I have two young boys and I have decided to pay attention to what they are learning in school. Beside basic writing and reading skills, mathematics feels like the next most important academic topic. I realize that relatively little has changed in 30 years regarding how mathematics is taught.
As a young kid, I liked mathematics but it became less enjoyable in high school. After a time, I learned to love algebra as it made it possible to solve difficult word problems more easily. To this day, I still do algebra to way I was initially taught, and I often get the right answers.
Later in my schooling, I came across the quadratic formula. That was quite a disappointment. I will not even reproduce it here because it is an ugly hack. Algebra all made sense up to this point, and then we had to memorize a complicated formula. I was not impressed and there were many students to complain about it too. I attended a Catholic school at the time, and I remember that the teacher told us to take it on faith.
Relying on faith to do mathematics did not square well with me. Maybe you can guess that I never did memorize the quadratic formula. Indeed, it is almost trivial and much more satisfying to derive it from first principes. Our teacher was kind enough to rush through the derivation on the blackboard once: it was too fast for any of us to understand, but sufficient for me to figure it out on my own later. How could he derive it at first and then ask us to take it on faith? I guess he thought that having us learn to figure it out on our own would take too long.
I would argue that there is little sense in having people who cannot derive the quadratic formula, memorize it. If they cannot derive it, it is very unlikely that they will grow up to become adults who use algebra in a non-trivial manner. And if they are never going to do algebra, why would they need the quadratic formula?
I think that most people at ease with algebra can derive the quadratic formula quickly. The first thing to note is that there is no reason to find the roots of a x2 + b x + c. You should divide throughout by a.
So, really, all you need to do is to figure out the roots of x2 + b x + c. It is good enough. It is already a more elegant problem.
Then you just need one more trick… it is called “completing the square” and it says that x2 + b x = (x + b/2) 2 – b2/4. Of course, there is no need to memorize the completion of the square formula…
From this, solving for the quadratic formula is easy… you start from…
x2 + b x = –c
and you complete the square…
(x + b/2) 2 = –c + b2/4
And that is all!
Admittedly, it can take me longer to solve for the roots of a quadratic polynomial using this derivation than someone who has memorized the formula. I would guess that it takes me an extra 5 seconds. But how often do you have to solve for the roots of such a polynomial without the assistance of a computer?
I will also reiterate my argument against memorizing multiplication tables… Reasoning out that 6 times 8 is 48 is more important than the fact itself. Similarly, figuring out the binomial formula from first principles might help you approach a wider range of problems with confidence…
To be fair to contemporary education, the American Common Core seems to recommend doing exactly as I did as a kid regarding the quadratic formula.
Of course, the quadratic formula was only the first of many disappointments to come. Next we learned trigonometry, and then we had to suffer through analytic geometry…
The pattern would repeat itself endlessly: “You have to learn that that the square of the secant minus the square of the tangent is 1 [replace by your favorite piece of trivia].” Then I would find a way to “route” around the problem. For example, in trigonometry, I eventually figured out that most things could be derived from Euler’s formula. That formula itself was intuitive enough if you could remember that the cosine and sine give you the coordinates of a point on the unit circle given an angle. In fact, almost all of trigonometry can be derived from this simple observation.
We still insist on presenting mathematics as a collection of definitions, facts and routines to be memorized. Thankfully, the emphasis is somewhat lesser than in my days… but if you scratch beyond the surface, too little has changed.
Evidently, the purpose of such mathematics in schools is primarily to rank students. It is not very different from the Imperial examination of the Song dynasty. We are teaching to the test because only the test results matter… the test only serves to separate the “good students” from the bad… under a pretence of “merit”.
And then, wait for it, we find out that many of our graduates lack skills that would make them employable! It does not matter because employers only look for degrees and diplomas… except when they do not or cannot.
My boys love to play Minecraft. They build all sorts of crazy devices. By my estimation, this is probably a better preparation for the “real world” they will face in 10 years than the math that they are learning in school. Sadly, I am quite serious.
Further reading: Several people have pointed out that many of my comments are related to a Mathematician’s Lament by Lockhart.