The number one murderer of truth is ideology. Ideology: we expect it in politics and religion, government and propaganda. We do not expect ideology in mathematics. Mathematics is supposed to be the one last refuge for absolute truth in a world battered by storms of confusion and uncertainty, right? At least that's the standard soundbite. But there is one particular ideological weed which has taken root deep in the mathematics classroom. "Rote memorization is evil." Time and again, math lecturers inure their students to the idea that memorization is to be avoided at all costs, that it is somehow mutually exclusive with deeper understanding.
As a math PhD student, teaching business calculus to pay my way, I've recently been infuriated by this anti-rote dogma. The week before a midterm which would cover multivariable implicit differentiation (among other things), I showed my students a formula which reduces the process from a 10-minute algebra-hell into a 30-second breeze. Then the weekend before the midterm, with almost no more time to talk to my students, the department sends me an email stating unambiguously that the students MUST do the work the hard way. Seeing red, I shot off an email demanding why they insist on this ridiculous policy. The response: "...memorizing a formula does not make sense to me."
The situation really stirs up righteous fury in me. When I don my math educator hat, my job is to teach students how to do mathematics. The official talking point is, "you don't use rote memorization in math". However, this is a lie. And I refuse to go before students and teach them a lie. I'm certainly not going to lie just to cover my ass.
In case you don't believe that the talking point is a lie, this very same calculus course, where the department said "memorizing a formula does not make sense", the students are required to memorize the formula for the sum of the first n integers (1+2+...+n) and for the sum of the first n squares (1+4+9+...+n^2). I'm not calling the talking point a lie in some vague philosophical "what is memory... pass the bong, man" way. I'm saying that it's a bald-faced, impeach-him-for-breach-of-oath, 1+1=3 whopper. It is fantasy. And it's being fed to students as gospel, students who are paying good money to eat it, students who are placing faith and trust in their professors to teach them what is true.
MEMORIZATION VS. UNDERSTANDING
When I learned calculus, it was self-taught out of a calculus book my parents bought me for my 14th birthday. That gave me the freedom to go at my own pace and understand everything as I went. In the process, I memorized certain things like derivative rules semi-naturally, without really trying. Some things, like the quotient rule, I eventually had to memorize. (Math dork trivia: the quotient rule is actually unnecessary, since you can pull the fraction upstairs and give it a -1 exponent, then use the product rule and chain rule) Therefore, to the best of my knowledge, learning math and memorizing things are interwoven processes.
"But that's obviously not the memorization they're talking about," comes the rejoinder. The memorization which we're really talking about here is the type using flashcards and repetition. Conventional wisdom says that this type of memorization should never be used in the classroom. Conventional wisdom also said that a black man with middle name "Hussein" could never be elected President of the United States. Sometimes conventional wisdom is wrong.
The pro-rote argument has two main prongs. First, students are going to use it anyway. To assume otherwise is not noble or principled, it is delusional. Pretending that this isn't the case, and teaching the class as if it isn't the case, is an exercise in delusion. If I stand there and tell my students "you shouldn't try to memorize any of this", it paints me as totally out-of-touch and makes me look like I don't know what I'm talking about.
The other prong is, in the standard university freshman calculus course, the material is simply covered too fast for the students to gain the type of deep wonderful understanding which makes rote memorization unnecessary. I would have trouble keeping up with it if I didn't know it all already, and I'm some kind of bulging-cranium math-genius freak! If there's any true ideology in math, it's the "snowball" ideology, that math builds on itself and if you fall behind, it starts snowballing out of control until you go to lecture and it sounds like the professor's speaking Ugaritic. This kind of makes it really important not to fall behind. I would even say that (GASP!) keeping up with the material is more important than having some sort of pure godlike rote-free understanding of it. If a freshman devotes all the time it takes to deeply understand Riemannian integration (something many mathematicians don't really accomplish until grad school), then they're going to fail the course because they didn't have any time left over to learn anything else.
The preceding argument is a kind of "lesser of two evils" argument, but I'll go even further and say there's a place for rote memorization even in an ideal self-paced program. Look at language, for example. Trying to learn a language with nothing but deep understanding would take a really freakin' long time.
SO DOES THIS MEAN GLOWING FACE MAN WILL QUIT MATH?
I'm stepped across a pretty serious line in this article. Taking a pro-rote stance in math education is like nailing Martin Luther's manifesto to the doors of the catholic church. So, I'm a math heretic. Hey, that sounds pretty cool, actually. Does that mean I'm quitting math?
Quite the opposite. The passion I've been feeling lately about math education, means that it still inspires some kind of growth and change in me. The day I stop caring and just put in the time, that's the day I'll quit teaching math. If I don't get fired first for being a mathemaverick.
FURTHER READING
This isn't the only bone I have to pick with math education. Read my article "Problems" In Mathematics to learn why freshmen shouldn't encounter many "problems" at all. They should encounter exercises and questions.
I'm pretty rare among pure math majors as someone who really likes teaching. I'm passionate about it. Look, I even wrote an article called, How To Be A Better Teacher.
If you sympathized with the article you just read, you might also sympathize with "How To Be Better At Math". I promise it doesn't contain any of the usual b.s. talking points you'd expect.
As a math PhD student, teaching business calculus to pay my way, I've recently been infuriated by this anti-rote dogma. The week before a midterm which would cover multivariable implicit differentiation (among other things), I showed my students a formula which reduces the process from a 10-minute algebra-hell into a 30-second breeze. Then the weekend before the midterm, with almost no more time to talk to my students, the department sends me an email stating unambiguously that the students MUST do the work the hard way. Seeing red, I shot off an email demanding why they insist on this ridiculous policy. The response: "...memorizing a formula does not make sense to me."
The situation really stirs up righteous fury in me. When I don my math educator hat, my job is to teach students how to do mathematics. The official talking point is, "you don't use rote memorization in math". However, this is a lie. And I refuse to go before students and teach them a lie. I'm certainly not going to lie just to cover my ass.
In case you don't believe that the talking point is a lie, this very same calculus course, where the department said "memorizing a formula does not make sense", the students are required to memorize the formula for the sum of the first n integers (1+2+...+n) and for the sum of the first n squares (1+4+9+...+n^2). I'm not calling the talking point a lie in some vague philosophical "what is memory... pass the bong, man" way. I'm saying that it's a bald-faced, impeach-him-for-breach-of-oath, 1+1=3 whopper. It is fantasy. And it's being fed to students as gospel, students who are paying good money to eat it, students who are placing faith and trust in their professors to teach them what is true.
MEMORIZATION VS. UNDERSTANDING
When I learned calculus, it was self-taught out of a calculus book my parents bought me for my 14th birthday. That gave me the freedom to go at my own pace and understand everything as I went. In the process, I memorized certain things like derivative rules semi-naturally, without really trying. Some things, like the quotient rule, I eventually had to memorize. (Math dork trivia: the quotient rule is actually unnecessary, since you can pull the fraction upstairs and give it a -1 exponent, then use the product rule and chain rule) Therefore, to the best of my knowledge, learning math and memorizing things are interwoven processes.
"But that's obviously not the memorization they're talking about," comes the rejoinder. The memorization which we're really talking about here is the type using flashcards and repetition. Conventional wisdom says that this type of memorization should never be used in the classroom. Conventional wisdom also said that a black man with middle name "Hussein" could never be elected President of the United States. Sometimes conventional wisdom is wrong.
The pro-rote argument has two main prongs. First, students are going to use it anyway. To assume otherwise is not noble or principled, it is delusional. Pretending that this isn't the case, and teaching the class as if it isn't the case, is an exercise in delusion. If I stand there and tell my students "you shouldn't try to memorize any of this", it paints me as totally out-of-touch and makes me look like I don't know what I'm talking about.
The other prong is, in the standard university freshman calculus course, the material is simply covered too fast for the students to gain the type of deep wonderful understanding which makes rote memorization unnecessary. I would have trouble keeping up with it if I didn't know it all already, and I'm some kind of bulging-cranium math-genius freak! If there's any true ideology in math, it's the "snowball" ideology, that math builds on itself and if you fall behind, it starts snowballing out of control until you go to lecture and it sounds like the professor's speaking Ugaritic. This kind of makes it really important not to fall behind. I would even say that (GASP!) keeping up with the material is more important than having some sort of pure godlike rote-free understanding of it. If a freshman devotes all the time it takes to deeply understand Riemannian integration (something many mathematicians don't really accomplish until grad school), then they're going to fail the course because they didn't have any time left over to learn anything else.
The preceding argument is a kind of "lesser of two evils" argument, but I'll go even further and say there's a place for rote memorization even in an ideal self-paced program. Look at language, for example. Trying to learn a language with nothing but deep understanding would take a really freakin' long time.
SO DOES THIS MEAN GLOWING FACE MAN WILL QUIT MATH?
I'm stepped across a pretty serious line in this article. Taking a pro-rote stance in math education is like nailing Martin Luther's manifesto to the doors of the catholic church. So, I'm a math heretic. Hey, that sounds pretty cool, actually. Does that mean I'm quitting math?
Quite the opposite. The passion I've been feeling lately about math education, means that it still inspires some kind of growth and change in me. The day I stop caring and just put in the time, that's the day I'll quit teaching math. If I don't get fired first for being a mathemaverick.
FURTHER READING
This isn't the only bone I have to pick with math education. Read my article "Problems" In Mathematics to learn why freshmen shouldn't encounter many "problems" at all. They should encounter exercises and questions.
I'm pretty rare among pure math majors as someone who really likes teaching. I'm passionate about it. Look, I even wrote an article called, How To Be A Better Teacher.
If you sympathized with the article you just read, you might also sympathize with "How To Be Better At Math". I promise it doesn't contain any of the usual b.s. talking points you'd expect.
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2 comments:
I couldn't agree with you more, glowing face man. Throughout my school career, my teachers demonized "rote memorization" as though it were some form of child abuse.
I don't know about you, but I think that in the west -- at least England and the U.S. -- there's an arrogant attitude towards rote memorization as opposed to "thinking things through yourself." The attitude seems to be: "Unlike those people in other countries, we actually use our brains instead of mindlessly memorizing lots of trivia."
For instance, I had a classmate in my Japanese class who had a lot of trouble with the fact that it was necessary to memorize passages of text in order to pass the tests. He kept going on about how he preferred the "Anglo" style of education to the "East Asian" style. I wanted to say, "Man, it's a language. It's something you memorize, not something you figure out on your own."
I've also found that I can understand why you say a certain thing in a certain way after I've memorized it. I'm sure you can say something similar about math. Also, it's nice to be able to do things quickly and not have to reinvent the wheel every time.
Perhaps in East Asia they have gone too far in the other direction, but that doesn't make rote memorization totally evil. After all, you need some foundational knowledge before you can begin to think about it.
Perhaps related to the anti-memorization trend there's also too much emphasis on discussion in schools. Maybe you don't experience that as much in math class, but I experienced it all the time in English classes. Teachers would say things like, "I'll try not to lecture too much." I always though, "You'd darn well better lecture! If I just wanted to have discussions, I could do that for a few dollars at a coffee shop." Actually, the best English class I had was one where the teacher devoted two out of three classes to lecture. Discussion has its place, but if there's too much of it, I begin to wonder what I'm paying my professors for.
Anyway, sorry for the longwinded comment, but you wrote a very thought-provoking post!
Well, I have to say that I can see both sides of this issue. I’m probably about 15 years older than you, and when I was young, rote memorization was in still in style and it frustrated me to no end. I remember in high school having a pop quiz where I was unable to remember the quadratic formula, so I spent the first ten minutes deriving it, and then didn’t have time to do all the problems. Remembering any formula was a serious challenge for me, and I always figured if I knew where to find it and how to use, there was really no point in storing it in my head, except to pass a test.
By the time I got to graduate school in the early nineties and was taking statistics, things had improved considerably. There were a few formulas we had to memorize (I know I had to calculate regression coefficients by hand), but most of the focus was on being able to choose the appropriate way to analyze a given data set to answer various research questions, and that was something I could do quite well.
On the other hand, now I have an 8 year old daughter who is learning multiplication, and I am very annoyed by the current approach of showing kids three different conceptual ways to solve a problem, and then quickly moving on, with very little hands on practice. I still think the multiplication table is something you should know like the back of your hand. If you’re trying to work through a complicated problem, and you have to stop to add up 7 eight times, you’re going to be very distracted from the overall process.
But I was a little confused by the situation you described—if you gave your students a formula to use, did it follow that they then had to memorize it? Why couldn’t they just write it down and take it out as needed to do the calculations?
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