Why does it take so long to learn mathematics?(tonysmaths.blogspot.co.uk)
tonysmaths.blogspot.co.uk
Why does it take so long to learn mathematics?
http://tonysmaths.blogspot.co.uk/2012/10/why-does-it-take-so-long-to-learn.html
10 comments
I have also made the observation that coming back to mathematics after a few months makes it seem much easier. Several famous mathematicians pointed out this effect as well (I can only recall Bertrand Russell saying it at the moment). It even works on a day to day basis when it comes to solving problems.
In university, there were several times where I would spend all night working on some proofs and would go to sleep having made almost no progress. I would have dreams about the proofs (this made for restless sleep) but upon waking I would have many new insights into the problem and would be able to prove it. Conscious thinking about problems only seems to be reliable if you have faced (and solved) a similar problem before.
If there were some way to exploit this effect to maximum benefit, that would be awesome.
In university, there were several times where I would spend all night working on some proofs and would go to sleep having made almost no progress. I would have dreams about the proofs (this made for restless sleep) but upon waking I would have many new insights into the problem and would be able to prove it. Conscious thinking about problems only seems to be reliable if you have faced (and solved) a similar problem before.
If there were some way to exploit this effect to maximum benefit, that would be awesome.
This is a rather silly question.
Why is math hard? It's not all hard, some math is simple enough for 1st graders to understand.
Why is the math I am currently studying hard? Because you already learned the easy stuff and moved on to the hard stuff.
Why does it take a long time to understand all the different things that can happen when you combine simple ideas in arbitrary ways? Because there are a lot of different ways to combine them.
Why is the body of knowledge so large? Because people have been contributing to it for thousands of years.
Why is math hard? It's not all hard, some math is simple enough for 1st graders to understand.
Why is the math I am currently studying hard? Because you already learned the easy stuff and moved on to the hard stuff.
Why does it take a long time to understand all the different things that can happen when you combine simple ideas in arbitrary ways? Because there are a lot of different ways to combine them.
Why is the body of knowledge so large? Because people have been contributing to it for thousands of years.
>> Why is math hard? It's not all hard, some math is simple enough for 1st graders to understand.
I will try to share my experience regarding this question. Please note I am neither an educator or a mathematician. Take this opinion with a grain of salt. (Also, I hope this was not just some rethorical question)
1. I'd say "basic math" is a fairly complex beast. The difference being that children in elementary school are taught how to do math, not how or why math works. It is one thing to be set to memorize a collection of established facts (like multiplication tables), shown how to use those to solve some highly constrained problem (the typical pen and paper multiplication algorithm) and then set to practice the method (lots of multiplication problems for homework). It is a very different thing to be shown the definition of a concept (Abelian groups) and then thrown into a bunch of theorem proofs that "ought to make sense" on first sight (derive the properties of basic, grade school, algebra by the rules of abstract algebra).
2. An orthogonal point is that elementary school teachers have a more polished "culture" of how to transfer knowledge to students. They take many years to teach a relatively modest body of knowledge, starting from the most simple cases (addition and subtraction of natural numbers), and then extending to more general concepts (plus multiplication, plus division, plus squares&square roots, etc) and at the same time deepening the usages (natural numbers, integers, fractions, decimals, etc). Besides the core theme of arithmetic, additional concepts are introduced as permitted by the level of progress in this field (geometry, history and theory of numbers, numeric bases) and the foundations of the next big theme for middle tier education (algebra) are layered down.
Compare that to the ridiculous expectation of undergraduate level math classes, where every field is to be self contained within one (or at most two) term courses, and every session ought to introduce a new concept. University professors are able to get away with this because those courses are not a graduation requirement for most of the students, so most of those are allowed to fall from the back of the train and only the most talented, persistent and courageous ones succeed.
3. Then there is the students themselves. In general, even though individual dedication of children will vary according to both self curiosity and parental enforcement, it can be said that all children are expected to more or less grasp what is this arithmetic thing about. Children either do apply themselves or not, but they do not tend to question the need to memorize all those facts or practice those exercises. Once and adult student reaches university level math courses, it has been instilled in their brain that "memorization" is a debased form of intellectual pursuit and that they ought to be trying and looking for the underlying patterns in their subject of study. Armed with this false belief and under the pressure of unreasonable deadlines, the natural thing to do is to weasel out of the hard work required to truly appropriate a piece of knowledge and try to find some way to "get it" without actually "going through it all".
I will try to share my experience regarding this question. Please note I am neither an educator or a mathematician. Take this opinion with a grain of salt. (Also, I hope this was not just some rethorical question)
1. I'd say "basic math" is a fairly complex beast. The difference being that children in elementary school are taught how to do math, not how or why math works. It is one thing to be set to memorize a collection of established facts (like multiplication tables), shown how to use those to solve some highly constrained problem (the typical pen and paper multiplication algorithm) and then set to practice the method (lots of multiplication problems for homework). It is a very different thing to be shown the definition of a concept (Abelian groups) and then thrown into a bunch of theorem proofs that "ought to make sense" on first sight (derive the properties of basic, grade school, algebra by the rules of abstract algebra).
2. An orthogonal point is that elementary school teachers have a more polished "culture" of how to transfer knowledge to students. They take many years to teach a relatively modest body of knowledge, starting from the most simple cases (addition and subtraction of natural numbers), and then extending to more general concepts (plus multiplication, plus division, plus squares&square roots, etc) and at the same time deepening the usages (natural numbers, integers, fractions, decimals, etc). Besides the core theme of arithmetic, additional concepts are introduced as permitted by the level of progress in this field (geometry, history and theory of numbers, numeric bases) and the foundations of the next big theme for middle tier education (algebra) are layered down.
Compare that to the ridiculous expectation of undergraduate level math classes, where every field is to be self contained within one (or at most two) term courses, and every session ought to introduce a new concept. University professors are able to get away with this because those courses are not a graduation requirement for most of the students, so most of those are allowed to fall from the back of the train and only the most talented, persistent and courageous ones succeed.
3. Then there is the students themselves. In general, even though individual dedication of children will vary according to both self curiosity and parental enforcement, it can be said that all children are expected to more or less grasp what is this arithmetic thing about. Children either do apply themselves or not, but they do not tend to question the need to memorize all those facts or practice those exercises. Once and adult student reaches university level math courses, it has been instilled in their brain that "memorization" is a debased form of intellectual pursuit and that they ought to be trying and looking for the underlying patterns in their subject of study. Armed with this false belief and under the pressure of unreasonable deadlines, the natural thing to do is to weasel out of the hard work required to truly appropriate a piece of knowledge and try to find some way to "get it" without actually "going through it all".
>1. I'd say "basic math" is a fairly complex beast. The difference being that children in elementary school are taught how to do math, not how or why math works.
Same thing happens with 99% of "easy" stuff.
E.g if you find cooking easy, it's because a cook only learns how to _do_ cooking, not how or why it works (which would involve chemistry, physics, physiology, math, biology and other stuff).
Same thing happens with 99% of "easy" stuff.
E.g if you find cooking easy, it's because a cook only learns how to _do_ cooking, not how or why it works (which would involve chemistry, physics, physiology, math, biology and other stuff).
There are several very important factors:
1. A lot of math teaching is terrible, full of teaching formulas and abstract concepts, instead of the why's and how those abstract concepts relate to real life. A former girlfriend of mine was amazed when I taught her to treat math as something conceptual, rather than memory-based -- that she almost never needed to memorize formulas, that if she understood their meaning, she could usually just derive them on the spot. Her grades suddenly skyrocketed. No teacher had ever explained that to her.
2. Math is hard because it is exact. Fuzzy thinking is tolerated, even encouraged, in literature, political science, etc., so people can slide by with a lot less talent. Math doesn't let you do that. So, it's "harder", and can take a lot more time to learn.
3. Concepts in math build upon each other much more rigorously than in many other disciplines. You can understand the basic concepts of postmodern literary criticism in just a few minutes, without even knowing about medieval criticism. But you can't understand the basic concepts of differential equations without a massive background in mathematical concepts up to that point.
So basically, in math there's a ton of stuff (similar to other disciplines) but it's all insanely exact (unlike many other disciplines) and most of the little bits depend on having a solid understanding of all the bits that come before it (unlike many other disciplines).
1. A lot of math teaching is terrible, full of teaching formulas and abstract concepts, instead of the why's and how those abstract concepts relate to real life. A former girlfriend of mine was amazed when I taught her to treat math as something conceptual, rather than memory-based -- that she almost never needed to memorize formulas, that if she understood their meaning, she could usually just derive them on the spot. Her grades suddenly skyrocketed. No teacher had ever explained that to her.
2. Math is hard because it is exact. Fuzzy thinking is tolerated, even encouraged, in literature, political science, etc., so people can slide by with a lot less talent. Math doesn't let you do that. So, it's "harder", and can take a lot more time to learn.
3. Concepts in math build upon each other much more rigorously than in many other disciplines. You can understand the basic concepts of postmodern literary criticism in just a few minutes, without even knowing about medieval criticism. But you can't understand the basic concepts of differential equations without a massive background in mathematical concepts up to that point.
So basically, in math there's a ton of stuff (similar to other disciplines) but it's all insanely exact (unlike many other disciplines) and most of the little bits depend on having a solid understanding of all the bits that come before it (unlike many other disciplines).
> full of teaching formulas and abstract concepts
"A lot of math teaching is terrible, full of teaching formulas and abstract concepts, instead of the why's and how those abstract concepts relate to real life."
Well, I think your terrible description shows how the situation is more like "teaching math is hard".
One thing about math is that once you get it seems easy and so it's easy to assume the teach who failed to teach you before this was doing it wrong. The problem is different students fall off the wagon at different points. Too much abstraction? Too little abstract? Not enough "real world applications", too many "real world applications" (the horrid "word problems" etc).
I'll give you that one-to-one teaching is especially useful since there are so many ways to misunderstand maths.
Well, I think your terrible description shows how the situation is more like "teaching math is hard".
One thing about math is that once you get it seems easy and so it's easy to assume the teach who failed to teach you before this was doing it wrong. The problem is different students fall off the wagon at different points. Too much abstraction? Too little abstract? Not enough "real world applications", too many "real world applications" (the horrid "word problems" etc).
I'll give you that one-to-one teaching is especially useful since there are so many ways to misunderstand maths.
I should apologize for saying "terrible". That was mean-spirited. The description seems a little foggy but it is hard to explain. I'd say math is full of teachers facing multiple pressures that results in them seeming terrible. They aren't terrible in scheme of things either.
There's been a variety of papers using MRI scans to show the brain restructuring itself during learning. This takes time. Mathematics and CS deal with abstraction upon abstraction, as well as counter-intuitive systems (negative numbers, unreal numbers, infinity just as a start), both of which I would suspect require more intense rewiring. I've taught CS to English grads, and entire classes can be spent trying to establish a single, simple (to me) concept.
But theresdefinitely more effective techniques than just exposure and time. Finding the smallest new knowledge beyond the students current (ie one thing at a time) can make a big difference. As does finding the right metaphor from within their current concepts, when adding a new one. But such things are harder to scale, being student centric, and most educations rely on some amount of scaling
But theresdefinitely more effective techniques than just exposure and time. Finding the smallest new knowledge beyond the students current (ie one thing at a time) can make a big difference. As does finding the right metaphor from within their current concepts, when adding a new one. But such things are harder to scale, being student centric, and most educations rely on some amount of scaling
I had a similar (or identical?) experience with engineering. I "learned" much of it in school. Some of it was really challenging, and I quite frankly, did not understand it well. What's odd is that I put in a lot of effort in school. But it didn't stick. I just couldn't get it all.
Fast forward 16 years. I spent 6 years as a practicing engineer, followed by 10 doing other things. But now when I look back at a book that I had trouble with in school, it seems very straightforward. I've forgotten some of the math, but in general, everything is easier now.
Is this because learning this sort of thing takes time to sink in? (I did not actively study engineering after quitting my engineering job). Is it a matter of context? Things to compare the concepts to? Or even biological age?
Also interesting that I have not found this to be at all the case with programming, which I learn much like anything else - a bit at a time with no "leaps" in understanding.
Fast forward 16 years. I spent 6 years as a practicing engineer, followed by 10 doing other things. But now when I look back at a book that I had trouble with in school, it seems very straightforward. I've forgotten some of the math, but in general, everything is easier now.
Is this because learning this sort of thing takes time to sink in? (I did not actively study engineering after quitting my engineering job). Is it a matter of context? Things to compare the concepts to? Or even biological age?
Also interesting that I have not found this to be at all the case with programming, which I learn much like anything else - a bit at a time with no "leaps" in understanding.
I couldn't agree more that is takes time to understand mathematics. Mathematics is akin to riding a bicycle. It looks very difficult at first to someone untrained and unpracticed. Yet the more and more repetition, and getting up after falling down, it gradually becomes easier and a better grasp of the mechanics.
For those wanting to learn mathematics I strongly urge use it everyday in a small way. The preferably way to do that would be to include it in your "work". When you apply the mathematics to the numbers you already know and understand then it becomes clearer over time.
For those wanting to learn mathematics I strongly urge use it everyday in a small way. The preferably way to do that would be to include it in your "work". When you apply the mathematics to the numbers you already know and understand then it becomes clearer over time.
The problem with math is that very abstract concepts are introduced to students just because it's part of a curriculum. It would be the equivalent of teaching Turing machines to accountants.
Math is full of rules of thumb. Some concepts rest on top of more fundamental ones (e.g. infinitesimal calculus), but often this deeper reasoning is glossed over or ignored completely.
To really learn math is hard. To pass exams though, all you need to do is memorize rules. Not everybody is comfortable with that, some people need more concrete examples, or a better understanding of the big picture, to really grasp a subject.
Math is full of rules of thumb. Some concepts rest on top of more fundamental ones (e.g. infinitesimal calculus), but often this deeper reasoning is glossed over or ignored completely.
To really learn math is hard. To pass exams though, all you need to do is memorize rules. Not everybody is comfortable with that, some people need more concrete examples, or a better understanding of the big picture, to really grasp a subject.
Time is a factor in understanding almost any complicated concept but its far from the main reason.
I think schools (or atleast the schools I was exposed to) have always focused on solving theoretical puzzles without much context or understanding of how it would be applied.
If more time was spent on understanding concepts and less time mindlessly applying recipes and equations, perhaps learning math wouldn't be so difficult.
I think schools (or atleast the schools I was exposed to) have always focused on solving theoretical puzzles without much context or understanding of how it would be applied.
If more time was spent on understanding concepts and less time mindlessly applying recipes and equations, perhaps learning math wouldn't be so difficult.
I suspect that we could reorder some class offerings to improve the "throughput" of a math program. I can divide my math education career into before and after the semester I took the course based on Smith/Eggen/St. Andre's "A transition To Advanced Mathematics." Mathematics is proofs, and knowing how to speak them is completely critical. After that course, I could do mathematics much more easily than before. I feel that this book is fundamental to learning mathematics. And the majority of that is being blooded in the art of proof by contradiction and proof by induction.
Changing habits of thinking is a very long process.) It is as if you're re-training some of your inner neural networks. Training takes time.)
"Learning mathematics takes time, and it has always astonished me how much better I understand material when I go back to it, months or years later, than when I first studied it.
. . . .
"Certainly I had little understanding of how an area of mathematics fitted together: my learning at University consisted of reading strings of definitions and theorems, with little idea where it was all going, making sure I understood each result before going on to the next one, until, perhaps, in the last lecture of the course the lecturer would say something like "and so we have now classified all Lie algebras" and I would suddenly find out what the point of it all had been."
Fields medalist Terence Tao has quite a few good articles on the work of a research mathematician on his blog. Searching around just now (while looking for something else, which Google didn't turn up), I found his post "On writing,"
http://terrytao.wordpress.com/advice-on-writing-papers/
which links to several articles by other mathematicians on how they learn and think about mathematics to write it up for other mathematicians to read.