working memory, attention, and mathematical problem solving: a longitudinal study of grade 1...
TRANSCRIPT
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.