On math and memorization
posted on 2017-05-09 with tags [math] [education]

Some thoughts on the current state of math education and how it could be improved.

Two weeks ago, my friends and I were talking with each other (online, which as a side effect created a record of all that was said), and the topic of finals came up. One of my friends mentioned their apprehension for their math final, citing the amount of memorization required. As is typical for the group of friends in question, this resulted in an hour-long discussion/debate which I think is worth preserving elsewhere, rather than being forever relegated to the history of the online group.

The friend (who will remain unnamed for anonymity purposes) had an aversion to trigonometry in particular, and went on to explain their system for memorizing the signs of trigonometric functions. As it turned out, a number of people had similar methods, all of which were some variant of an acronym for "ASTC"—all positive in quadrant I, sine positive in quadrant II, tangent in III, and cosine in IV.

Having never heard of this acronym, my immediate reaction was, and I quote: "that's stupid and bad."

If you've known me for a long enough time, then you certainly also know that I have some rather strong opinions on how math should be taught. And this... this abomination was the very epitome of everything I had against the current state of math education! (Sidenote: if you haven't known me well enough, you're about to become very well acquainted with the burning hatred I have for learning and teaching math poorly.)

I tried to explain an intuitive method for achieving the same task through an extremely poorly drawn diagram. For those following along with the actual math, it was something like this:

From this point on, it's going to be easier to write commentary on the ensuing discussion, which will remain unedited save for names in the interests of accuracy.

meit literally takes like 2 seconds of thought
Ai bet i can say astc in faster than 2 seconds
Aits already engrained into my mind

This is exactly the mindset that I find actively harmful to actually learning math. The response after I asked what people hoped to achieve by memorizing things like this that don't actually help anyone learn anything was:

Ai understand exactly why they are what they are
Abut its faster not to do that on a test
BI mean he's not wrong
BI also learned the acronym
BI know how the math works
Band don't really bother with it

Now, this really bothers me. The issues with the math curriculum are an entirely different rant, but the way the school system works is partially to blame here, and I've only rarely ever seen a (middle school / high school level) course that fully addresses the issue of memorization's being the optimal "strategy" for students for whom grades are all that matter. By no means do I imply that doing so is nontrivial or even easy, but I do think that schools and teachers should certainly put more focus into fixing this problem.

Cif you don't have good mathematical intuition
methis is something that can easily be learned and understood
Cno it can't
meit doesn't rely on "intuition"
Cit's really hard to get good mathematical intuition
Clike you can teach the individual case easily
Cbut I bet there are tons of these things that would bother you that you just don't know about
Cand the solution isn't to teach them the individual case
Cit's to try to develop the thinking skills that make them do this on their own
methat is what I am attempting to achieve by yelling at A (sorry A if you feel like you are the sole target of my anger, which just exists in general :P)
mehe clearly knows how triangles work
mehe is just opting to memorize the acronym instead of using that knowledge

C's point, which I agree with in retrospect, was that schools should teach students how to use their existing knowledge to figure out new concepts rather than teaching rules to memorize such as the one that started this discussion. While I am in complete agreement, I also think the students have a part here as well. A math student has to be willing to take this extra step; math students must challenge the reasoning behind what they know. The teacher holds some responsibility for teaching "shortcuts" instead of concepts and ideas, but it is also partially the students' responsibility to ask themselves if they can explain why a certain fact is true. If they can't, if they don't have a solid mental model of how what they know actually works, all of their higher level knowledge will be brittle and lack foundation.

Abecaues it is faster!
Ai just dont see the problem if i understand the derivation
Aand if its faster for me
Ashould i be trying to do it the "right way" if it takes longer for me? not on a test
Cnot for him
Cor for me
Abut for me
Cyeah and that's the issue I would want to fix
Cthe acronyms are faster for you because you don't have properly developed mathematical intuition
mewell ok, you're blaming the school system for this and it's maybe partially that but I also think if you were willing to try it, you would find that it really doesn't take as much longer as you think or even longer at all

Again, the issue with the math class itself here is that it rewards "I can do this fast" over "I can understand what I'm doing," which is not an easy problem to solve but most definitely a solvable problem. Teachers should ensure that students know where the facts come from, not just that they can memorize the facts. Tests should consist of subjectively graded, open-ended questions such as derivations, not questions that only determine whether the test-taker can arrive at the right answer. From personal experience, some students' obsession with only knowing the right answer, regardless of how it might have been arrived at, is already bad enough.

Cit's easy for you because you can see problems and kinda see what the answers are going to be like
Cwhich makes it so that you don't usually make big mistakes
Cwhereas that might now be the case for others
meI maintain this isn't a case of lack of intuition, it's lack of willingness to use it
Cbut it's less accurate/quick for him
BI think it's he understands why it works but it still takes him longer to get to the logic than just acronym it
BWhich is clearly a lack of intuition
mewhat no, if he's capable of getting to the logic then he has the intuition

"I memorize things because I can't do it any other way" is, I think, an entirely counterproductive attitude. So many people to whom I've tried to explain some concept have given up despite clearly being capable of understanding it simply because they've convinced themselves that they're not. Nobody understands everything immediately upon seeing it, and nobody is totally incapable of understanding anything either. I believe that there is no "maximum" amount that a given student can learn, but memorizing shortcuts instead of learning concepts will only take away underlying and necessary understanding for the future.

Aand if i ever falter i know that that can be used and probably will use it because that is what i have been taught to do rather than think about the graph of sin or the unit circle
meI would rather rely on logical steps that I can clearly see to be right, than rely on my memorization and hoping that I remembered some random letters correctly
Abut the more important thing imo is it got us to the same point
Awhere its now 95% of the time instantaneous
methe end result is not indicative of the methods used to get to it... do you expect to remember astscttswhatever for the rest of your life? is solving every problem like that not absurdly unsustainable? how can you be sure that you're really learning?

Let's say I want to draw a picture. This picture has a dog, so I buy a dog-shaped stamp and use it to stamp a dog onto the paper. Later, I need to draw a cat and a rabbit too, so I go buy two more stamps. Soon enough, I end up with a mountain of stamps when a simple pencil would suffice if I just learned how to draw these things. And what if I want to draw a dog with two tails (adapt a concept for another use, following along with the reality corresponding to the analogy)? Or just the head of the cat? Memorizing the solution to a problem means failing to be able to break it down into its key parts, a crucial skill for any type of problem solving.

meI think he's clearly capable of doing it, but he just sticks to rote memorization and that is what I am trying to change
Cbecause the other way is slower and less accurate for him
meare rigorous, provable steps not always more accurate than recalling some acronym?
Cnot if you can't tell when you messed up a proof
Asee the problem is the steps
Awith each step my odds of doing something wrong increase
meyou DEFINITELY can't tell when you messed up a "memorize and recall"
Cright but if the memorize is more accurate in the first place it doesn't matter
meit does matter though cause then you're not doing things based on knowing why you're doing them, you're just doing them because The Acronym told you to
Ai just dont want to do the whole process so frequently when tehre is an easier way
methe "whole process" is really just like one step though in this case, and in general choosing an "easier way" over a meaningful way is just harmful to everyone
mehim, the teacher, my sanity

Understanding fosters more understanding, and lack of it perpetuates. Math is innately cumulative in nature, so if you can't prove where one thing comes from, everything that builds on it is in turn baseless. Even if you understand steps 1 through 200, if you don't get step 201, then there's no way you'll fully understand steps 202 through 5000 without deliberately making the effort to grasp that 201st step. And by the same token, if you can't tell when you mess up a basic concept, then you'll certainly be lost when you mess up something that's built 10 levels on top of that concept and have no idea what you've done wrong.

Cright but it's faster for him the memorize
mewhy is the underlying assumption that we are optimizing for speed?
meok so again that's partially a problem with the education system but that should not be the sole thing you are focused on in math class. the primary goal should still (theoretically, hopefully) be to learn
Ai wish it was but it dont
meok that's a far worse problem of the perspective you have on why you're even doing these things
mememorizing special cases will not help you in the future
melearning how to think and... well learning how to learn will

No, you're not learning about quadratic equations for the sole purpose of their being practically useful later in your life. And please don't tell me you're learning about them "because they're gonna be on the test." The real reason you're in math class is to learn how to learn, so to speak. To quote Ben Orlin: "Math is the playground of reason." Math class can teach you one of the most valuable skills in your life, even if you don't plan to study math at all after finishing high school: math teaches you how to think, how to extrapolate information from what you know, how to make logically sound conclusions. And memorizing mnemonics ruins all of that.

Ethis is a non-primitive action andy
mefair point. but he's *capable* of doing it, he's just choosing not to, which irritates me because that's just throwing away the whole reason behind all of the things he's doing
Cwell if he wants to put in more work yeah
Cbut that's more work
meis it really that much more work though
meisn't memorizing gibberish also work
mebut meaningless work instead
Cfor some people
Cmaybe not for you
Cbut yes, for most people
meI really still think it's not an issue of lack of intuition, or time constraints, or amount of work, but just general apathy towards learning and the reasons behind it
methat was not meant to sound as harsh as it does
me"apathy" was poorly chosen but you get the idea lol

This is the problem with extrinsic motivation. When students are motivated only by an external force such as grades, they are less invested, they start to care less. The impetus behind learning should be intrinsic motivation, where students learn not for any reward from some outside agent but because they genuinely want to learn. I'm not trying to accuse anyone of not wanting to learn—everyone has some measure of intrinsic motivation. But when external factors start to compete with or even overcome internal factors in terms of a student's drive for learning, that student may begin to prioritize goals that aren't necessarily conducive to achieving the student's best interests.

Dpersonally i dont like to use acronyms and identities that much because it seems more likely that I'd remember something wrong than make a mistake in my algebra
BThose are important/useful
Dyeah but why memorize them when they're easy to derive
Di'm not doubting that they're useful
Aderivation is more error prone and slower for me personally
Di'm just saying that memorizing it isn't useful
DI see. For me it is the memorization. Everyone is different though and entitled to their own method so long as it brings good results

Yes. You are ultimately allowed to do whatever you want with your learning process, and at the end of the day I can do absolutely nothing to stop you. You might even get the same results as those you would get with a different, more effort-intensive method. And if the end result is all you care about and you simply won't be convinced otherwise, then I have nothing more to say. However, I sincerely hope that everyone reading this can find that spark of curiosity, that intellectual drive that serves as motivation to actually learn, standing in solidarity against all the external factors that tempt hapless students into relying on fragile and meaningless shortcuts.

Abecause i undesrtand 100% the reasoning and just use a method that is faster for me when i need to know quickly
Adue to it being more efficient
Afrom a time standpoint and an effort standpoint
mebut if you're not *learning*, what's the point of any time and effort
Elearning can come when it isn't penalized

Ah, don't you love the modern world?

Fcan somebody provide a sumary?
Cmemorize or derive
Foh lol
Fmemorize/derive to what extents?
Cspecific case?
Cwhether ASTC for trig positivity should be memorized
Cgeneral case is memorizing or deriving any kind of easy formula
Fi feel like deriving every time takes time from when you could be doing higher level stuff
Fto an extent

The conversation had basically ended at this point, so F's consideration was left mostly unconsidered. It's worth considering, though, so I'll do so now: yes, I acknowledge that rederiving formulae takes longer on tests, where time is of the utmost importance. However, for one thing, that 10 extra seconds really isn't going to make a tangible difference in the grand scheme of things, and additionally, if a student were to happen to forget one of the myriad shortcuts floating around in their flimsy, precarious web, the student who has not made provisions to build this information from scratch will be hopelessly lost. (And I say this last thing with my tongue firmly lodged in my cheek, so please don't murder me, actual teachers: you'll save time overall because you'll spend less studying!)

Well, I've ranted about a lot of different facets of things I find wrong with the current state of math education based on this one hour-long conversation/debate. I've talked about the problems with the school system (or whatever nebulous entity that's supposed to refer to), about how some school environments condition students to optimize for speed rather than understanding, about the slew of outside factors that warp people's motivation for learning math, and about the reason we require everyone to take math classes in the first place.

None of us can control any of these factors directly, but everyone—every last person in the world—has the capacity to care about math. To appreciate its beauty, to actually think for once. To make it a worthwhile endeavor, not some fuzzy pile of vague high school memories. To stop asking "how," and start asking "why." And not only that—everyone stands to benefit from caring about math. Telling someone that a square number can only have a remainder of 0, 1, or 4 when divided by 8 is one thing, but having people actually discover why on their own gives them an entirely different experience, an experience that develops problem solving skills and an "intuition" that can be applied virtually anywhere. Math class teaches math, but most importantly, math class teaches thinking. It's intended to push the boundaries of what we know.

If math really is "the playground of reason," then memorization is the stereotypically featureless detention room made for the singular purpose of boring the children confined to it. So go, play, explore, discover, learn. It's what the playground was built for.