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.
ARGUMENTS AGAINST MEMORIZING
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??|
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.
ARGUMENTS FOR MEMORIZING
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|
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.|