Working Memory, Attention, and Mathematical Problem Solving:

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• Working Memory, Attention, and Mathematical
  Problem Solving
• A longitudinal study of Grade 1 Children at Risk
  and Not at Risk for Serious Math Difficulties
• H. Lee Swanson
• University of California-Riverside
• June, 2010

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Key Contributors

• Dr. Margaret Beebe-Frankenberger, Project
  Director
• Bev Hedin Project Management-School Liaison
• Doctoral Students Diana Dowds, Rebecca Gregg,
  Georgia Doukas,James Lyons, Olga Jerman, Kelly
  Rosston,Xinhua Zheng, Krista Healy
• Funded by the U.S. Department of Education,
  Institute of Education Sciences/Cognition and
  Student Learning

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General Significance Mathematics and Learning
Disabilities

• Students at risk for mathematical disabilities
  are a large segment of the public school
  population
• There is a need to know the processes that
  underlie problem-solving difficulty in such a
  large population.

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• Previous studies as well as our own have shown
  that a significant proportion of the variance
  related to solution accuracy in word problems is
  related to WM, but the specific sources of
  variance and its relationship to growth have not
  been clearly identified.

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Assumptions

• To comprehend and solve mathematical word
  problems one must be able to keep track of
  incoming information. This is necessary in order
  to understand words, phrases, sentences, and
  propositions that, in turn, are necessary to
  construct a coherent and meaningful
  interpretation of word problems. We assume that
  this keeping track of information draws upon WM.

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Research Questions

• Which components of WM (central executive,
  phonological loop, visual-spatial sketch pad) are
  most directly related to components of word
  problem solving (e.g., problem representation,
  solution planning, solution execution) ?
• Specifically,we will determine whether growth in
  WM moderates growth in components of problem
  solving and how these relationships vary within
  and between ability groups.

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Research Question 2

• 2. What cognitive mechanisms and academic skills
  underlie the relationship between WM and problem
  solving accuracy?
• Specifically, we explore the role of several
  processes (e.g., distractibility, controlled
  attention, phonological processing) and knowledge
  base (e.g., calculation, reading, knowledge of
  word problem solving components) in moderating
  growth in WM and word problem solving.

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Research Question 3

• 3. Does growth in WM have varying effects on word
  problem solving as a function of MD vs. Non MD
  groups?
• We explore if growth in problem solving is
  isolated to growth in specific components of WM.

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Sample

• Participants were selected from both public and
  private schools from grades 1 -two groups were
  identified.
• Children who score above the 40th percentile on
  standardized measures of mathematical
  problem---such children were not considered as at
  risk for math difficulties
• Children who score below the 25th percentile
  (below a scale score of 8) on the measures of
  word problem solving and number naming speed were
  considered at risk and eligible for further
  screening.

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Grade 1 Classification Data
Total Sample Total Sample Math Disabled Math Disabled Average Achievers Average Achievers
Variable N Mean SD N Mean SD N Mean SD
Age (MOS) 127 79.63 8.11 42 80.21 3.88 85 79.34 9.54
Fluid Intelligence (Raven) 127 107.61 15.08 42 101.43 12.46 85 110.66 15.39
Computation (Math-WISC-III) 127 9.61 4.01 42 5.12 2.3 85 11.82 2.54
Rapid Digit Naming (CTOPP) 127 9.87 2.06 42 8.57 1.95 85 10.51 1.80
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Latent Class Analysis

• 1. Because our classification criteria differ
  considerably from studies that focus primarily on
  calculation abilities, we determined the
  stability of our classification.
• 2. We performed a latent transitional class
  analysis on the two classification tasks
  (arithmetic subtest of WISC-III, digit naming
  speed from CTOPP) utilizing the SAS LTA (Latent
  Transitional Analysis) program (Lanza, Lemon,
  Schafter, Collins, 2008).
• 3. The latent transition probability that latent
  class membership was maintained at the next point
  in time (year 3) contingent on latent class
  membership at grade 1 was 1.00. The estimated
  probability that a child was assigned to the
  correct latent class at grade 3 based on the
  WISC-III was 1.0, whereas the estimated probably
  was .89 for the digit naming speed task.
• 4. Because the literature suggests that math
  disabilities and reading disabilities are
  comorbid, children meeting or not meeting SMD in
  grade 1 were further divided into subgroups of
  children yielding relatively low or high reading
  scores (lt or equal 35th percentile vs. gt than
  the 35th in word recognition on the WRAT-3).
  The latent transition probability for children
  with math disabilities-alone at grade 1 sharing
  both math and reading difficulties at grade 3 was
  .16.
• Point. There does not appear to be support in
  this data set for the notion that children with
  SMD at grade 1 reflect children with late
  emerging reading difficulties

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Assessments Administered to Students Each Year
(30 measures)

• Arithmetic (WRAT-3, WIAT)
• Raven Progressive Matrices Test (fluid
  Intelligence)
• Random Letter and Number Generation (inhibition)
• Battery of STM and WM tasks
• Fluency (speed at naming words that with letter B
  and animals)
• Updating

• Word problems
• Components of Word Problems
• Computation and Computation fluency skills (CBM)
• Phonological Awareness (Real word, Pseudo-word
  Efficiency from the TOWRE, Elision-CTOPP)
• Rapid naming speed from the CTOPP
• Word attack,identification, and comprehension
  subtests (WRMT-R)
• Connors Behavior Rating Scale

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Composite Scores

• Knowledge baseCalculation (WIAT, WRAT), Reading,
  Knowledge of Problem Solving Component
• Controlled AttentionRandom Generation,
  Fluency-inhibitioncategorization and words
• Distractibility Connors Teacher Rating
• Speedrapid naming of letters and numbers
• STM-Forward Digit, Words, Nonwords
• Visual-WMMatrix, Mapping Directions
• ExecutiveUpdating, Listening Span, Conceptual
  Span

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Regression Model Predicting Grade 3 Problem
Solving Accuracy from Grade 1 Latent Measures

• WM onlyModel 1
• Attention/inhibition measures -Model 2
• Phonological/Storage-Model 3
• General Reading-Ability-Model 4
• Mathematical Knowledge Base-Model 5

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Prediction of Problem Solving at Grade 3 from Grade 1 Latent Measures Prediction of Problem Solving at Grade 3 from Grade 1 Latent Measures
Model 1 B SE ß
R2.50, F(3,96)32.45, p lt .001 R2.50, F(3,96)32.45, p lt .001 R2.50, F(3,96)32.45, p lt .001
WM-Phon. 2.08 0.21 0.95
WM-Visual -0.28 0.18 -0.11
WM-Exec 1.55 0.85 0.85
Model 2-Attention
R2.51, F(6,84)13.75, p lt .001 R2.51, F(6,84)13.75, p lt .001 R2.51, F(6,84)13.75, p lt .001
Inattention -0.009 0.009 -0.04
Random 0.05 0.25 0.02
Inhibition -.41 0.19 -0.21
WM-Phon. 2.13 0.28 0.95
WM-Visual -0.22 0.21 -0.09
WM-Exec 1.48 0.21 0.82
Model 3-Reading/Naming Speed
R2.55, F(5,94)22.66, p lt .001 R2.55, F(5,94)22.66, p lt .001 R2.55, F(5,94)22.66, p lt .001
Reading 0.35 0.25 0.18
Naming Speed -.45 0.2 -0.2
WM-Phon. 1.62 0.27 0.78
WM-Visual -0.57 0.22 -0.22
WM-Exec 1.64 0.18 0.9
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Model 4-Phonological Processes B SE ß
R2.57, F(4,95)31.67, p lt .001
Raven -.004 .02 -.01
Phonological .28 .28 .15
Naming Speed -.46 .20 -.20
WM-Phon. 1.63 .31 .78
WM-Visual -.49 .20 -.18
WM-Exec 1.61 .19 .88
Model 5-Knowledge Base
R2.61, F(8,91)18.07, p lt .001
Calculation (Grade 3) -.07 .19 -.03
Raven -.01 .02 -.04
Reading .53 .28 .29
Inhibition -.68 .16 -.32
Naming Speed -.66 .19 -.29
WM-Phon. 1.73 .29 .83
WM-Visual -.35 .19 -.13
WM-Exec 1.74 .18 .95
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Hierarchical Model of Growth

• Hierarchical Linear Modeling---Focus on Growth
  and Random Effects
• Key points in the interpretation---
• Intercepts centered at wave 3
• Random Effects are related to wave 1 classroom
  instruction

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Growth Modeling Results related to Fixed Effects
At-risk At-risk     Not at Risk Not at Risk Not at Risk
  Estimate Estimate SE SE Estimate Estimate SE F-ratio F-ratio
Problem Solving                
Intercept 0.71 0.71 0.1 0.1 1.20 1.20 0.07 8.14 8.14
Growth 0.76 0.76 0.06 0.06 0.39 0.39 0.04 13.42 13.42
Math
Intercept 1.75 1.75 0.21 0.21 3.02 3.02 0.15 12.20 12.20
Growth 1.11 1.11 0.08 0.08 1.43 1.43 0.05 5.94 5.94

Reading
Intercept 1.18 1.18 0.12 0.12 1.78 1.78 0.08 8.82 8.82
Growth 0.87 0.87 0.04 0.04 0.7 0.7 0.03 5.78 5.78

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Growth Modeling Results related to Fixed Effects
At-risk SMD   Not at Risk Not at Risk Not at Risk
  Estimate Estimate SE Estimate SE F-ratio F-ratio
Phon-loop (STM)
Intercept 0.20 0.20 0.04 0.33 0.03 3.84 3.84
Growth 0.18 0.18 0.02 0.23 0.01 2.72 2.72
Sketchpad
Intercept 0.62 0.62 0.08 0.89 0.06 3.64 3.64
Growth 0.43 0.43 0.05 0.58 0.03 3.44 3.44
Executive
Intercept 0.38 0.38 0.06 0.69 0.04 9.42 9.42
Growth 0.28 0.28 0.03 0.38 0.02 3.92 3.92
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Growth Modeling-Unconditional Means Model For
Problem Solving Accuracy

• Unconditional Means Model
• Random Effects
• Parameter Variance SE
• Intercept 0.24 0.07
• Growth 0.06 0.03
• Residual 0.24 0.03
• Fit Statistics
• Deviance 700.6
• AIC 712.6
• BIC 729.7
• Fixed Effects
• Effect Estimate SE
• Intercept 1.04 0.06
• Growth 0.51 0.03

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Unconditional Mean Model Unconditional Mean Model Unconditional Mean Model Conditional Means Model Conditional Means Model Conditional Means Model Reduced Means Model Reduced Means Model Reduced Means Model
Fixed Effects Fixed Effects
Parameter Estimate SE Parameter Estimate SE Parameter Estimate SE
Intercept 1.04 0.06 Intercept 1.00 0.06 Intercept 1.00 0.06
Growth 0.51 0.03 Inhibition 0.03 0.05 Inhibition - -
Speed 0.08 0.1 Speed - -
WM-Ph. .23 0.06 WM-Ph. .21 0.06
WM-Vis 0.003 0.05 WM-Vis - -
WM-Exec .20 0.06 WM-Exec .19 0.06
Growth .52 0.13 Growth .48 0.04
Inhibition -.12 0.04 Inhibition -.12 0.03
Speed .11 0.04 Speed .08 0.03
WM-Ph. 0.09 0.07 WM-Ph. - -
WM-Vis 0.03 0.03 WM-Vis - -
WM-Exec -.11 0.05 WM-Exec - .08 0.04
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Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model Growth Modeling for Unconditional, Conditional and Reduced Model

Unconditional Mean Model Unconditional Mean Model Unconditional Mean Model Unconditional Mean Model Unconditional Mean Model Conditional Means Model Conditional Means Model Conditional Means Model Conditional Means Model Conditional Means Model Reduced Means Model Reduced Means Model Reduced Means Model Reduced Means Model Reduced Means Model
Random Effects Random Effects Random Effects Random Effects
Parameter Parameter Variance Variance SE Parameter Parameter Variance Variance SE Parameter Parameter Variance Variance SE

Intercept Intercept 0.24 0.24 0.07 Intercept Intercept 0.15 0.15 0.05 Intercept Intercept 0.15 0.15 0.05
Slope Slope 0.06 0.06 0.03 Slope Slope 0.04 0.04 0.02 Slope Slope 0.04 0.04 0.02
Residual Residual 0.25 0.25 0.03 Residual Residual 0.23 0.23 0.03 Residual Residual 0.23 0.23 0.03
Fit Statistics Fit Statistics Fit Statistics Fit Statistics Fit Statistics Fit Statistics Fit Statistics Fit Statistics Fit Statistics Fit Statistics
Deviance Deviance 700.6 700.6 Deviance 532.2 532.2 532.2 Deviance 535.1 535.1 535.1
AIC AIC 712.6 712.6 AIC 564.2 564.2 564.2 AIC 557.1 557.1 557.1
BIC BIC 729.7 729.7 BIC 606.4 606.4 606.4 BIC 586.1 586.1 586.1

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Explained Variance

• What is the reduction in random effects related
  to classroom on problem solving when individual
  differences in cognitive processes are taken into
  consideration?
• (Focus is on Explainable Variance)
• Between Level of Performance Differences nested
  within Classroom (Intercept)
• Problem solving (.24-.15)/.2438
• Between Growth Differences nested within
  Classroom (Slope)
• Problem solving (.06-.04)/.0633

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• Problem Solving--Intercept 1.0
• Problem Solving-Slope .52
• WM-Exec--Intercept .20
• WM-Exec -slope -.08
• Interpretation-
• 1.0 estimates problem solving when predictors are
  set to zero
• Children who differ by 1 point on WM-Exec
• differ by .20 points on problem solving
• .52 estimates growth for each testing session in
  Problem Solving
• The parameter estimate of -.08 related to the
  slope indicates that children who differed by 1.0
  with respect to WM-Exec have growth rates that
  differ by -.08 (higher levels of WM yield smaller
  growth rates ?)

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Summary

• 1. Ability group differences emerged across the
  majority of cognitive measures
• ---classification criteria robust at final
  wave-classification holds on measures (wave 1 and
  3)
• 2. Of the wave 1 cognitive predictors, WM,
  Inhibition and naming speed uniquely predicted
  Wave 3 problem solving Accuracy.
• 3. Growth in Executive System of WM, naming
  speed, and Inhibition moderated Growth in Problem
  Solving Accuracy

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Summary Cont.

• 4.Not merely a function of low order skills--- WM
  contributes unique variance to problem solving
  beyond the contribution of fluid intelligence,
  reading and computation skill, phonological
  processing, STM, and processing speed.
• 5. Not merely a function of specific executive
  activities identified in this study--- WM
  contributes to problem solving beyond measures of
  inhibition and activation of LTM (measures of
  math and reading skill)---processes related to
  executive processing.

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Caveats

• 1. Some measures not behaving as they do with
  adults.
• 2. Collinearity related to some measures (e.g.,
  correlation between latent measures highe.g.,
  STM and WM-EX, .83, Phon. Awareness Reading
  .95)
• 4. Reconsidering Digit Naming classification
  criteria (naming speed for numbers may not be
  stable)
• 5. Not instigating a direct intervention on WM
  (currently in progress)
• 6. Results are correlational---must be followed
  up with causal models
• 7. Have not isolated the source of variance
  related to the WM residual.