BetterExplained is a really great resource for any math that they cover. It's the kind of explanations of math that I think Khan Academy was originally going for but lately has seemed to drop in quality.
I really hope math teachers and professors can start to move more towards this kind of "intuition" training as opposed to the arbitrary instruction found in most textbooks. It won't get as good of grades on tests but it's much more helpful in a students long term to have "knowledge" as opposed to "instruction".
It's been frustrating because my high school Calculus teachers were really good about teaching in that way. In college the teachers have been really dismal at reaching any kind of usefulness. They've long abandoned any kind of real understanding for the short term benefit of "just remember this formula" on (their own) tests.
Thanks, really glad the site is resonating. My goal is to ask the genuine (and slightly uncomfortable) question of "Did I experience the concept beyond the technical definition?"
For something like Calculus, I could parrot out the formal definition of the derivative, using limits. But could I describe it as "X-Ray vision", where I look at a curve and see a sequence of tangent lines? A filled-in circle and see the constituent rings?
I think we've forgotten to let people hear the song, not just memorize the sheet music.
Came here to say "Thank You!". My interest in math got rekindled due no small measure to your site. Now as I try to learn more math, I try very hard to look for an "intuitive" meaning.
That's awesome to hear, thanks. I realize learning is more than knowledge transfer, you have to keep up the momentum to learn (for me, the joy of an intuition keeps me cranking away at tough topics).
Great job Kalid. I try to do this with kids. Simple ideas like why anything to power zero is 1 ? why do we need complex numbers ? Why is limit of something tending to zero/infinity even an interesting and useful concept ?
Thanks. It's great exploring these "What if?" questions with kids (or adults). Math becomes more like a world to explore than something we're supposed to "learn".
One of the analogies i gave to the kids was that why did man have to learn to travel by water. Because walking on land wouldn't get them to the other island. Complex numbers sometimes sometimes are used to get to islands that aren't accessible from working with regular numbers. I was obviously referring to finding roots to cubic equations using complex numbers. Using complex numbers is in some cases like walking on water :-). These are not exact but they make kids excited. The fact that Math is replete with hacks just like real world is something that needs to be made more vivid and interesting
Convergent oscillation between formal definition and aesthetic intuitions is my teacher's goal. Smoothly going up to pure theory so you can forget it and go back to playing music.
I've noticed the same attitude towards math instruction in college, but sometimes I can't help sympathizing with the professors. Usually when you teach a course you expect to learn something in the process, but that's simply not going to happen for a math researcher teaching Calc 1. It's a miserable situation if you don't enjoy teaching, and they're there to research, not to teach. No wonder many of them spend as little time on the class as possible and teach straight from the textbook. I think "instruction without intuition" is another way of saying "bad instruction".
I agree that this lesson helps to drive a more applicable idea into the mind.
As for the comment regarding university math courses I have experienced the same. Could the abandonment of teaching real understanding come from the fact that university students are increasingly being treated as customers rather than pupils?
My impression is that the student isn't a customer or a pupil - he or she is a conduit by which government tuition subsidy is added to the school's budget.
Great book. Although it can be a bit handwavey and a bit light on the math. Honestly I think it's best supplemented with a good signals and systems text.
I like "Signal Processing and Linear Systems" by Lathi and "Digital Signal Processing : principles, algorithms and applications" by Proakis.
I really hope math teachers and professors can start to move more towards this kind of "intuition" training as opposed to the arbitrary instruction found in most textbooks. It won't get as good of grades on tests but it's much more helpful in a students long term to have "knowledge" as opposed to "instruction".
It's been frustrating because my high school Calculus teachers were really good about teaching in that way. In college the teachers have been really dismal at reaching any kind of usefulness. They've long abandoned any kind of real understanding for the short term benefit of "just remember this formula" on (their own) tests.