"Maths is the music of reason"
This beautifully written lament takes some powerful swipes at school maths, textbooks, our suppression of the drama of maths history, our collective cultural ignorance of maths (we think we know but we don't) and supplies some great examples of real maths teaching (the triangle in a box problem, the sum and difference of two numbers problem)
Simplicio and Salviato were two characters used by Galileo in his polemic against the Church. Paul Lockhart uses the same characters to construct a modern day polemic about the organised, uninspiring religion of standardised School textbook maths.
Simplicio is a "back to basics", instructionist, consumer-oriented, career-oriented defender of the traditional maths curriculum. Salviato is a passionate advocate of exploration and discovery - maths as an art form.
SIMPLICIO: All right, I understand that there is an art to mathematics and that we are not doing a good job of exposing people to it. But isn’t this a rather esoteric, highbrow sort of thing to expect from our school system? We’re not trying to create philosophers here, we just want people to have a reasonable command of basic arithmetic so they can function in society.
SALVIATI: But that’s not true! School mathematics concerns itself with many things that have nothing to do with the ability to get along in society — algebra and trigonometry, for instance. These studies are utterly irrelevant to daily life. I’m simply suggesting that if we are going to include such things as part of most students’ basic education, that we do it in an organic and natural way. Also, as I said before, just because a subject happens to have some mundane practical use does not mean that we have to make that use the focus of our teaching and learning. It may be true that you have to be able to read in order to fill out forms at the DMV, but that’s not why we teach children to read. We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas. Not only would it be cruel to teach reading in such a way— to force third graders to fill out purchase orders and tax forms— it wouldn’t work! We learn things because they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math.
SIMPLICIO: But don’t we need third graders to be able to do arithmetic?
SALVIATI: Why? You want to train them to calculate 427 plus 389? It’s just not a question that very many eight-year-olds are asking. For that matter, most adults don’t fully understand decimal place-value arithmetic, and you expect third graders to have a clear conception? Or do you not care if they understand it? It is simply too early for that kind of technical training. Of course it can be done, but I think it ultimately does more harm than good. Much better to wait until their own natural curiosity about numbers kicks in.
SIMPLICIO: Then what should we do with young children in math class?
SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.
SIMPLICIO: It seems like we’d be taking an awful risk. What if we de-emphasize arithmetic so much that our students end up not being able to add and subtract?
SALVIATI: I think the far greater risk is that of creating schools devoid of creative expression of any kind, where the function of the students is to memorize dates, formulas, and vocabulary lists, and then regurgitate them on standardized tests—“Preparing tomorrow’s workforce today!”
SIMPLICIO: But surely there is some body of mathematical facts of which an educated person should be cognizant.
SALVIATI: Yes, the most important of which is that mathematics is an art form done by human beings for pleasure! Alright, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises. You learn things by doing them and you remember what matters to you. We have millions of adults wandering around with “negative b plus or minus the square root of b squared minus 4ac all over 2a” in their heads, and absolutely no idea whatsoever what it means. And the reason is that they were never given the chance to discover or invent such things for themselves. They never had an engaging problem to think about, to be frustrated by, and to create in them the desire for technique or method. They were never told the history of mankind’s relationship with numbers— no ancient Babylonian problem tablets, no Rhind Papyrus, no Liber Abaci, no Ars Magna. More importantly, no chance for them to even get curious about a question; it was answered before they could ask it.
SIMPLICIO: But we don’t have time for every student to invent mathematics for themselves! It took centuries for people to discover the Pythagorean Theorem. How can you expect the average child to do it?
SALVIATI: I don’t. Let’s be clear about this. I’m complaining about the complete absence of art and invention, history and philosophy, context and perspective from the mathematics curriculum. That doesn’t mean that notation, technique, and the development of a knowledge base have no place. Of course they do. We should have both. If I object to a pendulum being too far to one side, it doesn’t mean I want it to be all the way on the other side. But the fact is, people learn better when the product comes out of the process. A real appreciation for poetry does not come from memorizing a bunch of poems, it comes from writing your own.
SIMPLICIO: Yes, but before you can write your own poems you need to learn the alphabet. The process has to begin somewhere. You have to walk before you can run.
SALVIATI: No, you have to have something you want to run toward. Children can write poems and stories as they learn to read and write. A piece of writing by a six-year-old is a wonderful thing, and the spelling and punctuation errors don’t make it less so. Even very young children can invent songs, and they haven’t a clue what key it is in or what type of meter they are using.
SIMPLICIO: But isn’t math different? Isn’t math a language of its own, with all sorts of symbols that have to be learned before you can use it?
SALVIATI: Not at all. Mathematics is not a language, it’s an adventure. Do musicians “speak another language” simply because they choose to abbreviate their ideas with little black dots? If so, it’s no obstacle to the toddler and her song. Yes, a certain amount of mathematical shorthand has evolved over the centuries, but it is in no way essential. Most mathematics is done with a friend over a cup of coffee, with a diagram scribbled on a napkin. Mathematics is and always has been about ideas, and a valuable idea transcends the symbols with which you choose to represent it. As Gauss once remarked, “What we need are notions, not notations.”
SIMPLICIO: But isn’t one of the purposes of mathematics education to help students think in a more precise and logical way, and to develop their “quantitative reasoning skills?” Don’t all of these definitions and formulas sharpen the minds of our students?
SALVIATI: No they don’t. If anything, the current system has the opposite effect of dulling the mind. Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.
SIMPLICIO: Fair enough. But what about those students who are interested in pursuing a career in science or engineering? Don’t they need the training that the traditional curriculum provides? Isn’t that why we teach mathematics in school?
SALVIATI: How many students taking literature classes will one day be writers? That is not why we teach literature, nor why students take it. We teach to enlighten everyone, not to train only the future professionals. In any case, the most valuable skill for a scientist or engineer is being able to think creatively and independently. The last thing anyone needs is to be trained.
I love this essay but also have a few brief critical comments:
(1) In the above dialogue in response to Simplicio's point that children cannot rediscover the Pythagorean theorem unaided, Lockhart, speaking through Salviato responds:
Let’s be clear about this. I’m complaining about the complete absence of art and invention, history and philosophy, context and perspective from the mathematics curriculum. That doesn’t mean that notation, technique, and the development of a knowledge base have no place. Of course they do. We should have both. If I object to a pendulum being too far to one side, it doesn’t mean I want it to be all the way on the other side. But the fact is, people learn better when the product comes out of the process.I agree with what Lockhart is saying here but I don't think he sticks to this position consistently throughout his essay. In his passionate enthusiasm for maths as an art form he does let the pendulum swing too far one way. I would say he more or less denies the importance of behaviourist learning (see Dennett) and doesn't grasp that what works for the creative student does not work for all students.
Open ended discovery learning is another possible road to purgatory. To draw an example from the language wars. Whole language techniques may work well for many students but other techniques (phonics) are essential for the other 25%. According to Kevin Wheldall, "25 per cent of low-progress readers will fail to learn to read if they do not have systematic instruction using phonics" (source)
(2) Lockhart is wrong to imply that other subjects are not butchered by School
(3) Papert's constructionist use of logo programming does open up a possible pathway to solve some of the problems identified by Lockhart but this is not even mentioned