yes!!!

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.