p lo t po in ts in the first q u ad ran t o f a C artesian p lane u s ing w hole num ber o rdered p a irs [€ , CN , V]□ label th e axes of the first quad ran t o f a C artes ian p lane and identity ' th e o rig in□ p lo t a po in t in th e first q u ad ran t o f a C artesian p lane, g iven its o rdered p a ir□ m atch po in ts in the first q u ad ran t o f a C artes ian p lan e w ith the ir co rre spond ing o rd ered pa ir□ plot points in the first quadrant of a Cartesian plane w ith intervals of 1 ,2 ,5 or 10 on its axes, given whole num ber ordered pairs□ d raw shapes o r designs, g iven o rd ered p a irs in the first q u ad ran t of a C artesian p lane□ d e te rm ine the d is tance b e tw een po in ts a long ho rizon ta l an d vertical lines in the first q u ad ran t of a C artes ian p lane□ d raw shapes o r designs in th e first q u ad ra n t o f a C artes ian p lane an d iden tify the p o in ts u sed to p roduce them[C] C om m un ica tion [ME] M entalM athem atics an d[PS] P roblem Solv ing m TechnologyjCNj C onnec tions E stim ation [RI R eason ing [V] V isualization- L A D O I L A H G M V C L B O U Y i P30Prescribed Learning Outcomes Suggested Achievement IndicatorsC9 perfo rm an d describe single tran sfo rm ations of a 2-D shape in th e first q u ad ran t of a C artes ian p lane (lim ited to w hole num be r vertices){C, CN. PS, X V]□ iden tify the coo rd inates of the vertices of a g iven 2-D shape (lim ited to the first q u ad ran t o f a C artes ian plane)□ perfo rm a tran sfo rm ation o n a g iven 2-D shape a n d identify th e coo rd inates o f the vertices o f the im age (lim ited to th e first quad ran t)□ describe the positional change of the vertices of a g iven 2-D shape to the co rrespond ing vertices of its im age as a resu lt of a tran sfo rm ation (lim ited to first quad ran t)S b y - L A D O I L A H G M V C L B O UAppendix B Transformation Geometry VocabularyTransformation Geometry Vocabulary2-D shape - a shape that has the dimensions of length and widthaxis - a horizontal or vertical line on a coordinate grid, labeled with numbers to show what the point on the graph means centre o f rotation - a fixed point around which other points in a shape rotate in aclockwise or counter clockwise direction. The centre o f rotation may be inside or outside a shape.clockwise - the direction a clock’s hands move. The abbreviation is cw.congruent - identical in size and shapecoordinate grid - a grid with each horizontal and vertical line numbered in order, starting at zeroC oordinate G ridV6 ........ j . . . . . . . . . . j................. .......5 .......... ]..............4 ....... !......"|............|— 4 -............. ..................3 ............ 1......... |.........|.... |............ ...............2 ........... ........ ........ I | ! | 1 ........ ...... ....... J | 0 1-----------1----------1-----1--------------1-------1 ► x ■0 1 2 3 4 5 633counterclockwise - the opposite direction to clockwise. The abbreviation is ccw.yline o f reflection - a line that falls exactly halfway between the points o f a shape and the matching points o f its mirror imageline o f symmetry - a line that divides a 2-D shape in half so that if you fold the shape on this line, the halves will matchordered pair - a pair o f numbers that describes a point on a coordinate grid; the first number denotes the position on the x-axis and the second number denotes the position on the y-axis34origin - the point on a coordinate grid at which the horizontal and vertical axes meet; the ordered pair of the origin is (0, 0) plot - to locate and draw on a coordinate gridreflection - the result o f flipping a 2-D shape across a line o f reflection; each point in a 2-D shape flips to the opposite side of the line o f reflection, but stays the same distance from the line (Line AB reflects the image CDEFGH creating the image C' D' E' F' G' H')rotation - the result of turning a shapeCenter of Rotationsymmetrical - a way of describing a 2-D shape with at least one line o f symmetrytransformation - the result o f moving a shape; translations, rotations, and reflections are transformationstranslation - the result o f sliding a shape along a straight line (Shape ABCDE is translated horizontally to create the Shape A'B'C'D'E')translation rule - a way of describing a translation with using the directions up or down and left or right a certain number o f units on a coordinate grid x-axis - the horizontal axis in a coordinate grid y-axis - the vertical axis in a coordinate grid36Appendix C Transformation Geometry Unit TestPracticeLesson 11. a) What rule will translateAABC so vertex C touches the horizontal axis? b> What rule will translate AABC so vertex A touches the vertical axis? c) What rule will translate A ABC so vertex C touches the horizontal axis and vertex 4 touches the vertical axis?2. a) Draw shape DEFGH on a grid, b) List the coordinates of D, E, F, G, and H. e) Translate DEFGH using the rule (R4, U6).Label the image D'E’F'G'H',List the coordinates of D', E', F , G', and H'. e) Are DEFGH and D'E’F'G'H' congruent?How do you know?Lesson .33. a) Rotate rectangle ABCDI turn ccw around vertex A.Label the image rectangle A'B'C'D'.b) Why did you need only the position of one of S', C , or D' to know how to draw the image rectangle?c) Why does vertex B move less than vertex D during the rotation?4. a) Which shapes are translations of A? State thetranslation rules.b) Which shapes are reflections of A? Describe the lines of reflection.c) Which shapes is a rotation of A? Describe the rotation.d) How do you know E is not a reflection of F?157U4.3) : : J CA ... j5 10
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The effect of front-loading mathematical vocabulary prior to teaching a unit on transformation geometry Rockel, Hallie Nov 30, 2010
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Title | The effect of front-loading mathematical vocabulary prior to teaching a unit on transformation geometry |
Creator |
Rockel, Hallie |
Date Issued | 2010-11 |
Description | It has been my personal observation that students are more comfortable with solving mathematical problems when only numbers are involved. However, to be mathematically literate, a student must also understand the unique and sometimes confusing vocabulary of mathematics. This paper addresses the definition of mathematical literacy, the importance of being mathematically literate, and how students can become mathematically literate. After reviewing the research, I decided to implement the strategy of front-loading vocabulary prior to teaching the concept of transformation geometry in an effort to find out, to what extent does pre-teaching of math vocabulary impact students' ability to grasp an understanding of a math concept. The end of the unit assessment indicated that many of the students met the prescribed learning outcomes for transformation geometry as they were able to communicate their understanding using appropriate mathematical vocabulary. It is my belief that the strategy of front-loading vocabulary is an effective strategy that can help students acquire mathematical literacy. |
Genre |
Graduating Project |
Type |
Text |
Language | eng |
Series | University of British Columbia. LLED 590 |
Date Available | 2014-07-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0078028 |
URI | http://hdl.handle.net/2429/48550 |
Affiliation |
Education, Faculty of Language and Literacy Education (LLED), Department of |
Peer Review Status | Unreviewed |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
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