Did you know that the area of a circle is Pi times its radius squared? Did you know the area of a triangle is one half of its base times its height? Did you know for a right triangle (a triangle with one angle of 90 degrees or pi/2 radians) the sum of the squares of the two sides adjacent to the right angle is the same as the square of the side opposite the right angle, the hypotenuse (did you know that I, a mathematician, still cannot spell hypotenuse?)? More importantly, do you know why?
Earlier today I posted a Facebook blasting the mainstream method of mathematics education. This was out of various frustrations, including my frequent interactions with people in the late stages of chronic math apathy, and conversations with colleagues on the state of our students. I have colleagues, how weird is that! Much of my despair boils down to our dogmatic adherence to a system which is fairly seriously flawed. I don't claim to have THE solution, I have thoughts that I would like to see tried out, but mostly, I want to see us have the courage and vibrancy to experiment boldly with our curriculum, changing it to try to get out of this malaise.
I believe I was unduly harsh, as math is not the only subject that suffers from some deep problems endemic to our education system. One horribly harmful practice is the passing along of students who have not sufficiently mastered current materials to higher level courses. Because math is such a rich and useful subject, attempting to understand a subject without a thorough grasp of the prerequisite courses is akin to the canonical fool attempting to build a sturdy foundation upon ground made of sand. Although factoring polynomials (mostly quadratics) and negative numbers are two fine examples of things one ought master before proceeding, as they end up in fairly frequent use, by biggest concern is fractions.
I cannot understand what part of our system so scars students that they stay scared of fractions long through their math careers. If one simply wishes to work with fractions, two simple rules suffice, common denominators to add, multiply straight across. This will not convey an understanding of the nature of fractions, but it will provide you with all the "special" tools that you need to manipulate them, they won't be things of beauty, but at least they ought no longer be feared. However, as I indicate at the end of the first paragraph, the essential point of math is NOT an isolated scattering of lifeless, lonely facts, but an intricate, intertwined network of collaborating reasons why things are true.
Were I to be given an island on which to implement my social experiments, my math curriculum would, broadly, look as follows (and mothers could keep their children...). An introduction to the positive integers, addition, and the number one. These topics are deeply connected, as you can think of one, the unit, as the starting place for all mathematics, without it there is only nothing, then addition miraculously conjures the rest of the whole numbers, as one is joined by one for two, and so forth. Here we introduce the miracle of zero, which leaves numbers unaltered Should we ever experience diminishing, as happens when one departs, we become familiar with negative numbers (not subtraction, which is a misleading myth). Now, one might find it useful to repeatedly add a number to itself, such as if students arrive on buses each carrying 32 children, 32 children after the first, 32+32 after the second, 32+32+32 after the third, and thus is multiplication formed from addition.
Now things become a little tricky, but I have faith in our citizenry to persevere. Just as zero played a special role for addition, leaving things unaltered, our unit plays the same role for multiplication. And just as the addition of -n undoes addition by n, because together they add to 0 which leaves things untouched, multiplication by 1/n undoes multiplication by n, because they multiply together to zero. Thus we run into the first fractions as reciprocals to the whole numbers, and we noticed that division too is an illusion. Consider 0*n, which we know to be n added to itself zero times, when no buses have arrived, there are no students, no matter how many are on each bus, so 0*n=0. Now suppose you are discovering this for the first time, what must you suspect about 1/0? Well, one expects multiplying by 1/0 to undo multiplication by 0, because that is the role of 1/n to undo n. So, 1/0 times 0*n should be n, no matter what n is. However, no matter what n is, 0*n always arrives at 0, so 1/0*0 needs take on more than one value to satisfy our system, since it cannot, and indeed must be 0 since anything times 0 is zero, we discover that 1/0 cannot be included in our system without breaking the patterns that we have established. Because there is no division, and n/0 is instead n*(1/0), we must conclude that, should zero occupy a denominator, we have left our reasonable system of numbers, so this is to be avoided. This has gone on long enough, but we may quickly build exponentiation from multiplication the same way we build multiplication from addition, and roots now are un-exponents, the same way negatives are un-addition and fractions are un-multiplication.
Of course, this is a bit of a mess when skated through in 2 paragraphs, rather than over the course of a year or two, but the point is that it is all interrelated. This is where the beauty of mathematics comes from, it is a mosaic, and a puzzle, and a fractal laden, dew adorned spiderweb on a cool morning. Addition and multiplication are different, not separate, and subtraction and division no longer exist, much simplifying the order of operations. Recall that these operations have the counter intuitive property that they do not commute (that is, 3-2 is not 2-3, and 2/3 is not 3/2). Since these operations no longer exist, this strange fact, an artifact of a formulaic approach to math, need no longer be considered. Now 3+(-2) and (-2)+3, as well as 2*(1/3) and (1/3)*2 can be swapped as much as is desired.
The WHY is beautiful, without it math is just an exercise in memorization and rote manipulation, two things that are fairly boring and easily done by computer. With why, math becomes a rich environment for exploring, every new piece of information not a fact, but a new piece of the puzzle which must be slotted into an awaiting rough edge. The easiest method to determine the area of a circle requires calculus, but I can prove the Pythagorean theorem on a napkin in under 5 minutes (if we are in the same place, feel free to ask me to do so, it is a beautiful proof). How amazing is that, these two facts, that we casually teach side by side in an introduction to geometry, have vastly different backgrounds and stories. Personally, I think the area of a triangle is the easiest of the three, and accordingly, I leave it as an exercise for the reader, should you be in the mood to explore. It may help to take right triangles as your point of departure, but remember that not all triangles are right triangles.
As I posted on Facebook, learning math the traditional way is akin to learning to write poetry by taking lines from the poems of the greats and sticking them next to each other in varying orders. Or, to be less absurd, it is learning about history by rote memorization of dates and events, with no thought paid to the overall correlation and causality, the beautiful, interconnected, vastness of human experience. It is learning a foreign language sentence by sentence, hardly paying heed to individual words in the phrases blindly committed to memory, let alone the system of syntax and grammar. Overall, I have to say that what is a very important question to answer, but only because you need to establish the what before you can sail off into the sunset after your elusive why's...