# 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: 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. #### 15 responses to “Memorization or Learning?”

• Ryan

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.

• educationrealist

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.

• Momof4

“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.

• Michael Watts

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 can write the fact down. Those aren’t the same thing.

• educationrealist

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.

• Michael Watts

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:

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.

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.

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.

• Michael Watts

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

• educationrealist

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.

• Michael Watts

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 “the two 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.

• educationrealist

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.

• Michael Watts

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. (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.)

 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.

• educationrealist

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.

• Michael Watts

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

• Mark Roulo

What separates memorization from learning is a sense of meaning … when
you learn a fact, it’s bound to others by a web of logic.

I’d say that there is a lot of truth in this. The way that *I* would describe it is that knowledge is a framework of facts, relations, and rules.

As some examples to illustrate what I’m getting at:
Multiplying large integers (234 x 56) is the *SAME* as multiplying polynomials. If we finish up algebra I and the kid doesn’t see this, then there is some knowledge that has been missed. This is true even if the kid knows how to multiply both large integers and polynomials.
Invert and multiply” for dividing fractions is much the same The technique is fine. But the *reason* why this works, how this fits into the *system* is often missing. My guess is that the folks teaching this to the students actually *DO* show the kids why it works. Once. Then the kids do a bunch of practice problems with “ours is not to reason why, just invert and multiply!” The kids have been *taught* why this works, but they haven’t *learned* it and have just memorized a text manipulation technique.

For math, this can easily result in a mindset that I describe as Math as a collection of magic spells. To divide integers, you have one technique. To divide decimals you have another (this is the one where you move the decimal points in unison until the “left” number is an integer). To divide fractions and mixed numbers you have yet another technique. There is very little in common, and multiplying is an entirely different collection of techniques. The kids essentially memorize a collection of text manipulation
recipes and a collection of heuristics for when to apply each. Bright kids can get pretty far along the math trail before this catches up with them!

I have a difficult time seeing how to improve this, however, given (a) the current intelligence/talent of the average student and (b) the desire to have a *pace* that gets the stereotypical kid who takes Algebra in 9th grade ready for Calculus as a college freshman.
My son, as an example (that I’ve mentioned before on your blog), took TWO f*cking years to completely nail down multiplying mixed numbers. He isn’t particularly stupid (he’s taking junior college classes now as part of a dual enrollment program and has a 4.0 at the JC) and he’s not particularly bad at math, either (his SAT math was a 630 … and could have easily have been over 700 if he’d be willing to slow down and be less sloppy). I think that it just takes a lot longer to *understand* than to memorize a bunch of text manipulating
short cuts. And I insisted on what I consider understanding for most things. The current US HS pace is difficult to maintain without leaning on the short cuts.

Physics has similar issues.

Newtonian physics is basically ALL ABOUT CONSERVING:
Energy
Momentum (both linear and angular)
Mass

That is it! In theory, once you can set up the appropriate equations describing these four quantities and the learn how to manipulate them (algebra and calculus) you are done!!!!

But that’s not how we teach physics. Gravity problems tend to be set up such that momentum is NOT conserved. Magic spell for *that* sort of problem! Balls bouncing off walls has another set of equations. Teaching the *system* takes longer than teaching the short cuts and the standards won’t be met (by most students) taking the long way. So .. memorize these formulas, practice a bunch of problems, next!

History has less “system” and requires more memorization because lots of pieces *ARE* pretty arbitrary. Henry IV of England followed Richard II, but Henry’s father could easily have named him something else. There isn’t much of a system here, so you just have to learn the name. But … if you know a fair amount *about* Henry IV, then the name isn’t so hard to remember for the same reason that remembering a friend’s name isn’t such a chore.

But learning something (maybe a lot) about the following guys takes longer than just learning their names:
Richard II
Henry IV
Henry V
Henry VI
Edward IV
Edward V
Richard III
I do NOT expect that very many US high school history teachers would be asking their students to memorize English king lists for a test, but Dennis Fermoyle (who used to blog at publideducationdefender.blogspot.com), a US history teacher, had a 2007 post on topics the Minnesota State Legislature expects him to cover. I’ll include the relevant bits:

Th

Thee Minnesota State Legislature and a bunch of ivory tower academic gurus
have put together a list that my sophomore American history students are
supposed to know by the end of the year that I have them. … Here is a
list of what the state of Minnesota says my kids should understand about
[the period from the end of Reconstruction (1877) up to the beginning
of our involvement in World War I (1917)] (numbering provided by me, MJR):

Morrill Land Act,
Plains Indian Wars,
Dawes Act of 1887,
Wounded Knee,
Carlisle Indian Industrial School,
White Earth Reservation,
industrial mining in the southwest and midwest,
the Bessemer Steel Process,
barbed wire,
:
100) Alice Paul

Last year we spent twelve school days on this period, including tests.
Now, I suppose I can throw all this stuff at my kids, but if I do that,
how interesting do you think it will be? How much of it do you think any
of them will really understand?

To *learn* the inter-relationships between historical events and people requires
spending a lot more time on the individual events and people than *I* remember
spending in school

So, the kids wind up “memorizing” for the tests, then forgetting. They won’t remember
by using this stuff and they don’t have a framework for the pieces to support each other
and they mostly don’t care, so … memorize and forget.

But even the ones who *CARE* don’t see this as much more than disconnected facts.

Memorization’s defenders are right: It’s a mistake to downplay factual
knowledge … but memorization’s opponents are right, too. Memorized
knowledge isn’t half so useful as knowledge that’s actually understood.

So I agree with this 🙂

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?”

My answer is that memorizing SOHCAHTOA is perfectly fine because deriving the *names* of these relationships from 1st principles is … not math. The students don’t usually know Latin nor the Sanskrit that led to the word and its use here.

But I would expect the kids to *memorize* it, not have it on the cheat sheet.

I am fine with the quadratic formula being on a cheat sheet, even though one *can* derive it from 1st principles. Maybe there is a hint as to my rule here?

I don’t want the kids to have a cheat sheet for the sine/cosine of 30, 45, 60 degrees. They should know this (memorize) and/or be able to reconstruct this from SOHCATOAH and the appropriate triangles plus some geometry.

I guess I view the relationships in the basic Trig as much more foundational than the relationships that eventually generate the quadratic formula.

“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.”

My basic approach to math as well.

One-on-one you can avoid checking out by giving enough problems that
they will get wrong (mostly) for most zombification. In practice, this
can be tough …

I don’t practically know how to do this for a class. But at some point
the kids just aren’t going to care about what they don’t care about. Then
what do you do?

But what if repeated use isn’t happening in part because of the pause that
occurs when memory should kick in?

Instead of “invert and multiply,” my son does mixed number division like this:

A) The problem:
3 1/3 divided by 2 1/2
B) He “knows” the following “facts”:
1) N/N = 1 (for N not 0)
2) N/1 = N
3) N/1 * 1/N = N/N = 1
4) Mixed-number to improper fraction conversion and back again
C) So he wants to get rid of the denominator. He has to remember this strategy.
D) Write 2 1/2 = 5/2 as an improper fraction, now we have this:
3 1/3 divided by 5/2
E) We can multiply anything by 1 without changing the value, so
(3 1/3 divided by 5/2) * (2/5 divided by 2/5)
F) We now have:
(3 1/3 * 2/5) divided by (5/2 * 2/5)
G) So:
(3 1/3 * 2/5) divided by 1
H) So:
(3 1/3 * 2/5)
I) So:
(6/5 + 2/15)
J) So:
(18/15 + 2/15)
K) So:
(20/15) = 1 1/3

If the kid stumbles on any of these steps, I’m not sure what to do other
than continue to work on the more basic stuff??

Most of the memory that I’m seeing here is times-tables (which I *DO* favor
drilling until the kids just ‘knows it’) and basic integer multiplication,
division and addition. If *these* are the problem, then the foundation is
the problem, no?

No question that this approach takes longer to get an answer, but I’m not sure
that I see a memory failure. Maybe you have a better example than mine?

So did they learn, or did they memorize?

I think a reasonable heuristic is:

Can they manipulate the thing in ANY way that isn’t just recall of
a problem they’ve seen before?

For history:

Can they see a relationship that they haven’t been explicitly taught?