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Second Graders’ Understanding of Constant Difference and the Empty Number Line

Second Graders’ Understanding of Constant Difference and the Empty Number Line. Gwenanne Salkind EDCI 726 & 858 May 10, 2008. Introduction.

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Second Graders’ Understanding of Constant Difference and the Empty Number Line

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  1. Second Graders’ Understanding of Constant Differenceand the Empty Number Line Gwenanne Salkind EDCI 726 & 858 May 10, 2008

  2. Introduction • The NCTM Standards (2000) state that prekindergarten through grade 2 students should “develop and use strategies for whole number computations, with a focus on addition and subtraction” (p. 78) • Second graders typically have difficulty understanding and solving two-digit subtraction problems that require regrouping.

  3. Review of Literature • Children can solve two-digit subtraction problems strategically (Carpenter, et al., 1999; Carroll & Porter, 2002). • Representations can be powerful tools for learning (NCTM, 2000; Goldin, 2003). • The empty number line is a visual representation that has been used to develop conceptual understanding of subtraction strategies (Bobis, 2007; Klein, Beishuizen, & Treffers, 1998) • Constant difference is a “powerful strategy for subtraction because messy, unfriendly problems can easily be made friendly” (Fosnot & Dolk, 2001, p. 148).

  4. The Empty Number Line 24 + 27 = ? 53 – 27 = ?

  5. Constant Difference • Adding or subtracting the same number to both the subtrahend and the minuend in a subtraction problem does not change the answer. 50 – 25 = 25 49 – 24 = 25

  6. Research Questions Do second grade students who were taught using empty number lines: • Use a constant difference strategy to solve subtraction problems more frequently? • Have better mental computation skills? (speed, accuracy) • Have greater procedural competence?(accuracy)

  7. Participants – Second graders • Treatment Group • 8 boys, 6 girls • 36% Asian, 21% black, 14% white, 14% Hispanic, 14% multi-racial • Control Group • 7 boys, 8 girls • 40% Asian, 33% Hispanic, 13% multi-racial, 7% black, 7% white

  8. Similarities & Differences in Instruction • Both groups • Two-week unit (6 lessons) • Two-digit subtraction • Constant difference • Number lines • Strings, T/F, Story Problems • Treatment group only • Empty number lines

  9. Example of number line used during instruction (both groups)

  10. Examples of empty number lines used during instruction (treatment group only) True or False? 49 – 24 = 50 – 25 True or False ? 35 – 30 = 36 – 29

  11. Data Sources Used to Answer Each Research Question

  12. Analyses • Quantitative • Individual student scores were determined for mental speed tests, written subtraction tests, and interviews. • T-tests were used to compare means between treatment and control groups. • Qualitative • Student written work samples, written subtraction tests, and notes from student interviews were analyzed for evidence of the use of the constant difference strategy. • True/False equations (interviews) were coded according to students’ solution strategies: invalid strategy (I), guess (G), solved both sides (S), and used relational thinking (R).

  13. Mean Scores of Pre/Posttests

  14. Mean Scores of Interview Subtests

  15. Use of Constant Difference Strategy • There was no evidence that a student changed a subtraction problem into an easier problem using a constant difference strategy. • Students did use the constant difference strategy to find given differences and to solve true/false equations.

  16. Example of Using a Constant Difference Strategy to Find Given Differences

  17. Examples of Using a Constant Difference Strategy to Solve True/False Equations

  18. Key Findings • A high percentage of students used a constant difference strategy to find given differences in both classes. • Only students in the who were taught using empty number lines used a constant difference strategy to solve true/false equations.

  19. Key Findings • There were no statistically significant differences in mental computation speed or accuracy between students taught with an empty number line and those who were not. • There were no statistically significant differences in procedural competence between students taught with an empty number line and those who were not.

  20. Limitations • The instructional unit was too short. • There was not enough difference in instruction between the two treatment groups.

  21. References Bobis. J. (2007). The empty number line: A useful tool or just another procedure? Teaching Children Mathematics, 13(8), 410-413. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Carroll, W. M., & Porter, D. (2002). Invented strategies can develop meaningful mathematical procedures. In D. L. Chambers (Ed.), Putting research into practice in the elementary grades (pp. 16-20). Reson, VA: The National Council of Teachers of Mathematics. Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann. Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 275-285). Reston, VA: NCTM. Klein, A. S., & Beishuizen, M., & Treffers, A. (1998). The empty number line in Dutch second grades: Realistic and gradual program design. Journal for Research in Mathematics Education, 29(4), 443-464. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

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