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I would like to have a comparison or a big picture of how and why the approach for teaching math varies from primary (or pre primary) to middle to higher classes.

I understand at every level one need to learn concepts and procedures, so why I see so much compartmentalization for teaching math at different levels?/

I accept that in pre-primary classes focus should be on learning how to learn and learning using 5 senses. But is there anything more than this that nessicitates a different approach especially when moving to upper primary, then to middle, secondary and higher and finally to university? I would like to know about those approaches and contrast the same to have a big picture.

I am new to maths teaching, so kindly bear with me if you find this question too naive.

Thanks and regards

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It's a very broad question (to do the comparison you want). And made even more difficult when you consider that best pedagogy for any of the stages is not agreed on.

My personal opinion is that much of the methodology should be similar, because humans are similar. Much more than people think. There's probably some social sensibilities that are different. For example, teaching skiing I can't treat adults or pre-schoolers how I treat 5th graders (10 year old boys are the bomb...I would semi-literally go to war with them). But a lot of the drills and practice required are same (much more than different).

I went to a school that taught college classes "just like high school": similar class sizes, teachers rewarded for teaching not research, homework drill problems rather than project-style questions, being called to the board, more frequent tests versus just a midterm/final, etc. In talking with sisters that went to normal schools, I thought it was a total gyp what they were getting. From a student perspective, this sort of support/attention/methodology was much more time efficient in training us than the college lecture halls at Enormous State Universities or non-native English speaking grad students whose real priority was their thesis work.

I think a lot of the rationale for "not spoonfeeding", etc. ignores human behavioral traits. They are generally, instead, excuses for the economic setup involved in US colleges. (Which is why I recommend to kids not to go to an R1 school but go to a liberal arts school on a ROTC scholarship instead.) It's even worse in graduate school courses in terms of the "go teach yourself" excuses, grade inflation, etc.

P.s. "It never gets easier. You just go faster." -Greg LeMond.

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