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Decimal Fractions. Which of these is the smallest number? Explain why you think your answer is correct. 0.625 0.25 0.375 0.5 0.125. Source: TIMSS 1999, Middle School, B-10, p-value = 46%. Key Knowledge for Understanding Decimal Fractions. Place value system Partitioning numbers

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Decimal Fractions

  • Which of these is the smallest number? Explain why you think your answer is correct.

    • 0.625

    • 0.25

    • 0.375

    • 0.5

    • 0.125

Source: TIMSS 1999, Middle School, B-10, p-value = 46%


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Key Knowledge for Understanding Decimal Fractions

  • Place value system

    • Partitioning numbers

    • Representing values less than a whole

  • Fraction

    • Meaning of denominator and numerator (unitizing)

    • Equivalent fractions (reunitizing)

  • Strategies showing conceptual understanding

    • Common unit (use smallest place value)

    • Composite units (use each place value separately)


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Students’ Decimal Fraction Strategies

Adapted from Stacey & Steinle (1999)

  • Shorter is Smaller

    • Using whole number reasoning

    • Decimal-Fraction connection not established

  • Longer is Smaller

    • Misunderstanding of decimal-fraction connection, particularly denominator and numerator relationship (e.g., 0.35 means 1/35)

  • Apparent-Expert Behavior

    • Follow correct rules without understanding why:

      • Equalizing with zeros  CAUTION: Reinforces whole number reasoning

      • Comparing digits from left to right




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Longer is Smaller (1)

Always give answer of 0.625


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Longer is Smaller (2)

“The squares in the 2nd one are much smaller than in the first.”


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Equalizing Length with Zeros

Always give correct answer of 0.125


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Comparing Digits

Always give correct answer of 0.125


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Place Value Understanding

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


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Strategies for Developing Decimal Fraction Understanding

  • Use multiple representations

    • Number line model (placement and reading)

    • Fraction notation

    • Real-life context (e.g., money; volume)

  • Emphasize fraction-decimal connection


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Strategies for DevelopingDecimal Fraction Understanding

  • Discuss role of zero

    • When does zero affect a number’s value? How does it affect the value?

  • Example:

    • Starting with the number 23.5, place a 0 so that the new number is:

      • Equivalent

      • Larger

      • Smaller


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Diagnosing Student (Mis)Conceptions of Decimals

  • Which is smaller, 0. or 0.    ?

  • Write two decimals between 0.4 and 0.5

  • Explain why 0.5 is equal to ½.

  • Which of these is the smallest number? Explain why you think your answer is correct.

    • 0.625

    • 0.25

    • 0.375

    • 0.5

    • 0.125


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Multiplication and Division with Decimal Fractions

  • 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?

    • Patterns:

      • 8 x 50 = 12 / 20 =

      • 8 x 5 = 12 / 2 =

      • 8 x .5 = 12 / .2 =

      • 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?


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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.

  • 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.

  • 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.


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