Also, math education in general has a huge problem around teaching solutions without teaching the problem. (Hence the pervasive question of what will I ever need this for...) I will always know what a limit does, because my precalculus teacher taught it by finding the area under a curve using rectangles. We started with large rectangles, were shown the error. Cut the rectangles in half, hey error goes down. Make them really really small, error goes down even more! What happens if we make them zero? Oops, equation explodes because zero-width rectangles also have zero area... The function is discontinuous at zero. So let's learn limits to work around discontinuous points of a function!
https://en.wikipedia.org/wiki/Montessori_education
Also, math education in general has a huge problem around teaching solutions without teaching the problem. (Hence the pervasive question of what will I ever need this for...) I will always know what a limit does, because my precalculus teacher taught it by finding the area under a curve using rectangles. We started with large rectangles, were shown the error. Cut the rectangles in half, hey error goes down. Make them really really small, error goes down even more! What happens if we make them zero? Oops, equation explodes because zero-width rectangles also have zero area... The function is discontinuous at zero. So let's learn limits to work around discontinuous points of a function!