Working Memory, Attention, and Mathematical Problem Solving: A longitudinal study of Grade 1 Children at Risk and Not a

1. 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

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. Grade 1 Classification Data

11. 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

12. 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 Assessments Administered to Students Each Year (30 measures)

14. 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

15. 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

18. 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

20. Growth Modeling: Results related to Fixed Effects

21. Growth Modeling: Results related to Fixed Effects

22. 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

26. 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%

27. 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 ?)

28. 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

29. 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.

30. 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.