The future is computers, and computers work through tagged languages and symbolic systems. Everybody needs to understand formal symbolic systems. Math is the best way that I know of to do this.
It used to be that to be a composer, first you learned the piano. The piano was thought to give one an appreciation of the nature and nuance of music. From there you could much more easily learn other instruments and then learn how to put them together into something beautiful.
Math is the new piano. Learn symbolic manipulation -- and I'd say that means first-year calculus -- and you can work with the kinds of symbolic systems you'll find everywhere else in the modern world.
I used to do some tutoring when I was a kid, and I think this level of understanding is achievable by 90% of the population given the right environment. The real question is "why are the structures and environments for education that we create so unable to accomplish this?"
Dumbing down the system isn't the answer. It's not like you can dumb down the world to make up for your inability to prepare students for it.
Why is first year calculus the standard for symbol manipulation? Algebra is also pure symbol manipulation, and the applications of it are more universal, whereas calculus is symbol manipulation over 2 functions, plus algebra; but those two functions have a lot of conceptual baggage that is not nearly as useful.
I think abstract algebra (group, ring, field) is a better way to teach symbol manipulation. Or even logic. A proof of well-ordering theorem beats integration by parts if your goal is to teach deduction and symbol manipulation.
Let's be honest: calculus is taught for its application in engineering, not to teach deduction and symbol manipulation. If that is your goal, there are more suitable maths to teach.
You need to change the way math classes think of proofs. Every times I have seen a proof in math class, they managed to take a beautiful and elegant argument, and distill it down to the least number of characters that formally proves the statement.
Presently, I am doing research work in (applied) chryptography, which as you might imagine involves reading a fair amount of modern math papers. All but one of the papers I read were easier to understand than the textbook proof for Pythagorean's thuerom.
I agree. More motivations, intuitions, insights behind fewer proofs would be more useful than lots of proofs, if the goal is to teach mathematical thinking, not to teach particular math results.
I think proofs are compressed because of lack of margins. Don't laugh! Authors seem to think they need to cover lots of essential results, and if you do motivations, intuitions, insights for all of them, already long math books will be impossible to hold in your hands. In my opinion the obvious solution is to omit most proofs and do far more non-formal discussion around important proofs. Which won't happen because their goal is to teach particular math results, not mathematical thinking, contrary to what they say.
No, all functions do that. Consider the functions f(x)=x^2, and g(x).
f(2)=4, but f(g(x))=g(x)^2. A more intuitive example would be abs(x)=|x|. For any numeric input, you get a numeric output, but for a functional input, g(x), you get g`(x) as an output, where g`(x) is a new function simmilar to, but possibly different from, g(x).
> Everybody needs to understand formal symbolic systems.
Everybody does not need to understand this and I dare (with fingers crossed) say a good percentage of those on HN don't either. There needs to be rationalization within the educational system for what is taught, with relevance to education for the sake of knowledge and for how material can be used for future employment prospects & general life skills. That may play to the lowest common denominator or the student mean, but that's largely and unfortunately what public education has become over the past fifty years.
But is it the best way educators know of to do this? Because at the end of the day, whether math is a good basic skill that pays dividends in logical thinking and symbolic manipulation (dividends beyond what other methods could deliver) is an empirical question. What you say sounds plausible, but theories can be plausible and dead wrong.
I'm dreaming of a curriculum where students start to program their own computer algebra system in 9th grade (or earlier), say, using Python or Clojure. Starting by teaching it to solve simple equations for x, then proceed over the next 4 years by extending it to know about binomial formulas, solving quadratic euations, perform gaussian elimination, symbolic differentiation and integration, etc. I guess the world isn't ready for that idea, yet ;)
Automatic differentiation by dual number! Really, after I learned it, I wondered why symbolic differentiation isn't taught this way by default. Good use case of operator overloading, by the way.
It used to be that to be a composer, first you learned the piano. The piano was thought to give one an appreciation of the nature and nuance of music. From there you could much more easily learn other instruments and then learn how to put them together into something beautiful.
Math is the new piano. Learn symbolic manipulation -- and I'd say that means first-year calculus -- and you can work with the kinds of symbolic systems you'll find everywhere else in the modern world.
I used to do some tutoring when I was a kid, and I think this level of understanding is achievable by 90% of the population given the right environment. The real question is "why are the structures and environments for education that we create so unable to accomplish this?"
Dumbing down the system isn't the answer. It's not like you can dumb down the world to make up for your inability to prepare students for it.