1. Developing the Mathematical Knowledge Necessary for Teaching in Content Courses for Elementary Teachers David Feikes Purdue University North Central AMTE Conference January 27, 2006

2. Copyright © 2005 Purdue University North Central

3. Connecting Mathematics for Elementary Teachers (CMET) NSF CCLI Grants: DUE-0341217 & DUE-0126882 Focus on How Children Learn Mathematics Focus on How Children Learn Mathematics Mathematical Content Courses for Elementary Teachers Mathematical Content Courses for Elementary Teachers

4. 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. CMET Supplement CMET includes 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: how children come to know number addition as a counting activity how manipulatives may embody (Tall, 2004) mathematical activity concept image (Tall & Vinner, 1981) in understanding geometry For example, we discuss how linking cubes may embody the concept of ten in understanding place value. At a more sophisticated level of mathematical thinking Base Ten Blocks (Dienes Blocks) may be a better embodiment of the standard algorithms for addition and subtraction. In addition, the CMET supplement contains: problems and data from the National Assessment of Educational Progress (NAEP) our own data from problems given to elementary school children questions for discussion

6. CMET is not a methods textbook. CMET is not a methods textbook. CMET is not a mathematics textbook. CMET is not a mathematics textbook. CMET is a supplement for mathematical content courses for elementary teachers. CMET is a supplement for mathematical content courses for elementary teachers.

7. Mathematical knowledge necessary for teaching is fundamentally different, but very much related to: 1) Mathematics Content Knowledge-- a textbook understanding of mathematics. 2) Pedagogical Content Knowledge-- how to teach mathematics. Mathematical knowledge necessary for teaching can be developed through knowledge of children’s mathematical thinking.

8. Survey Items: 1. Children learn mathematics through an understanding of set theory. 2. The concept of 1-1 correspondence is an important concept in the development of children’s ability to count. 3. Number is an easy concept to teach young children. 4. If children can count, they understand the concept of number. 5. Initially addition is a counting activity for children. 6. The concept of ten is the basis for place value. 7. For children first learning multiplication, multiplication is repeated addition. 8. Understanding multiplication will be easier for children who have developed the concept of ten. 9. Children in second and even some first graders can solve division problems by partitioning or using a repeated operation.

9. 10. In upper primary grades children often think of division as the opposite of multiplication to divide. 11. Children initially do not see that addition is commutative because addition is a counting activity and they are counting differently depending on the order of the numbers. 12. Initially, some children understand negative numbers as a quantity representing a deficit. 13. Initially, some children understand negative numbers as a location on a number line. 14. Sometimes children understand negatives in one context and not the other. 15. Children without an understanding of negative numbers frequently indicate that -7 > -3 16. The concept of repeated addition can be used to explain the multiplication of a positive number times a negative number. 17. Patterns are one way to illustrate a negative number times a negative number e.g., -2 x 1 = -2, -2 x 0 = 0 therefore –2 x –1 = 2 and so on.

10. 18. Symbolic notation of fractions should be introduced immediately with young children. 19. Teachers should always predivide shapes when children are pictorially representing fractions. 20. In a fraction, the numerator and denominator are related multiplicatively. 21. In a fraction the numerator and denominator are related additively. 22. Some children believe 6/7 = 8/9. 23. Children are likely to cross multiply to solve ratio and proportion problems. 24. Point, line, and plane are undefined terms to children. 25. Some children believe that perpendicular lines must be horizontal and vertical. 26. Many children believe that parallel lines that are not lined up are not parallel. 27. Mental imagery is essential to learning geometry. 28. Children use their concept images to determine if a given shape is a triangle. 29. Learning area is about learning formulas.

11. Conclusions: One way to help preservice teachers construct both mathematical knowledge and the mathematical knowledge necessary for teaching is by focusing on how children learn and think about mathematics. We believe preservice teachers will be more motivated to learn mathematics as they see the context of applicability (Bransford et al. 1999). If they see how children think mathematically and how they will use the mathematics they are learning in their future teaching, then they will be more likely to develop richer and more powerful mathematical understandings. Using knowledge of how children learn and think about mathematics will also improve preservice teachers’ future teaching of mathematics to children.