[schoen]

Your review is a very interesting take. It reminds me of someone's observation (I forgot whose) that mathematics is often presented as a set of successful proofs and derivations, without an explanation of the motivation behind them or how they were discovered. So proof tactics may seem somewhat magical, even though they might in fact be a result of a mathematician's tinkering and blundering around, including alternative approaches that didn't work.

An interesting example that someone gave from elementary mathematics is the derivation of the quadratic formula.

ax²+bx+c=0 (a≠0)

4a(ax²+bx+c)=4a(0)=0

4a²x²+4abx+4ac=0

4a²x²+4abx+4ac+b²=b²

4a²x²+4abx+b²=b²-4ac

(2ax+b)(2ax+b)=b²-4ac

2ax+b=±√(b²-4ac)

2ax=-b±√(b²-4ac)

x=(-b±√(b²-4ac))/2a

Someone discussing this pointed out that this derivation is easy to follow, but extraordinarily mysterious in terms of how we knew to do various things at various steps, such as mysteriously multiplying both sides by 4a or mysteriously adding b² to each side at some point. How did we know to do that?

Of course there are different explanations of the underlying motivations and the history of how people discovered this proof, but it's easy to be given the proof without any of that context, and that kind of thing is in fact the rule rather than the exception in many parts of math study.

Again, I think this was someone else's observation but I don't remember where I came across it.

[romwell]

>An interesting example that someone gave from elementary mathematics is the derivation of the quadratic formula...

A tangential point: this is indeed a horrible way to derive the formula, for the reasons you mentioned.

If anyone is curious, here's a better way to think about it. The graph of ax² + bx + c is just the graph of ax² translated. Keeping that in mind, let's investigate.

First, consider a very easy problem: find roots of ax² = 0. The graph intersects the x-axis at x=0, done.

Now, let's shift the whole graph down by Q, and solve the problem again. The equation for that graph is ax² - Q, and it intersects the x-axis at ±√(Q/a). Still easy.

Now, let's shift the whole graph again to the right by R. The equation for that new graph is a(x-R)² - Q.

What of the roots? Oh, we don't need to do much work here! The places where the graph intersects the x axis simply shifted to the right by R. So the roots are R±√(Q/a).

So, to recap: the roots of a(x-R)² - Q = 0 are R±√(Q/a).

What if our equation is written in the form ax² + bx + c? Well, now is the time for algebra. Open up the parentheses:

a(x-R)² - Q

=a(x² - 2xR + R²) - Q

= ax² + (-2aR)x + (aR² - Q)

= ax² + bx + c

Solve the following system for Q and R:

-2aR = b

aR² - Q = c

Obtain:

R = -b/2a

Q = b²/4a - c

Now plug these Q and R into the formula we already have: R±√(Q/a) - to obtain the all-familiar result

x=(-b±√(b²-4ac))/2a

What the formula is hiding is the simple idea that the roots of a parabola are easily found if you know where the vertex is. So assume you do, and work backwards from there.

A deeper idea is solving an easier version of the problem, and then changing the problem back to the more general original question, refining the solution on each step.

And this is, in fact, how mathematics is often done.

>and that kind of thing is in fact the rule rather than the exception in many parts of math study.

There's work done to change it[1]. Note that in the argument above, I could have left out all the "work", leaving only the questions, and many people would still be able to do the work. And with the right preparation, the student would be led to ask the same questions.

[1]https://en.wikipedia.org/wiki/Inquiry-based_learning

[fjsolwmv]

And that's just a special case of Completing the Square (but has a very handy visualization).

https://en.m.wikipedia.org/wiki/Completing_the_square

[romwell]

Not quite. "Completing the square" is an algebraic step; the argument is geometric (translation of the parabola). One can learn how to complete the square (symbolic manipulation) and be completely unaware of the geometry.

If you never learned the algebraic trick, would you invent it when solving this problem? It's not immediately clear from the algebra that a quadratic polynomial ax^2 + bx + c even can be rewritten in the form a(x-Q)^2 - R.

Conversely, while this technique involves algebra that amounts to completing the square, it exhibits a general technique.

If you want to say "..and that's just a special case of..", homotopy continuation methods[1][2] -

- because the idea here is transforming the simpler polynomial x^2 - C into a more complicated one, and seeing what happens to the roots.

[1]https://en.wikipedia.org/wiki/Numerical_continuation

[2] https://en.wikipedia.org/wiki/Numerical_algebraic_geometry#H...

[jacobolus]

Personally I prefer to break things into three steps:

First put the equation into the form

x² = 2ax + b

Now complete the square:

(x – a)² = a² + b

Finally,

x = a ± √(a² + b)

[romwell]

Sure, it's short. But it doesn't get to why one would do these steps if you don't know they are going to lead to a solution. Completing the square is a non-obvious step to make.

The argument in my previous comment attempts to provide motivation for such a step, starting from simpler questions. It uses the geometry of the problem, and builds up from solving a simpler problem first.

That approach also uses the notion of transformation and invariance (seeing what happens to the roots when we move the graph around, and noticing that the distance between the roots doesn't change we shift horizontally).

Again, the important part here, is that you could lead someone to ask the same questions and have them answer them themselves.

"Let's solve this equation. Looks complicated. Can we solve a simpler problem first? What would a simpler problem be? Now how can we make it a little more complicated, and how does it affect the answer?".

And that's how math is done.

After all is said and done, one can extract "completing the square" as a shortcut technique. But that's what it is - a shortcut through the woods. Learning a shortcut won't teach you how to walk in the forest on your own.

[schoen]

Thanks! It's quite possible that my example came from Paul Lockhart or a proponent of somewhat similar ideas.

[romwell]

Indeed, and I hope these ideas gain more ground. Mathematics as a magic trick needs to end.