Second Graders’ Understanding of

Constant Differenceand the Empty Number

Line

Second Graders’ Understanding of

Constant Differenceand the Empty Number

LineGwenanne Salkind

EDCI 726 & 858

May 10, 2008

IntroductionIntroduction

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.

Review of LiteratureReview 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).

The Empty Number LineThe Empty Number Line

24 + 27 = ?

53 – 27 = ?

Constant DifferenceConstant 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

Research QuestionsResearch 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)

Participants – Second gradersParticipants – 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

Similarities & Differences in InstructionSimilarities & 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

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?

Example of number line used during instruction (both groups)

Example of number line used during instruction (both groups)

Examples of empty number lines used during instruction (treatment group only)

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

Data Sources Used to Answer Each Research QuestionData Sources Used to Answer Each Research Question

Research Questions

Data Sources 1 2a 2b 3

Mental Speed Tests

Written Subtraction Tests

Student Interviews

Student Work Samples

AnalysesAnalyses 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).

Mean Scores of Pre/PosttestsMean Scores of Pre/Posttests

Treatment

n = 14

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.

Mean Scores of Interview SubtestsMean Scores of Interview Subtests

Treatment

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 1.14Note: There were no statistically significant differences between means.

Use of Constant Difference StrategyUse 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.

Example of Using a Constant Difference Strategy to Find Given Differences

Example of Using a Constant Difference Strategy to Find Given Differences

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

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

Percent of Students Who Used a Constant Difference Strategy on Written Classwork

71%

14%

87%

0%0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Finding GivenDifferences

Solving True/FalseEquations

Per

cen

t o

f S

tud

ents

Treatment Group(n = 14)

Control Group(n = 15)

Number of Students Who Used a Constant Difference Strategy on the

True/False Interview Task

0

1

2

3

4

5

6

7

Treatment Group Control Group

Nu

mb

er o

f S

tud

ents

Pre

Post

Key FindingsKey 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.

Key FindingsKey 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.

LimitationsLimitations

The instructional unit was too short. There was not enough difference in

instruction between the two treatment groups.

ReferencesReferencesBobis. 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.