Intelligence and Algebra

A few years back, I was reading up on an interesting theory about working memory, as it seemed hopeful that working memory, not IQ alone, had some predictive value in learning algebra. I periodically google for new results in any field I’m interested in—not interested enough to pay for access to the papers, but I can still find out quite a bit just from google.

Recently, I stumbled across this paper: Do Measures of working memory predict academic proficiency better than measures of intelligence?

If I understand the paper correctly, working memory has been shown to be more predictive than IQ in “hierarchical regression analyses conducted with observed variables”. In this study, by contrast,

we examined the relationships between measures of intelligence, working memory and academic proficiency using latent variables in structural equation models. One advantage in using latent versus observed variables is that measurement errors are modelled explicitly. It is widely acknowledged that measures of working memory and executive functioning are not pure (e.g., Rabbitt, 1997; Miyake et al., 2000). In regression analyses, measurement errors are confounded with true measures of the constructs in question. By using multiple indicators, extracting their common variance, and modelling measurement errors explicitly, latent level analyses provide a more precise examination of relationships between conceptual constructs.

This sort of stuff always makes my head hurt. But I believe they are saying that they used a different method that doesn’t rely on observed variables to see if working memory is actually more predictive than IQ.

They tested three models:

  1. “Model 1 assumed independence between the working memory and intelligence constructs.”
  2. “Model 2 is analogous to the hierarchical regression.” (that is, working memory is more predictive”
  3. “Model 3 postulated a direct path from the working memory latent variable to intelligence and the path from working memory to algebraic proficiency was fixed at zero.”

Their findings:

In summary, our analyses of the data from all three studies show good support for Model 3. Only intelligence has a significant direct path to algebraic proficiency. At best, working memory has only an indirect effect on algebraic proficiency.

Okay, so here’s the thing: whether it’s intelligence or working memory notwithstanding, algebra proficiency is linked to cognitive ability.

The researchers are using “only” in the context of the comparison with working memory, but it’s still an amazing statement. Of course, there’s no follow-up research to determine the depth of algebra proficiency’s link to cognitive ability. We don’t know if there’s a basement to the cognitive ability needed to learn algebra We haven’t investigated whether different curriculum or instruction methods are needed for high vs. low cognitive ability students.

Any such research might explain the achievement gap in a most compelling and entirely unsatisfactory manner. So we won’t do the research.

Only intelligence has a significant direct path to algebraic proficiency.

I understand the genuine difficulties in acknowledging reality. I understand the fear of potential outcomes. But we’re spending billions, wasting lives, and causing tremendous unhappiness.

Only intelligence has a significant direct path to algebraic proficiency.

I’ve said it before: I don’t teach groups. I teach individuals. And I see so many individuals who are lost at school, who have given up. It’s not the teachers. It’s not the students. It’s the expectations.

Only intelligence has a significant direct path to algebraic proficiency.

So it is written. But it can’t be said.

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4 responses to “Intelligence and Algebra

  • Steve Sailer

    I don’t have enough working memory anymore to understand the abstract. I blame Lipitor.

    Working memory is clearly a sizable chunk of IQ, but IQ kind of cheats by testing for a lot of different things, then coming up with a g factor that is hard to define, but also kind of the kind of thing that you know when you see it in a person.

    • educationrealist

      I know what you mean about the abstract. What the hell is a structural equation model when it’s at home, anyway? I have a colleague who is an actual math major and two days younger than my son, rot his soul. He tells me that this is a pretty common type of overview done to cast doubt on exactly the sort of claims it’s casting doubt on here.

      There’s no question in my mind that working memory is a big chunk of IQ, and I was very ready to believe that it had an impact on math proficiency independent of IQ. That is, I could believe that someone with a low IQ of, say, 95 but high working memory could succeed in algebra more than someone with an IQ of 115 but low working memory capacity. But I suppose it’s not to be.

      I see the impact of working memory limitations all the time in math. Multi-step equations come to mind. Kids lose track of what the hell they are doing. They did one long, complicated step (distributed on both sides of the equation, for example) and now they are looking at a totally different equation. What were they doing? It takes them a 30 second to evaluate the same problem and get back to where they were and what they were doing and realize that they need another step. Then they “cancel out” on both sides (3x+5+8x = 9+5x-2 they will subtract 3x from 8x and add 2 to 9), but I don’t think I can blame that on working memory.

      So what I’ve done to try to offset that limitation is give them practice at solving equations at every different possible point I can think of. So when they look at the next step, they don’t have to think of it as the same problem, but a new one–one they also know how to do. That’s helped.

  • C

    Interesting read…i am curious because my hearing impaired son who has very high intelligence scores but low working memory and processing speed, has developed remarkably in math over the past two years. I am curious how someone with reportedly low WM can handle complex algebraic expressions as he is. There are so many rules and so many steps! I suppose our scenario supports the research? Who’s to say the working memory issues were true LD though, may have matured a little differently and misdiagnosed due to being hard of hearing?

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