Paul Lockhart’s A Mathematician’s Lament: How School Cheats us of our Most Fascinating and Imaginative Artform is a short but valuable book, easily finished in an afternoon.
Lockhart’s basic take is that most of us have math backwards. We approach (and thus teach) it as useful but not fun–something to be slogged through, memorized, and then avoided as much as possible. By contrast, Lockhart sees math as more fun than useful.
I do not mean that Lockhart denies the utility of balancing your checkbook or calculating how much power your electrical grid can handle, but most of the math actual mathematicians do isn’t practical. They do it because they enjoy it; they love making patterns with numbers and shapes. Just because paint has a very practical use in covering houses doesn’t mean we shouldn’t encourage kids to enjoy painting pictures; similarly, Lockhart wants kids to see mathematics as fun.
But wait, you say, what if this loosey-goosey, free-form, new math approach results in kids who spend a lot of time trying to re-derive pi from first principles but never really learning algebra? Lockhart would probably counter that most kids never truly master algebra anyway, so why make them hate it in the process? Should we only let kids who can paint like the Masters take art class?
If you and your kids already enjoy math, Lockhart may just reinforce what you already know, but if you’re struggling or math is a bore and a chore, Lockhart’s perspective may be just what you need to turn things around and make math fun.
For example: There are multiple ways to group the numbers during double-digit multiplication, all equally “correct”; the method you chose is generally influenced by things like your familiarity with double-digit multiplication and the difficulty of the problem. When I observed one of my kids making errors in multiplication because of incorrect regrouping, I showed them how to use a more expanded way of writing out the numbers to make the math clearer–promptly eliciting protests that I was “doing it wrong.” Inspired by Lockhart, I explained that “There is no one way to do math. Math is the art of figuring out answers, and there are many ways to get from here to there.” Learning how to use a particular approach—“Put the numbers here, here, and here and then add them”–is useful, but should not be elevated above using whatever approach best helps the child understand the numbers and calculate the correct answers.
The only difficulty with Lockhart’s approach is figuring out what to actually do when you sit down at the kitchen table with your kids, pencil and paper in hand. The book has a couple of sample lessons but isn’t a full k-12 curriculum. It’s easy to say, “I’m going to do a free-form curriculum that requires going to the library every day and uses every experience as a learning opportunity,” and rather harder to actually do it. With a set curriculum, you at least know, “Here’s what we’re going to do today.”
My own personal philosophy is that school time should be about 50% formal instruction and 50% open-ended exploration. Kids need someone to explain how the alphabet works and what these funny symbols on the math worksheet mean; they also need time to read fun books and play with numbers. They should memorize their times tables, but a good game can make times tables fun. In short, I think kids should have both a formal, straightforward curriculum or set of workbooks (I have not read enough math textbooks to recommend any particular ones,) and a set of math enrichment activities, like tangrams, pattern blocks, reading about Penrose the Mathematical Cat, or watching Numberphile on YouTube.
(Speaking of Penrose, I thought the chapter on binary went right over my kids’ heads, but yesterday they returned all of their answers in math class in binary, so I guess they picked up more than I gave them credit for.)
YouCubed.org is an interesting website I recently discovered. So far we’ve only done two of the activities, but they were cute and I suspect the website will make a useful addition to our lessons. If you’ve used it, I’d love to hear your thoughts on it.
That’s all for now. Happy learning!
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I’m leaning quite heavily on a more traditional math curriculum, partly because it’s easier for me (I’m not the most self motivated teacher), and partly because it seems to be working. I’m doing the exact opposite with English, because it’s easy to just shove books at my kids and ask them to explain what they’ve read and iterate from there.
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Having seen the “lament” several times before, my big beef is that it misrepresents art and music, especially the latter. Professional musicians spend hours a day practicing, and I’m not just talking about classical musicians. That jazz pianist improvising for an hour non-stop? He practices nothing but scales for an hour a day, plus has memorized hundreds of pop and showtune standards, and has had to learn music theory and chord progressions… And if you want to do painting or sculpture that doesn’t depend on a political statement but is just something people like to look at, you have to practice drawing over and over. And so on with sports and dance, etc… I mean, there’s definitely a large place for enjoying the masters of a field, I mean, there’s no point to art (barring some philosophical quibbling, I suppose) if you don’t have people enjoying it. And there’s pleasure to be had from good, deliberate, repetitive practice for some people. But I think it’s the same as my complaints about Science! where they hope that kids who otherwise wouldn’t do science will think it’s cool and fun and, oops, oh, look, physics is really fricking hard and biology and chemistry involve hundreds of hours in a lab doing grunt work and there’s really no way to change that… (I suppose it happens in a lot of places… It’s fun to toss a ball around, and there’s glory in being a football star, but the reality of training camps in August are probably less fun for most people…)
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I am in partial agreement. Doing anything competently does involve a lot of work that most people–even people who enjoy the activity–regard as difficult or even boring. But we don’t introduce music to children by making them practice scales nor art by making 5 yr olds practice figure drawing for hours on end.
This is why I hew to my 50/50 rule. 50% direct instruction–“here’s how to draw a face” and 50% open-ended exploration/play with the activity “now draw your own silly faces.” Play is important for children and a big part of how they process what they’ve learned, but they still need to actually learn something.
Math seems like an area where people swing back and forth. One decade it’s “we’re going to drill all of the basics” and the next its “we need open-ended exploration without getting caught up in algorithms.” IMO, both sides are important. Kids need to memorize those multiplication tables, but they also need a chance to–if you’re lucky–play with the numbers and have fun.
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It can definitely go too far either way. I think the problem with drill isn’t too much of it per se, but starting it without understanding either psychological research or being aware of traditional pedagogy for the field (for example, if you read 19th century math primers, they’re not all that different from accepted practice now–kids use small items to count things and do basic addition, the math is connected to the real world, etc.). And from a psychological standpoint, introducing too many new topics at once should be a big red flag to anyone who knows anything about working memory, and going for 100% recall before moving on is almost definitely counterproductive. A related point is that some teachers will focus on skills that aren’t directly related, for example, the child is writing some numbers backwards, therefore can’t move on in math, even if that child is doing the arithmetic operations successfully verbally or with manipulatives. The appropriate thing would be to offer focused handwriting work, or even occupational therapy. (Obviously, this is something that is fairly trivial when homeschooling–big reason I’m going that route with my kids.)
In the other direction, well, the whole “make it interesting” thing without underlying curriculum and pacing will lead to just… no math. At least, that’s what I experienced when I went to a progressive elementary school. (or, worse, the teacher gets to decide which students would benefit right now from more math, but not using any objective tests to decide who gets to learn some math…) Basically, like the worst stereotypes of unschoolers, except with the teachers deciding to do this rather than the parents… (And this was a public school…)
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One of my kids had atrocious handwriting at the age of 5. Could do plenty of math, but I never could get the teachers to *notice* because the numbers were always written backwards and upside down and on the back of the paper…
For me, I find it much easier to teach a subject when I feel like I have a kind of “complete concept” of it in my head. From the concept, I can derive each individual piece that needs to be taught (might take a little time, but it can be done) but without the concept, I can’t be sure I’m heading in the right direction. This was the difficulty I had with Social Studies for a while–I lacked a unifying concept of the field.
With math I’m on a firmer footing, but I think the inverse is true for a lot of other homeschooling parents–they’ve got the social studies down pat, but math is something they often feel less comfortable with. So even if I disagree with Lockhart’s “Let’s just have kids rederive everything from scratch!” idea as a weee bit too optimistic, I still think the book offers a valuable insight into what math *is* for people struggling to move beyond just seeing it as a set of algorithms to be memorized.
As for topics, I agree. I am often frustrated by the lack of depth a topic is covered in; for example, a book might introduce fractions, but only get so far as adding fractions with like denominators before switching over to angles. We’re not talking about books aimed at 6 year olds, for whom “Two halves of a pizza make one whole pizza” is a new insight. I think we can progress a bit further, at least. But luckily I can just pull out the next book in the series and keep going. Flexibility is nice. :)
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