> None of them had a good answer! These were the top mathematicians in the world.
I can believe that.
Why?
I'm not one of "the top mathematicians in the world", but I do hold a Ph.D. in applied math from a world class research university and have published peer-reviewed research in math.
My view is that (A) the problem, solve for x in x^3 = 27, (B) the request "show your work", and (C) the objective of the work of the OP to "simplify" some algebraic expressions are at best flawed introductions to math as pure/applied mathematicians do it and in our educational system as efforts in, call it, pedagogy.
IMHO the goal of "simplify" an algebraic expression is especially flawed; that began to dawn on me in high school, and I so concluded in college and since.
Why? To "simplify" an algebraic expression is mostly a matter of style often without clear criteria or a unique answer; such simplification can at times be to illustrate something particular to the context but not be general.
Really, in math, we manipulate/leave algebraic expressions in whatever form is useful for what we are doing with the expressions and, IMHO, essentially never much for a goal of mere style or simplification.
E.g., an important manipulation of algebraic expressions was taking the algebra for discrete Fourier transforms and, essentially, manipulating it to illustrate how to do the calculations of the fast Fourier transform (FFT) -- work mostly of J. Tukey, supposedly at a US Presidential Science Advisers meeting to answer a question of R. Garwin. The FFT is darned important, and curiously the main point can be discovered and illustrated just by manipulating the algebraic expression to be in one of several particular forms.
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.
E.g., currently a big problem and hot topic in applied/research math is over fitting. Well, hush, don't tell anyone, but in some important cases can make some surprisingly good progress on over-fitting, realliy, get rid of the concerns, by essentially rewriting some of the algebra and just looking and observing. How 'bout that! No, I don't offer to fill in the details! Uh, in some cases, this work can also be a great way around some really nasty numerical stability problems.
But, right, simplifying some algebra can be important when have an important objective in mine, and style is not such an objective and, really, is not a good guide to what would be a simplification useful for some important objective.
Or, with the FFT and over-fitting, I've given two cases where there is an important objective for simplifying an algebraic expression -- alas, in both cases, without the important objective in mind, neither simplification would be seen to have better style!
> 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.
> IMHO the goal of "simplify" an algebraic expression is especially flawed
I teach calculus occasionally at a local university, and always make a point to highlight this fact to my class. In their previous algebra courses, the "objective" was more often than not to factor something into the smallest expression possible.
But in calculus, you generally want to expand an expression in to more terms to take advantage of linearity properties of operations like differentiation, integration, etc.
The concept of "simplify this" isn't very well defined, and I tended to not be a stickler for the final form of most things.
I can believe that.
Why?
I'm not one of "the top mathematicians in the world", but I do hold a Ph.D. in applied math from a world class research university and have published peer-reviewed research in math.
My view is that (A) the problem, solve for x in x^3 = 27, (B) the request "show your work", and (C) the objective of the work of the OP to "simplify" some algebraic expressions are at best flawed introductions to math as pure/applied mathematicians do it and in our educational system as efforts in, call it, pedagogy.
IMHO the goal of "simplify" an algebraic expression is especially flawed; that began to dawn on me in high school, and I so concluded in college and since.
Why? To "simplify" an algebraic expression is mostly a matter of style often without clear criteria or a unique answer; such simplification can at times be to illustrate something particular to the context but not be general.
Really, in math, we manipulate/leave algebraic expressions in whatever form is useful for what we are doing with the expressions and, IMHO, essentially never much for a goal of mere style or simplification.
E.g., an important manipulation of algebraic expressions was taking the algebra for discrete Fourier transforms and, essentially, manipulating it to illustrate how to do the calculations of the fast Fourier transform (FFT) -- work mostly of J. Tukey, supposedly at a US Presidential Science Advisers meeting to answer a question of R. Garwin. The FFT is darned important, and curiously the main point can be discovered and illustrated just by manipulating the algebraic expression to be in one of several particular forms.
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.
E.g., currently a big problem and hot topic in applied/research math is over fitting. Well, hush, don't tell anyone, but in some important cases can make some surprisingly good progress on over-fitting, realliy, get rid of the concerns, by essentially rewriting some of the algebra and just looking and observing. How 'bout that! No, I don't offer to fill in the details! Uh, in some cases, this work can also be a great way around some really nasty numerical stability problems.
But, right, simplifying some algebra can be important when have an important objective in mine, and style is not such an objective and, really, is not a good guide to what would be a simplification useful for some important objective.
Or, with the FFT and over-fitting, I've given two cases where there is an important objective for simplifying an algebraic expression -- alas, in both cases, without the important objective in mind, neither simplification would be seen to have better style!