Revisiting Math

Euclid in the School of Athens, Raphael Sanzio

Some call it an 'aha' moment. I can't think of it as anything other than Divine Providence. In any case, it happened today after finding this gem of a Parent's Review Article entitled "Home Arithmetic" by Ms. May Everest Boole, written for our discovery no less than 117 years ago. I'm ashamed to admit I had to use the handy desktop calculator to figure that out without counting on my fingers. Which is precisely why I bought a complete math curriculum for the past three years to do the job of teaching my children the subject for me. It was, at the time, the path of least resistance.

But today, as we worked through a workbook page of 3 digit, 3 column addition, I couldn't help but feel the senselessness of it all. Yes, she knows how to round to the nearest hundred, thousand, ten thousand. She knows how to add those numbers for an 'estimate'. She knows to start adding from the right column and carry the number over to get the exact answer, but... does she really know anything more than a set of rules to operate by? Why do we round, why do we estimate, why do we carry? WHY does any of it matter?

Then I read this from the article:

...it is almost impossible for a pupil to extract out of a mathematical operation what it has to teach about the abstract, after he has once learned to perform that operation mechanically. It is therefore desirable that a child should learn from each operation all that it can give of knowledge of Laws of Thought, before he ever sees it performed in the class-room, where skill in manipulation is being cultivated. Each operation should be introduced to the child first by a series of play-lessons, in which no eagerness is aroused, on which no examination depends, and during which he is free to soak up all it contains of Philosophic Truth, unhampered by the need for following any prescribed method.

and this...

Beware of writing, in play-lessons, anything which does not represent some process actually going on in the child's mind. E.g. It is natural to a child to count the more valuable coins or counters before the less valuable ones; allow him to do additions in that order, till he discovers the inconvenience of doing so. The first few examples of each operation should involve no "carrying" and therefore no inconvenience from beginning at the "wrong" end. When he begins to "carry," let him still work in the wrong order, and correct his results. If it does not soon occur to him to spare himself this trouble, you may suggest it to him; but for some time after you have suggested it, make him do each sum in both ways, the clumsy and the convenient way, and become accustomed to see the identity of results.

and this!!!

When he can do an easy addition, of about three columns and three rows, slowly, but without effort, beginning indifferently at either end, and can explain the rationale of each process, addition may pass to the stage of "work." Subtraction should them be taken up for play-lesson, the same principle being observed as in play addition.

When he can easily do a short division by one small digit, let him do one on the top of the slate and leave it; prepare the rest of the slate as for long division, and set the same sum over again in long division form. Tell him to write down all the steps (multiplication and subtraction) by which he got his successive remainder. Repeat this process during several weeks, or even months: do not set any division by more than one digit till the child quite realizes that long and short division are identically the same process; that the former helps memory when the divisor is large, but gives needless trouble in writing when it is small. This will forestall the difficulty which many, even clever, children experience in understanding long division. Remember too that no time is wasted which serves to impress on the child's consciousness that abstract truth has a sanctity and authority of its own, independent of special method; and that our choice between methods, equally valid abstractly, is to be decided by human convenience.

I was convinced immediately, because I'd already seen the fruits otherwise; dull, dry, imposed, meaningless. Don't get me wrong, the curriculum could still be very useful, I'm just realizing that, had she grappled with the wrong way for a while, then discovered the truth of it, with me 'scaffolding' as necessary, then truth would have unfolded itself to her in a much more remarkable way. There would have been understanding, ownership, conquest, interest, and so much more meaning to it all.

And to know a bit of why things have come to be as they are:

When you use notation, shew him that ten was chosen as our "carrying" standard, because savages counted on their fingers; make him realise early, that ten has no special value as a standard except what is given to it by the conformation of man.

It is fascinating really.

Why on earth isn't it?

I think this critique of K-12 math education, "A Mathematician's Lament" by Paul Lockhart, a research mathematician, does a great job explaining why. He is impressed: (emphasis all mine, link courtesy of Jimmie via Squidoo)

Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education.

I bet he'd agree with Charlotte Mason in that 'The Child is a Person'

Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them, that we think it’s too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon, why not about triangles?

And this just took my breath away...

All this fussing and primping about which “topics” should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake— it’s like rearranging the deck chairs on the Titanic! Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, {@$&#} it. Remove this from mathematics and you can have all the conferences you like; it won’t matter. Operate all you want, doctors: your patient is already dead.

The saddest part of all this “reform” are the attempts to “make math interesting” and “relevant to kids’ lives.” You don’t need to make math interesting— it’s already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That’s why it’s so fun!

Here it is, this theme again; that knowledge is intrinsically valuable, in and of itself, without need of spicing up, and knowledge striven for remains in a much more meaningful way. While I would disagree on the one point of the glory of it, it's refreshing to see others coming to the same truth as Charlotte Mason - and oh. so. passionately. Lockhart beautifully gives several examples bringing math alive within the article, worth looking at for yourself. And let's face it; his case is quite convincing.

I believe we are on track for an awakening of sorts in math this next year.

This post is part of the Carnival of Homeschooling.