Modifying arithmetic practice to promote understanding of mathematical equivalence - PowerPoint PPT Presentation

About This Presentation
Title:

Modifying arithmetic practice to promote understanding of mathematical equivalence

Description:

Mathematical equivalence is a fundamental concept in algebra ... Modified arithmetic practice will help. The account makes specific predictions ... – PowerPoint PPT presentation

Number of Views:53
Avg rating:3.0/5.0
Slides: 56
Provided by: nicolem98
Learn more at: https://ies.ed.gov
Category:

less

Transcript and Presenter's Notes

Title: Modifying arithmetic practice to promote understanding of mathematical equivalence


1
Modifying arithmetic practice to promote
understanding of mathematical equivalence
Nicole M. McNeil University of Notre Dame
2
Seemingly straightforward math problem
3 5 4 __ 3 5 __ 2 3 5 6 3 __
Mathematical equivalence problems
3
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

4
Most children in U.S. do not solve them correctly
16
of children who solved problems correctly
Study
5
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

6
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 __
7
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 __
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

9
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 __
10
Operations on left side problem format
11
Operations on left side problem format
12
Operations on left side problem format
13
Equal sign as operator
Child participant video will be shown
14
Add all the numbers
Child participant video will be shown
15
Recap
12 8
2 2 __
12 8
2 2 __
3 4 5 3 __
16
Recap
12 8
2 2 __
Internalize narrow patterns
12 8
2 2 __
17
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 __
18
The account makes specific predictions
  • Performance should decline between ages 7 and 9
  • Traditional practice with arithmetic hinders
    performance
  • Modified arithmetic practice will help

19
The account makes specific predictions
  • Performance should decline between ages 7 and 9
  • Traditional practice with arithmetic hinders
    performance
  • Modified arithmetic practice will help

20
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

21
Performance as a function of age
Percentage of children who solved correctly
Age (yearsmonths)
22
The account makes specific predictions
  • Performance should decline between ages 7 and 9
  • Traditional practice with arithmetic hinders
    performance
  • Modified arithmetic practice will help

23
The account makes specific predictions
  • Performance should decline between ages 7 and 9
  • Traditional practice with arithmetic hinders
    performance
  • Modified arithmetic practice will help

24
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

25
3 4 5 3 __
Set
Ready
Solve
26
Performance by practice condition
Percentage of undergrads who solved correctly
Practice condition
27
The account makes specific predictions
  • Performance should decline between ages 7 and 9
  • Traditional practice with arithmetic hinders
    performance
  • Modified arithmetic practice will help

28
The account makes specific predictions
  • Performance should decline between ages 7 and 9
  • Traditional practice with arithmetic hinders
    performance
  • Modified arithmetic practice will help

29
Performance by elementary math country
Percentage of undergrads who solved correctly
Elementary math country
30
Interview data
  • Experience in the United States
  • Experience in high-achieving countries

31
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)

32
Smack it (traditional format)
9 4 __
7 8 __
2 2 __
4 3 __
33
Smack it (traditional format)
9 4 __
7 8 __
7
2 2 __
4 3 __
34
Smack it (nontraditional format)
__ 9 4
__ 7 8
7
__ 2 2
__ 4 3
35
Snakey Math (traditional format)
36
Snakey Math (nontraditional format)
37
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)

38
Summary of sessions
homework
homework
homework
homework
39
Understanding of math equivalence by condition
Arithmetic practice condition
40
Follow-up performance by condition
Arithmetic practice condition
41
Computational fluency by condition
42
Computational fluency by condition
43
Interview data
  • Experience in the United States
  • Experience in high-achieving countries

44
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)

45
Traditional grouping
4 6 __
4 5 __
4 4 __
4 3 __
In this example 4 n
46
Nontraditional grouping
6 4 __
5 5 __
4 6 __
3 7 __
In this example sum is equal to 10
47
Understanding of math equivalence by condition
Arithmetic practice condition
48
Follow-up performance by condition
Arithmetic practice condition
49
Computational fluency by condition
50
Computational fluency by condition
51
Summary
  • Performance declines between ages 7 and 9
  • Traditional practice with arithmetic hinders
    performance
  • Modified arithmetic practice helps

52
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

53
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)

54
2 2 ? 4 8
55
What other types of input might matter?
Write a Comment
User Comments (0)
About PowerShow.com