modifying arithmetic practice to promote understanding of mathematical equivalence nicole m. mcneil...
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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%
% o
f ch
ildre
n w
ho
solv
ed
pro
ble
ms
corr
ect
ly
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)
Perc
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corr
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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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nd
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olv
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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
Perc
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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?