Tag Archives: memory palace

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.

Memory Palace for Thee, but not for Me

Should we teach kids how to memorize?, asks Greg Cochran. It’s a worthwhile question, and I have some thoughts, but got halfway through that post and hit some snags.

The comments, though, got me thinking about memory in general.

Back in high school, I used to write out all the acting Oscar nominees in order, lefthanded, to keep me at least somewhat focused in math class. During college, I’d write out the Roman emperors, English rules from William I to Elizabeth II, or the US Presidents, again, left-handed, to keep myself focused during college. I outgrew this habit at some point, probably when people asked me what I was writing; by my 40s, in grad school, I know I just doodled. I rarely set out to memorize things, and get no pleasure from the knowledge. What I do enjoy is the memories that come back with the data. So 1934 was It Happened One Night, with Clark Gable and Claudette Colbert, which got me thinking of Roscoe Lee Karns and Cliff Edwards and all the other reporters in His Girl Friday. Then the throwaway movies until Gone with the Wind, all hail Olivia (who outlived her feuding sister), and thinking about what a great decade the 40s was, oops, losing track of the lecture, get back to writing names. Recalling information keeps me focused, but the information itself doesn’t give me much enjoyment.

So if I know all the actors who’ve won best supporting Oscar and then best acting Oscar, and I didn’t set out to learn it, have I memorized that information? Huh. I realized I didn’t know, so I went and Wiki’ed up on memory. This was a helpful read, although I’m sure I have some of it wrong. Take my descriptions at your own risk.

Thus my echoic memory is much better than my iconic; both being short-term memory associated with a sense (hearing and sight). My students get a kick out of the fact that I look at geometry figures probably five times while drawing them. I can forget which way a right triangle faces in the time it takes me to turn 180 degrees from the book to the board.

In practice, this means my short-term auditory memory would be termed eidetic if it were vision-based, while I suck at that game where they give you 30 seconds to look at a tray of items. I’ve known this for a while, that hearing is extremely important to my short-term memory. I can easily maintain five or six conversations at once (very useful in teaching). However, five days later, my recall isn’t better and often worse than anyone else.

Fun example of this: a few months ago, Steven Pinker tweeted a language and memory study. I started to take it and then ran into a hitch.

The problem was, for me, that the practice wasn’t anything like the test. In the practice, up comes four groups of two letters situated around a cross. Then the screen blanks out, and occasionally you’ll see an arrow pointing to the position to remember. Some time later, a letter pair appears in the space and you indicate if it’s the same pair or a different one.

So I went through 20 or so practices, and did great, getting them all right. Then comes the actual test format, and I fall out of my chair, howling with laughter:


No more letters.

Until today, I didn’t even have the words to describe this. But in the practice, I literally vocalized the four pairs, saying “yn, qg, ds, hm”. The arrow would come up, and I’d say “okay, that’s qg”. Up comes “qg”, it’s the same. The whole test, I did with my short-term auditory memory, the echoic.

I guess if I were Chinese, I could do the real test that way. But the minute it came up, I realized I couldn’t rely on my auditory memory, and that’s game over. I’ve come back to the test since he closed it for research (didn’t want to screw up his results), and practiced two forms of memory. In the first, I say them aloud: “x on top, 7 on left, double T on right” and that helps, but I can’t say it fast enough and the image blanks out. Then I try it using my iconic memory (as I now know it), and if I focus really hard, I can usually see three of them before it blanks out. But it’s really hard.

I wondered if maybe the researcher planned it, but wouldn’t the practice be part of the actual test? I guess for most people it’s not a huge change.

Anyway, it’s a great example of how I use auditory memory instead of iconic. Auditory and visual long-term memory, if there is a such a thing, reverses in strengths for me. I can only vaguely recall the names of my high school history teachers, but their faces are quite clear in my mind. I likewise remember student faces much better than names, which is weird given that my short-term visual memory (iconic) is dreadful, but I guess they aren’t linked. For names, I need not only appearance but position—I can be completely discombobulated if a student changes seats. Periodically, I will randomly screw up names. I went a month calling an Anthony Andrew, despite having taught him two years in a row. So it appears that long-term, I rely more on visual than auditory, whereas short-term it’s reverse. When I think of the word “capybara”, I visualize the page of Swiss Family Robinson, kids’ edition, where I saw it at seven, the picture and the words on the page. The memories of books I recall in this essay are all strongly visual.

Long-term memory breaks down into into explicit/declarative memory and implicit memory (also known as procedural memory). Explicit memory is composed of episodic (autobiographical) and semantic (factual and general knowledge).

So my semantic memory is outstanding. My episodic memory, not anything special, particularly if it’s not autobiographical (there is some difference there).

Reading all this made something very clear that’s bothered me for thirty years or more: I have a terrible time with implicit memory if it involves moving parts not under my direct control. Specifically: driving, horseback riding, skiing, tying shoelaces. I didn’t drive until I was 22, because the act of learning was just so unnerving. Ran away like a ninny from skiing, never bothered with horseback riding. My younger brother finally shamed me into learning to tie my shoes, although which brother, and our relative ages, changes as the years go by, the better to embarrass me. If everything’s under my control, no problem: cooking, typing.

This gave me an interesting new way to think about how I learn. I’ve thought of my memory as a database since I first knew what a database was. Every so often the database goes wonky. Sometimes it’s a random switch of names. Sometimes I just can’t get the name. Once, on a contract, I remember complaining about the fact that I couldn’t remember the name of the Vietnamese PC guy, the one with the really heavy accent. “Which one?” asked my boss. “There are two?” “Sure. Pham and Tran.” “Crap. I had a duplicate data key.”

And every so often I just file away a false fact. For a good decade or so, my memory said that Richard Nixon had been governor of California. It’s not that I thought he was. I knew he wasn’t. But my memory thought he was, so if I were writing out the presidents who had been governors, Dick Nixon would be on the list. Took me years to find and fix that key.

In most cases, I effortlessly add information to my semantic database, creating links and keys between “data” fields and update references—that is “learning”, without really thinking of it as such. Until I was 25 or so, I had no idea how to learn if the process couldn’t be added to the database. As described in the “learning math” essay, I was completely helpless in those cases. When I was younger, I was often told I was exceptionally bright, and while I didn’t disagree, there was always this nagging concern—if I’m so bright, why are some things utterly incomprehensible to me?

Thanks to my first real job out of college, I finally figured out that, when the learning process wasn’t invisible, I had to learn through an insane series of trial and error tasks in which I traverse the landscape like Wile Coyote waiting for something to blow up in my face—this is how I learned programming and most of the math that I know.

So, put into my new terminology (probably inaccurately), if the new information has no link to my current knowledge, if I have no way to store and access it, I have to go out and acquire a metric ton of episodic memories to create a database table for my semantic memory, to build the connections and cross references. I have always known that this is somewhat unusual; I’ve watched many folks listen to lectures and get right up to speed while I’ve been sitting around unable to focus. I’ve tentatively concluded that my data fields have far more attributes—more metadata, if you will (but not metamemory, which is different), which makes the initialization more labor-intensive, but more useful in the long run.

Obviously, storage and recall is a whole field of memory study, and in some way everyone struggles with the process I describe. But for me, it was a very clear gap between the information I could or couldn’t learn, and I had no way of bridging that gap for the first 25 years of my life. For a long time, whole areas of learning weren’t under my conscious control and I had no way of anticipating what they might be. The trial and error, Wile Coyote process was a huge breakthrough that changed my life and expanded my career paths.

So when people talk about memory palaces, like in The Mentalist or Sherlock, I’ve got no frame of reference. My memory is not spatial, but associative. It’s a database that I retrieve, not a room where I put things. Do not tell me that it’s just a simple technique that anyone can learn. I couldn’t. Full stop.

End the investigation into Ed’s memory.

I imagine that during ed school, I read something about semantic memory. Lord knows Piaget must mention it somewhere. I probably dozed through a Willingham post on it, or read it in a book. But you know me, it doesn’t get put into semantic memory until I have a data table and some crosstabs.

I am not trying to become an expert on memory, nor am I unaware of the fact that there’s probably all sorts of reading I could do simply to discuss it more intelligently.

But all this leads me to a few observations/questions.

First, it seems that the myth of ‘they’ve never been taught’, the problem of kids forgetting what they learned, could be framed as the difference between episodic memory and semantic memory. That is, the kids I’m describing remember the topic as an episode in their lives, not as reference information, and since it wasn’t a very interesting episode they lose it quickly.

Next, I’ve remarked before, and will do again, that many smart kids (in my work, almost entirely recent Asian immigrants) can regurgitate facts and learn procedures to a high degree of accuracy but retain none of them and even while knowing them have no idea what to do with them outside a very limited task set. Whenever you see kids screaming “we have to have the test today” they are kids who know they will lose the information. So these kids may be relying entirely on episodic memory? If this is true, our reliance on test scores as a knowledge indicator becomes, er, unnerving, particularly since this behavior seems so strongly linked to one demographic here in the US. And I speak as someone who likes test scores.

Then the opposite case: in math, at least, I’ve noticed that fact fluency is not required for understanding of higher math, and that it’s not at all unusual to see kids who are fact fluent but can’t grasp any abstraction. It may be that these kids are filing math facts into implicit memory, as tasks. Or maybe that’s always the case in math.

It goes without saying that memory is linked to cognitive ability, right? Oops, I said it anyway.

I’ve also noticed that teaching, like police work, is a profession with limited need for semantic memory (the content fact base, rules of the job) and a tremendous need for episodic memory (what worked last time). This may be why teaching isn’t given much respect as a profession, and also why smarts, past a certain level, doesn’t appear to play much of a role in teaching outcomes.

And so, Greg Cochran asks whether we should teach kids to memorize.

Realize too that the memorization battles are just another front in the skills vs. knowledge debate. The skills side, touted primarily by progressives and, separately, many teachers themselves, emphasize the need for students to know how to do things—think critically, problem solve, analyze information. The knowledge side, headed by the great E. D. Hirsch, complain about the skills stranglehold and want to emphasize the need for students to know things—facts organized into a logical curriculum. Those pushing for memorization are squarely on the knowledge side of things, and often mock teachers for being too stupid to understand the importance of knowledge.

I have not entered this debate because until now I haven’t had a framework for my answer of “it depends”. Do we want to reward bright kids for memorizing content knowledge without a clue about what it means and little ability to use it, as is de rigueur in many Asian immigrant populations? I submit that we don’t. Do we want our kids of middle ability or higher to memorize math facts and general content knowledge, the better to improve their reading comprehension and understanding of advanced math? I submit that we do. And how much memorization, exactly, can we expect and demand from our low ability kids? I submit that the answer is “We don’t know, and are scared to find out”.

In other words, memorization requirements, like everything in education, is ultimately set by student cognitive ability, which we aren’t allowed to discuss in any meaningful way. But teachers like me, who are required to deal with a 3-4 year range in ability levels, with a canyon-sized gap in content knowledge from high to low, have to make decisions about skills vs knowledge debate every day. Those on the outside should realize that teachers have many good reasons for pushing back on the memorization point, given the students they teach and the expectations forced on them by those who don’t know any better.