1. Using Children’s Mathematical Thinking to Promote Understanding David Feikes Purdue University North Central NCTM Annual Meeting and Exposition April 29, 2006

2. Copyright © 2005 Purdue University North Central

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

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 Materials: 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 ten in understanding place value and 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 to these descriptions the CMET materials contain:  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. 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. The Concept of Measurement

7. 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). 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!

8. 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?

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

10. 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).

11. Mary says the wall is 10 steps long. Sara says the wall is 13 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.

12. Andrew measured his pencil and says it is 5 paperclips long. Aaron says this wall is 8 steps long. Connie who is working with Aaron says the wall is 8 and ½ steps long. Children mix units to measure an object such as using both small and large paperclips or they use both inches and centimeters. When asked to make their own rulers, children will draw hash marks at uneven intervals. Children use both inches and centimeters to measure a length.

13. Accumulation of Distance A third key concept is accumulation of distance. Accumulation of Distance is the result of placing the unit end to end alongside of it to measure the object signifies the distance from beginning to end.

14. Children may also think that the 4 on the ruler represents 4 hash marks – not 4 equal units.

15. A child is counting his footsteps out loud to measure the length of a table when the teacher stops to ask him what does the 7 he just said mean. He responds, “7 means the space covered by the seventh foot,” not the total distance of covered by 7 feet.

16. Relationship between Number and Measurement Relationship between number and measurement is illustrated in that different numbers using different sized units may represent the same distance. Relation between number and measurement means measuring is counting the number of iterations you make; however simply counting is not enough. Length is not just a matter of counting but realizing that different sized units can be used to represent the same length. Children must establish a correspondence between the units and the attribute such as knowing that one unit on a ruler represents 1 inch or A common misconception in understanding area is a rectangle with a side of 7 inches has an area of 7. These relationships may seem transparent to adults but may not be easy for children. As a further illustration, when children mix both inches and centimeters, measurement is not significantly different than counting.

17. Ted and Martha measured opposite walls in the room which are the same length. Ted says that the front of the room must be shorter than the back wall because Martha counted more steps. Mandy said that she and Mark had the same problem on the side walls. She measured one wall and got 18 steps and Mark measured the opposite wall and got 15 steps. Mandy said since I got more steps and the walls are the same my feet must be bigger. Is she right?

18. What does this illustration say about the child’s understanding of fraction? Matt says it is 7 and ½ steps long.

19. Summary  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 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.  Our intent is to improve the teaching of mathematics to children!  The Connecting Mathematics for Elementary Teachers Project is using knowledge of children’s mathematical thinking to help preservice elementary teachers understand mathematics more deeply and develop more specific mathematical knowledge necessary for teaching.