The way I see it is that most (including me) can't begin to think creatively with maths without the basics being automatic, e.g. what do do with terms in order to solve an algebra equation and how to add, multiply, divide and subtract fractions. If all your mental effort is directed at figuring out the basics then there is not enough 'power' left over to do the creative stuff. So in my view repetition has its place in maths, but it should not be everything in maths. I guess maths teachers could be more innovative in how they get students to do the laborious repetitive practice. Basketball coaches, for example, don't seem to have much of a problem in getting their charges to stand in the key and shoot ball after ball after ball until they are consistently get the basketball through the hoop. Such training is as much of a social event as repetitive practice. Perhaps that is the secret with maths - working out a way to make repetitive exercises more of a social event.
I'm a math(s) teacher and could not agree more. If you have to halt your progress in the 'flow' of a problem because you have to compute 7x13 by hand instead of in your head, you're not gonna have the 'headspace' (=mental power) to think about what needs to be done next. Or at least that how I see it. If students could be taught how to solve a problem symbolically first, either by algebraically or with propositions so you can do the calculations all at once at the end, it could work. But the teaching prepositions is more difficult than the algebra (because they have to be good at language as well), and if they could learn that, they can learn anything. So it is a catch 22. The side step I have actually consistently seen in 5 years of teaching is when the kid has the times tables down. Like snap of the fingers 12x6. Then division becomes easy, because that is just about searching for multiplicative factors. Then factorising in algebra becomes easy because you can see how the coefficients decompose and then all of advanced algebra become accessible because one can factorise. So yeah, teach the journey, but you can't make the trek would the survival basics (strong mental addition and multiplication skills).
> working out a way to make repetitive exercises more of a social event. Make it less appealing to introverts? Make it possible for people to be judged not just by the teacher but also their peers? Increase anxiety until morale improves? > I guess maths teachers could be more innovative in how they get students to do the laborious repetitive practice Whatever technique works for army conscripts may also work for school conscripts (in all subjects). If people are forced to attend whether motivated or not, don't expect them to be motivated. > Basketball coaches, for example, don't seem to have much of a problem in getting their charges to stand in the key and shoot ball after ball after ball until they are consistently get the basketball through the hoop. Sounds like torture to me. Having not been instructed on any concepts that would help me plan a better shot, to me this would feel like rolling the dice until I had rolled enough 6s while not getting better at anything (how do you get better at randomness?) except somehow for mysterious reasons other people roll enough 6s faster than me. The torture would be double if other people were watching me. I guess if people are the opposite of me, if they see no patterns or rhyme or reason in math and somehow understand how to throw balls, they would hate math the same way I would hate throwing basketballs.
I have been a long time fan of the aops users IDMasterz and XmL. I just recently found out that the Liang-Zelich theorem paper was written by you two and I agree, I was completely dumbstruck
Teaching math creatively gives a non-mathed-up person the hibber-jibbers because it is often unnecessarily detailed and condescending like “new math” was! You are mathed when you know what to do each time mathematically. It’s a game in which you get to make beautiful large patterns, whether they are infinitely tiny items or giant, and you are able to move these shapes rapidly in a sublime pattern. You only need the rules given simply in a pattern that itself, moves smoothly upon itself in ways that multiply or divide easily enough for the user to maintain the language for these principles and the game rules that the language represents. Do that, and we will be happy. Yes, even if we began in an un-mathed state.
Why does a man who is sigularly intrigued with math think that his approach is going to be generally applicable? By his account, he's an amazing and talented man, but his interest is not general and I didn't hear any evidence that he has understanding of the vast majority of students.
I have heard this argument many times, but no one has defended it better than him.
The way I see it is that most (including me) can't begin to think creatively with maths without the basics being automatic, e.g. what do do with terms in order to solve an algebra equation and how to add, multiply, divide and subtract fractions.
If all your mental effort is directed at figuring out the basics then there is not enough 'power' left over to do the creative stuff.
So in my view repetition has its place in maths, but it should not be everything in maths.
I guess maths teachers could be more innovative in how they get students to do the laborious repetitive practice.
Basketball coaches, for example, don't seem to have much of a problem in getting their charges to stand in the key and shoot ball after ball after ball until they are consistently get the basketball through the hoop. Such training is as much of a social event as repetitive practice. Perhaps that is the secret with maths - working out a way to make repetitive exercises more of a social event.
I'm a math(s) teacher and could not agree more. If you have to halt your progress in the 'flow' of a problem because you have to compute 7x13 by hand instead of in your head, you're not gonna have the 'headspace' (=mental power) to think about what needs to be done next. Or at least that how I see it. If students could be taught how to solve a problem symbolically first, either by algebraically or with propositions so you can do the calculations all at once at the end, it could work. But the teaching prepositions is more difficult than the algebra (because they have to be good at language as well), and if they could learn that, they can learn anything. So it is a catch 22. The side step I have actually consistently seen in 5 years of teaching is when the kid has the times tables down. Like snap of the fingers 12x6. Then division becomes easy, because that is just about searching for multiplicative factors. Then factorising in algebra becomes easy because you can see how the coefficients decompose and then all of advanced algebra become accessible because one can factorise. So yeah, teach the journey, but you can't make the trek would the survival basics (strong mental addition and multiplication skills).
> working out a way to make repetitive exercises more of a social event.
Make it less appealing to introverts? Make it possible for people to be judged not just by the teacher but also their peers? Increase anxiety until morale improves?
> I guess maths teachers could be more innovative in how they get students to do the laborious repetitive practice
Whatever technique works for army conscripts may also work for school conscripts (in all subjects). If people are forced to attend whether motivated or not, don't expect them to be motivated.
> Basketball coaches, for example, don't seem to have much of a problem in getting their charges to stand in the key and shoot ball after ball after ball until they are consistently get the basketball through the hoop.
Sounds like torture to me. Having not been instructed on any concepts that would help me plan a better shot, to me this would feel like rolling the dice until I had rolled enough 6s while not getting better at anything (how do you get better at randomness?) except somehow for mysterious reasons other people roll enough 6s faster than me.
The torture would be double if other people were watching me.
I guess if people are the opposite of me, if they see no patterns or rhyme or reason in math and somehow understand how to throw balls, they would hate math the same way I would hate throwing basketballs.
That's Vedic Math. Ancient Indians used it and now it is only known by few.
I have been a long time fan of the aops users IDMasterz and XmL. I just recently found out that the Liang-Zelich theorem paper was written by you two and I agree, I was completely dumbstruck
yes!!!
Teaching math creatively gives a non-mathed-up person the hibber-jibbers because it is often unnecessarily detailed and condescending like “new math” was! You are mathed when you know what to do each time mathematically. It’s a game in which you get to make beautiful large patterns, whether they are infinitely tiny items or giant, and you are able to move these shapes rapidly in a sublime pattern. You only need the rules given simply in a pattern that itself, moves smoothly upon itself in ways that multiply or divide easily enough for the user to maintain the language for these principles and the game rules that the language represents. Do that, and we will be happy. Yes, even if we began in an un-mathed state.
Absolutely.
I totally agree - i rarely say that.
100% agree.
yes
agreed.
This was great explanations.
Same problem with me I had problem in geometry this demotivated me
wow great point honestly
❤️
Why does a man who is sigularly intrigued with math think that his approach is going to be generally applicable? By his account, he's an amazing and talented man, but his interest is not general and I didn't hear any evidence that he has understanding of the vast majority of students.
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