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I think the more pervasive problem in schools is a lack of mastery in basic foundational skills before moving on to more advanced topics. Too many students lac

I think the more pervasive problem in schools is a lack of mastery in basic foundational skills before moving on to more advanced topics. Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem involves perhaps a dozen simple algebraic manipulations. These basic things have to be drilled and practiced A LOT before moving on to more advanced topics. There is no way around that. No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

throwawayjava 9y ago
> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics. Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms. In a typical non-proof-based calculus sequence, tricky u substitution and integration by parts problems are the two places where you really see "I was good at math in high school" students start to struggle because they lack a good conceptual foundation in trig and logarithms. Up until that point the students with trig identities etched into their skulls by practice and drilling do just fine. But from that point onward, they really start to struggle because they need a bit of geometric intuition and some practice with more effective problem solving techniques than "blindly apply equations until something works". Really, both are needed -- drilling and practicing, as well as a solid (and preferably proof-backed) intuition about elementary properties of trig funcations and logarithms.
- crispyambulance 9y ago
  I agree, you need both. I am not optimistic about the state of mathematics education. So many things have to "go right" for a student to master mathematics.

throwawayjava 9y ago

> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics. Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms. In a typical non-proof-based calculus sequence, tricky u substitution and integration by parts problems are the two places where you really see "I was good at math in high school" students start to struggle because they lack a good conceptual foundation in trig and logarithms. Up until that point the students with trig identities etched into their skulls by practice and drilling do just fine. But from that point onward, they really start to struggle because they need a bit of geometric intuition and some practice with more effective problem solving techniques than "blindly apply equations until something works". Really, both are needed -- drilling and practicing, as well as a solid (and preferably proof-backed) intuition about elementary properties of trig funcations and logarithms.

crispyambulance 9y ago
I agree, you need both. I am not optimistic about the state of mathematics education. So many things have to "go right" for a student to master mathematics.