Since I’ve written before about the evils of memorization, it’s now time to put in a good word on its behalf. Specifically, I’d like to share the rhyming catch-phrase I just invented:

A little baby-voiced, I realize. But hear me out.

First off: Knowledge matters. I mean names, dates, facts, formulas—they’re essential. They’re valuable. They’re part of what it means to be educated. Anyone who denigrates factual recall, or who touts Wikipedia as a fitting replacement for the human brain, is missing something big about the nature of expertise.

“Critical thinking” isn’t the opposite of “factual knowledge.” Critical thinking *relies* on factual knowledge. Critical thinking is the intelligent manipulation of facts—knowing what to emphasize and what to discard, how to synthesize and draw connections, when to seek out new data and what data, exactly, to seek.

If critical thinking is the ability to build towers, then facts are the bricks. You kinda need ‘em, and memorization can help.

The problem isn’t memorization, per se.

The problem is when memorization spreads like a weed, and begins to substitute for reason. The problem is “ASTC,” “FOIL,” and other mnemonic shortcuts that circumvent actual mathematical reasoning. The problem is when all of algebra or calculus is reduced to chugging through formulas whose origins and purpose you don’t understand. That’s when math stops being math, and becomes Following Recipes 101, a far less meaningful and worthwhile class.

What’s the solution? Memorize at the beginning of the year… and never, ever again. Memorization (whether by rote or by mnemonic device) serves us best when it gives us a starting place. It’s much easier to build well-integrated, highly connected knowledge structures when you’ve got a few basic facts to build off of.

If you’re taking a chemistry class, memorize the first few rows of the periodic table.

If you’re taking a course on American history, then memorize the major wars and the years that they occurred.

If you’re taking a course on trigonometry, memorize SOHCAHTOA.

Memorize this stuff on Day #1. It’ll give you a skeleton, and you’ll spend the rest of your time adding flesh and muscle to this bone-bare framework. But don’t keep memorizing more than you need to. For example, don’t memorize that sin(π/2) = 1. Reason out *why* sin(π/2) = 1, based on more elementary facts, and then remember it because it’s part of an interconnected structure of knowledge.

Memorization isn’t the essence of learning. But it’s a useful tool for getting us started.

One of the reasons that rote memorization has been so loudly demonized is that ‘facts’ memorized out of context can be completely misunderstood. So, it does not make too much sense for a student in Grade 5 to spout Sohcahtoa unless the rest of the trig program is present and in use. But worse, in the past, people were wont to memorize untruths, many of which were eventually exposed and swept away by the Enlightenment (think of the standard procedure for determining if one had caught a witch) and more recently by the scientific method.

If there is one field in which memorization is generally acceptable, if not outright necessary, it is Mathematics. The beauty of mathematics is that ‘facts’ are indeed facts. There are numerous ways of showing that 7×8 is 56. But why demonstrate that every time? Why not just know it? That way, when you are asked to simplify 56/96 you can quickly suspect that 8 might be the common factor. Similarly, my progress in math is far greater when I actualy know the facts instead of working them out every time. The greater connectedness of the math facts makes it easier to reconstruct any that one may have forgotten. And that is why math is so comforting. Much of what we need and use can be derived, if in a time-consuming reconstrution of the main points. But if I have forgotten the exact date that Constantinople fell to Mehmet the Conqueror, I will have to open Wikipedia or an actual encyclopedia, even if I am on vacation, far from a library and an internet connection. If only I had stored that fact in my brain!

Definitely – that interconnectedness is the essence of mathematical truth. The word “memorize” itself can also be a hang-up – I tend to use a pretty narrow definition, referring only to rote repetition of a fact, or memorization using a mnemonic or other memory trick (which excludes contextualized practice and learning via connections to other facts).

One thing I’ve learned after taking a few math courses is that it’s very helpful to memorize some things. I used to think that I could do it without memorization–that I could just figure things out on the fly, but I started suffering on exams because of the extra time I took in deriving things (such as the distance formula in algebra) that I could’ve had memorized. To get a good grasp of maths I think it’s necessary to be able to derive formulas and to know where these formulas come from, but we’d better have such formulas memorized come exam time.

You’re kidding. You can’t be called that in real life. It’s statistically wildly improbable. (Are you by any chance a hitch-hiker by hobby?)

You’re right, I’m not called that in real life 🙂

I’m just another lowly human awaiting his Ford Prefect.

Now… that is a relief.

Yeah, memorization is often indispensable for exams. It raises the (entirely separate) question of whether a well-designed exam should reward memorization at all. But it’s hard to argue that many currently do.

I believe that a well-designed assessment rewards BOTH quick recall of facts (AKA Memorization) and cleverness/creativity/application of ideas. Since we all have a spectrum of abilities ourselves and a spectrum of ability/interest level in our students in class, we should make sure that there are places where any student can legitimately feel successful for at least part of what we evaluate them on.

I think the problem is that facts are much much *harder* to memorize out of context. My calculus students spent semesters memorizing algebra and they still don’t know the basics it until they actually use them to do something. Their memory is generally pretty faulty and you have to keep reminding them as you go that, in fact, (a+b)^2 is not the same as a^2 + b^2. Even in my own experience, memorization in math comes as a byproduct of practice in context. But I would be interested to see how well your idea fares in practice.

Yeah, I certainly wouldn’t advocate “memorizing” that (a+b)^2 is a^2 + 2ab + b^2. I’d lump that in with FOIL, ASTC, and other nefarious memorization.

As a rough rule of thumb, I’d say the only things that should be memorized in math class are definitions, axioms, etc. – things that are truly logically primary and cannot be deduced from other facts.

Yes, definitions definitely need to be memorized. But again, you memorize them by working with them. Nobody memorizes all the definitions at once.

,I fear this still fails to get to the heart of the matter. The problem is the use of the words “memorize” and “memorization.” To most people, there is an automatic assumption of “rote” brought to bear whenever these words appear. But “memorize” and “memorization” would better be replaced by “know” and “knowing.” I KNOW the quadratic formula, but never sat down to memorize it. I know the formulas for sine and cosine angle sums and differences, but not through rote. And while I don’t “know” in the same way the double angle formulas for sine and cosine, I can derive them in seconds from the previously-mentioned formulas.

My calculus II teacher in NYC, who had a Ph.D in mathematics from Columbia, hadn’t memorized the derivatives for tangent, cosecant, secant, or cotangent, but he “knew” them the same way I know the double angle formulas for sine and cosine: he could very quickly derive them from previously known things.

I recommend that mathematics students memorize as little as possible, know just as much as they feel they should (but they’ll always in fact know more than that), and not sweat about what hasn’t yet stuck. Because the reality is that anyone can know without much effort facts that they need to know on a regular basis. Teach algebra a few times and you’ll KNOW the quadratic formula, or the point-slope formula, or how to find the perpendicular to a line through a given point not on the line.

I could write more extensively about this subject, but I doubt I’ll convince any true believers in the virtues of rote memorization that it’s mostly an unsystematic waste of time that causes many kids more frustration than it’s ever worth. If you teach effective mnemonic systems (see, for example, THE MEMORY BOOK, by Harry Lorayne and Jerry Lucas, or books by Bruno H. Furst), as I have, you can help kids not only know things in math and other subjects that have some logical structure to them, but even arbitrary facts like dates, names, terminology, notation, and other often random sorts of things. Then, students can decide for themselves when it’s worthwhile to put in time to apply the methods to a given set of facts.

Absent giving kids the tools for “rote” learning, I think it’s both hypocritical and somewhat sadistic to insist that they spend time memorizing the facts YOU claim are vital to their lives.

Yeah, what you just said is the gist of the piece I wrote for The Atlantic. I’d say that’s a good first-order approximation of my feelings on memorization; this blog post represents the quadratic term in the Taylor polynomial of my beliefs.

Thus any error you perceive error around this localised point of discussion is small?

Sorry……certainly an ectra error in my comment!

My teachers in high school railed against SOH CAH TOA and taught the Winding Function instead. Later, when doing my practicum to become a teacher myself, I was shocked to see SOH CAH TOA being used to introduce trigonometry. It was difficult for me to deal with it. I wrote a blog post about it here: http://mathsocietyclub.wordpress.com/2010/05/19/soh-cah-toa-vs-the-winding-function/ .

I think it’s valuable to teach both. I teach “sohcahtoa” (what I call “triangle trig”) to 9th graders, and the “winding function” (what I call circle trig) to 11th graders. The former is computationally easier, and gets at the nice truth that side-length ratios remain constant in families of similar triangles. But the latter is conceptually deeper, and of course opens the door to most of what trigonometry is really about.

By teaching both, students can see how a relatively intuitive notion (triangle ratios) is generalized in an oblique but powerful way (a common trend in mathematics; think of the complex numbers as an extension of the reals, or of rational exponents as an extension of integer ones).

For what it’s worth, also, the triangle notion of sine really is a function, too, just with a smaller domain and range. You input an angle strictly between 0 and pi/2; output the ratio from the corresponding right triangle.

I learned SOHCAHTOA first, in grade 8 or 9, changed schools after grade 10 and when in grade 11 we were introduced to trigonometric functions I couldn’t make the formulas stick. I memorized them, practiced, thought I knew them and whenever I didn’t use them for a week or two I forgot them. I asked my teachers and they told me I should just memorize and practice more. In the end I got a friend to explain it to me – once he mentioned the unit circle/winding function it took me about 5 seconds to understand, and never forget it since. Apparently, my senior high school teachers expected us to know about the winding function and couldn’t understand my difficulty with the conceptualization.

FOIL helped me out, but our class never got into the whys. Only the mechanics of solving equations. I only ever had maths 101 in my career. For some reason I enjoyed algebra and have come to appreciate stats, and even more so my lack of knowledge of stats.

My daughter is an ME major, I don’t even ask.

One nice thing about stats is how clear the purpose is. It’s easy for a class to teach the mechanics of algebra without providing any motivation; much harder with statistics.

Over the years, I have put much thought into the intersection between mathematics and memorization.

For me, memory work is an essential part of my daily mathematical practice. Usually about a half hour each morning, plus a little extra time each day. Aside from any practical considerations, this is simply time that I enjoy very much. It is a meditative time to look back at all of the mathematical nuggets that are stuck in the nooks and crannies of my head. It is a time to make associations between branches of math.

Within mathematics, memorization is simply the acknowledgement that there is a large body information that is worthwhile to maintain within your mind indefinitely. The basic definitions, theorems, and proofs of any discipline fall within this purview. Also basic problem solving techniques. We all put a lot of labor into learning calculus. Suppose that I don’t do any calculus for a few years, and then find that I need to calculate the area under a curve. Well, I have some nice integration techniques tucked away in my memory. Every month or so, a simple integration by parts problem comes up in my morning memory work. So, I am pretty confident that I will always know how to integrate by parts, or by substitution, or by partial fractions.

Memorization, to me, is kind of like warm up practices. It is like a musician practicing scales, or a basketball player practicing shots. I enjoy being competent at mathematics, and memorization assures that I remain competent in the areas that I have worked so hard.

( I should say here that my memorization work does not look like what most people think memorization looks like. Whenever I find a mathematical fact that is worth committing to memory, I turn it into 10 or 20 memorization items. Some of these will be classic recall items, and some will be very simple problems that use that mathematical fact, or very simple numerical examples. So, one is also learning to recognize that fact when it appears in the wild. )

[ This way of memory work is known as spaced repetition. Look it up. ]

{ One can use this technique in a math class by giving comprehensive quizzes regularly. }

🙂

Sounds like a very solid set of practices!

One of the difficulties with “learning” is that people are not all the same. I have an appalling short term memory, and I got into math (maths, when you get to Brum) because it didn’t need much memory. If stuff makes sense then I don’t have to “commit it to memory”, it gradually gets in there unaided. My wife, on the other hand, who is a classical ballet choreographer, has a totally visual memory, and still can only deal with halves and quarters. “I don’t do thirds” she said once.

So some people will never make the connections, never cope with the symbolism and the abstractions. It seems to me that math particularly is not for everyone, and a lot of futile effort is expended on them. The closest I can see to this elsewhere is tone-deafness with music.

Needless to say, I am now following your blog……

Thanks for reading! I think the tone-deafness analogy is a good one, largely because I don’t think tone-deafness is quite as binary or incurable as most people assume. Although some people obviously have a gift for music, the rest of us can develop greater musicality with practice. Not saying everyone can learn to write symphonies or conduct mathematical research, but with the right lessons and practice, everybody can make strides.

Having studied in South Asian schools for the first eleven years of my life, I can safely say that the entire education system relies on the memorising facts and vomitting them on the exam paper. Memorising isn’t the problem – it’s the part where the facts just remain as meaningless facts and not being used for anything useful.

England, on the other hand, has a very different outlook on education, where memorising is nowhere near as extensive as it was in South Asia. However, as much as I like this change, I find it easier just to memorise facts.

For instance, algebra is something I’m good at because the equations (which I don’t understand) stick to my brain. But when it comes to some logical questions which requires us to link things to the real world, I’m useless – and I feel uesless now because I couldn’t give you an appropriate example. But do you have any tips on how I can get rid of this habit? I finish secondary school in exactly one year and I’ll pretty much fail all my exams if I don’t get rid of it.

In brief, very brief, you should ask yourself “What does it say?”, in words, that is. Explain it to someone who doesn’t know maths, or algebra. here is a simple example:

a^2 – b^2 = (a – b)(a + b) says “The difference between the squares of two numbers is equal to the difference of the numbers times the sum of the numbers. For this one, draw a square of side 5 and in one corner draw a square of side 2. Now look at the rest of the bigger square. Break it into two bits, rearrange them – bingo!

Draw a picture, plot a graph, guess the answer. Ask yourself “What does it mean?”.

Oh, and in England kids ask the teacher, a habit you have to learn.

Those both strike me as good pieces of advice – ask “what does it mean?” and more generally, ask!

Where I struggle – and I think that Michael Paul Goldenberg hovers around this – is to understand what the distinction is (if there even is one) between what I have memorized and what I know. Have I memorized the driving route from my home to my children’s school, or do I just KNOW how to get there? Have I memorized the fact that 7 * 8 = 56 or do I just know this? Back when we used to dial phones there were a handful of phone numbers I could dial without really thinking at all about what I was doing. After teaching for 27 years I feel that I can just recall so many facts simply from repetition and recognition. Since I get hung up on semantics when constructing certain arguments for my students I wish that I had a better idea of where this difference is.

In my understanding of English, memory is knowledge. To know something is to remember it, to be able to examine it in the mind. To learn is to store knowledge in this way. On the other hand, memorization is the act of putting selected facts and processes into memory. Memorization is done on purpose but most of the knowledge we have is accidental – listening to the news, the teacher and our parents can deposit enormous quantities of knowledge (some of it less than reliable) in our memory. Upon analysis, one can see that in fact, memorization might lead to sounder knowledge. After all, the news on television and our parents opinions are not always objective. Memorization can nevertheless be prone to the bias of subjectivity in most realms: the winner’s perspective continues to colour the ‘facts’ of History. Mathematics and science are generally less likely to offer lies for consumption (although a long ago dispute that I had with my high school Physics teacher was not aided by the school library’s dearth of knowledge following the introduction of ether as an explanation for the propagation of light in space) and it should generally be safe to assume that the facts on offer are true.

Ben seems to be referring, in this essay on memorization, to the process of using mnemonics to better recall some processes. He implies that mnemonics are deplorable in general but useful in expediting the rapid retention of necessary knowledge. It might be more even-handed to recognize that sometimes the mnemonic is more easily retained than the original concept. Take SOHCAHTOA. Many people who have no idea what this acronym refers to have heard of it and know it. But without the context, it is completely useless. And what I see is that many students have recall of the acronym but not of the process to which it refers. Is that the fault of the mnemonic or has there been a breakdown elsewhere in their learning? More likely the latter.

Those who claim not to have memorized much are suggesting that due to some happy circumstance (in my case. loads of homework and a combination of excellent curricula and teachers) they simply retained the new material with only the aid of their own abilities and the assigned homework. This certainly worked for me all the way through the end of Grade 12. (Thanks to excellent curricula and excellent teachers). But at some point, attending the lectures and doing the assignments is not enough. At that point, learning has to become more active. The student has to study – memorize definitions, redo harder problems and seek out more of the same type. At that stage, most people go home because they believe that the extra effort indicates tht they are not good at math any more. Ask anyone who has continued in Math and they will tell you about a course or a concept that taught them both humility and perseverance. And just as often, they will have discovered, as I did, the value of selecting basic knowledge to memorize in an effort to learn the material. It is entirely possible that we just know some concepts because of older siblings and family activities (cribbage, anyone) but real learning requires an effort, a conscious decision to process the concepts and to retain them. Developing neural pathways. It might be that that is sometimes memorization.

Interesting comment. I wouldn’t say that I struggle around that issue so much as that I make a distinction (which may not be important to others) between that which I make a conscious effort to get into long-term memory and that which simply winds up there without my being particularly concerned with having that happen. And I make another distinction between rote memorization (just repeated exposure to some particular fact in an effort to have it stick, like in the use of flash cards for math facts or vocabulary words), and things I attempt to learn with some sort of conceptual framework (if I can find one). And there are also things that are too random and arbitrary to find a “natural” structure to them (at least for me) that I may try to learn using various mnemonic systems (e.g., knowing that the 20th president of the US was James Garfield or that Martin van Buren was the 8th, pretty much ‘on demand, something I can do only because I made a conscious effort with a mnemonic system back around 1980. On the other hand, I already knew, for instance, that Lincoln was the 16th president from repeated exposure to that fact as a school kid. I don’t think I “tried” to learn that: it just became embedded.

What always concerns me regarding math education is the idea of “memorizing/knowing” formulas and facts when that is seen as the rote, flash card “memorizing/knowing” and no other possibilities are considered. That seems strange to me, particularly with formulas, as they usually have meaning that is at least as important as the formula itself. I don’t like unnecessary emphasis on rote, and I don’t like what can be foolish ignoring of structure and meaning that might both have a lot of value and actually contribute to the ability to quickly recall the “fact.”

Certainly, when it comes to learning arithmetic tables, there are structural elements in those tables kids can use to help build up their mastery. But traditionally, that’s not the way kids are told to try to learn/memorize those facts. I suppose it shouldn’t be a war to the death between those approaches: both can be useful and I’m for giving students options. But at the same time, I hope teachers of elementary math (and parents of young kids) don’t deny children the chance to develop some of the key number sense concepts they can learn, use, and strengthen by applying them to those tables. There is so much richness there that is unlikely to be touched on by the pure rote approach.

The fear that “kids don’t know their facts” quickly drives some parents and teachers to insist that there be drill and practice. Lacking omniscience, I can’t claim that the rote approach is unnecessary for any child or all children, but I really want to see more balance in early math instruction, and would like to keep to an absolute minimum teaching procedures without any concern about why they work. I think we do students a terrible disservice if we don’t have them explore some of the algorithms for arithmetic we demand that they “know.”

It used to be described as short-term and long-term memory, but I prefer shallow and deep memory. Stuff in deep memory is knowledge, whether it is any use or not. This is the stuff that you know that you know, even if you cannot get it out all the time. The phone number you just looked up and succeeded in not forgetting on the way to the phone never gets into deep memory, and it’s just as well! If you do forget the phone number you have to look it up again, it really has gone.

Cognitive scientists do not have much to say on this that is enlightening.

Great article! My mom is a big advocate of memorizing (her school used to make her memorize everything) and made me memorize all the things that you just said not to. It did actually make the process quicker and I was able to solve problems quicker than other kids… when I recalled the information correctly. And it took time away from studying other subjects, like when I was still doing trigonometry others would already have started calculus. I think both memorizing and problem-solving skills are important but it’s a matter of balancing things correctly.

I’ve come to this page late, but I was wondering if anybody has any tips on encouraging Calculus students in high school to memorize things like the Quadratic Formula and the unit circle. I am an independent tutor, and one of my students is a high school senior, and so she has a lot on her plate already with other AP subjects and sports. I actually was rather taken aback that she doesn’t have this memorized already, since she seems like she’s a good student in general, but a little obstinate perhaps. Should I recommend a certain amount of time per day, a certain number of days? Any advice would be helpful!

I’m not sure why anyone really needs to memorize the quadratic formula, though with enough exposure, it has a funny way of getting stuck in one’s memory eventually. I’m increasingly fond of solving by completing the square, though not necessarily as traditionally taught (which is mechanically, with no understanding, making it, unsurprisingly, like using the quadratic formula with no idea of where it comes from). I recommend strongly looking at James Tanton’s approach to quadratics and completing the square, available free at his site gdaymath.com under the course tab.

As for the unit circle, that’s hard to avoid, really. How does a student get INTO calculus without having had to master basic trigonometry reasonably well? And that does mean more than a passing familiarity with the unit circle. However, there are ways to build up one’s knowledge of the information in it through a combination of understanding the coordinate plane, knowledge of special triangles and how the relationships therein are derived from the Pythagorean Theorem, and getting a handle on how sine and cosine change as the y- and x-coordinates change, respectively, for even the first quadrant. The rest is pretty easily extrapolated. Again, I prefer to have students build up this sort of information from what they know already about the basic principles entailed, and then start taking advantage of various symmetries and patterns that are NOT random or arbitrary at all, hence ripe for easy memorization.

Whether one thinks it useful to offer mnemonic devices of varying kinds for this sort of thing is a matter of personal preference, taste, and style. I’m not big on such things when the material isn’t just a matter of convention or nomenclature. Concepts are concepts, and while there are important facts that ground them, of course, I’d like there to be a healthy interplay between concepts and facts whenever possible. But that’s just me. Others have conflicting philosophies, I well know.

It might be instructive (for both you and your pupil) to explore with her why she’s chosen not to learn these things. There’s clearly been a choice made, conscious or not, and it serves a purpose for her not to know them. Perhaps just getting her to acknowledge that will help her break through the barrier she’s put up against bothering to do so, and if you approach her with some options about how to do it besides only rote, you might have a winning strategy.

How does memorizing SOHCAHTOA help knowing that sin(pi/3) is sqrt(3)/2?