Modifying arithmetic practice to promote understanding of mathematical equivalence

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Modifying arithmetic practice to promote
understanding of mathematical equivalence
Nicole M. McNeil University of Notre Dame
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Seemingly straightforward math problem
3 5 4 __ 3 5 __ 2 3 5 6 3 __
Mathematical equivalence problems
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Why we care about these problems

• 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

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Most children in U.S. do not solve them correctly
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of children who solved problems correctly
Study
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Why dont 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 isnt arithmetic a building block?

• Knowledge of arithmetic should help, right?
• Childrens 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

12 8
2 2 __
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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

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

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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

3 4 5 __
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Operations on left side problem format
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Operations on left side problem format
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Operations on left side problem format
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Equal sign as operator
Child participant video will be shown
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Add all the numbers
Child participant video will be shown
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Recap
12 8
2 2 __
12 8
2 2 __
3 4 5 3 __
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Recap
12 8
2 2 __
Internalize narrow patterns
12 8
2 2 __
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Recap
12 8
ops go on left side
2 2 __
means get the total
add all the numbers
Internalize narrow patterns
12 8
2 2 __
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

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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

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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
Percentage of children who solved correctly
Age (yearsmonths)
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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

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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

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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 __
Set
Ready
Solve
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Performance by practice condition
Percentage of undergrads who solved correctly
Practice condition
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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

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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

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Performance by elementary math country
Percentage of undergrads who solved correctly
Elementary math country
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Interview data

• Experience in the United States
• Experience in high-achieving countries

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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)

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Smack it (traditional format)
9 4 __
7 8 __
2 2 __
4 3 __
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Smack it (traditional format)
9 4 __
7 8 __
7
2 2 __
4 3 __
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Smack it (nontraditional format)
__ 9 4
__ 7 8
7
__ 2 2
__ 4 3
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Snakey Math (traditional format)
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Snakey Math (nontraditional format)
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Assessments

• 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)

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Summary of sessions
homework
homework
homework
homework
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Understanding of math equivalence by condition
Arithmetic practice condition
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Follow-up performance by condition
Arithmetic practice condition
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Computational fluency by condition
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Computational fluency by condition
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Interview data

• Experience in the United States
• Experience in high-achieving countries

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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)

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Traditional grouping
4 6 __
4 5 __
4 4 __
4 3 __
In this example 4 n
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Nontraditional grouping
6 4 __
5 5 __
4 6 __
3 7 __
In this example sum is equal to 10
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Understanding of math equivalence by condition
Arithmetic practice condition
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Follow-up performance by condition
Arithmetic practice condition
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Computational fluency by condition
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Computational fluency by condition
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Summary

• Performance declines between ages 7 and 9
• Traditional practice with arithmetic hinders
  performance
• Modified arithmetic practice helps

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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 shouldnt 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?