Tag Archives: working memory

Memorization or Learning?

I originally started to write a post on a memorization technique I’m using for the unit circle, and went looking for representative jeremiads both pro and con. Instead, I found Ben Orlin’s piece When Memorization Gets in the Way of Learning (from five years back):


…which is the opposite of a standard, boring piece and serves as a good counterpoint to explain some recent shifts in my pedagogy.

It’s a good piece. In many ways, the debate about memorization runs parallel to the zombie problem–students regurgitate facts without understanding. Ben’s against that. Me, too. Ben says that testing requirements create tensions between authentic learning and manageable tests; I have various means of ensuring my students understand the math rather than just hork it up like furballs of unknown origin, so am less concerned on that point.

But I don’t agree with this sentiment as much as I probably did a decade ago: Memorizing a list of prepositions isn’t half as useful as knowing what role a preposition plays in the language. 

Not in math, anyway.


A couple years ago, after I’d taught trigonometry two or three times, I suddenly noticed that at the end of the year, my students were very fuzzy on their unit circle knowledge. (It’s no coincidence that Ben’s article and my observations are both focused on trigonometry, a branch of math with a significant fact base.) When working trig equations, they’d factor something like the equation above, use the Zero Product Property, solve for sin(x)…and then stop.

“You’re not done,” I’d point out. You’ve only solved for sin(x). What is the value of x?”

Shrug. No recognition. My tests are cumulative. Many students showed significant recall of concepts. They were using ratios to solve complex applications; they were sketching angles on the coordinate plane–both concepts we hadn’t revisited in months. They could sketch the unit circle from memory and eventually figure out the answer. But they had no automatic memories of the unit circle working backwards and forwards, even though I had emphasized the importance of memorizing it.

Upsetting, particularly at the end of the year. The name of the class is Trigonometry, after all. Solving for sin(x) requires not one tiny bit of trig. It’s all algebra. Trigonometry enters the picture when you ask yourself what angle, in radians or degrees, has a y to r ratio of 1 to 2.

The sine of π/2 is not among [the important things to memorize]. It’s a fact that matters only insofar as it connects to other ideas. To learn it in isolation is like learning the sentence “Hamlet kills Claudius” without the faintest idea of who either gentleman is–or, for what matter, of what “kill” means.

Well, okay, but….if a student in a Trig class can’t work a basic equation without a cheat sheet, what exactly has he learned? He already knew the algebra. Does the same standard hold for SOHCAHTOA, or can I still assume the student has successfully learned something if he needs a memory aid to remember what triangle sides constitute the sine ratio? What else can be on the cheat sheet: the Pythagorean Theorem? The ratios of the special rights?

Ben describes memorization as learning an isolated fact through deliberate effort, either through raw rehearsal or mnemonics, both of which he believes are mere substitutions for authentic learning. He argues for building knowledge through repeated use.

Sure. But that road is a hard one. And as Ben knows much better than I, the more advanced math gets, the more complex and numerous the steps get. Most students won’t even bother. Those who care about their grades but not the learning will take the easier, if meaningless route of raw rehearsal.

So how do you stop students from either checking out or taking the wrong road to zombiedom?

I’ve never told my students that memorization was irrelevant, but rather that I had a pretty small list of essential facts. Like Ben, I think useful memorization comes with repeated use and understanding. But what if repeated use isn’t happening in part because of the pause that occurs when memory should kick in?

So I’ve started to focus in on essential facts and encouraged them to memorize with understanding. Not rote memorization. But some math topics do have a fact base, or even just a long procedural sequence, that represent a significant cognitive load, and what is memorization but a way of relieving that load?

The trick lies in making the memorization mean something. So, for example, when I teach the structure of a parabolas, I first give the kids a chance to understand the structure through brief discovery. Then we go through the steps to graph a parabola in standard form. Then I repeat. And repeat. And repeat. And repeat. So by the time of the first quiz, any student who blanks out, I say “Rate of Change?” and they reflexively look for the b parameter and divide by 2. Most of them have already written the sequence on their page. The memorization of the sequence allows them repeated practice.

But it’s not mindless memorization, either. Ask them what I mean by “Rate of Change”, they’d say “the slope between the y-intercept and the vertex”. They don’t know all the details of the proof, but they understand the basics.

I take the same approach in parent function transformations, after realizing that a third of any class had drawn parent functions for days without ever bothering to associate one graph’s shape with an equation. So I trained them to create “stick figures” of each graph:stickfigures

I drew this freehand in Powerpoint, but it’s about the same degree of sloppiness that I encourage for stick figures. They aren’t meant to be perfect. They’re just memory spurs. Since I began using them a year ago, all my students can produce the stick figures and remind themselves what graph to draw. They know that each of the functions is committed on a line (to various degrees). Most of them understand, (some only vaguely), why a reciprocal function has asymptotes and why square root functions go in only one direction.

So did they learn, or did they memorize?

I haven’t changed my views on conceptual learning. I believe “why” is essential. I’m not power pointing my way through procedures. I am just realizing, with more experience, that many of my students won’t be able to use facts and procedures without being forced to memorize, and it is through that memorization that they become fluid enough to become capable of repeated use.

Like Ben, I think a zombie student with no idea that cosine is a ratio, but knows that cos(0) = 1, has failed to learn math. I just don’t think that student is any worse than one who looks at you blankly and has no answer at all. And addressing the needs of both these students may, in fact, be more memorization. Both types of students are avoiding authentic understanding. It’s our job to help them find it.

So I’ll give an example of that in my next post.

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.