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

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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.

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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).

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

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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

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Research Questions Do second grade students who were taught using empty number lines: 1. Use a constant difference strategy to solve subtraction problems more frequently? 2. Have better mental computation skills? (speed, accuracy) 3. Have greater procedural competence? (accuracy)

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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

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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

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Strings 12 – 6 = 13 – 7 = 14 – 8 = 50 – 25 = 51 – 26 = 52 – 27 = 49 – 24 = True or False? 15 – 7 = 16 – 8 35 – 30 = 36 – 29 29 – 17 = 30 – 19 32 – 20 = 33 – 21 30 – 22 = 29 – 23 Story Problems Aaron is 31 years old. Fahim is 18 years old. What is the difference in their ages? Sara is 43 years old. Tom is 8 years younger than Sara. How old is Tom?

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Example of number line used during instruction (both groups)

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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

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Data Sources Used to Answer Each Research Question Research Questions Data Sources12a2b3 Mental Speed Tests Written Subtraction Tests Student Interviews Student Work Samples

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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).

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Mean Scores of Pre/Posttests Treatment n = 14 Control n = 15 TestsPrePostPrePost Mental Speed (10)1.642.713.272.87 Written Subtraction (8)3.213.573.80 Note: There were no statistically significant differences between means.

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Mean Scores of Interview Subtests Treatment n = 7 Control n = 7 SubtestsPrePostPrePost True/False (10)2.435.861.432.86 Differences (12)5.296.14 5.43 Story Problems (3)0.861.711.571.14 Mental Computation (4)0.711.430.571.14 Note: There were no statistically significant differences between means.

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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.

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Example of Using a Constant Difference Strategy to Find Given Differences

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

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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.

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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.

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

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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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