modifying arithmetic practice to promote understanding of mathematical equivalence nicole m. mcneil...

Modifying arithmetic practice to promote understanding of mathematical

equivalence

Nicole M. McNeilUniversity of Notre Dame

Seemingly straightforward math problem

Mathematical equivalence problems

3 + 5 = 4 + __

3 + 5 = __ + 2

3 + 5 + 6 = 3 + __

Theoretical reasons Good tools for testing general hypotheses about

the nature of cognitive development E.g., transitional knowledge states, self-

explanation, etc.

Practical reasons Mathematical equivalence is a fundamental

concept in algebra Algebra has been identified as a “gatekeeper”

Why we care about these problems

Most children in U.S. do not solve them correctly

16%

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Study

Why don’t children solve them correctly?

Some theories focus on what children lack Domain-general logical structures Mature working memory system Proficiency with “basic” arithmetic facts

Other theories focus on what children have Mental set, strong representation, deep attractor

state, entrenched knowledge, etc. Knowledge constructed from early school

experience w/ arithmetic operations

But isn’t arithmetic a building block?

Knowledge of arithmetic should help, right?

Children’s experience is too narrow Procedures stressed w/ no reference to = Limited range of math problem instances

Children learn the regularities Domain-general statistical learning mechanisms

that pick up on consistent patterns in the environment

2 + 2 = __ 12+ 8

Overly narrow patterns

Perceptual pattern “Operations on left side” problem format

Concept of equal sign An operator (like + or -) that means “calculate

the total”

Strategy Perform all given operations on all given numbers

3 + 4 + 5 = __

Overly narrow patterns

Perceptual pattern “Operations on left side” problem format

Concept of equal sign An operator (like + or -) that means “calculate

the total”

Strategy Perform all given operations on all given numbers

Overly narrow patterns

Perceptual pattern “Operations on left side” problem format

Concept of equal sign An operator (like + or -) that means “calculate

the total”

Strategy Perform all given operations on all given numbers

3 + 4 = 5 + __

“Operations on left side” problem format

“Operations on left side” problem format

“Operations on left side” problem format

Equal sign as operator

Child participant

video will be shown

Add all the numbers

Child participant

video will be shown

Recap

2 + 2 = __ 12+ 8

2 + 2 = __ 12+ 8 3 + 4 + 5 = 3 + __

Internalizenarrow patterns

Recap

2 + 2 = __ 12+ 8

2 + 2 = __ 12+ 8

Internalizenarrow

patterns

Recap

2 + 2 = __ 12+ 8

2 + 2 = __ 12+ 8

add all the numbers

ops go on left side

= means “get the total”

2 + 7 = 6 + __

The account makes specific predictions

Performance should decline between ages 7 and 9

Traditional practice with arithmetic hinders performance

Modified arithmetic practice will help

The account makes specific predictions

Performance should decline between ages 7 and 9

Traditional practice with arithmetic hinders performance

Modified arithmetic practice will help

Performance should get worse from 7 to 9

Why? Continue gaining narrow practice w/ arithmetic Strengthening representations that hinder

performance

But… Constructing increasingly sophisticated logical

structures General improvements in working memory Proficiency with basic arithmetic facts increases

Performance as a function of age

Age (years;months)

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The account makes specific predictions

Performance should decline between ages 7 and 9

Traditional practice with arithmetic hinders performance

Modified arithmetic practice will help

The account makes specific predictions

Performance should decline between ages 7 and 9

Traditional practice with arithmetic hinders performance

Modified arithmetic practice will help

Traditional practice with arithmetic should hurt

Why? Activates representations of operational patterns

But… Decomposition Thesis “Back to basics” movement Practice should “free up” cognitive resources for

higher-order problem solving

3 + 4 + 5 = 3 + __SetReadySolve

Performance by practice conditionPerc

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Practice condition

Performance should decline between ages 7 and 9

Traditional practice with arithmetic hinders performance

Modified arithmetic practice will help

The account makes specific predictions

Performance should decline between ages 7 and 9

Traditional practice with arithmetic hinders performance

Modified arithmetic practice will help

The account makes specific predictions

Performance by elementary math country

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Elementary math country

Interview data

Experience in the United States

Experience in high-achieving countries

1 + 1 = 21 + 2 = 31 + 3 = 4…

2 + 1 = 32 + 2 = 42 + 3 = 5…

9 + 1 = 109 + 2 = 119 + 3 = 12…

1 + 3 = 44 = 1 + 32 + 2 = 4

2 + 4 = 66 = 2 + 46 = 1 + 5

9 + 3 = 1212 = 9 + 38 + 4 = 12

Effect of problem format

Participants 7- and 8-year-old children (M age = 8 yrs, 0 mos;

N = 90)

Design Posttest-only randomized experiment (plus follow

up)

Basic procedure Practice arithmetic in one-on-one sessions with

“tutor” Complete assessments (math equivalence and

computation)

Smack it (traditional format)

9 + 4 = __ 7 + 8 = __

2 + 2 = __ 4 + 3 = __

Smack it (traditional format)

7

9 + 4 = __ 7 + 8 = __

2 + 2 = __ 4 + 3 = __

Smack it (nontraditional format)

7

__ = 9 + 4 __ = 7 + 8

__ = 2 + 2 __ = 4 + 3

Snakey Math (traditional format)

Snakey Math (nontraditional format)

Understanding of mathematical equivalence Reconstruct math equivalence problems after

viewing (5 sec) Define the equal sign Solve and explain math equivalence problems

Computational fluency Math computation section of ITBS Single-digit addition facts (reaction time and

strategy)

Follow up Solve and explain math equivalence problems (with

tutelage)

Assessments

Summary of sessions

Week 1 Week 2 Week 3 Weeks 4-6

Traditionalformat

Practice Session 1

Practice Session 2

10 min practice

Assessments

Follow up

Nontraditionalformat

Practice Session 1

Practice Session 2

10 min practice

Assessments

Follow up

Control Assessments PracticeSessions

homework

homework

homework

homework

Understanding of math equivalence by condition

Arithmetic practice condition

Follow-up performance by condition

Arithmetic practice condition

Computational fluency by condition

Measure Control Traditional Nontraditional

Accuracy% correct (SD) 86 (26) 90 (25) 92 (14)

Reaction timeM (SD) 9.16 (6.80) 6.98 (3.86) 7.64 (4.08)

ITBS scoreM NCE (SD) 52.65

(20.14)53.00

(20.35)53.32

(18.08)

Computational fluency by condition

Measure Control Traditional Nontraditional

Accuracy% correct (SD) 86 (26) 90 (25) 92 (14)

Reaction timeM (SD) 9.16 (6.80) 6.98 (3.86) 7.64 (4.08)

ITBS scoreM NCE (SD) 52.65

(20.14)53.00

(20.35)53.32

(18.08)

Interview data

Experience in the United States

Experience in high-achieving countries

1 + 1 = 21 + 2 = 31 + 3 = 4…

2 + 1 = 32 + 2 = 42 + 3 = 5…

9 + 1 = 109 + 2 = 119 + 3 = 12…

1 + 3 = 44 = 1 + 32 + 2 = 4

2 + 4 = 66 = 2 + 46 = 1 + 5

9 + 3 = 1212 = 9 + 38 + 4 = 12

Effect of problem grouping/sequence

Participants 7- and 8-year-old children (N = 104)

Design Posttest-only randomized experiment (plus follow

up)

Basic procedure Practice arithmetic in one-on-one sessions with

“tutor” Complete assessments (math equivalence and

computation)

4 + 6 = __

4 + 5 = __

Traditional grouping

4 + 4 = __

4 + 3 = __ In this example:4 + n

6 + 4 = __

5 + 5 = __

Nontraditional grouping

4 + 6 = __

3 + 7 = __ In this example:sum is equal to 10

Understanding of math equivalence by condition

Arithmetic practice condition

Follow-up performance by condition

Arithmetic practice condition

Computational fluency by condition

Measure Control Traditional Nontraditional

Accuracy% correct (SD) 94 (10) 94 (11) 98 (6)

Reaction timeM (SD) 5.30 (2.60) 5.56 (2.59) 4.30 (1.56)

ITBS scoreM NCE (SD) 33.26

(14.22)50.35

(17.69)50.86

(13.49)

Computational fluency by condition

Measure Control Traditional Nontraditional

Accuracy% correct (SD) 94 (10) 94 (11) 98 (6)

Reaction timeM (SD) 5.30 (2.60) 5.56 (2.59) 4.30 (1.56)

ITBS scoreM NCE (SD) 33.26

(14.22)50.35

(17.69)50.86

(13.49)

Performance declines between ages 7 and 9

Traditional practice with arithmetic hinders performance

Modified arithmetic practice helps

Summary

Implications

Theoretical Misconceptions not always due to something

children lack Limits of Decomposition Thesis Learning may not spur conceptual reorganization

Practical Early math shouldn’t be dominated by traditional

arithmetic May be able to facilitate transition from

arithmetic to algebra by modifying early arithmetic practice

Special thanks

Institute of Education Sciences (IES) Grant R305B070297

Members of the Cognition Learning and Development Lab at the University of Notre Dame

Martha Alibali and the Cognitive Development & Communication Lab at the University of Wisconsin

Administrators, teachers, parents, and students

Curry K. Software (helped us adapt Snakey Math)

2 + 2 4 + 8

What other types of input might matter?