22 June 2016

Over time, while observing people trying to help others with math problems, I've noticed a common pattern that is, in my opinion, both wrong and extremely misleading. When describing a solution, people often use the phrasing that you "have to" do a particular step to solve a problem. Here's an artificial example (unfortunately, I don't have any real examples off the top of my head).

Question: How do I solve the equation $$x^2 - 7x + 10 = 0$$?

Answer: This is a quadratic equation. You have to use the quadratic formula $$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ which gives the answers $$x = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm 3}{2} = 2, 5$$.

What's the problem here? The formula is correct, and the resulting numbers are correct as well. People with experience with math contests (especially the high school variety) should notice it. Why would you use the quadratic formula for this equation? Certainly it's an option, but by no means is it something that you have to do.

For example, $$u$$ and $$v$$ are the roots of $$x^2 - bx + c$$, exactly when $$u + v = b$$ and $$uv = c$$. So you might try out a few factors of 10 and notice that 2 and 5 happen to sum to 7. Now you have the answer! If you're only checking for rational roots, there's only a couple possibilities to check, so the entire process takes no more than a few seconds.

Alternatively, if you want a more algorithmic method, you could complete the square. $$(x - 3.5)^2 = x^2 - 7x + 12.25$$, so $$x^2 - 7x + 10 = (x - 3.5)^2 - 2.25$$. Therefore, if $$x^2 - 7x + 10 = 0$$, we have $$(x - 3.5)^2 = 2.25$$, from which we get $$x - 3.5 = \pm 1.5$$, so $$x$$ is either 2 or 5. People familiar with the derivation of the quadratic formula will realize that this process is essentially pushing through the derivation instead of jumping straight to the formula, but for someone just learning about quadratic equations, it looks entirely different.

Describing the quadratic formula as something you "have to" do encourages the mindset that formulas are opaque incantations that are to be memorized. After all, plugging values into the formula is what you have to do, right? That mindset carries forward into their later education, and they keep looking for and memorizing incantations that solve problems.

However, the truth is that high level math is a creative endeavor. Memorizing formulas to do math is like memorizing poems to become a poet. I don't have personal experience here, but I expect that great poets have poems memorized, not because memorizing poems makes a great poet, but because they made a personal connection and the memory formed naturally. That's exactly how my experience with math has been. I remember many problems, solutions, and solution techniques, but in only very few cases did I set out explicitly to memorize them.

Rather than prescribing a formula as the solution, it is better to emphasize that it is just one of potentially many solutions. Different solutions have different strengths and weaknesses. In the case of solving quadratic equations, the quadratic formula works for any quadratic, but can involve time consuming and error-prone computations. Conversely, trying factors of the constant term is inconsistent, but significantly easier and faster when it works. Exploring the space of possible solutions is important to learning mathematics, and so it is important to not accidentally discourage such exploration.