1. Using Knowledge of How Children Learn Mathematics in Mathematics Content Courses David Feikes Keith Schwingendorf Purdue North Central Purdue North Central Using Knowledge of How Children Learn Mathematics in Mathematics Content Courses David Feikes Keith Schwingendorf Purdue North Central Purdue North Central

2. Copyright © 2007 Purdue University North Central

3. Connecting Mathematics for Elementary Teachers (CMET) Connecting Mathematics for Elementary Teachers (CMET) NSF CCLI Grants DUE-0341217 & DUE-0126882 The views expressed in this paper are those of the authors and do not necessary reflect those of NSF.

4. Table of Contents Chapter 1Problem Solving Chapter 2Sets Chapter 3Whole Numbers Chapter 4Number Theory Chapter 5Integers Chapter 6Rational Numbers - Fractions Chapter 7Decimals, Percents, and Real Numbers Chapter 8Geometry Chapter 9More Geometry Chapter 10Measurement Chapter 11Statistics/Data Analysis Chapter 12Probability Chapter 13Algebraic Reasoning

5. Mathematical Content Courses for Elementary Teachers Focus on How Children Learn Mathematics! Methods Courses Graduate Courses

6. CMET Materials: Descriptions, written for prospective elementary teachers, on how children think about, misunderstand, and come to understand mathematics. Descriptions, written for prospective elementary teachers, on how children think about, misunderstand, and come to understand mathematics. These descriptions are based on current research and include: These descriptions are based on current research and include: how children come to know number how children come to know number addition as a counting activity addition as a counting activity how manipulatives may embody (Tall, 2004) mathematical activity how manipulatives may embody (Tall, 2004) mathematical activity concept image (Tall & Vinner, 1981) in understanding geometry concept image (Tall & Vinner, 1981) in understanding geometry In addition to these descriptions the CMET materials contain: In addition to these descriptions the CMET materials contain: problems and performance data from the National Assessment of Educational problems and performance data from the National Assessment of Educational Progress (NAEP) Problems and performance data from the Third International Mathematics and Science Study (TIMSS) Problems and performance data from the Third International Mathematics and Science Study (TIMSS) our own data from problems given to elementary school children our own data from problems given to elementary school children questions for discussion. questions for discussion.

7. The Concept of Measurement In elementary school, measurement has traditionally been presented as procedures and skills. However, a more careful analysis indicates that measurement is a concept. Teaching measurement is more than teaching the procedures for measuring, it is also helping children understand the concept of measurement

8. Over seventy-five percent of the fourth grade children missed this question. Most children who missed this question answered 8 or 6. Why 6?

9. Iteration/Repeating a unit One of the most important underlying concepts of measurement is the building-up activity of iteration or repeating a unit (e.g., paperclip, inch, or centimeter). One of the most important underlying concepts of measurement is the building-up activity of iteration or repeating a unit (e.g., paperclip, inch, or centimeter). Measurement involves learning to repeat a unit and the mental ability to place the unit end to end to measure or represent the length of the object being measured (unit iteration). The unit can be reused! Measurement involves learning to repeat a unit and the mental ability to place the unit end to end to measure or represent the length of the object being measured (unit iteration). The unit can be reused!

10. Nancy measured her pencil and got 6 inches. Why do you think she started measuring from 1?

11. Partitioning/Subdividing A second key concept is the breaking-down activity of partitioning or subdividing. Partitioning is the mental activity of “slicing up” an object into the same- sized units. Children frequently struggle creating units of equal size (Miller, 1984).

12. The Concept of Measurement In elementary school, measurement has traditionally been presented as procedures and skills. However, a more careful analysis indicates that measurement is a concept. Teaching measurement is more than teaching the procedures for measuring, it is also helping children understand the concept of measurement. 1996 National Assessment of Educational Progress, (NAEP) Over seventy-five percent of the fourth grade children missed this question. Most children who missed this question answered 8 or 6. Why 6? Mary says the wall is 10 steps long. Michael is asked to measure the length of a table in pencils. He says he cannot do it because he does not have enough pencils.

13. Sara says the wall is 13 steps long.

14. Preservice Teachers  Preservice elementary teachers’ mathematical knowledge, beliefs, and efficacy about teaching and the learning of mathematics can be developed by focusing on how children learn and think about mathematics in content courses.

15. Methods The following analysis compares the data from the treatment courses, which used CMET and focused on knowledge of children’s mathematical thinking with control courses, which were taught without using the CMET materials or an emphasis on children mathematical thinking. The following analysis compares the data from the treatment courses, which used CMET and focused on knowledge of children’s mathematical thinking with control courses, which were taught without using the CMET materials or an emphasis on children mathematical thinking.

16. The Likert items possible responses ranged from Strongly Agree to Strongly Disagree, responses were given numerical values accordingly from 5 to 1. The Likert items possible responses ranged from Strongly Agree to Strongly Disagree, responses were given numerical values accordingly from 5 to 1. The negatively worded questions are in bold. The negatively worded questions are in bold. Positively worded questions with higher scores up to 5 and negatively worded questions with scores closer to 1 are the closest to our theoretical position or an indication of an adherence to the beliefs and knowledge we believe are most important. Positively worded questions with higher scores up to 5 and negatively worded questions with scores closer to 1 are the closest to our theoretical position or an indication of an adherence to the beliefs and knowledge we believe are most important.

17. Knowledge of Children’s Mathematical Thinking Control Mean SD (n) Treatment Initially addition is a counting activity for children 4.07.741 (55) 4.66.479 (58) Children who can count, say the numbers in order, understand the concept of number. 2.84 1.058 (56) 2.27.997 (59) The concept of ten is the basis for place value. 3.79.680 (56) 4.17.699 (59)

18. Knowledge of Children’s Solution Methods Control Mean SD (n) Treatment Children are likely to cross multiply to solve ratio and proportion problems. 3.88.715 (56) 2.88 1.076 (24)

19. Knowledge of Children’s Geometric Thinking Control Mean SD (n) Treatment Children first understand shapes as a ‘concept image’, i.e., a shape is a rectangle because it looks like a door. 4.14.787 (37) 4.57.502 (37) Children frequently look at the lengths of the rays or the distance between the arrows to determine which angle is larger or smaller. 4.08.894 (37) 4.57.502 (37) Mental imagery is essential to learning geometry. 3.78.947 (37) 4.43.728 (37)

20. Teachers - Learning I liked the part where it includes the exact verbiage that a child said in regards to the decimals. It gives teachers an opportunity to be inside the head of a child. I liked the part where it includes the exact verbiage that a child said in regards to the decimals. It gives teachers an opportunity to be inside the head of a child. This chapter refreshed why we count the number of decimal factors and why we move the number of decimal places in the divisor to make a whole number. I had forgotten why. It was just how it is suppose to be. It is just the rule. It reminded me so that I can explain it to the children. When there is an explanation to give to the children, they believe it and trust it. This chapter refreshed why we count the number of decimal factors and why we move the number of decimal places in the divisor to make a whole number. I had forgotten why. It was just how it is suppose to be. It is just the rule. It reminded me so that I can explain it to the children. When there is an explanation to give to the children, they believe it and trust it.

21. Concluding Comments by a Teacher …you’re not teaching in the CMET how or when to teach concepts, but rather giving some insight as to how children think about and learn these. This chapter made me realize that this is the first text I have ever read that aims to help the reader understand what children have been and will be going through when attempting to learn math. This is such a neat idea because not only will pre-service teachers and parents understand their children’s mathematical thinking better, but they will also have more empathy for them. …you’re not teaching in the CMET how or when to teach concepts, but rather giving some insight as to how children think about and learn these. This chapter made me realize that this is the first text I have ever read that aims to help the reader understand what children have been and will be going through when attempting to learn math. This is such a neat idea because not only will pre-service teachers and parents understand their children’s mathematical thinking better, but they will also have more empathy for them.

22. Summary The CMET Project is premised on using children’s mathematical thinking to promote mathematical understanding for preservice teachers. The CMET Project is premised on using children’s mathematical thinking to promote mathematical understanding for preservice teachers. We believe this approach will enhance teacher’s learning of mathematics. We believe this approach will enhance teacher’s learning of mathematics. We are also linking research to practice. We are also linking research to practice. As part of our evaluation, we are attempting to both describe and assess the mathematical knowledge necessary for teaching. As part of our evaluation, we are attempting to both describe and assess the mathematical knowledge necessary for teaching. Our intent is to improve the teaching of mathematics to children! Our intent is to improve the teaching of mathematics to children!

23. Focusing on knowledge of children’s mathematical thinking raises prospective teachers’ efficacy to understand and teach mathematics as well as having an impact on their beliefs about mathematics and its teaching. Focusing on knowledge of children’s mathematical thinking raises prospective teachers’ efficacy to understand and teach mathematics as well as having an impact on their beliefs about mathematics and its teaching. Initial qualitative evidence suggests that using this approach also influences teachers and parents in their teaching and work with children. Initial qualitative evidence suggests that using this approach also influences teachers and parents in their teaching and work with children.

24. Looking to the Future We are looking for faculty who might be interested in attending a faculty workshop or using CMET in either your content or methods courses for our next grant! We are seeking a funding source to support the development of parent resources and parent research.