> IMHO the goal of "simplify" an algebraic expression is especially flawed
This, too, has occurred to me. I am curious as to what heuristics tools like Mathematica use when you ask them to simplify an expression.
> Too much in math as commonly taught in K-12 and early college isn't really close to math as done by people really using math in, say, the STEM fields but is stuffed in there by the teachers as part of pedagogy or having a source of exercises and test questions.
> In response, generally it would be good to lower the emphasis on such make work pedagogy, get the students through it (minimize it and have lenient grading of it), and get on to what is important in math and its applications, research, etc.
I wish that I encountered proof-based math much earlier, and not the weird two-column proof thing they teach in geometry in high school. When I started working with proofs, math made a lot more sense to me.
Once again, like "show your work," nobody is told precisely what "simplify" means. It would be preferable to teach about "form," and then about manipulating expressions to convert them from one form to another.
For at least one class, I noticed that my daughter's textbook had replaced "simplify" with "show in standard form," where they had been told what standard form is.
I was lucky to go through a K-12 math curriculum that used proofs. And I agree that the two column format is awkward. I'm reminded of Edward Tufte's critique of PowerPoint, that a restrictive template makes it harder to express ideas. I wrote my proofs and derivations the same way that they were presented in the textbook, and in class: In a conversational style. This had the added benefit of being able to learn that style by example. When we talk about "using" math later in life, it's not just using math to get an answer, but being able to explain and justify that answer to other people.
> It would be preferable to teach about "form," and then about manipulating expressions to convert them from one form to another.
Teaching, even defining, form would not be so easy, either. In some cases, maybe for partial fractions decomposition or completion of the square, but generally, no.
Instead there is an easier approach, plenty effective: Just present the student with two algebraic expressions that are equal and have the student show that the two are equaly. So, the student gets practice in manipulating algebraic expressions; the goal, show that the two expressions are equal, is clear; and there are no issues of style or form.
Of course a lot of the work in a common course in trigonometry is of this form. So, sure, when a student gets to manipulating trig functions in calculus, they have lots of practice manipulating trig expressions, maybe even more than commonly needed! :-) Or, maybe a good trig course would trim back some of the manipulation exercises and, instead, move on to some of the trig applications, especially to signal processing, Fourier transforms (just the finite versions if want to avoid calculus), power spectra, etc. Heck covering just overtones in music, how a violin or organ is tuned, would be both good and fun.
This, too, has occurred to me. I am curious as to what heuristics tools like Mathematica use when you ask them to simplify an expression.
> Too much in math as commonly taught in K-12 and early college isn't really close to math as done by people really using math in, say, the STEM fields but is stuffed in there by the teachers as part of pedagogy or having a source of exercises and test questions.
> In response, generally it would be good to lower the emphasis on such make work pedagogy, get the students through it (minimize it and have lenient grading of it), and get on to what is important in math and its applications, research, etc.
I wish that I encountered proof-based math much earlier, and not the weird two-column proof thing they teach in geometry in high school. When I started working with proofs, math made a lot more sense to me.