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![Page 1: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/1.jpg)
Modifying arithmetic practice to promote understanding of mathematical
equivalence
Nicole M. McNeilUniversity of Notre Dame
![Page 2: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/2.jpg)
Seemingly straightforward math problem
Mathematical equivalence problems
3 + 5 = 4 + __
3 + 5 = __ + 2
3 + 5 + 6 = 3 + __
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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
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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
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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
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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
![Page 7: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/7.jpg)
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 = __
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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
![Page 9: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/9.jpg)
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 + __
![Page 10: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/10.jpg)
“Operations on left side” problem format
![Page 11: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/11.jpg)
“Operations on left side” problem format
![Page 12: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/12.jpg)
“Operations on left side” problem format
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Equal sign as operator
Child participant
video will be shown
![Page 14: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/14.jpg)
Add all the numbers
Child participant
video will be shown
![Page 15: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/15.jpg)
Recap
2 + 2 = __ 12+ 8
2 + 2 = __ 12+ 8 3 + 4 + 5 = 3 + __
![Page 16: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/16.jpg)
Internalizenarrow patterns
Recap
2 + 2 = __ 12+ 8
2 + 2 = __ 12+ 8
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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 + __
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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
![Page 19: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/19.jpg)
The account makes specific predictions
Performance should decline between ages 7 and 9
Traditional practice with arithmetic hinders performance
Modified arithmetic practice will help
![Page 20: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/20.jpg)
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
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Performance as a function of age
Age (years;months)
Perc
en
tag
e o
f ch
ildre
nw
ho s
olv
ed
corr
ect
ly
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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
![Page 23: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/23.jpg)
The account makes specific predictions
Performance should decline between ages 7 and 9
Traditional practice with arithmetic hinders performance
Modified arithmetic practice will help
![Page 24: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/24.jpg)
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
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3 + 4 + 5 = 3 + __SetReadySolve
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Performance by practice conditionPerc
en
tag
e o
f u
nd
erg
rad
sw
ho s
olv
ed
corr
ect
ly
Practice condition
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Performance should decline between ages 7 and 9
Traditional practice with arithmetic hinders performance
Modified arithmetic practice will help
The account makes specific predictions
![Page 28: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/28.jpg)
Performance should decline between ages 7 and 9
Traditional practice with arithmetic hinders performance
Modified arithmetic practice will help
The account makes specific predictions
![Page 29: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/29.jpg)
Performance by elementary math country
Perc
en
tag
e o
f u
nd
erg
rad
sw
ho s
olv
ed
corr
ect
ly
Elementary math country
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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
![Page 31: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/31.jpg)
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)
![Page 32: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/32.jpg)
Smack it (traditional format)
9 + 4 = __ 7 + 8 = __
2 + 2 = __ 4 + 3 = __
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Smack it (traditional format)
7
9 + 4 = __ 7 + 8 = __
2 + 2 = __ 4 + 3 = __
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Smack it (nontraditional format)
7
__ = 9 + 4 __ = 7 + 8
__ = 2 + 2 __ = 4 + 3
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Snakey Math (traditional format)
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Snakey Math (nontraditional format)
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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
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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
![Page 39: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/39.jpg)
Understanding of math equivalence by condition
Arithmetic practice condition
![Page 40: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/40.jpg)
Follow-up performance by condition
Arithmetic practice condition
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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)
![Page 42: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/42.jpg)
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)
![Page 43: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/43.jpg)
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
![Page 44: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/44.jpg)
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)
![Page 45: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/45.jpg)
4 + 6 = __
4 + 5 = __
Traditional grouping
4 + 4 = __
4 + 3 = __ In this example:4 + n
![Page 46: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/46.jpg)
6 + 4 = __
5 + 5 = __
Nontraditional grouping
4 + 6 = __
3 + 7 = __ In this example:sum is equal to 10
![Page 47: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/47.jpg)
Understanding of math equivalence by condition
Arithmetic practice condition
![Page 48: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/48.jpg)
Follow-up performance by condition
Arithmetic practice condition
![Page 49: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/49.jpg)
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)
![Page 50: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/50.jpg)
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)
![Page 51: Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e225503460f94b0e6e5/html5/thumbnails/51.jpg)
Performance declines between ages 7 and 9
Traditional practice with arithmetic hinders performance
Modified arithmetic practice helps
Summary
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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
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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)
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2 + 2 4 + 8
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What other types of input might matter?