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The key to math education is practice. Even drilling, maybe. Other fields deal with concepts more or less mapped to the real world. Physics is about real world

The key to math education is practice. Even drilling, maybe. Other fields deal with concepts more or less mapped to the real world. Physics is about real world, more or less (right until you get to quants, then the level of abstraction rises dramatically). Same goes for biology, and even computer science in general. There, you can rely on words, which usually convey meaning. In math, you can't rely on words. You'll never understand even relatively simple things like complex analysis or Fourier transform just by reading about it -- words are never enough to transfer the knowledge to you. You need to play with it, solve actual problems, understand in practice how various "moving parts" are related to each other, and then accept the naming convention (which is almost an afterthought, born as a mean of reference, not as a way to describe things). Therefore, relentless practice and solving abstract problems (a lot of them) is the only way to teach (or learn) mathematical concepts. Some teachers want to make math more accessible with bringing it "down to earth", mapping mathematical concepts to more concrete problems. It is theoretically possible, and I was a supporter of this approach until very recently. However, math just doesn't work this way. Math is pure abstraction; linking the abstractions to earthly affairs too early shuts down mathematical thinking (creates biases that prevents applying mathematical insights to other fields that are different from the one learned). Mathematical abstractions cannot be transferred by words and formulas alone; they need to be internalized by practice and drilling.

Kenji 10y ago
Yes, and notation is key. If there is even just one symbol that the student can't understand, then the entire understanding of the paragraph, or even entire book, is at stake. However, I do think that examples and applications help a great deal. Sometimes, a mere description of first principles doesn't do it for me, and then, when I see an actual example, I suddenly understand it and the theory along with it.
Swizec 10y ago
Would you say that words and formulas are leaky abstractions then?
ArkyBeagle 10y ago
I think they key is being able to jump between the narrative ( story ) domain and the formal symbols domain. At first, it is learning a language. Later, it is learning how to learn that language dynamically.
DanBC 10y ago
> The key to math education is practice. Even drilling, maybe. Do you have any research to support that? Because from what I've seen we currently think that drilling does nothing to help with understanding, and that one of the problems people have with maths is applying the wrong technique to a problem because they don't understand the problem. > You'll never understand even relatively simple things like complex analysis or Fourier transform just by reading about it Very few 8 year olds are grappling with fourier transforms.
- vixen99 10y ago
  Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over? Understanding equates to learning the rules and when and where they can be applied. That's where drilling comes in.
  - dylanfw 10y ago
    > Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over? W&R proved 1+1=2 in their axiomatic system. It wasn't done to prove once and for all that 1+1 is in fact 2, but to show that their system produced mathematical truths. The truth of 1+1=2 was already assumed and understood since they were kids, and that's why it was necessary that their system also produce it. Of course PM was obliterated by Godel shortly after.
    - dwheeler 10y ago
      Godel did not destroy formalization. He showed that there would always be true things that cannot be proved. It's true that the axioms used by Whitehead and Russell are not the axioms usually used today, but PM is still an important book that has influenced much work through today and Beyond. You might find this website interesting, which formalizes axioms and then proves many things from them: http://us.metamath.org/index.html http://us.metamath.org/index.html
LeanderK 10y ago
i don't agree completely. I think math also has a point where, until you you pass it, can make the connection to real-life. Adding, Subtracting, simple equations. But then you have to let go and resist the urge to "justify" the material by providing real-life assignments. But i agree that the key is practice. At university, where the pace picks up dramatically, you figure out that going to the lecture is not as important as finishing the assignments. Always finish the assignments.
snarfy 10y ago
This very much reminds me of learning to play a musical instrument. You don't need to learn a bunch of music theory to get better. You just need to practice a lot. The theory does no good if you can't actually play, and it makes much more sense to learn after you can play since you know what it already sounds like when reading about it.
- compiler-guy 10y ago
  The same goes for youth sports. You go to practice for a couple of hours a couple of times a week, and do the same thing or very close variations of it over and over and over. Eventually you get pretty good, but there is just no substitute for good repetitions with a good coach correcting your form and technique. Then you scrimmage where you put those into practice perhaps at a slower pace and get a chance to be creative, and then you put them into a game where things happen full speed. And after the game you evaluate what went well and what didn't. It baffles me that anyone thinks mathematical learning (or any other) can be done any other way. Reps matter.
cmdrfred 10y ago
I always considered myself bad at math in school. For me the drilling as the problem. I have trouble focusing on arbitrary tasks. After I left high school and started to teach myself programming I became much better at it. Having an actual problem to solve, instead of a worksheet created by a random number generator, is key.
krmboya 10y ago
Reminds me of this interview by Knuth [1], where he says, because of his initial insecurity that he was not good enough, he started by out doing twice as many math exercises as his classmates. He at first put in a lot of effort, but eventually he was just coasting through ahead of his classmates. He attributes it to the drilling he put in initially. [1] http://www.webofstories.com/play/donald.knuth/9 http://www.webofstories.com/play/donald.knuth/9
- kaitai 10y ago
  But doing twice as many problems is not the same as drilling. When people say "drilling" in American education, they generally mean doing simple and mechanical problems for speed. For instance, the multiplication drills at some schools involve playing a voice on a CD or mp4 that speaks multiplication problems at a certain rate. Now, doing many exercises is a different thing! one that I encourage! In most textbooks, exercises range in difficulty and approach. I write math problem sets for students, and I build them to go from mechanical to sophisticated. The last few problems involve proofs, applications, links between different areas of mathematics and statistics. Students in my class who just do probability drills will not be able to do the last problem on the problem set. They don't have any practice at problem-solving in this context. Students who do all the exercises, and more, can do amazing things.
taneq 10y ago
I agree with your statements about needing to 'play with' mathematics to properly understand it. I don't think that's a good explanation for why Americans suck at it, though, because the same goes for software development and it's really hard to argue that you guys suck at software.
- pinaceae 10y ago
  no - it's largely immigrants from eastern europe and asia running tech. regions where math levels are higher due to parents pushing for it. "proper" americans are great in design, marketing and sales. mix the two systems and you get SV.
jostylr 10y ago
The key is "you need to play with it". The way to get this to work in practice with elementary mathematics is something like Guesstimation. To my knowledge, this is not taught in most math classes. And the reason is simple: it is messy. They might try something that looks similar with word problems, but that is a far cry from the majesty of investigating the world with simple arithmetic. The problems they come up are contrived and students are no fools. Problems must come from within, not be handed to people. Drilling vs understanding is not an either..or. Most learning starts with messing around, trying to figure something out. It is slow and will be painful if the interest is not there. At some point, understanding is sufficient with a foundation in place and then drilling comes in to get a mastery, but what "drilling" is and needs to be will vary from person to person. Too little, and the speed fails to materialize inhibiting understanding of higher level concepts. Too much and it gets boring shutting off critical interest. All of this suggests that teaching mathematics in a one-size-fits-all is difficult. Also, imagine trying to grade 120 guesstimations for weeks on questions the students come up with. This is the essential difficulty of our current teaching mindset, namely the single authoritarian figure blessing or cursing the work of others. The key to true learning is excitement and motivation and little external judgement. Interested people will largely self-correct with a few subtle pointers here and there. Even at the level of Fourier transforms, most students probably fail to understand the problem. So how can it possibly make sense? Math is about working around hard problems, but the teaching of mathematics leads one to believe that there are simple algorithms to apply and we are done. A lower level example of this is Newton's method. Solving f(x) = 0 in their experience (quadratic formula) is easy and graphing shows the zeros anyway. What is the problem it solves? But what if you just give them a graphical piece of the curve and ask where they think a zero is? Then the problem becomes clear and the solution can make sense. Then it gets translated into steps (draw a line that fits well, find its root, find out what it looks like around there, repeat). And then to master it, one can do a number of drills. Good luck doing this, however, with people who do not care one iota about it and live in a culture where the majority of Americans have disdain for math (probably born out of a mixture of shame of how poorly they did in math class and bitterness that this most wonderfully human tool for exploring the world has been denied them).
nav123 10y ago
I agree with you completely. Has anyone really good practice books and other literature to practice and drill? That's something I find really hard to find.
Fluid_Mechanics 10y ago
Not so sure about that; the act of thinking about math is a form of "practice", so in my experience reading about it was sufficient for most concepts.
libeclipse 10y ago
I disagree with this. I remember when I was in primary school and I was confused about how to multiply decimals. At that level I didn't even know what it would mean to multiply a decimal: how can you have 5, 0.2 times? I asked the teacher and instead of explaining the concept properly, i.e. 0.5 is half etc., I was just told to multiply the other way around, by adding up 0.2, 5 times - and do drills with that. It's this kind of teaching that makes good robots and terrible mathematicians. (Thankfully I'm neither the former not the latter.)
- aswanson 10y ago
  That was a good way to teach you commutativity, though.
- blahblah3 10y ago
  Well properly understanding why certain number systems work (fractions, negative numbers) requires some understanding of set theory, and the great mathematicians of the past struggled with a lot of these concepts we take for granted (negative numbers, complex numbers). It's unrealistic to expect high school teachers to understand these foundations.
Jemmeh 10y ago
I whole-heartedly disagree with this. A.) For the obvious not everyone learns the same way B.)Drills and practice are important, but being able to actually apply your math skills is also important. I remember in high school when there were math word problems so many people HATED them. These were people in honors/college level classes and they struggled to actually apply the math to situations. Find how fast the train is moving, how long the ladder is, and Geometry proofs all made many of them get confused. They didn't understand the point of the math they knew. Personally, I got SO bored with drills/memorization. And personally for me, that if that's the -only- method of learning used it's awful for long term retention. The math skills that I've retained the longest were taught in several different ways-- a mix of drills/practice, visualization, mixing it with other types of problems so that I could see how the puzzle pieces fit together, etc. I understand the point of what I'm doing, it's not just some abstracted concept that I happened to write 50 times. I agree that past a certain point you'll have to accept abstraction, but especially in early years up until probably Algebra, it's a useful skill to be able to use math in the real world. Seriously, I've met so many people who can't calculate a tip, understand fractions enough for simple cooking, do their budgeting, or begin to understand economics and taxes enough to be even decently informed citizens.
- clifanatic 10y ago
  > I got SO bored with drills/memorization. That's because you're lazy. I know you really genuinely think you aren't, but you are... I'm dealing with this same mindset in my 13-year-old son right now. That's like saying you want to get in shape, but the repetitious method of lifting weights up and down is just SO boring.
  - Jemmeh 10y ago
    That's like saying I'm lazy because I prefer sometimes doing a dance class over just running on the treadmill. There's more than one way to do things. That said, drills are important. You do have to practice at some point. But if it's the -only- thing that's used, as it is in some classes I've taken, it's boring and harder to retain the information. And in my personal experience, I've seen that sticking with -only- that method can sometimes be counterproductive. It's avoiding the depth of understanding needed for actually applying the math.
alistproducer2 10y ago
In a way I don't disagree. It depends on the person though. For a person that struggles with the abstractions, drilling is necessary because it forces one to spend enough time with the subject until it "clicks."
dleibovic 10y ago
This comment suggests you did not read the article. The article directly contradicts your assertions. In talking about how Japanese teaching methods are better than American teaching methods, it states: "Similarly, 96 percent of American students’ work fell into the category of “practice,” while Japanese students spent only 41 percent of their time practicing."

Kenji 10y ago

Yes, and notation is key. If there is even just one symbol that the student can't understand, then the entire understanding of the paragraph, or even entire book, is at stake. However, I do think that examples and applications help a great deal. Sometimes, a mere description of first principles doesn't do it for me, and then, when I see an actual example, I suddenly understand it and the theory along with it.

Swizec 10y ago

Would you say that words and formulas are leaky abstractions then?

ArkyBeagle 10y ago

I think they key is being able to jump between the narrative ( story ) domain and the formal symbols domain. At first, it is learning a language. Later, it is learning how to learn that language dynamically.

DanBC 10y ago

> The key to math education is practice. Even drilling, maybe. Do you have any research to support that? Because from what I've seen we currently think that drilling does nothing to help with understanding, and that one of the problems people have with maths is applying the wrong technique to a problem because they don't understand the problem. > You'll never understand even relatively simple things like complex analysis or Fourier transform just by reading about it Very few 8 year olds are grappling with fourier transforms.

vixen99 10y ago
Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over? Understanding equates to learning the rules and when and where they can be applied. That's where drilling comes in.
- dylanfw 10y ago
  > Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over? W&R proved 1+1=2 in their axiomatic system. It wasn't done to prove once and for all that 1+1 is in fact 2, but to show that their system produced mathematical truths. The truth of 1+1=2 was already assumed and understood since they were kids, and that's why it was necessary that their system also produce it. Of course PM was obliterated by Godel shortly after.
  - dwheeler 10y ago
    Godel did not destroy formalization. He showed that there would always be true things that cannot be proved. It's true that the axioms used by Whitehead and Russell are not the axioms usually used today, but PM is still an important book that has influenced much work through today and Beyond. You might find this website interesting, which formalizes axioms and then proves many things from them: http://us.metamath.org/index.html http://us.metamath.org/index.html

LeanderK 10y ago

i don't agree completely. I think math also has a point where, until you you pass it, can make the connection to real-life. Adding, Subtracting, simple equations. But then you have to let go and resist the urge to "justify" the material by providing real-life assignments. But i agree that the key is practice. At university, where the pace picks up dramatically, you figure out that going to the lecture is not as important as finishing the assignments. Always finish the assignments.

snarfy 10y ago

This very much reminds me of learning to play a musical instrument. You don't need to learn a bunch of music theory to get better. You just need to practice a lot. The theory does no good if you can't actually play, and it makes much more sense to learn after you can play since you know what it already sounds like when reading about it.

compiler-guy 10y ago
The same goes for youth sports. You go to practice for a couple of hours a couple of times a week, and do the same thing or very close variations of it over and over and over. Eventually you get pretty good, but there is just no substitute for good repetitions with a good coach correcting your form and technique. Then you scrimmage where you put those into practice perhaps at a slower pace and get a chance to be creative, and then you put them into a game where things happen full speed. And after the game you evaluate what went well and what didn't. It baffles me that anyone thinks mathematical learning (or any other) can be done any other way. Reps matter.

cmdrfred 10y ago

I always considered myself bad at math in school. For me the drilling as the problem. I have trouble focusing on arbitrary tasks. After I left high school and started to teach myself programming I became much better at it. Having an actual problem to solve, instead of a worksheet created by a random number generator, is key.

krmboya 10y ago

Reminds me of this interview by Knuth [1], where he says, because of his initial insecurity that he was not good enough, he started by out doing twice as many math exercises as his classmates. He at first put in a lot of effort, but eventually he was just coasting through ahead of his classmates. He attributes it to the drilling he put in initially. [1] http://www.webofstories.com/play/donald.knuth/9 http://www.webofstories.com/play/donald.knuth/9

kaitai 10y ago
But doing twice as many problems is not the same as drilling. When people say "drilling" in American education, they generally mean doing simple and mechanical problems for speed. For instance, the multiplication drills at some schools involve playing a voice on a CD or mp4 that speaks multiplication problems at a certain rate. Now, doing many exercises is a different thing! one that I encourage! In most textbooks, exercises range in difficulty and approach. I write math problem sets for students, and I build them to go from mechanical to sophisticated. The last few problems involve proofs, applications, links between different areas of mathematics and statistics. Students in my class who just do probability drills will not be able to do the last problem on the problem set. They don't have any practice at problem-solving in this context. Students who do all the exercises, and more, can do amazing things.

taneq 10y ago

I agree with your statements about needing to 'play with' mathematics to properly understand it. I don't think that's a good explanation for why Americans suck at it, though, because the same goes for software development and it's really hard to argue that you guys suck at software.

pinaceae 10y ago
no - it's largely immigrants from eastern europe and asia running tech. regions where math levels are higher due to parents pushing for it. "proper" americans are great in design, marketing and sales. mix the two systems and you get SV.

jostylr 10y ago

The key is "you need to play with it". The way to get this to work in practice with elementary mathematics is something like Guesstimation. To my knowledge, this is not taught in most math classes. And the reason is simple: it is messy. They might try something that looks similar with word problems, but that is a far cry from the majesty of investigating the world with simple arithmetic. The problems they come up are contrived and students are no fools. Problems must come from within, not be handed to people. Drilling vs understanding is not an either..or. Most learning starts with messing around, trying to figure something out. It is slow and will be painful if the interest is not there. At some point, understanding is sufficient with a foundation in place and then drilling comes in to get a mastery, but what "drilling" is and needs to be will vary from person to person. Too little, and the speed fails to materialize inhibiting understanding of higher level concepts. Too much and it gets boring shutting off critical interest. All of this suggests that teaching mathematics in a one-size-fits-all is difficult. Also, imagine trying to grade 120 guesstimations for weeks on questions the students come up with. This is the essential difficulty of our current teaching mindset, namely the single authoritarian figure blessing or cursing the work of others. The key to true learning is excitement and motivation and little external judgement. Interested people will largely self-correct with a few subtle pointers here and there. Even at the level of Fourier transforms, most students probably fail to understand the problem. So how can it possibly make sense? Math is about working around hard problems, but the teaching of mathematics leads one to believe that there are simple algorithms to apply and we are done. A lower level example of this is Newton's method. Solving f(x) = 0 in their experience (quadratic formula) is easy and graphing shows the zeros anyway. What is the problem it solves? But what if you just give them a graphical piece of the curve and ask where they think a zero is? Then the problem becomes clear and the solution can make sense. Then it gets translated into steps (draw a line that fits well, find its root, find out what it looks like around there, repeat). And then to master it, one can do a number of drills. Good luck doing this, however, with people who do not care one iota about it and live in a culture where the majority of Americans have disdain for math (probably born out of a mixture of shame of how poorly they did in math class and bitterness that this most wonderfully human tool for exploring the world has been denied them).

nav123 10y ago

I agree with you completely. Has anyone really good practice books and other literature to practice and drill? That's something I find really hard to find.

Fluid_Mechanics 10y ago

Not so sure about that; the act of thinking about math is a form of "practice", so in my experience reading about it was sufficient for most concepts.

libeclipse 10y ago

I disagree with this. I remember when I was in primary school and I was confused about how to multiply decimals. At that level I didn't even know what it would mean to multiply a decimal: how can you have 5, 0.2 times? I asked the teacher and instead of explaining the concept properly, i.e. 0.5 is half etc., I was just told to multiply the other way around, by adding up 0.2, 5 times - and do drills with that. It's this kind of teaching that makes good robots and terrible mathematicians. (Thankfully I'm neither the former not the latter.)

aswanson 10y ago
That was a good way to teach you commutativity, though.
blahblah3 10y ago
Well properly understanding why certain number systems work (fractions, negative numbers) requires some understanding of set theory, and the great mathematicians of the past struggled with a lot of these concepts we take for granted (negative numbers, complex numbers). It's unrealistic to expect high school teachers to understand these foundations.

Jemmeh 10y ago

I whole-heartedly disagree with this. A.) For the obvious not everyone learns the same way B.)Drills and practice are important, but being able to actually apply your math skills is also important. I remember in high school when there were math word problems so many people HATED them. These were people in honors/college level classes and they struggled to actually apply the math to situations. Find how fast the train is moving, how long the ladder is, and Geometry proofs all made many of them get confused. They didn't understand the point of the math they knew. Personally, I got SO bored with drills/memorization. And personally for me, that if that's the -only- method of learning used it's awful for long term retention. The math skills that I've retained the longest were taught in several different ways-- a mix of drills/practice, visualization, mixing it with other types of problems so that I could see how the puzzle pieces fit together, etc. I understand the point of what I'm doing, it's not just some abstracted concept that I happened to write 50 times. I agree that past a certain point you'll have to accept abstraction, but especially in early years up until probably Algebra, it's a useful skill to be able to use math in the real world. Seriously, I've met so many people who can't calculate a tip, understand fractions enough for simple cooking, do their budgeting, or begin to understand economics and taxes enough to be even decently informed citizens.

clifanatic 10y ago
> I got SO bored with drills/memorization. That's because you're lazy. I know you really genuinely think you aren't, but you are... I'm dealing with this same mindset in my 13-year-old son right now. That's like saying you want to get in shape, but the repetitious method of lifting weights up and down is just SO boring.
- Jemmeh 10y ago
  That's like saying I'm lazy because I prefer sometimes doing a dance class over just running on the treadmill. There's more than one way to do things. That said, drills are important. You do have to practice at some point. But if it's the -only- thing that's used, as it is in some classes I've taken, it's boring and harder to retain the information. And in my personal experience, I've seen that sticking with -only- that method can sometimes be counterproductive. It's avoiding the depth of understanding needed for actually applying the math.

alistproducer2 10y ago

In a way I don't disagree. It depends on the person though. For a person that struggles with the abstractions, drilling is necessary because it forces one to spend enough time with the subject until it "clicks."

dleibovic 10y ago

This comment suggests you did not read the article. The article directly contradicts your assertions. In talking about how Japanese teaching methods are better than American teaching methods, it states: "Similarly, 96 percent of American students’ work fell into the category of “practice,” while Japanese students spent only 41 percent of their time practicing."