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

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 HealyFunded by the U.S. Department of Education, Institute of Education Sciences/Cognition and Student Learning

Key Contributors

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.

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.

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.

Research Questions

1. 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.

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.

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.

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.

Total Sample Math Disabled

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

FluidIntelligence(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

Grade 1 Classification Data

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 (< or equal 35th percentile vs. > 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

Assessments Administered to Students Each Year (30 measures)

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

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

Composite Scores

Knowledge base=Calculation (WIAT, WRAT), Reading, Knowledge of Problem Solving Component

Controlled Attention=Random Generation, Fluency-inhibition—categorization and words

Distractibility =Connors Teacher Rating Speed=rapid naming of letters and numbers STM-Forward Digit, Words, Nonwords Visual-WM=Matrix, Mapping & Directions Executive=Updating, Listening Span,

Conceptual Span

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

WM only—Model 1 Attention/inhibition measures -

Model 2 Phonological/Storage-Model 3 General Reading-Ability-Model 4 Mathematical Knowledge Base-

Model 5

Prediction of Problem Solving at Grade 3 from Grade 1 Latent Measures

Model 1 B SE β

R2=.50, F(3,96)=32.45, p < .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 < .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 < .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

Model 4-Phonological Processes B SE β

R2=.57, F(4,95)=31.67, p < .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 < .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

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

CONSTANT1.0

Intercept

Slope

wave 1 wave 2 wave 3

Storage

Attention control

rword1 psword1 digf1 CatF1 LetF1

lisspan1 lisspan2 lisspan3Conspan1 Conspan2 Conspan3update1 update2 update3

0.680.23

0.140.00

0.78 0.94

0.83* 0.49* 0.36* 0.46* 0.64* 0.42* 0.60* 0.65* 0.40*

0.32*0.55*0.60*0.57* 0.47*

0.88

0.03*

1.00*

-0.11*

0.20*

Figure X: EQS 6 growthall Chi Sq.=89.50 P=0.05 CFI=0.94 RMSEA=0.05

0.54*0.83*

0.77*

0.680.23

0.140.00

0.78 0.94

0.83* 0.49* 0.36* 0.46* 0.64* 0.42* 0.60* 0.65* 0.40*

0.32*0.55*0.60*0.57* 0.47*

0.88

0.03*

1.00*

-0.11*

0.20* 0.54*0.83*

0.77*

At-risk   Not at Risk

  Estimate SE Estimate SE F-ratio

Problem Solving        

Intercept 0.71 0.1 1.20 0.07 8.14**

Growth 0.76 0.06 0.39 0.04 13.42**

Math

Intercept 1.75 0.21 3.02 0.15 12.20***

Growth 1.11 0.08 1.43 0.05 5.94***

Reading

Intercept 1.18 0.12 1.78 0.08 8.82***

Growth 0.87 0.04 0.7 0.03 5.78**

Growth Modeling: Results related to Fixed Effects

At-risk SMD   Not at Risk

  Estimate SE Estimate SE F-ratio

Phon-loop (STM)

Intercept 0.20 0.04 0.33 0.03 3.84*

Growth 0.18 0.02 0.23 0.01 2.72

Sketchpad

Intercept 0.62 0.08 0.89 0.06 3.64*

Growth 0.43 0.05 0.58 0.03 3.44*

Executive

Intercept 0.38 0.06 0.69 0.04 9.42***

Growth 0.28 0.03 0.38 0.02 3.92*

Growth Modeling: Results related to Fixed Effects

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

Unconditional Mean Model Conditional Means Model Reduced Means Model

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

Working Memory and Problem Solving

-1

-0.5

0

0.5

1

1.5

Wave 1 Wave 2 Wave 3

Testing Waves

Z-s

core

s MD-WM

NMD-WM

MD-PS

NMD-PS

Growth Modeling for Unconditional, Conditional and Reduced Model

Unconditional Mean Model Conditional Means Model Reduced Means Model

Random Effects

Parameter Variance SE Parameter Variance SE Parameter Variance SE

Intercept 0.24*** 0.07 Intercept 0.15** 0.05 Intercept 0.15** 0.05

Slope 0.06* 0.03 Slope 0.04** 0.02 Slope 0.04** 0.02

Residual 0.25*** 0.03 Residual 0.23*** 0.03 Residual 0.23*** 0.03

Fit Statistics Fit Statistics Fit Statistics

Deviance 700.6 Deviance 532.2 Deviance 535.1

AIC 712.6 AIC 564.2 AIC 557.1

BIC 729.7 BIC 606.4 BIC 586.1

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)/.24=38% Between Growth Differences nested

within Classroom (Slope) Problem solving (.06-.04)/.06=33%

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-Execdiffer by .20 points on problem solving

.52 estimates growth for each testing session in Problem SolvingThe 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 ?)

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

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.

Caveats

1. Some measures not behaving as they do with adults.

2. Collinearity related to some measures (e.g., correlation between latent measures high—e.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.