Where And When Should Math Interventions Be Targeted?

How much of your math achievement later in life is due to your stable underlying characteristics like general cognitive abilities and personality, and how much is due to your prior math competency or knowledge? The answer to this question is important, because it can help researchers, educators, and policy makers know how to direct where and when math interventions might be targeted for the greatest payoff.

A large body of research shows that individual differences in math achievement are fairly stable from the time when kids enter school to when they exit it. In other words, early math achievement consistently predicts later math achievement, even much later achievement. And yet, another large body of research on early math interventions shows a “fadeout” effect: the effects of these interventions decrease quickly over time. So what might account for this seeming contradiction in the research literature?

Like many advances in science, it turns out that it has to do with research methods.

In a new paper just published in Psychological Science titled “State and Trait Effects on Individual Differences in Children’s Mathematical Development,” researchers Drew Bailey, Tyler Watts, Andrew Littlefield, and David Geary sought to solve this seeming paradox:

“We hypothesized that individual differences in children’s later mathematical knowledge are more an indicator of stable, underlying characteristics related to mathematics learning throughout development than of direct effects of early mathematical competency on later mathematical competency. We tested this hypothesis in two longitudinal data sets, by simultaneously modeling effects of latent traits (stable characteristics that influence learning across time) and states (e.g., prior knowledge) on children’s mathematics achievement over time. Latent trait effects on children’s mathematical development were substantially larger than state effects. Approximately 60% of the variance in trait mathematics achievement was accounted for by commonly used control variables, such as working memory, but residual trait effects remained larger than state effects.”

Basically, the authors uncovered that stable personal traits had a larger influence than prior knowledge on later math achievement. I asked the first author of the study, Drew Bailey, if he could give some insight into how this new “state-trait” method might influence not only the research literature, but also potential policy decisions when it comes to math interventions. For example, he thinks that these findings may suggest that later math interventions may be needed.

JON: What are the research method implications of your paper? For example, you note that nearly all longitudinal studies “have been confounded by unmeasured traits that may affect children’s mathematics achievement across development.”

DREW: I think the study highlights some methodological concerns of which we were already aware. Most importantly, it is very difficult to precisely estimate the effect of any one factor on skill development, especially without an experimental study. One problem is that so many factors contribute to differences in children's math learning. Previous math knowledge is important: try teaching someone how to add before they know how to count, or teaching them how to add fractions before they know how to add whole numbers. However, this previous knowledge, along with later learning, is likely influenced by many cognitive skills - including working memory, attention, and processing speed - and skills often labeled as non-cognitive, such as motivation or conscientiousness, not to mention effects of schools, teachers, and other external environmental influences. It's difficult to account for all of these factors, which you'd probably want to do if you want to estimate the effect of increasing children's early math achievement on their later math achievement. Even if we knew all of the skills that contributed to children's math learning (we don't), it is difficult to collect reliable and valid measures of all of them for the same children at several different times.

This is not an argument against trying to understand the factors that contribute to children's math learning; understanding how children learn math has important practical and theoretical implications, and it's one of my primary research interests. However, I think it is important to keep in mind that our account of the development of children's math learning is incomplete. When experimental and non-experimental data give different answers about the likely effect of improving early math skills on later math outcomes, resolving these discrepancies may further our understanding of how children learn math.

What are the educational policy implications of your paper (especially in regards to interventions)?

I am hesitant to make any policy recommendations based on this project. There's still a lot we don't know. For example, do some types of interventions affect the relatively stable factors underlying children's long term math development? It seems worth finding out, and some previous work suggests that the answer might be yes, at least for the most disadvantaged children.

Alternatively, perhaps children at risk for persistently low math achievement will need continuing support after the conclusion of a successful early intervention to maintain their relative math achievement advantage. I think it's important to clarify that I'm not arguing against early intervention. There are some very well thought out and effective early math interventions, and lots of work suggests that it is probably easier for low achieving children to catch up with their peers early in schooling than later, when children are asked to learn an increasingly complex set of information. My point is that a plausible solution to the problem of diminishing effects of earlier effective math instruction over time is more effective later math instruction.

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My hope is that the general theoretical framework for understanding children's math achievement trajectories that we presented in the paper will be useful for making predictions about how to structure interventions to have the largest effects on children's long-term math achievement outcomes. I hope that the knowledge gained through testing these predictions will have helpful policy implications.

© 2014 by Jonathan Wai

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