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Is it really? I think repetition is dangerous, it might give you the impression that you understand something when in fact you only learned how to follow a spec

Is it really? I think repetition is dangerous, it might give you the impression that you understand something when in fact you only learned how to follow a specific formula. What you need in math is lots of unique problems to solve, not repetition of the same equation with different constants.

dagw 7y ago
it might give you the impression that you understand something when in fact you only learned how to follow a specific formula. Learning to follow the specific formula by rote is a good thing. For example you really should know your multiplication table up to 10 basically by heart. And the algorithms for doing basic arithmetic on large numbers needs to be practiced until you don't have to think about them. As should the algorithms for simple algebraic manipulation. If you try to move on without having those and similar skills under control you will never get to the understanding part of mathematics since you'll spend all your time dealing with the mechanics of arithmetic. What you need in math is lots of unique problems to solve, not repetition of the same equation with different constants. Repetition of the same problem with different constants is how you learn to spot patterns and develop techniques you can build upon. Like if you find that f(x)=7, f(3x)=21, f(5x)=35, f(9x)=63 then you might spot a pattern that you can explore and that will make it really easy to solve the rest of the problems.
- XorNot 7y ago
  Rote learning of multiplication tables was the most useless waste of my entire education. Whereas once I was older (as in like 20) I actually got the hang of them from real application, and then realised my struggles with memorisation were easily solved by finding simpler combinations to do so.
bitwize 7y ago
Unless you are a math prodigy, drills are paramount to understanding the concepts, especially before about high school. Almost no one needs deep understanding why three and four make seven, or why long division works. But if they cannot call upon these elementary tools on demand, in an instant, they will be flummoxed in later math classes. Hence, drilling until they've nailed it.
- username90 7y ago
  But the problem is that we spend ~9 years on those drills which is more time than a typical engineering student spend on algebra and above. How come we can teach so much higher math in such a short amount of time? It is not possible from scratch, the reason we can do it is because some bored students played around with the drills instead of just doing them and thus learned a lot of related concepts which are cached in on in later years. The losers are those who just spent years learning how to do the drills, getting good grades but didn't learn anything worthwhile at all. It would be good if we could catch those and correct their behavior before they had wasted 9 years on it, but as is we don't notice that most students fail to learn anything before we test them on more interesting problems in high-school. Note: I am not saying that we shouldn't teach arithmetic, on the contrary I'd argue that arithmetic is one of the most important things to learn in math since everything else is built on top of that intuition. We just have to realize that we practice arithmetic to build intuition and not to become a human calculator.
  - adrianN 7y ago
    An engineering student can learn more advanced math in a shorter time because a) they're older, b) they have mastered the elementary stuff already.
    - dagw 7y ago
      Also let's be honest, how many engineers actually learn the more advanced math. Most of them cram it into their brain for the test and then promptly forget it.
  - abdullahkhalids 7y ago
    > How come we can teach so much higher math in such a short amount of time? 1. Because, higher ed students are a subset of school students who are on average better than and more interested in solving math problems. 2. Because, the mental micro-skills needed for higher math are many and diverse and learning one during school doesn't make you better at any other. You have to learn all of them, which is why initial training takes longer. Once you have most of the micro-skills trained, you can use them to cover new material faster. Micro-skills include fast mental arithmetic, application of logical tautologies (If A=>B, then NOT B=> NOT A), visualization skills, the hand-eye coordination to draw and recognize complex symbols. There are also macro-life-skills like being able to sit and listen to a teacher for an hour without getting bored.
watwut 7y ago
You need both. You cant solve more complex problem if you are deriving quadratic formula from first principles each time. You need to "see" it quickly in order to recognize it in more complex situation. Complex problems are composed of simple problems and if simple problems are something you are figuring out each time, you got problem. Obviously, there is also such a thing as too much repetition. But understanding of how it works alone and being able to prove the formula is not enough for ability to solve problems.
- pingyong 7y ago
  Nobody even learns how to derive any formula in high school. The teacher might show it once (if even that), but then everyone just forgets it and applies the formula like a robot. Same with chain rule, integration by substitution, or anything really. People only learn how to apply it, and essentially never the idea behind it. In my class (which was focused on math) I had to explain to someone how the derivatives relate to the speed and acceleration of a car, something that should have been understood for 2 years now. And yet, she did completely fine in classes. Fairly good grades - not the best, because she had trouble with the later tasks that required more thinking, but the first 80% of every test she aced because she just practiced doing the same thing over and over again, without actually understanding any of it. Do I agree that this is a completely fine strategy for getting good grades? Sure. But you're gonna have a lot of catching up to do if you want to apply this knowledge to anything that isn't a text book task that was made up specifically for you without any context.
  - watwut 7y ago
    We did formula derivation and proves - and we exactly sucked in solving exercises. The understanding of how it works did not translated to being able to solve problems. When I went to college and was ambitious in math class that required calculations, it sucked and I was loosing. Until someone recommended me book full of exercises and I started doing them including boring ones. That is why I say you need both. Knowing that derivation is related to speed or that integral is area under the curve does not help to solve the integral at hand.

dagw 7y ago

it might give you the impression that you understand something when in fact you only learned how to follow a specific formula. Learning to follow the specific formula by rote is a good thing. For example you really should know your multiplication table up to 10 basically by heart. And the algorithms for doing basic arithmetic on large numbers needs to be practiced until you don't have to think about them. As should the algorithms for simple algebraic manipulation. If you try to move on without having those and similar skills under control you will never get to the understanding part of mathematics since you'll spend all your time dealing with the mechanics of arithmetic. What you need in math is lots of unique problems to solve, not repetition of the same equation with different constants. Repetition of the same problem with different constants is how you learn to spot patterns and develop techniques you can build upon. Like if you find that f(x)=7, f(3x)=21, f(5x)=35, f(9x)=63 then you might spot a pattern that you can explore and that will make it really easy to solve the rest of the problems.

XorNot 7y ago
Rote learning of multiplication tables was the most useless waste of my entire education. Whereas once I was older (as in like 20) I actually got the hang of them from real application, and then realised my struggles with memorisation were easily solved by finding simpler combinations to do so.

bitwize 7y ago

Unless you are a math prodigy, drills are paramount to understanding the concepts, especially before about high school. Almost no one needs deep understanding why three and four make seven, or why long division works. But if they cannot call upon these elementary tools on demand, in an instant, they will be flummoxed in later math classes. Hence, drilling until they've nailed it.

username90 7y ago
But the problem is that we spend ~9 years on those drills which is more time than a typical engineering student spend on algebra and above. How come we can teach so much higher math in such a short amount of time? It is not possible from scratch, the reason we can do it is because some bored students played around with the drills instead of just doing them and thus learned a lot of related concepts which are cached in on in later years. The losers are those who just spent years learning how to do the drills, getting good grades but didn't learn anything worthwhile at all. It would be good if we could catch those and correct their behavior before they had wasted 9 years on it, but as is we don't notice that most students fail to learn anything before we test them on more interesting problems in high-school. Note: I am not saying that we shouldn't teach arithmetic, on the contrary I'd argue that arithmetic is one of the most important things to learn in math since everything else is built on top of that intuition. We just have to realize that we practice arithmetic to build intuition and not to become a human calculator.
- adrianN 7y ago
  An engineering student can learn more advanced math in a shorter time because a) they're older, b) they have mastered the elementary stuff already.
  - dagw 7y ago
    Also let's be honest, how many engineers actually learn the more advanced math. Most of them cram it into their brain for the test and then promptly forget it.
- abdullahkhalids 7y ago
  > How come we can teach so much higher math in such a short amount of time? 1. Because, higher ed students are a subset of school students who are on average better than and more interested in solving math problems. 2. Because, the mental micro-skills needed for higher math are many and diverse and learning one during school doesn't make you better at any other. You have to learn all of them, which is why initial training takes longer. Once you have most of the micro-skills trained, you can use them to cover new material faster. Micro-skills include fast mental arithmetic, application of logical tautologies (If A=>B, then NOT B=> NOT A), visualization skills, the hand-eye coordination to draw and recognize complex symbols. There are also macro-life-skills like being able to sit and listen to a teacher for an hour without getting bored.

watwut 7y ago

You need both. You cant solve more complex problem if you are deriving quadratic formula from first principles each time. You need to "see" it quickly in order to recognize it in more complex situation. Complex problems are composed of simple problems and if simple problems are something you are figuring out each time, you got problem. Obviously, there is also such a thing as too much repetition. But understanding of how it works alone and being able to prove the formula is not enough for ability to solve problems.

pingyong 7y ago
Nobody even learns how to derive any formula in high school. The teacher might show it once (if even that), but then everyone just forgets it and applies the formula like a robot. Same with chain rule, integration by substitution, or anything really. People only learn how to apply it, and essentially never the idea behind it. In my class (which was focused on math) I had to explain to someone how the derivatives relate to the speed and acceleration of a car, something that should have been understood for 2 years now. And yet, she did completely fine in classes. Fairly good grades - not the best, because she had trouble with the later tasks that required more thinking, but the first 80% of every test she aced because she just practiced doing the same thing over and over again, without actually understanding any of it. Do I agree that this is a completely fine strategy for getting good grades? Sure. But you're gonna have a lot of catching up to do if you want to apply this knowledge to anything that isn't a text book task that was made up specifically for you without any context.
- watwut 7y ago
  We did formula derivation and proves - and we exactly sucked in solving exercises. The understanding of how it works did not translated to being able to solve problems. When I went to college and was ambitious in math class that required calculations, it sucked and I was loosing. Until someone recommended me book full of exercises and I started doing them including boring ones. That is why I say you need both. Knowing that derivation is related to speed or that integral is area under the curve does not help to solve the integral at hand.