On Fri, Sep 28, 2012 at 3:13 PM, kirby urner <email@example.com> wrote: > > > On Fri, Sep 28, 2012 at 11:36 AM, Paul Tanner <firstname.lastname@example.org> wrote: >> >> >> But when I said that we teach fraction addition/subtraction using the >> LCD such that the algorithm is too long and complicated and clunky to >> be written elegantly and concisely as a single equation, this single >> equation above is not the single equation I had in mind. In fact, this >> equation above is only part of this long and complicated and clunky >> algorithm. That is, on the right side we are not yet at an expression >> that is a single noncomplex fraction. >> > > Yes, you make a good point, plus you've already jumped ahead to a point > where m, the LCM, is already a bird in the hand. How did you get it in the > first place? I gave LCM(a,b) = (a*b)/GCD(a,b) with Euclid's Method for GCD. > You didn't give an algorithm, so the conceit of doing it "all in one line" > involves not explaining your techniques before and after. >
First, what I'm talking about is about making it quicker and easier once we have the least common denominator (LCD), so that once we have an LCD, we can in one written step go straight to a single noncomplex fraction in fraction addition/subtraction, just as we can with fraction multiplication/division.
And again, one of the main reasons we teach LCD-based fraction addition/subtraction is because we want students to already have (in line with Devlin calling mathematics "the science of patterns" - see the title of his book published by Scientific American) a pattern by which they can, by hand (using no machine), transform "algebraically" a sum of rational functions into a single rational function.
Lou Talman points out that mathematical literacy is, at its heart, fluency at transforming expressions into other, more useful expressions, including, very and I think most importantly, abstract expressions, which means being able to do so using no machine.
And being able to do this means being able to handle expressions of which fractional forms are a part. So my ultimate motivation here is about making this aspect of algebra easier. Yes, you have your CS thing, but I have my thing which is "What can you do when all you have is mind, something to write with, and something to write on?"
Side note: Don't knock this last point - Terence Tao himself does not use computers in his work, which means that for him essentially the only tools he uses apart from his mind are pencil and paper (chalk and blackboard) - the only tools needed are mind, something to write with, and something to write on. How do I know this? At roughly 43:20 in
he is asked by an audience member whether he uses computers in his work and he answered that he essentially does not.
The idea I'm putting forth here to you is that those who say that everything is now computers are just wrong - there is still massive amounts of theoretical work in mathematics and mathematical science that needs not computer skills but the skills that express themselves only with the tools of mind, something to write with, and something to write on. That will never go away.