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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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.