Friday, 21 June 2013

MAT: Memorize This

Ah yes, the debate about memorizing concepts versus enduring understanding. I've had opinions about that for a while. There even seems to be immediate relevance on Twitter (hi @PaiMath!), so I'll spill this out into a post for you, the reader, to think about. Even though I should be marking papers.


1) IT'S UNNECESSARY. This is valid. For many reasons. The primary one being Google. But yes, why should I have to memorize the times tables when I have a calculator sitting right in front of me??

Here's the thing. It takes time to punch something into a search engine, or a calculator. "Sure!", you counter, "all of three seconds!" Consider that a student will need to multiply numbers many times over the course of a week. Let's conservatively say 10 times. 10x40 weeks means 400 times in a year. Which is 1200 seconds. Which is 20 minutes of your life WASTED to verify "4 times 3 is 12".

Why does the student not just know that after the 20th time? Alternatively, maybe they DO, but it's become so rote, so ingrained to just "punch it in", that they'll do it even though they're pretty sure they know the answer! By not memorizing, we're making people second guess themselves - if they were even bothering to think in the first place.

There's also this problem.

2) DIFFERENTIATION. Also valid. There isn't only one way to solve a problem, and we should be able to learn how to attack something from different angles. As long as we're right in the end, who cares what algorithm we use to get there, right?

Counterpoint. To find the vertex of a parabola, there are a bunch of methods, including graphing, completing the square, and partial factoring. So who cares, right? Jump to the following year - you need to change to standard form for a conic. So just complete the square... what do you MEAN you don't know how to complete the square?!

In Grade 10, we built this wonderful second floor, and all the students designed their own floor plans. And it was awesome. But now we're making the third floor, and some of those plans are MISSING a supporting wall. That's a building code violation! "But," comes the protest, "I don't NEED that wall!" Not if you were done building, no, but you're still going.

For those who don't know conics, consider simply comparing two fractions. Standard practice is to use a common denominator. Conventional wisdom is that a common numerator works just as well. Let's move along to adding fractions. How exactly does a common numerator help here?

SIDEBAR: I think this 'differentiation' is what's causing education "gaps" from year to year. It's not that the material WASN'T taught, it's that something that turned out to be rather key wasn't emphasized. Why? Well, teachers aren't psychic, students are forgetful... and no one had to memorize it.

(Disclaimer: Ontario high schools don't teach conics any more. So the above metaphor may be equal parts invalid and ironic.)

3) IT'S TEDIOUS AND BORING. Not to mention probably a bit painful too. Nothing sucks the joy out of anything faster than having to simply memorize a bunch of mathematical facts - for apparently no reason.

Don't you know your history??
Yet there's that old saying, "Those who cannot remember the past are condemned to repeat it". (George Santayana, often paraphrased as 'those who do not read history are doomed to repeat it'.) It applies here. Because if you don't remember "how to multiply by 4", that's 20 minutes of your life gone, per year, which probably amounts to at least a couple hours through twelve years of school - and we're not even getting to using math when shopping at the grocery store.

Sometimes, a little tedium is necessary. Again with proverbs, "Short term pain for long term gain". If you memorize the algorithm for completing the square NOW, you will find things a lot easier down the road! Trouble is, humans are notoriously BAD at helping out their future selves; case in point, a teacher blogging instead of doing their marking.

4) NO LEARNING IS OCCURRING. So true. Machines memorize stuff! Shouldn't we be better than that?

Yes, yes, we should.


Honestly, I don't have a good counterpoint this time. Aside from the fact that you are learning something - you're learning how to memorize. There's something to that, given all the damn passwords our machines require of us these days.

Which is a weak protest. Moreover, since the point to education IS learning, we'd better kill all the memorization, never mind any of those other counter-arguments! Or so the story goes.

But not so quick, memorization lovers. You're on shaky ground too.


1) EFFICIENCY. "If you wish to make an apple pie from scratch, you must first invent the universe." (Carl Sagan) Screw that, I've got all the ingredients memorized already, lets bake the sucker.

By doing that, you're missing out on so much. The universe, in fact. You're also stuck eating apple pie forever, unless you take a step back to see there might be other fruit options available. Or what if an ingredient is missing, temporarily forgotten? Do you know enough to substitute?

Turning it back to maths, there was a student some years ago who was really good at remembering things. This may have been his go-to method... until Grade 12, when there was finally too much to commit to memory. And without seeing how all the pieces interlocked along the way, what was there to fall back on? For that matter, was there even an alternative method available to the student?

2) USEFULNESS. You will need this information later in life! Remember, if you don't know completing the square, you'll be in trouble when we hit conics!

Because yeah, every student in the world is going to take the Grade 12 course that has conics in it. (I grant that many parents FORCE that decision, but let's try to be realistic.) If you can predict exactly what a Grade 5 student is going to need five years down the road, either you've got their life mapped out for them (you're doing it wrong), or you somehow know them better than they know themselves (people change).

Pictured: Not a fertility goddess
Back to the metaphors, if a building is only going to be one story high, you don't need to worry about installing a staircase, no matter how "useful" it would be. "But," comes the protest, "What if you later realize you want more floors??" Well, then it might not be a bad idea to consider a redesign of the ground floor too. Because displaying that life-size statue of a fertility goddess by the entrance may no longer be as high of a priority either.

3) FLUENCY. If you don't know the basics, how can you understand anything beyond that? If you want to build high, you need a solid foundation!

Okay. So at what point is it no longer a foundation? More to the point, when do we actually build the damn house? Because, oh my God, if counting is the foundation for addition is the foundation for multiplication is the foundation for area is the foundation for geometric properties is the foundation for algebraic proofs, kill me now.

That's not fun. Maths IS fun, and playing around with things. Or it's supposed to be.

I wonder if part of the social stigma on maths is that, after 50 years of memorization, people have forgotten it even can be fun. You wouldn't hear people say, "Yeah, I never got the hang of having fun" - hahaha, big joke.

Frankly, if the only real justification we can provide for memorizing is "You need to know the fundamentals!": 1) no you don't, see above; 2) apparently you're psychic, see above again; 3) true motivation has to be internal, not external.


At the risk of overgeneralizing:

Memorization comes from a time when we couldn't look things up very easily, when complex calculations necessitated a slide rule, and when we were assured that knowing math facts would get you a good job. People bought into that because, well, I suspect it was mostly true. But that time has passed.

Standards based mathematics now exists because calculators and computers have made things easy if all you want is an answer, because society has made "math sucks" a thing, and because there is no guarantee that knowing math will help you become employed. People (more importantly, students) won't blindly buy into memorizing any more.

And really, in the end, IS it necessary? No. But is it HELPFUL? Oh yes. Thus the best plan would seem to be, have the students learn through standards, and then once they've got the understanding, commit that stuff to memory.

Good thing there's absolutely no chance of people getting mixed messages.

Something I tweeted earlier today.


  1. Am I allowed to make a small change to your metaphor? It's not a foundation. It's scaffolding. Memorization is needed to help build the understanding needed for the higher concepts, just like scaffolding is there to provide support to a building under construction until the walls and floors are there and can support themselves. At some point, the scaffolding can be removed (the need to memorize is gone once the concepts are understood). If the building isn't going to be tall (if the student won't be taking higher level maths), the scaffolding is temporary ("unneeded" math gets forgotten through lack of use) but the one-storey house remains. Those planning on a skyscraper (students heading to post-doctoral work with maths) need the scaffolding until the tower is complete (memorize until the concepts are as natural to them as speaking) or the design is re-done (the students redefine mathematics through their studies and become hailed as heroes.)

    I believe I have a comeback for "No Learning Is Occurring" - machines won't always be around. At some point, you'll be in a situation where either the cell phone's battery is dead or the computer has blue-screened (become sad, if I understand the Mac world...). At this point, it's either pull out pen and paper or fall back on memorized multiplication tables. Remember, machines are subject to the laws of physics, which includes entropy.

    1. Except that doesn't work. If I follow you, you're saying the memorizing would come FIRST, followed by the understanding. No - once a concept is memorized, we tune out, oblivious to the fact that said scaffolding is a crutch that may not always be there.
      "This is why a negative times a negative is a positive, it's..."
      "I don't care, I have a rule."
      "This is how a computer works, it has a binary..."
      "Whatever, just fix it so it works like before."

      There's no drive, no interest in going deeper. Worse, people start memorizing rules then using them where they DON'T APPLY (I hate cross multiplying so much I can't even say). The understanding needs to be there first, which comes from discovery and learning. From there, we shift to adding it to the bag of tricks (the memorizing) in order to move onwards and upwards - assuming that's even the goal.

      As to the comeback, I meant to imply that memorizing is turning us into machines. So yes, our tools may break down, but that's no reason for us to act just like them. Learn to memorize, and all you've learned is how to memorize, you'll have no hope with pen and paper. Learn for understanding, and yes, if the machines break down, you have something to fall back on, which actually supports the point rather than goes against it.

      Unless I'm wildly misinterpreting, which is possible given my current mental state.

    2. I think I may have gone beyond the borders of your metaphor. I was thinking, there are students who will drop math as soon as they can (the one-storey building, with or without scaffolding), and students who want to explore math (the towers and such).

      The lack of drive is disturbing. The problem seems to be that math isn't seen as integral to life as language is, so people (students and older) will do the bare minimum. Do that with language and you get people who have trouble with spelling and grammar. Do that with math, and you get people who can't estimate how much they're spending.

      I figure, with math, core items, like the times tables, ultimately need memorization just for speed. Some if it comes from doing the same equation over and over - if one needs to multiply 3x4 repeatedly, one quickly sees the equation and gets to 12 without having to think about it. Higher levels, beyond simple addition and multiplication, are like creative writing and poetry - sure, one can know the grammar (in the case of math, the operations), but putting together something coherent needs more than known 3x4=12.

      I think we're on the same plane.

      And, yes, I hate, "The computer is broken, fix it!" types. There's too much that can break to just wave a magic wand to get the computer running again. I do wish that basic computer operation classes included basic troubleshooting, like interpreting error messages and why a reboot is the first thing a tech does.

  2. I would only disagree with your comment that knowing math doesn't make you more employable. I've found that when you tell people you're good at math, they automatically assume you must be really smart. Surely that's a leg up? :)

    1. Heh - maybe they'll think you're overqualified! But fair enough, I grant that it could be a benefit. So long as the company doesn't find any suspicious Facebook photos while doing their data mining...