[Article] Rote Memorization in Mathematics
Discussion: [Article] Rote Memorization in Mathematics
From the forum: Sam's Essays
This thread was started by: Glowing Face Man.
Discussion start time: 2009-11-14 16:55:50.
From the forum: Sam's Essays
This thread was started by: Glowing Face Man.
Discussion start time: 2009-11-14 16:55:50.
From: Jimmy, from the old Blog Comments
Date: Mar 10 2009.
Date: Mar 10 2009.
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!
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!
I've found too many times that pure rote memorization was essential to gaining an understanding later. Example, I memorized 9 functional groups in chemistry for my biology course without learning any of their properties or anything like that.
It wasn't until after I had seen them used did I really put it together how they work.
Having the facts memorized means you can draw conclusions from the data in your head. I try to go into class with the information memorized and rely on the teacher to put it together for me and it helps so much.
It wasn't until after I had seen them used did I really put it together how they work.
Having the facts memorized means you can draw conclusions from the data in your head. I try to go into class with the information memorized and rely on the teacher to put it together for me and it helps so much.
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?
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?
From: John, from the old Blog Comments
Date: Oct 27 2009.
Date: Oct 27 2009.
Brilliant article! I’ve recently come across another
math major who teaches a Calculus class who told me
that “not much memorization is needed in my class”.
The idea of mathematical purity seems to be so
enamoring such that it’s somehow degrading to think
of it as a series of if:then processes that can be
memorized. It’s almost as if they don’t want the
“language of the universe” to be corrupted by
chopping it into memorizable chunks that make it
easier to use. The irony is of course that they too
had to go through this ’summation’ of learning to
get where they are on the understanding totem pole.
math major who teaches a Calculus class who told me
that “not much memorization is needed in my class”.
The idea of mathematical purity seems to be so
enamoring such that it’s somehow degrading to think
of it as a series of if:then processes that can be
memorized. It’s almost as if they don’t want the
“language of the universe” to be corrupted by
chopping it into memorizable chunks that make it
easier to use. The irony is of course that they too
had to go through this ’summation’ of learning to
get where they are on the understanding totem pole.
Interesting article.
I've been teaching / lecturing mathematics for thirty years, but cannot
get a job teaching in the UK because "You haven't passed 'English' high
school mathematics."
But in my humble experience, we use more rote (and "worse") than we realise.
For example, "What is 26 divided by 3?" Of course it is 8 with remainder 2.
But how did you get it? No, you did not "divide".
The definition of "division" is really to ask the question "How many
'threes' are there within 26, and how many extra remain left over after you
have removed the count of threes?" The only way to do this is to subtract 3
from 26, keeping count, until you have less than 3 remaining. This is in
fact what a computer does. (And 'smart' kids need to think twice before
contradicting me on this one!)
But this method is too tedious, so what we do instead is memorise the
multiplication tables and guess (yes, guess) an answer and back check;
like this:
Try 9 : from our memory of tables we have 3 x 9 = 27 oops, too big.
Try 7 : from our memory of tables we have 3 x 7 = 21 oops, that leaves (by
instant quick thinking subtraction) a remainder of 5, which is more than 3.
Try 8 : from our memory of tables we have 3 x 8 = 24, that leaves (by
instant quick thinking subtraction) a remainder of 2, which is less than 3,
so we must have the correct answer. A darned sight quicker than repeated
subtraction. We do it so fast that we kid ourselves that we are "doing division".
So, we do "division" by (1) guess, and (2) check.
Similarly, the derivative of x to the power of n is
n times ( x to the power of (n-1) ).
Pardon? What makes you think that that is true?
Oh, you "remember" it....
Of course, some would say one should "use a method".
You mean like finding the limit of the ratio of the difference between
function values to the infinitesimal difference?
You're an idiot: I already know the answer from memory so why do it over
every time?
And then there is integration?
Same old trick: Integration is the limit of a summation - too slow and hard.
Much easier to use our memory to (1) guess, and (2) check.
Try integrating the reciprocal of one minus x squared without memory nor check!
So, teaching mathematics is not so easy; and impossible without rote.
So one should specifically explain to students (even at Uni) why rote is
very important in Mathematics.
Colin
I've been teaching / lecturing mathematics for thirty years, but cannot
get a job teaching in the UK because "You haven't passed 'English' high
school mathematics."
But in my humble experience, we use more rote (and "worse") than we realise.
For example, "What is 26 divided by 3?" Of course it is 8 with remainder 2.
But how did you get it? No, you did not "divide".
The definition of "division" is really to ask the question "How many
'threes' are there within 26, and how many extra remain left over after you
have removed the count of threes?" The only way to do this is to subtract 3
from 26, keeping count, until you have less than 3 remaining. This is in
fact what a computer does. (And 'smart' kids need to think twice before
contradicting me on this one!)
But this method is too tedious, so what we do instead is memorise the
multiplication tables and guess (yes, guess) an answer and back check;
like this:
Try 9 : from our memory of tables we have 3 x 9 = 27 oops, too big.
Try 7 : from our memory of tables we have 3 x 7 = 21 oops, that leaves (by
instant quick thinking subtraction) a remainder of 5, which is more than 3.
Try 8 : from our memory of tables we have 3 x 8 = 24, that leaves (by
instant quick thinking subtraction) a remainder of 2, which is less than 3,
so we must have the correct answer. A darned sight quicker than repeated
subtraction. We do it so fast that we kid ourselves that we are "doing division".
So, we do "division" by (1) guess, and (2) check.
Similarly, the derivative of x to the power of n is
n times ( x to the power of (n-1) ).
Pardon? What makes you think that that is true?
Oh, you "remember" it....
Of course, some would say one should "use a method".
You mean like finding the limit of the ratio of the difference between
function values to the infinitesimal difference?
You're an idiot: I already know the answer from memory so why do it over
every time?
And then there is integration?
Same old trick: Integration is the limit of a summation - too slow and hard.
Much easier to use our memory to (1) guess, and (2) check.
Try integrating the reciprocal of one minus x squared without memory nor check!
So, teaching mathematics is not so easy; and impossible without rote.
So one should specifically explain to students (even at Uni) why rote is
very important in Mathematics.
Colin
On the subject of integration, it actually is standard here in US for engineering students to have to struggle through some simple integrals the "hard" way. For example, calculate the integral from 0 to 3 of x^2 dx. Thing is, in order to do this, you absolutely have to know the formula for the sum of the first n squares. Which isn't obvious, by any stretch of the imagination-- in fact, it's presented without proof and they must memorize it by rote!