A Look Inside the Math Matters Block of Courses.

Welcome to Math Matters On the first day of class Heaton and Lewis begin by distributing an overview of what they have planned for Math Matters. The purpose of this document is to communicate to the students the instructors’ high expectations for the course and the idea that it is a joint effort. Thus, while students register for specific courses for which Heaton or Lewis is the instructor of record, there are many assignments jointly made by Heaton and Lewis and for which students will be assessed as part of their grades in both their pedagogy and their math class.

Curriculum Materials Part of the goal for this NSF project is to adapt and implement appropriate NSF-funded curriculum materials. Heaton and Lewis have used materials from two different NSF funded curriculum projects.

Early Assessments In trying to understand the similarities and differences of their students to other elementary education majors, Heaton and Lewis are collecting information about their students’ mathematical abilities and beliefs. Instruments used early in the course include a Mental Math Quiz and a Mathematical Beliefs Survey. In each case, the instruments are also used with a reasonably comparable population. The Mental Math Quiz is given to all students in Math 200 and the Mathematical Beliefs Survey is given to all students in CURR 308. Students have 10 minutes to work the Mental Math Quiz reprinted below. They are instructed to avoid any paper and pencil computation, solving all problems mentally and then recording the answer. The quiz was first used in Louisiana as part of LASIP (the Louisiana Systemic Initiative Program). Lewis obtained the quiz from R. D. Anderson, Emeritus Boyd Professor of Mathematics at LSU and past president of the Mathematical Association of America. Future elementary school teachers have more difficulty than one might expect. Math Matters students missed an average of 6.5 questions while the control group missed an average of 8.6 questions.   Mental Math Quiz 1) 48 + 39 = 3) 14 x 5 x 7 = 5) 4 x 249 = 7) .25 x 9 = 9) 1/2 + 1/3 = 2) 113 ­ 98 = 4)  85/39 is closest to what integer? 6) 6(37 + 63) + 18 = 8) 12.03 + .4 + 2.36 = 10) 90% of 160 = 11) The sum of the first ten positive integers (1+3+5+ … +17+19) is equal to what integer? 12) If you buy items (tax included) at $1.99, $2.99 and $3.98, the change from a $10 bill would be? 13) To the nearest dollar, the sale price of a dress listed at $49.35 and sold at 25% off is ______? 14) The area of a square of perimeter 20 is ______? 15) The ratio of the area of a circle of radius one to that of a circumscribed square region is closest to? a) .5, b) .6, c) .7, d) .8, e).9 16) The average (artithmetic mean) of 89, 94, 85, 90, and 97 is ______? 17) If 4/6 = 16/x, then x = ______? 18) If 2x + 3 = 25, then x = ______? 19) The square root of 75 is closest to what integer? 20) To the nearest dollar, a 15% tip on a restaurant bill of $79.87 is ______? The mathematical beliefs survey is a 67-question survey developed by Heaton and Amy Spiegel (an evaluator for the Math Matters project). It contains some items from a belief survey originally developed by Fennema, Carpenter, and Peterson, as part of the Cognitively Guided Instruction project at the University of Wisconsin-Madison, adapted for use in this project. Each item offers a statement and asks the respondent to indicate that he/she Strongly Agree, Agree, Neither Agree nor Disagree, Disagree or Strongly Disagree. Responses are then given a numerical value (SA = 4; A = 3; D = 2; SD = 1) The response of “Neither” is treated as not expressing an opinion (as opposed to being between Agree and Disagree) and is not give a value. About half the statements are expressed in a way that anticipates that someone with a traditional approach to teaching mathematics will agree while someone with a reform approach will disagree. The other half anticipates that a traditionalist will disagree and a reformer will agree. Although data from 2001/2002 is still under review, there appear to be few differences in the beliefs of Math Matters students and other students pursuing a degree in elementary education. Heaton and Lewis also gave the survey to a small sample of faculty in the Department of Mathematics and Statistics (13) and in the College of Education and Human Sciences Elementary Teacher Education program (7). Math faculty were far more likely than Education faculty to respond “Neither Agree nor Disagree.” This may indicate that the mathematics faculty are less certain about issues related to teaching elementary school children, but it is also possible that the math faculty read questions more carefully and are reluctant to offer an opinion about a statement that has too many uncontrolled variables. The education faculty offered a more unified front in their responses than did the math faculty. On 34 occasions they had consensus (i.e. everyone responded SA/A or D/SD) while this occurred only 4 times for the math faculty. In general, mathematics faculty often agreed with education faculty. The average response for the Traditional Statements was M = 2.12; T = 1.71 while for the Reform Statements the average response was M = 3.07; T = 3.27. Below find a sample set of statements where the two faculties agree or disagree. Given the amount of agreement between the two faculties, it is interesting to speculate as to why there is such disagreement on these questions. A further analysis of the results of this study are forthcoming. Math Matters Beliefs Survey Sample Responses   Math Faculty = M (13)  College of Education and Human Sciences  Faculty = T (7) 4 Strongly Agree 3 Agree   Neither Agree nor Disagree 2 Disagree   1 Strongly Disagree Math Faculty and College of Education and Human Sciences Faculty Agree           6. When a student gets a problem right, it’s not necessary to ask how they got the answer.           M = 1.50          T = 1.29 SA A N D SD 9. A child’s answer could be reasonable even if it doesn’t match the teacher’s answer.           M = 3.31          T = 3.57 SA A N D SD 30. Planning a lesson does not require knowledge of students’ understanding.           M = 1.38            T = 1.14 SA A N D SD 31. A teacher needs to be a good listener to effectively teach mathematics.           M = 3.54          T = 3.71 SA A N D SD 41. Allowing children to discuss their thinking helps them to make sense of mathematics.           M = 3.58          T = 3.57 SA A N D SD 59. Improvisation is central to teaching.           M = 3.46          T = 3.50 SA A N D SD               Math Faculty and College of Education and Human Sciences Faculty Disagree           7. Teachers should let children work from their own assumptions when solving problems.           M = 2.25            T = 3.29 SA A N D SD 32. Mathematics assessment should occur every day.           M = 2.78          T = 3.86 SA A N D SD 35. Computation should be de-emphasized in elementary school.           M = 1.90          T = 2.67 SA A N D SD 50. Frequent drills on the basic facts are essential in order for children to learn them.           M = 3.22          T = 2.00 SA A N D SD 57. The use of key words is an effective way for children to solve word problems.           M = 2.14          T = 2.83 SA A N D SD 62. Time should be spent practicing computational procedures before children are expected to understand the procedures.           M = 2.70            T = 1.83 SA A N D SD

Sample Reflective Writings Heaton uses reflective writings to make connections within and across mathematical, pedagogical, and field experiences through writing. The reflective writings ask students to consider issues related to mathematics education fueled by their own experiences and things they have read. Lewis reads these and includes points from Heaton’s evaluation in his course grades. Here are several sample reflective writing assignments.

Examples of Early Mathematics Assignments Early each fall, Lewis makes certain assignments as an opportunity to set the expectations for the year. The first problem offers students an opportunity to make connections between mathematical content and pedagogy. Students who focus too much on creating an entertaining story and too little on the mathematical challenge (“help children understand how big the values really are”) are often quite surprised when their work receives a fairly low mark.

Write a children’s story that uses at least five quantities with large values in ways that will help children understand how big the values really are. Include references to places, things, and events that will make sense to them. The story should have between 500 and 1000 words. (#5, page 21, Number and Number Sense)

The next problem follows a class discussion of the number of grains of rice one gets if they start with one grain on the first square of a checkerboard, two on the second, etc., doubling the number for each subsequent square. The homework assignment is basically to gain some understanding of the size of the number, 2⁶⁴-1, and thus estimate the volume of that much rice. The problem is rich in opportunities for students to handle calculators, large numbers, converting from one unit to another, estimating an important but unknown piece of data (e.g., How many grains of rice are there in a cubic meter or in a pound of rice), etc. It also sends a clear message that some assignments are ill posed and will take a serious time commitment to complete. They learn that the expectations of Math Matters are significant and that not everything is nicely packaged, ready for memorization and repeating on a test.

Recall our discussion about the game of chess and how a humble servant for a generous king invented it. The king became fascinated by the game and offered the servant gold or jewels in payment, but the servant replied that he only wanted rice—one grain for the first square of the chess board, two on the second, four on the third, and so on with each square receiving twice as much as the previous square. In class we discussed how the total amount of rice was 2⁶⁴ grains of rice. (To be completely precise, it is this number minus one grain of rice.) Suppose it was your job to pick up the rice. What might you use to collect the rice, a grocery sack, a wheelbarrow, or perhaps a Mac truck? Where might you store the rice? (This assignment builds on #13, page 18 of Number and Number Sense.)

In 1994, Lewis received a letter from four elementary school students at a small school in central Nebraska. The students in Mrs. Thompson’s 1st and 2nd grade class have discovered a large number in one of the magazines their class receives and they wrote Lewis seeking the name of the number. Heaton and Lewis give their students a copy of the letter and challenge them to write a response pointing out that both will grade the assignment in their class. Heaton stresses that when she grades the assignment she will be looking for evidence that the future teachers have successfully used this as a teaching opportunity. Lewis indicates that his interest is in whether students find the mathematics in the letter and respond to the mathematics adequately. The letter was signed by the four students and the teacher. In the bottom right hand corner they had stapled the item they had cut out of Kid City magazine.

The Math Matters students have found this assignment to be quite challenging. Some focus narrowly on the question of what is the name of the number and they search the web for an answer. Several fail to count the number of zeros and notice that the students have miscounted the number. Very few of our students ever question the truth of the basic statement. After all, it appeared in a magazine. Here is the text of the letter from Mrs. Thompson’s class.

 

Dear Math Professors, We are 1st and 2nd graders in Wheeler Central Public School in Erickson, Nebraska. We love to work with big numbers and have been doing it all year! Every time we read something with a big number in it we try to write it. Then our teacher explains how to write it. We are getting pretty good at writing millions and billions! We have a problem that we need your help with. We were reading amazing ‘Super Mom’ facts in a Kid City magazine. It told how many eggs some animals could lay. We came across a number that we don’t know. It had a 2 and then a 1 followed by 105 zeros!! We wrote the number out and it stretches clear across our classroom! We know about a googol. We looked it up in the dictionary. A googol has 100 zeros. Then what do you call a number if it has more than 100 zeros? Is there a name for it? Another problem is that we learned about using commas in large numbers. In the magazine article they used no commas when writing this large number. That confused us. Also, if you write a ‘googol’ with 100 zeros, how do you put the commas in? It doesn’t divide evenly into groups of 3 zeros. There will be one left over. We appreciate any help you can give us solving this “big” problem. Thank you for your time. Sincerely, Mrs. Thompson’s 1st & 2nd graders Apple Of My Eye Megan Kansier Marcus Wittmann Mark Rogers Ashley Johnson The tiny female apple aphid is a champ as an egg layer. This insect can lay as many as 21000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 eggs in 10 months.

Sample Test Items for Math 300 Lewis’ exams have proven to be stressful for the Math Matters students. As indicated earlier, most of their other courses do not give exams, thus the experience stands out as different. Many students seem to expect the worst, i.e. to assume that test items may be similar to the most difficult homework assignments. Here are a few test items from the Number and Number Sense course.

Curriculum Project A highlight of the Fall semester is the Curriculum Project assigned jointly by Heaton and Lewis and counting for about 10% of the total points in each of their courses. Students are challenged to compare certain NCTM standards with material found in one elementary curriculum project and material typically taught at the college level to future elementary teachers. A sub theme is that teachers will need to continue learning mathematics after they become teachers if they are to stay abreast of the material they will need to teach. Here is the essence of the curriculum project in Fall 2000.   Math Matters Curriculum Project Fall 2000   It is impossible for Math Matters to help you learn all the mathematics that you will need to know in order to teach children. As a teacher, you will encounter unfamiliar areas of mathematics you need to learn and teach on your own. The goal of this project is to investigate a new mathematical topic area of the elementary mathematics curriculum and consider what teachers need to know to teach it well and what children need to learn to understand the topic in deep and meaningful ways suggested by The Principles and Standards for School Mathematics. You will work in groups of four on this project. The first step is for each group to choose a different area of mathematics to investigate. Your choices are: Data Analysis and probability; Geometry; Reasoning and Proof; Algebra. The project will have four parts:   Read, analyze, and synthesize the 3-5 grade band for your topic in Principles and Standards. What should be taught in this area at this grade level? What does a child need to know to be able to understand the topic in the ways suggested by the standards?   Pick a set of curriculum materials. These include the NSF curriculum materials (Everyday Math, Trailblazers, Investigations) or the LPS curriculum, Math Central. How does the curriculum present your topic? Do you find what you read about the topic in the NCTM standards? What similarities and differences exist between what you find in the curriculum materials and in the standards? Analyze and synthesize your findings.   Determine what you and your classmates need to know about this topic to teach it well. What makes the topic easy or hard to teach? What do teachers need to learn to teach the topic well to children? You will need to use additional resources such as books used in math courses and methods courses designed for future elementary school teachers.   Create and work out 5 sample math problems that would help teachers learn the mathematics they need to know to teach the topic. Indicate the important mathematics the problems would help teachers learn. Also, create and workout 5 sample math problems that would help children learn the topic in deep and meaningful ways. Indicate the important mathematics the problem would help children learn. Each group will turn in one project report. Use pictures, diagrams, graphs, tables, and charts to help you represent your ideas. It is expected that you will be communicating important mathematics in each part of this project.

 

Teaching a Math Lesson As part of their practicum experience at Roper Elementary School, Math Matters students are given many opportunities to learn to teach under the guidance of the master teachers who supervise their practicum experiences. Math Matters students teach math at least once a week both semesters and they are frequently asked to teach other subjects as well to individual students, groups of students or the whole class. Four times each semester they have a formal assignment to teach a lesson and prepare a report to Heaton on their teaching experience. Below find a example of one of these “formal lessons” from Fall 2000. It has been edited slightly for space considerations.   LESSON #4   This math lesson should connect to the curriculum in the classroom in which you are working. You are responsible for planning and doing the instruction for all students during this lesson. In addition to a learning objective(s) for the math topic you are teaching, you need one learning objective related to classroom management. You also need to make an attempt to make your teaching constructivist in nature. Push yourself to take risks beyond what you did for Lessons 2 and 3. Think carefully about the transitions within your lesson. Plan how you are going to move from one activity or part of the lesson to the next. What kinds of things do you need to consider? Your report should include your lesson plans, the videotape of your lesson, and your written reflections/analysis of the lesson. Every report should include the following items: EXPECTATIONS FOR THE LESSON PLANS MATHEMATICAL TOPIC GRADE MATHEMATICAL LEARNING OBJECTIVE(S) FOR TOPIC: What are the major math concepts or skills that students will work on in this lesson? What do you expect them to be able to do as a result of this lesson? How is this lesson connected to previous lessons? What do students already understand or find difficult with the topic. LEARNING OBJECTIVE(S) RELATED TO CLASSROOM MANAGEMENT: What will be required of you during the lesson to accomplish this objective? What are students working on in this lesson? What do you expect of yourself and students as a result of this lesson? YOUR OWN AND CHILDREN’S LEARNING OBJECTIVE(S) RELATED TO CONSTRUCTIVIST TEACHING: What feature or features of constructivist teaching are you going to try to use in your lesson? What will be new about this for you? What will be new for the children? MATERIALS: List all materials needed. LAUNCH: Describe how the lesson will begin. Your aim is to motivate students to get involved in the activities to follow and let them know what they will be studying. LESSON DEVELOPMENT: For each segment of the lesson, give a step-by-step description of what you will be doing and what students will be doing. In particular, provide examples of the types of questions you may ask. Include approximate times for each segment and describe your plans for transitions from one segment to the next. Describe the feature of constructivist teaching you are going to try to insert in your teaching practice and how you plan to do this. CLOSURE: Describe how you will bring the lesson to a close. Plan a way for students to reflect on what they learned. Include at least three questions you will ask students to help you know if the lesson was a success. GEARING UP AND GEARING DOWN: How will you modify the lesson if it turns out to be too easy for some students? How will you modify the lesson if it turns out to be too hard? ASSESSMENT: Identify data you will collect to help determine how well individual children understood the mathematics within your lesson and where they had trouble (classroom observations, written class work, homework, etc.). Prepare to gather data that will enable you to write specifically about what children do and do not understand about the mathematics at the end of your lesson. EXPECTATIONS FOR WRITTEN LESSON REFLECTIONS AND ANALYSIS:  Respond in detail to all of the following questions. What mathematical understandings or knowledge did you want to communicate through this lesson? Also, How well did you frame the lesson so that students would know what they would be studying? How effective were you at communicating the mathematics throughout the lesson? Give an example of when you thought you were effective. Give an example of when you thought you could have communicated about the mathematics more effectively. How well did you try to pull it all together again and help students reflect on what they had learned? Give an example of something you did at the end that went well. Give an example of something you did at the end that you could have done differently. From the standpoint of understanding the mathematics yourself, how well prepared were you to teach the lesson? What did you do to prepare to teach the mathematics? At this point, are there aspects of the mathematics you were uncertain of, not prepared for, or felt limited by to teach? How well do you think children understood what you wanted them to learn? Describe what happened for three different children in the lesson. If you were to teach the next lesson on this topic, what would you do next? Why?

 

Geometer’s Sketchpad Assignment Each spring we introduce our students to The Geometer’s Sketchpad. A colleague, Dr. David Fowler, gives a presentation to the Math Matters students offering a brief tutorial in using Sketchpad. Fowler has created several worksheets that allow individual students to learn to use the various Sketchpad commands and even more assistance is available on his web site. A follow-up class is held in a computer laboratory with each student working at a different machine. Fowler, Heaton and Lewis wander from student to student providing hints when a student gets stuck on a particular part of the tutorial. After two class periods, Fowler assigns a set of problems that the students are to work using The Geometer’s Sketchpad. The problems are chosen so as to practice various commands and (hopefully) use the dynamic nature of The Geometer’s Sketchpad to gain a valuable geometric insight. Below are two problems from the assignment given in Spring 2001.   The Geometer’s Sketchpad Assignment   1. A farmer wants to divide a rectangular strip of land among her four daughters, Lise, Emmy, Ada and Grace (named for famous women in math and science: Lise Meitner, Emmy Noether, Ada Lovelace and Grace Hopper). She wants to give Ada and Lise separate triangular corners of land. Emmy and Grace will farm the center region together. Is there a way to do this so that the sum of the areas of the triangular regions equals the area of the center region?     2. Every morning at half-past nine, the mail train rolls down the line. Jill, the mail clerk, tosses a bag off the slow-moving train at point B. Jack wakes up in his shack at point J, walks in a straight line to B and then in a straight line to the post office, at point P. The diagram is a bird's-eye view with the objects tilted on their sides, sort of a cubist bird's-eye view. With a little reflection, you should be able to identify a location for B so that the length of Jack's trip is minimized.

 

Geometry Assignments For the Spring semester the mathematical topic shifts to geometry using primarily the Shapes and Measurement materials developed by Judy Sowder et. al. at San Diego State University. In addition to the material contained in that book, Lewis likes to discuss Polya’s problem solving advice and to regularly give students problems that they can first seek to “solve” and then make the effort to give a “proof” of what they have discovered. Here are a few examples of the geometry problems Lewis has assigned to the Math Matters students.   How many different squares are there on a checkerboard? If the number of rows and columns of the checkerboard is doubled, how does the number of squares change?   Two red (i.e. non-black) squares are removed from opposite corners of a checkerboard. Given a set of 31 dominoes each the size of two squares, is it possible to cover the remaining 62 squares with the 31 dominoes?   Consider a large circle and pick “n” points on the circle. [Here n might be 2, 3, 4, and so forth.] Connect each pair of points with a chord. Notice that if n = 2 the circle is cut into 2 regions. If n = 3 the circle is cut into 4 regions and if n = 4 the circle is cut into 8 regions. How many regions do you get if n = 6? What about n = 8?   A domino can be thought of as two squares joined along one side. Similarly, a triomino might be a polygon formed by joining three squares together. (In each case two matching sides must fit together exactly.) Continue in this manner, define what is meant by tetromino and pentomino. How many pentominos are there? Once you have a conjecture as to the number of different pentominoes, can you provide an argument that your answer is correct? Note: We will say that two polygons are the same if one can be shifted, rotated and/or flipped to fit exactly onto the first shape.

Shapes from Four Triangles Lewis’ first exam in the geometry course, Math 301, included a bonus problem that he found among the supplementary materials for Shapes and Measurement. The problem asks how many shapes can be made using four congruent isosceles right triangles. Students could respond in class or work on the problem over night but they were told that work done outside of class would be held to a higher standard. The results were modest at best, with few of the students providing a correct solution and none of the students accepting the challenge of explaining why they had found a complete set of shapes. Heaton and Lewis decided to use the problem as the basis for a spring project that would involve solving a mathematics problem that had once been viewed as difficult, and then teaching the same material to an elementary school student. Working in pairs, the Math Matters students were “sent off into battle” without the benefit of any validation from Heaton or Lewis that they had a correct solution to the problem. As part of their report, they were to solve the mathematics problem and to use the problem as an opportunity to teach geometry to a single student. Finally, they were asked to reflect on their student’s understandings and on their own role in teaching geometry to elementary school children. Below find a copy of the Shapes from Four Triangle assignment.   Shapes from Four Triangles   We want to revisit the “Shapes from Four Triangles” problem. Think of this both as a mathematical task and as a mathematics lesson that can be taught to elementary school students. How can you present this task to the student you will interview? How can you set the stage for the student to gain an understanding of this problem? How far can the student go in exploring this problem? For your review, here is the basic problem: Given four congruent isosceles right triangles, how many different polygonal regions can you make, using all four triangles each time? When do we say that two shapes are truly different and when should we say that they are the same? When you believe you have found all the shapes, how do you give a mathematical argument that there are no more to be discovered and there are no duplications? Note: The first three shapes below are all the SAME polygonal region, since some rigid motion shows they are congruent. You may need to help your student understand “flips” (reflection about a line) and “rotations” (twisting about a pivot point). In case it is not clear, the intent is that a side of one triangle should fit exactly on a congruent side of an attached triangle. Making a shape involving something like the fourth shape below is not allowed (do you see that there would be an infinite number of answers if we permitted shapes like this one?):                      Remember that you want your student to discover as much as possible for himself (or herself). But there may be some critical points where you need to guide the student over an intellectual “bump” so that he (she) can move on to the next part of the problem. While you may want to adjust your “plan” after you begin to work with your student, here are some possible suggestions: Be sure that you have a statement of the problem that the student understands. Discuss what terms like “congruent” mean. Find some manipulatives for your student to use. For example, get four identical isosceles right triangles that the student can use. Invite the student to make various shapes and draw the outline on paper. Get card stock paper for your student to use in cutting out the shapes. Perhaps you should let the student(s) cut out lots of shapes including some that are duplications of each other. Then your student can twist and flip shapes trying to tell when two spaces are identical. Whenever he (she) finds two that are identical, be sure to “throw out” the extra shape. Once the student is close to having a fairly complete list, encourage the student to think about the pair of questions: Do I have all of the shapes? How can I explain to someone else why I am sure that I have all the shapes? Note: This last step is more abstract and hence more difficult in a subtle way. It is related to a shift from a pure “hunt” for different shapes and what you might call an organized “hunt.” Your student may not make much progress on #4 but you should make your best effort to answer these last two questions as part of your report.   The Shapes from Four Triangles Report   Your Shapes from Four Triangles report should include the following components. Analysis of Mathematical Task How does this task fit with the NCTM standards in geometry? Which standards do you see it addressing? What do you think is reasonable to expect of a child at your grade level on this task? Remember to hold high expectations. Analysis of Teaching this Task What did you do to prepare for your work with a child on this task? Describe a point in the interview when you felt confident about how you were working with the child. Explain why were you feeling this way. Describe a point in the interview when you were feeling uncertain in your role. Explain why you were feeling this way. Analysis of the Student’s Success How did your student do on the task? Did he or she do what you expected? Did anything surprise you? What do you think the child fully understands? What do you think the child has partial understanding of? What do you think the child has little understanding of? For each statement you make about what the child does and does not understand connect it to some evidence from the interview that indicates this. Explain what mathematics you think you are seeing or hearing or not seeing or hearing in the child’s work. Reflections on the Experience How do you think you could improve the learning experience if you repeated it with a new student? Consider the task and your preparation as well as your role and the student’s role during the interview. Videotape Submit a videotape of each time you worked with your student. Photographs Take at least five photographs that will help you and your student reflect on where he (she) is in understanding the problem. As the final thing you do with your student, ask them to write a reflection on one of the photographs. To help the student reflect on the photograph, you need to ask the student questions. Here are some suggestions: What were you doing at the time of the photograph? What part of the problem were you working on? Was the work you were doing in this picture hard or easy? Why? Turn in all of your student’s work. Include anything they make or draw. Label the photographs. Include something about what the student was doing at the time and what was going through your mind as you watched.   Futher Study of "Shapes from Four Triangles" Developed by undergraduate students in Mathematics Education with Ruth Heaton and Carolyn Edwards