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Second Graders

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Title: Second Graders


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
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?
10
Example of number line used during instruction
(both groups)
11
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
12
Data Sources Used to Answer Each Research Question
Research Questions Research Questions Research Questions Research Questions
Data Sources 1 2a 2b 3
Mental Speed Tests ? ?
Written Subtraction Tests ? ?
Student Interviews ? ? ?
Student Work Samples ?
13
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).

14
Mean Scores of Pre/Posttests
Treatment n 14 Treatment n 14 Control n 15 Control n 15
Tests Pre Post Pre Post
Mental Speed (10) 1.64 2.71 3.27 2.87
Written Subtraction (8) 3.21 3.57 3.80 3.80
Note There were no statistically significant differences between means. Note There were no statistically significant differences between means. Note There were no statistically significant differences between means. Note There were no statistically significant differences between means. Note There were no statistically significant differences between means.
15
Mean Scores of Interview Subtests
Treatment n 7 Treatment n 7 Control n 7 Control n 7
Subtests Pre Post Pre Post
True/False (10) 2.43 5.86 1.43 2.86
Differences (12) 5.29 6.14 6.14 5.43
Story Problems (3) 0.86 1.71 1.57 1.14
Mental Computation (4) 0.71 1.43 0.57 1.14
Note There were no statistically significant differences between means. Note There were no statistically significant differences between means. Note There were no statistically significant differences between means. Note There were no statistically significant differences between means. Note There were no statistically significant differences between means.
16
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.

17
Example of Using a Constant Difference Strategy
to Find Given Differences
18
Examples of Using a Constant Difference Strategy
to Solve True/False Equations
19
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20
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21
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.

22
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.

23
Limitations
  • The instructional unit was too short.
  • There was not enough difference in instruction
    between the two treatment groups.

24
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). Childrens
    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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