Sunday, March 09, 2008

what is maths? Paul Lockhart's Simplicio-Salviato dialogue

Lockhart's Lament (pdf 25pp) introduced by Keith Devlin

"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


Kaz Maslanka said...

I too appreciate what the author is trying to do by creating excitement and fun relative to practicing maths. However, I think his language is confusing as many others who think that math is an art. Yes one may metaphorically state that there is an art to math however, we may also say there is an art to plumbing. What is important is that maths has an aesthetic and it is the aesthetic that defines its beauty. Furthermore the aesthetics of mathematics are different from the aesthetics of art. I say let art be art and maths be maths and try to understand the difference so that we can clearly enjoy both. Even though they embody separate aesthetics one may fuse them to create a polyaesthetic experience. Polyaesthetic is what excites me however; I think it is important to delineate the aesthetics so that one may understand how to join them.

Anonymous said...

I have an interesting situation with regards to this. The students I teach are quite difficult at most times, and are generally quite behind in maths. They can often remember ways of doing things by rote, but have troubles generalising and get completely stuck if they find themselves in an unexpected situation. Probably for this reason, they really like worksheets and textbooks. They can get the answers (one way or another) - in fact there _is_ an answer. They have no idea of what this answer means, or, for that matter what the question means or why the hell they're doing it. But that doesn't really matter if they can get the answer. On the other hand, whenever I have tried more open, exploratory or play based activities, the students tend to revolt (in a manner of speaking). Not always, but often. There's no clear end in sight, and they have difficulties working independently, so things start going awry. A lot of this, no doubt, comes from their lives outside of school, and earlier schooling.

It's a conundrum - I can work through the state's curriculum via a textbook and tick all the boxes, and the administrators will be happy and so will the students (though they won't actually have learnt anything useful), or I can tear my hair out trying something different where the students get confused, I don't cover the expected material, and the outcome might not be much better. I'm tending towards the latter option, in the hope that over time the students will become more responsive...

Anonymous said...

I liked this post, because of its emphasis on "math as art", and "math as concepts". I would've liked being exposed to that more when I was in school, I think. We got exposed to it a little. I remember my Trig. teacher talking to us about some relationship between a Trig. formula, and e^x, and I think the Fibonacci sequence. Something of that sort. I remember he used the phrase "mathematically beautiful" to describe it. There was a nice symmetry to it, though I can't remember exactly what it was now.

I kind of got the experience of "math as art" while programming. In high school I got interested in trying out Trig. functions in Basic on a graphics screen. I was able to generate some nice patterns using "waves" that I generated using Sin and Cos.

I can very much relate to what's been discussed here about going through a math course, getting correct answers on tests, but not really learning a thing, because I was just going through the motions. I didn't know what it meant. I had that experience in some math courses. Thankfully I had some others where I "got it". Not in the same subjects, but I had the experience of really understanding a form of math at a conceptual level.

Reading your excerpt and description of the dialogue, I also had some trepidation about this approach. I think it's valuable to not only get the concepts, but to also have a mathematical vocabulary, like understanding algebraic expressions, for example, and the standard techniques that can be applied to them.

Also, if a student is planning on going into engineering or further study of math in school, they're going to need to know some standard formulas, even if by rote.

Anonymous said...

I am in the same boat as the earlier anonymous. My 9th grade (US) students have not learned much math and prefer the safety of worksheets. They actively resist being asked to deal with uncertainty and unfamiliar territory. They need to do this, after all, they must think for themselves. Lots of people prefer routine and simplicity and directed activity. Adults and children prefer not to have to think and prefer not to struggle to learn something new.

I was excited by Lockharts two problems, the triangle in a box and the two numbers problem. Is there a good source for problems like this that are rich. I could try to develop them on my own, but do not want to have to think too hard...

mokum von Amsterdam said...

Rading this blog let me to the whole essay and it just made my day.
Thank you Paul Lockhart.

Anonymous said...

"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,"

but where does lockhart say that all students should do math? Math is not for everyone, and you yourself make the point that it's for creative people. If you're not good at math, don't take it. But if you are good at it, there should be a system in place to encourage that.

Anonymous said...

I just discovered Lockhart's essay over the week-end, and though I am not even done reading it, I cannot wait to discuss it and I will shamelessly (but with your permission) use your blog to do so.

Mostly to say that I wholeheartedly agree with your point (1). Obviously, your mileage may vary with any instruction method. I remember my high-school days when a Nobel laureate in Chemistry had started a tour de France of schools to passionately argue that we needed more lab time in high school and less theory. And even at 18, it broke my heart to realize how short-sighted this great man was. He was basically saying: "This is what I wish I had, so this is what you must do." I had *no interest* in spending more time in the lab that we already were (2 hours *every* week in high school!)

I see the same problem, an unwillingness to allow for different students and different methods, with Lockhart's approach. As a "pure" mathematician, I am very passionate about the strong relationship between math and applications, and the discipline would be nothing if generations of mathematicians had not shamelessly plundered practical problems for ideas for the developpement of the subject. That does not mean that we are tied down by applications, btw: we do not hesitate to play with the original problem until we made it more to our liking and possibly impenetrable by its originators.
(If you think String Theory is only tenuously linked to any kind of reasonably concrete physics, or Not Even Wrong, you should see the math that is *inspired* by String Theory!)

Lockhart's "Math as an Art" position seems very intransigent in his writings, even though I would imagine that he has more common sense than what transpires in the essay and that he would be quite amenable to admit the importance of applications.

He boasts about teaching subversive mathematics at all levels of the curriculum, and I am impressed enough by his writings to believe he can get away with this, and that his students are very lucky to have him (and do I wish I'd been one of them!). But one can only shudder at the notion of ham-fisted administrators imposing such a method on an unwilling and unready professorate. As unhappy as I am with the current state of math education (in all the countries I experienced it), the damage wrought on the unfortunate students would surely be a thousand times greater under this scenario. New math anyone?

Unknown said...

I recently finished reading the book version of Paul Lockhart's essay, and I am activiely seeking out people with which to engage in meaningful discussion about the complex job of teaching, regardless of the content area. Also, I'd like to discuss certain philosophical revelations that certain parts of the text led me to think up! (And they address really difficult, complex, and somewhat taboo topics such as religion, murder, and free will, so maybe tread lightly if you are easily offended...) Anyway, I have one question for the original poster of this blog regarding the following section:

(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

What in the world is "Papert's constructionist use of logo programming"?

Bill Kerr said...

hi carol,

I have a couple of essays about Papert's methods on my website - a practical classroom based one (ISDP) and a more theoretical one, reviewing his first book (Papert)

Seymour Papert was a mathematician who worked with Piaget and later became an AI researcher who worked with Marvin Minsky at MIT. He has written 3 books about education, the best two are Mindstorms and The Children's Machine

David said...

Hi all,

We are hoping to extend the conversation about Paul Lockhart's work. As such, we are planning an online "book study" which you are all welcome to join.

See for details.

David Wees

Lee Milby said...

@Kaz Maslanka :

As a painter and programmer, i can attest that the modes of thought applied to art are actually as he says- the exact creative forces that inspire the exploration of logistics. Also, look back in history to the age of romanticism where science, astronomy, music, art AND math all bloomed together- often by the same hands (it was common to study all, as each topic informed the others and inspired new ways of thinking about EVERYTHING). Look even farther back and we see artist/engineers such as Leonardo. Keep in mind that Leonardo was not the only "renaissance man" who existed during the renaissance, he just happened to be extremely prolific and particularly passionate.

We need to bring the curiosity- the thirst for learning back to education systems.

Kaz Maslanka said...

Hi Lee,
I appreciate your statements and I see nothing wrong with them however, you are missing my point. I am not talking about the decision process and the logic involved in making aesthetic expressions. I am talking about the difference in the aesthetic s between math and art. At some level, every work of art can be analyzed mathematically and every visualization of math can be seen as visually aesthetic, however, what is important about these expressions or I should say it's importance is what defines them to be more in the realm of math or more in the realm of art. The mathematic aesthetic requires cognitive effort to experience its beauty. The aesthetics of math lies within the genre of thinking (with effort). The aesthetics of art lies in the realm of direct experience (no thinking) or I should say no effort in thinking. While one can ponder the decisions made while constructing a work of art, when one looks at a piece of art one 'gets' an immediate response with little or no effort. Math doesn't work that way - you have to study math to appreciate its beauty. Some people say that Euler's formula is the most beautiful equation; I am not going to debate its value, however, it means nothing without learning about natural logarithms, imaginary numbers and the circle. I fully believe that Lockhart's educational quest is worthy and correct however, he is confusing the word art with the word aesthetic. If math and art were the same thing then they would use the same word. The point that Lockhart and others need to make is that we need to teach math from an aesthetic viewpoint similar to the way art is taught. Saying that math = art only confuses the issue.