Saturday, July 24, 2010

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.


  1. What a beautiful article you have shared with us today. I had already been convinced of the truth of mathematical discovery being just as important than any other discovery by Paul Lockhart's article and my own instinct, but it was lovely to hear the PNU article voice this as well so beautifully. Have you seen "Mathematics A Way of Thinking"? It is a textbook, but it believes in the power of discovery over the rote working of sums. I think you might like it.

  2. Naomi,
    This is excellent! So exciting to read about awakenings in math. Please keep us posted as your year goes on. We are venturing into new territory for math this year, also. Off to read the Lockhart article...

  3. Phyllis - I haven't heard of Mathematics A Way of Thinking. It sounds interesting. Do you use it or know anyone that does? I've heard Right Start is a good math program, my friend has offered to loan me hers to try out.

    Nancy - I will. Please let me know if you find anything also. Specifically, I'm looking for a math book that tells me what to do to accomplish this all daily, which also introduces new concepts in a living way to excite discovery, leaving the effort to the child, touching on historical background pertinent to that concept, written with good language, without any modern bias, oh, and of course, all of it for a reasonable price. It must exist somewhere!

  4. Well, it took me three tries, but I finally read all of this (my children are having One of Those Days). Anyhow, thank you for writing all of this! What beautiful thoughts...

    I loved it when he said that mathematics is the music of reason. The ancients believed that music was numbers (ratios) in motion. It's like we're coming full circle.

  5. Well I read the Mathematician's Lament last Jan. and was left wondering - Now what? I hate the curriculum I have to work with now but since I'm in a half homeschooling/ half classroom private school I don't get to choose the math curriculum of my choice. I loved Math-U-See when I did it and am thinking about just doing it anyway as extra math for my kids. What are your thoughts? Would doing two completely different math programs at the same time confuse or enhance the experience?

  6. My thoughts exactly, now what!?! What I realized from the article (which I'm not saying is the right conclusion) is that it makes no sense to just teach my kids a set of rules and have them apply them mindlessly over and over without first having put them in touch with the beauty of its truth.

    He started the article with an example of music and how silly it would be to teach them everything without having heard a song first. It is senseless. We could imagine the same being done in nature, cooking, art, or any number of subjects. Why learn classification before holding a bug, a flower, a snail in your hand first. Why learn measurements and proper utensils first before appreciating a hot batch of chocolate chip cookies made from scratch with your own hands?

    You could, but why? Unless you didn't care about the nature of the learner.

    The Parent's Review article said to start with play-lessons first to soak up "philosophic truth" and "knowledge of Laws of Thought", then move to rote manipulation of that operation, or "work" as she called it.

    The question I have is where do I find a book that introduces addition in this way? Sums are interesting, I'm sure. I just haven't found the author who presents it in an interesting way. I don't mean to simply wrap some twaddly story around it, but to talk about its history, how people struggled with it before, how it is useful, ideas about it that we may not have considered otherwise. I think that's what Lockhart was trying to say, it is more interesting than we can handle already!

    Then let them struggle for some time trying to figure out a sum the wrong way, try to work it out on their own so that they can perceive and understand the value of the rules to be introduced. For example, let them add from left to right and get stuck with the extra tens and not know what to do with them before we teach them to add from right to left and to carry.

    This way, it makes sense and there is reason and meaning behind it, and they are able to try to find a solution on their own, rather than perform mechanical tasks.

    Once they've gone through this, then give them the pages of workbook practice, which is where some of these curriculums could come into play.

    So again, how to implement this? I haven't found the right books yet, but I'm sure they're out there and look forward to finding them. Will keep posted. Mabye we can email Lockhart and get his recommendation?

  7. This post is in the 239th Carnival of Homeschooling, history of home education edition, which now is up at The Common Room, http://tw0.us/LUJ My theme is 'the history of homeschooling in America.' It was very interesting to research and I learned some fascinating things along the way (do you know why we have age segregated classrooms in America?).

    Please pay us a visit, and reciprocate in the publicity the carnival brings you by passing along the link along so others can visit as well.

    Please consider other ways to spread the news about the carnival as well- the more people who visit the carnival, the more link-love you get- If you have a facebook account, you could pass on the link there (fb doesn't like tiny url links, so here's the long one: http://heartkeepercommonroom.blogspot.com/2010/07/carnival-of-homeschooling-239.html, and if you have a twitter account, please tweet!