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:

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.

October 30th, 2018 at 2:13 pm

[…] Source: Education Realist […]

October 30th, 2018 at 2:49 pm

I graduated from a Compsci/Math program, so as you might imagine I was quite derisive of memorization.

However learning a second language as a working adult has kicked my ass and caused me to be far more introspective about learning. I think every field is based on a set of facts, and if you memorize them understanding becomes more obvious or instinctual.

Efficient learning requires memorization then understanding.

October 30th, 2018 at 4:00 pm

Like you, I’m from a programming background, where memorizing isn’t that important. I also have a memory that works quite well with repeated use, so I memorize without conscious effort.

Not all learning requires memorization. Algebra, for example, requires relatively little. Trig, Geometry, and Calculus a lot more. History requires knowledge of a basic set of facts. English, less so.

November 1st, 2018 at 7:36 pm

“Every field is based on a set of facts, and if you memorize them understanding becomes more obvious or instinctual”. The Classical curriculum is based on that recognition that higher-order behaviors require background knowledge. In the grammar stage (grades 1-4), kids are taught the foundational language and facts associated with each discipline; in the logic stage (5-8), they deepen their knowledge and make connections and relationships; and in the rhetoric stage (9-12), they deepen knowledge further and enter into abstract reasoning and the construction and defense of evidence-supported argument.

October 30th, 2018 at 5:47 pm

What do you mean by “rate of change”? Interpreting it in the usual way as the slope of the line tangent to the parabola at a particular point, the definition “the slope between the y-intercept and the vertex” can’t possibly be right, since (a) that slope is constant while the rate of change of a parabola depends on the independent variable; and (b) the line between a parabola’s vertex and y-intercept intersects the parabola at two points, the vertex and the y-intercept, and therefore cannot be tangent to the parabola at any point.

Memorization through practice is an important part of learning math best reflected in the famous quote “in mathematics you don’t understand things. You just get used to them.” ( https://en.wikiquote.org/wiki/John_von_Neumann ) (I learned this quote in the form “Nobody understands mathematics. You just get used to it after a while.”) The difference is that, if you’re learning, you practice with the fact until you can

use it, whereas the zombie thinks they’re done after they canwrite the fact down. Those aren’t the same thing.October 30th, 2018 at 11:20 pm

It is the slope between the two points, isn’t it? Not the slope of the parabola. And rate of change is the slope between any two points, not the tangent.

October 30th, 2018 at 11:54 pm

Sure, “rate of change” is the slope of any line. But your post implies that a particular, constant rate of change is associated with any given parabola:

Those are the only two mentions of “rate of change” in the post. I can’t understand what they’re supposed to mean — the rate of change of a parabola is not a constant value (if it were, the parabola would be a straight line); you’d have to talk about the rate of change at a particular point on the parabola. And assuming the b parameter is the coefficient of x^1, dividing it by 2 has nothing to do with finding the rate of change anywhere. It’s part of finding the parabola’s vertex (which has x-coordinate b / (-2a)), but the location of the vertex isn’t relevant to finding any rate of change.

October 30th, 2018 at 11:56 pm

For the fairly common case where the parabola’s vertex

isthe y-intercept, what do you tell them “the rate of change” is?October 31st, 2018 at 6:15 am

Sorry, just saw this: “For the fairly common case where the parabola’s vertex is the y-intercept, what do you tell them “the rate of change” is?”

It’s zero, because b is zero, so half of b is zero.

Again, I teach my students in algebra 2 to graph parabolas like this:

Given, say, y=3x^2 + 12x -7, so a=3, b = 12, c =-7

The rate of change between y-intercept and vertex is half of b, or 6.

The line of symmetry for the parabola is the opposite of the rate divided by a, or -6/3, which is -2. So h=-2, or the horizontal distance from the y-intercept to the vertex.

Then you multiply rate of change times line of symmetry to find the vertical distance from the y-int to the vertex (aka the “completing the square value). So -2*6 = -12

Then you add that to c, which gives you k. so -7 + -12 =-19.

In vertex form, the parabola is 3(x+2)^2-19

This is easier for students who need a clear procedure–and also easier mathematically–than -b/2a for Los, then plug in to find y value.

Now, if b=0, they wouldn’t need to use that method, because if b=0 it’s a simple vertical shift down.

October 31st, 2018 at 12:06 am

Sorry, I misread your “It is the slope between the two points, isn’t it?” as “It is the slope between two points, isn’t it?”

You can draw a line between any two points, and that line will have a slope, and we also call that slope the rate of change (of y with respect to x, or what have you).

I’m not sure what you mean by “

thetwo points”. The vertex is a point of obvious geometric significance to a parabola, and is the unique point on the parabola where the rate of change is 0. The y-intercept is a point of basically no significance whatever.October 31st, 2018 at 4:48 am

The rate of change (slope) from the y-intercept to a parabola’s vertex is always half the linear term (b). So this allows kids to have a clear procedure to graph a parabola in standard form.

October 31st, 2018 at 5:50 am

Ah, that is true.

It’s strange to talk about “the rate of change” between two points though. Measuring the slope of a line that runs between two points on a curve gives you the average rate of change between those points, not the rate of change at any point.[1] (Just as, if you were to drive from your home to the grocery store, and then back from the store to your home, we wouldn’t say your speed during the trip had been 0 — we would instead say your average speed over the trip was 0. Your speed could have been anything at any time.)

[1] If the curve is continuous, the mean value theorem guarantees that there is a point within the interval with rate of change equal to the average rate of change over the interval.

October 31st, 2018 at 6:17 am

Well, this is algebra 2. In precalc, we talk about average rate of change. Here, I’m just introducing them to the concept.

In the case of the trip from the grocery store to home, I think we say your average rate was total distance over total time, not zero.

November 1st, 2018 at 6:45 am

Your directionless speed can use weird accounting. Your average velocity is 0, as your position doesn’t change.