1. Decimal Fractions  Which of these is the smallest number? Explain why you think your answer is correct. A. 0.625 B. 0.25 C. 0.375 D. 0.5 E. 0.125 Source: TIMSS 1999, Middle School, B-10, p-value = 46%

2. Key Knowledge for Understanding Decimal Fractions Place value system Place value system –Partitioning numbers –Representing values less than a whole Fraction Fraction –Meaning of denominator and numerator (unitizing) –Equivalent fractions (reunitizing) Strategies showing conceptual understanding Strategies showing conceptual understanding –Common unit (use smallest place value) –Composite units (use each place value separately)

3. Students’ Decimal Fraction Strategies Adapted from Stacey & Steinle (1999) Shorter is Smaller Shorter is Smaller –Using whole number reasoning –Decimal-Fraction connection not established Longer is Smaller Longer is Smaller –Misunderstanding of decimal-fraction connection, particularly denominator and numerator relationship (e.g., 0.35 means 1/35) Apparent-Expert Behavior Apparent-Expert Behavior –Follow correct rules without understanding why: Equalizing with zeros  CAUTION: Reinforces whole number reasoning Equalizing with zeros  CAUTION: Reinforces whole number reasoning Comparing digits from left to right Comparing digits from left to right

4. Shorter is Smaller (1)

5. Shorter is Smaller (2)

6. Longer is Smaller (1) Always give answer of 0.625

7. Longer is Smaller (2) “The squares in the 2 nd one are much smaller than in the first.”

8. Equalizing Length with Zeros Always give correct answer of 0.125

9. Comparing Digits Always give correct answer of 0.125

10. Place Value Understanding Always give correct answer of 0.125 and talk about comparing place values or relative size of numbers.

11. Strategies for Developing Decimal Fraction Understanding Use multiple representations Use multiple representations –Number line model (placement and reading) –Fraction notation –Real-life context (e.g., money; volume) Emphasize fraction-decimal connection Emphasize fraction-decimal connection

12. Strategies for Developing Decimal Fraction Understanding Discuss role of zero Discuss role of zero –When does zero affect a number’s value? How does it affect the value? Example: Example: –Starting with the number 23.5, place a 0 so that the new number is: a)Equivalent b)Larger c)Smaller

13. Diagnosing Student (Mis)Conceptions of Decimals Which is smaller, 0. or 0. ? Which is smaller, 0. or 0. ? Write two decimals between 0.4 and 0.5 Write two decimals between 0.4 and 0.5 Explain why 0.5 is equal to ½. Explain why 0.5 is equal to ½.  Which of these is the smallest number? Explain why you think your answer is correct. a) 0.625 b) 0.25 c) 0.375 d) 0.5 e) 0.125

14. Multiplication and Division with Decimal Fractions Mutiplication and Division break “rules” Mutiplication and Division break “rules” –Multiplying by a value between 0 and 1 makes the product smaller. –Dividing by a value between 0 and 1 gives a bigger quotient. How can you help students make sense of these outcomes? How can you help students make sense of these outcomes? –Patterns: 8 x 50 =12 / 20 = 8 x 50 =12 / 20 = 8 x 5 =12 / 2 = 8 x 5 =12 / 2 = 8 x.5 =12 /.2 = 8 x.5 =12 /.2 = 8 x.05 = 12 /.02 = 8 x.05 = 12 /.02 = –Contextualized problems: I have 5 meters of ribbon and each bow requires 0.5 meters of ribbon. How many bows can I make? I have 5 meters of ribbon and each bow requires 0.5 meters of ribbon. How many bows can I make?

15. References Martine, S. L., Bay-Williams, J. M. (2003). Investigating students’ conceptual understanding of decimal fractions using multiple representations. Mathematics Teaching in the Middle School, 8(5), 244-247. Martine, S. L., Bay-Williams, J. M. (2003). Investigating students’ conceptual understanding of decimal fractions using multiple representations. Mathematics Teaching in the Middle School, 8(5), 244-247. Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8-27. Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8-27. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362. Stacey, K., & Steinle, V. (1999). A longitudinal study of children’s thinking about decimals: A preliminary analysis. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education. (Vol. 4, pp. 233-240). Haifa: PME. Stacey, K., & Steinle, V. (1999). A longitudinal study of children’s thinking about decimals: A preliminary analysis. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education. (Vol. 4, pp. 233-240). Haifa: PME.