Do we learn, or memorize mathematics

[Ivan Seeking]

On one hand, from my point of view at least there is a general understanding, even perhaps a correct sense of logic that one acquires while studying math and the physical sciences. since I am convinced, and I believe that studies have now shown that all of that math homework actually makes us "smarter", I have long been a stong defender the study of math in particular at all levels. But at the same time, as I grow older and dumber, I have noticed how much of mathematics is memorized. I know this since I have noticed how much I have forgotten. Use it or lose it.

But buried in this is a more fundamental issue. I can remember that as I did my homework, many of those wrong answers made sense for a time. Then, as if someone hits a switch, in the midst of doing all of those problems, the correct way of thinking becomes perfectly clear. And what once seemed logical makes no sense at all. So what exactly has happened here? My sense is that the operations are forgotten but the logic somehow remains - as if something was permanently hard wired in the brain. But, AFAIK there is no such hard wiring in the brain. So, is logic \(\displaystyle really just a matter of memorization, or if not, what exactly is the distinction in this context between learning, and memorizing?\)

 

[neurocomp2003]

IMO...they are highly intertwined but the distinction between learning and memorizing i think is the ability to apply a concept. Memory I think is just the association of A to B that's why biology tests people say seem to be pure memorization...however once your able to apply the knowledge you have just memorized ie...when a kid learns teh concept of addition...first they begin to memorize 1+1=2 and 2+2=4...then sometime downt he future they are able to apply teh concept of addition(perhaps they have to grasp the word first). They have learned the meaning of addition.

What comes first I dont' know perhaps you can first learn(ie by counting your fingers)
before you memorize 1+1=2..

As for how the brain stores both processes, they may just be one and the same method of storage

 

[honestrosewater]

Can 'mechanical procedures' help to draw some lines? Regardless of the origin of a proof, isn't the proof itself basically just a 'purely mechanical' manipulation of symbols? Granted, determining the basic necessities doesn't tell you what is actually going on in people's brains, but it's a start.

 

[Kube]

I agree, I think both memorisation and comprehension come hand in hand. Simple rules are learned and stored in the brain, and then applied to create new rules, eg. addition leads eventually to multiplication. But that's my opinion, because usually, people with basic grounding in any subject can usually advance faster than others who memorise straight.

 

[joeboo]

Learning of all kinds involves memorization in some form. The question is, what is one memorizing when they learn?

Certainly, a man could memorize every mathematical theorem, and definition, and be able to interpret them. However, this does not make him a mathematician, in my opinion.

It's the understanding of the relations between mathematical facts that shows one has learned the subject.

A quick example to describe:
Having moved around a good deal as a child, I never stayed at one school too long. What with varying requirements for education, I never took a geometry course. However, I did take Calculus. When it came time to take the SATs, I had very little knowledge of area and volume formulae. Note this is the type of thing one who had memorized mathematics would be fluent in. However, I knew how to apply the calculus in order to calculate areas and volumes of geometric objects.

In closing, if one has specific knowledge ( ie memorization ) of a specific body of mathematics ( say plane geometry ) it offers him little when he is faced with a different body of mathematics ( non euclidean geometry ). However, if he understands the principles, relations and theory of that same body, he can attempt to solve problems in new fields by searching for similarities.

 

[Crosson]

"In mathematics we don't understand things, we just get used to them."

--John Von Neumann

 

[BicycleTree]

Whenever you are asked to solve a problem or produce a proof, you are relying on something more advanced than memorization. You need insight, understanding. Memorization only gives you the pieces of the puzzle; mathematics focuses on learning how to put the pieces together in meaningful ways.

That's odd, joeboo--I think of SAT-level plane geometry as much more intuitive than calculus. There are some formulas to learn, but they are only tools. The main thing is intuitively inspecting the problem to break it down nicely. For example, if you want to see if the information given is enough to completely specify a figure, the best way to do it is mentally to try to wiggle the figure and see if that's possible without altering any of the given angles or lengths.

 

[joeboo]

That's odd, joeboo--I think of SAT-level plane geometry as much more intuitive than calculus. There are some formulas to learn, but they are only tools. The main thing is intuitively inspecting the problem to break it down nicely. For example, if you want to see if the information given is enough to completely specify a figure, the best way to do it is mentally to try to wiggle the figure and see if that's possible without altering any of the given angles or lengths.

Perhaps I didn't explain myself properly ...
What you are describing is exactly what I suggest, except you don't go far enough. Consider the part of your quote I bolded. What is 'nicely" ? For you, it may involve dissecting the object into smaller parts whose areas you can calculate. For another, it may involve parameterizing the perimeter of the object and using Green's Theorem to evaluate the area. The key is, both are searching the problem at hand for notions that are familiar to them, and using tools with which they are comfortable.
Note that this is the very essence of mathematics. Take a statement, then using logic and axioms determine if the statement is true. Axioms are statements we are familiar with, and know the veracity of. Logic is a tool we are familiar with.
The question is simply: What notions you familiar with? What Tools are you comfortable with?
My original point was, the answer to those 2 questions does not matter if you are unable to take a problem and compare it to those notions and apply those tools. That is why I said it is the relations between those notions that represents an understanding of mathematics ( here I am suggesting that if you can't use the knowledge, then you cannot understand it ... a contestable claim for sure ).

 

[Ivan Seeking]

I don't know if this is the same that I heard reported some time ago, but it addresses a similar hypothesis. It also hits the nail on the head from my point of view. The problem solving skills acquired in college, and in particular the ability to solve new problems, are not only what I value most, they have proven to be the most valuable result of all of those long nights spent plodding through all of those homework problems.

IQ’s are rising all over the world. Maybe the development of mathematics education has something to do with it.

...Popular explanations of the Flynn effect often note improvements in nutrition and increased access to formal schooling, but these authors emphasize the changing nature of mathematics education and the possible effects on the prefrontal cortex. This is because fluid intelligence (the ability to reason and deal with unfamiliar problems) has rapidly improved in recent history. They suggest contemporary education requires kids to get a lot of practice with prefrontally-based fluid cognitive skills. [continued]

http://www.chriscorrea.com/archives/2004/math-education-and-the-flynn-effect/

 

[selfAdjoint]

BTW Ivan, two studies, in Norway and in Denmark, have shown that while IQ increased between 1930 and 1970, it has stayed flat and even possibly declined since 1970.

(Edit) Here's the link (it's a long one; hope it works. If not google Flynn effect Norway).
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V9F-4G7DXTD-3&_coverDate=05%2F23%2F2005&_alid=290424767&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=5897&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=6f59f0b16fb6ee4bdcb62dc31ced8ef9

 

[bross7]

I would argue that it is memorized more based on how few people actually can prove what they do. Lot's of people know how to take derivatives, but as them to prove it and many of them can't. I remember in high school that proofs always was the weakest unit for most people (as a non-math major in University I can't speak for the same there...I would hope that math majors are capable of proving what they solve). To me, that smacks of memorization instead of understanding.

 

[BicycleTree]

Perhaps I didn't explain myself properly ...
What you are describing is exactly what I suggest, except you don't go far enough. Consider the part of your quote I bolded. What is 'nicely" ? For you, it may involve dissecting the object into smaller parts whose areas you can calculate. For another, it may involve parameterizing the perimeter of the object and using Green's Theorem to evaluate the area. The key is, both are searching the problem at hand for notions that are familiar to them, and using tools with which they are comfortable.
Note that this is the very essence of mathematics. Take a statement, then using logic and axioms determine if the statement is true. Axioms are statements we are familiar with, and know the veracity of. Logic is a tool we are familiar with.
The question is simply: What notions you familiar with? What Tools are you comfortable with?
My original point was, the answer to those 2 questions does not matter if you are unable to take a problem and compare it to those notions and apply those tools. That is why I said it is the relations between those notions that represents an understanding of mathematics ( here I am suggesting that if you can't use the knowledge, then you cannot understand it ... a contestable claim for sure ).

Yes, I understand what you're saying, and I agree. I was mostly just replying to your sentence, "note this is the type of thing one who had memorized mathematics would be fluent in." Memorization without thought isn't any good in plane geometry either.

 

[arildno]

BTW Ivan, two studies, in Norway and in Denmark, have shown that while IQ increased between 1930 and 1970, it has stayed flat and even possibly declined since 1970.

I know; I was born in 1971..

 

[selfAdjoint]

Isn't there a pretty full set between memorizing "6X4 = 24", then inverting it to "24 has divisors 6 and 4", then going to "24 has prime factors 2³ and 3", then to "every number has a unique decomposition into prime power factors". And then when you come back to 6X4 all that other stuff comes up too in your mind and that is at least the beginning of "understanding".

 

[russ_watters]

I can remember that as I did my homework, many of those wrong answers made sense for a time. Then, as if someone hits a switch, in the midst of doing all of those problems, the correct way of thinking becomes perfectly clear. And what once seemed logical makes no sense at all. So what exactly has happened here?

That's the way it worked for me and my interpretation of that is that that's real learning. Math requires some pretty complex logic and the "click" is just when you finally make the connection and figure out the logic.

For the original question, the answer is both. However, different people are different. My memorization skills are terrible and as a result, I had to learn everything otherwise I'd never be able to pass a test. But at the same time, when there are a lot of complex steps, you also need to be able to memorize them. Eventually, I reached a pretty hard limit to the level of complexity of math that I could learn - though while most of my friends would commit things to short-term memory and then brain-dump after the test, what I learn is near permanent.

 

[Ivan Seeking]

That's the way it worked for me and my interpretation of that is that that's real learning. Math requires some pretty complex logic and the "click" is just when you finally make the connection and figure out the logic.

For the original question, the answer is both. However, different people are different. My memorization skills are terrible and as a result, I had to learn everything otherwise I'd never be able to pass a test. But at the same time, when there are a lot of complex steps, you also need to be able to memorize them. Eventually, I reached a pretty hard limit to the level of complexity of math that I could learn - though while most of my friends would commit things to short-term memory and then brain-dump after the test, what I learn is near permanent.

But what exactly differentiates learning from memorizing?

 

[russ_watters]

But what exactly differentiates learning from memorizing?

Memorizing means you know the Newtonian physics distance/speed/acceleration equations from memory. Learning means you know how to use them and integrate/differentiate to find them.

Ask a handful of calculus students to find a derivative and most will know how - ask them what a derivative is in the physical sense and I would suspect most won't know. Its the difference between knowing what to do and knowing why you are doing it.

Using myself as an example again, I did relatively poorly in my algebra and pre-calc classes, but calculus 1 was a piece of cake because I could see the physical significance of what I was doing. Along the same lines, my memory is much more visual that verbal. I remember things with physical significance muh better than my friends do - I'm always pulling out random things I learned years ago that most people wouldn't remember: I remember learning numerical integration in 9th grade because of its significance and carried that with me when I learned calculus. But I'm absolutely awful at remembering names.

 

[selfAdjoint]

That's the way it worked for me and my interpretation of that is that that's real learning. Math requires some pretty complex logic and the "click" is just when you finally make the connection and figure out the logic.

For the original question, the answer is both. However, different people are different. My memorization skills are terrible and as a result, I had to learn everything otherwise I'd never be able to pass a test. But at the same time, when there are a lot of complex steps, you also need to be able to memorize them. Eventually, I reached a pretty hard limit to the level of complexity of math that I could learn - though while most of my friends would commit things to short-term memory and then brain-dump after the test, what I learn is near permanent.

My son worked out in the second grade how to do multiplication by repeated addition. He could do it in his head for small enough factors. So when it came time to learn the times tables by rote, he was handicapped - he could gve the simpler answers as fast as the other kids who had memorized, but completely failed on the bigger factors. I couldn't persuade him to buckle down and memorize because it seemed trivial compared to his method based on understanding. This handicap shadowed his math work all the rest of his days in school.

 

[Ivan Seeking]

What about pattern recognition? Does anyone know if this ability is strictly innate, or can it be enhanced?

I have often noticed that this seems to play a role in mathematics.

 

[BicycleTree]

Every specific mental skill can be enhanced. For example, video gamers score extremely far above the general population on tests of visual acuity.

 

[selfAdjoint]

What about pattern recognition? Does anyone know if this ability is strictly innate, or can it be enhanced?

Pattern recognition is the basic skill tested by the Raven Matrices IQ tests. They are highly g-loaded, so I don't think they can be enhanced, at least at the high end. I do believe familiarity with pattern matching puzzles raised the apparent IQs of a lot of people after WWII, but not again at the high end.

 

[honestrosewater]

Does learning, as you mean it here, require you to make inductive inferences? If you're given enough information to memorize, it seems deductive reasoning doesn't require you to "understand" anything. Think of Searle's Chinese Room Argument:

Imagine a native English speaker who knows no Chinese locked in a room full of boxes of Chinese symbols (a data base) together with a book of instructions for manipulating the symbols (the program). Imagine that people outside the room send in other Chinese symbols which, unknown to the person in the room, are questions in Chinese (the input). And imagine that by following the instructions in the program the man in the room is able to pass out Chinese symbols which are correct answers to the questions (the output). The program enables the person in the room to pass the Turing Test for understanding Chinese but he does not understand a word of Chinese

- http://plato.stanford.edu/entries/chinese-room/

On the other hand, in inductive reasoning, can you ever be given enough information so that you don't have to "understand" what you're doing? Maybe the difference between "memorization" and "learning" is the type of reasoning involved? Am I totally missing the point here?

 

[BicycleTree]

Pattern recognition is the basic skill tested by the Raven Matrices IQ tests. They are highly g-loaded, so I don't think they can be enhanced, at least at the high end. I do believe familiarity with pattern matching puzzles raised the apparent IQs of a lot of people after WWII, but not again at the high end.

The patterns you are asked to recognize on the Raven test are very simple in nature, having to do with only a few basic properties such as symmetry and edge counting. The more properties may come into play in the pattern--and math has a great many of those--the more learnable it is, because familiarity with the properties involved is learnable.

Also, support your claim that the scores on the Raven test did not increase at the "high end" in the 20th century.

 

[selfAdjoint]

The patterns you are asked to recognize on the Raven test are very simple in nature, having to do with only a few basic properties such as symmetry and edge counting. The more properties may come into play in the pattern--and math has a great many of those--the more learnable it is, because familiarity with the properties involved is learnable.

I think this learning of patterns does go on, but the talent for it is not generally learnable. Individual patterns can be learned, even categories of patterns, but the ease and speed with which they are learned does vary between mathematicians. Some are inherently better at it than others, and the difference is highly g-loaded, and IQ by Raven matrices is a good surrogate for g.

Also, support your claim that the scores on the Raven test did not increase at the "high end" in the 20th century.

The Flynn effect is stated in terms of means. There is no evidence that the long term upper limit of IQ, somewhere around 200, has been breached. You hear reports, but they often sound fishy, someone took a single test once and scored above 200. Can they do it repeatedly? Can their scores be translated into the equivalent value in g?

 

[BicycleTree]

I think this learning of patterns does go on, but the talent for it is not generally learnable. Individual patterns can be learned, even categories of patterns, but the ease and speed with which they are learned does vary between mathematicians.

OK, that's a point.

Some are inherently better at it than others, and the difference is highly g-loaded, and IQ by Raven matrices is a good surrogate for g.

Because of the Flynn effect, it must be said that while Raven's progressive matrices may predict g, performance on them is learnable. Whether this learning involves only that the testees have more patterns memorized, or whether they also use different mental algorithms to process the test, is the question.

The Flynn effect is stated in terms of means. There is no evidence that the long term upper limit of IQ, somewhere around 200, has been breached. You hear reports, but they often sound fishy, someone took a single test once and scored above 200. Can they do it repeatedly? Can their scores be translated into the equivalent value in g?

First, I think that the so-called "upper limit" becomes meaningless. You stop measuring how well the testee's brain is constructed and start measuring the biological limits on any human's brain. Biological limits mean that at the very highest level of math, say the level of Karl Gauss, it's probably not possible to learn better mathematic pattern-recognition skills. But it says nothing about whether it is possible for a mathematician significantly below Gauss, but still at a high level, to learn better mathematic pattern-recognition skills.

Also, at a high tested IQ, test-specific mental quirks start playing a greater role. Is someone with an IQ of 190 smarter than some person with an IQ of 170? The IQ 190 person has a brain more adapted to IQ tests, but is there any real-world activity that the IQ 170 person would be less able to excel at than the IQ 190 person, assuming neither has a special talent for it? I doubt it; evidence is that while mathematicians and scientists have high IQs, I don't think it's the case that the very highest-IQ mathematicians and scientists are also the very most accomplished.

Here is a report on the video-game study I alluded to:
http://news.nationalgeographic.com/news/2003/05/0528_030528_videogames.html
The SciAm article I read also stated that when the tests were first used, the video gamers mostly got perfect scores. When the tests were made more difficult so that the video gamers did not always score perfectly, the performance of the non-gamers was indistinguishable from chance.

 

[Ivan Seeking]

There was a physics professor at OSU who lost his sight some years ago. I once asked him if he thought a blind man [visually impaired] can learn complex mathematics. I am such a visual learner that it was hard for me to imagine learning math without having vision. He was admittedly quite perplexed by this question and said that he didn't know; he would have to think about it. But since then I have seen that at least a few people who have been blind from birth have obtained advanced degrees in mathematics and physics. But how they do it is a mystery to me.