A Dialogue on Mathematics and Literature: |
PDT and XYZ
Reexamine everything you learned in school and reject whatever insults your soul.
PDT: Maybe a good place to start is with the appalling observation, that keeps nagging at me, that somewhere between grade 6 and university graduation -- somewhere along the way -- we manage to completely alienate two thirds of the population. Go to any gathering, and wait till mathematics comes up in the conversation, and listen to what people say.
XYZ: "I could never do math."
PDT: And if they happen to be in a situation in which they have to cope with math?
XYZ: --A lot of anxiety.
PDT: For sure. Now in math-ed sessions over the past years we've certainly focused on these attitudes and on this anxiety -- making the classroom more friendly, more student-centred.
XYZ: Any results?
PDT: Hard to say, but I would think that we've made almost no progress on that one. I often think that most math teachers are so used to this state of affairs that they take it as an inevitable consequence of how people are made -- that two thirds of us simply won't be able to think mathematically, and there's really nothing to be done about it. But I totally reject that idea. I think it's a terrible situation -- really shocking.
XYZ: What can be done?
PDT: I look back at all my struggles with this problem, and I figure my real lack has been one of imagination more than anything else. I've been a mathematician for many years, and have taught the subject in many different contexts and at many different levels, and I certainly understand math well enough to make some progress on this problem. What I lack is not knowledge but something more like vision.
XYZ: And how can you get this "imagination," this "vision"?
PDT: Maybe it often comes from the intersection of two different worlds, when one world strikes a deep chord in another. For example, my research life is spent with biologists, and in interacting with them and seeing what they do with mathematics and what it means to them, the question of what they should have been seeing and doing in the math classroom when they were in school is always on my mind. And over time I think I've got some important insights.
XYZ: Is that going to let you see how to write this high school text book you were talking about earlier?
PDT: It will help, but it doesn't yet make the crucial imaginative leap that I feel is so badly needed.
XYZ: So where does that come from?
PDT: There is another world that has been engaging me more and more. And that's the world of literature, especially poetry and drama.
XYZ: For example, your math and poetry course?
PDT: That's a big part of it. I've been involved in that for 14 years now, and what I've found happening over the years is that I've become more intrigued with the apparent difference and the real similarities between the study of math and the study of literature. And increasingly I'm discovering that in my struggle to find the right way to teach mathematics, and it has been a very perplexing struggle for me, I can get extraordinary insights by transposing my questions over to the other side of the math/literature analogy, and trying to answer them there.
XYZ: Hmmm. Sometimes you must get answers that seem very strange.
PDT: And that's exactly where things get interesting. You're tempted to dismiss them by saying, well math is different from literature, but the more I try to pin that down, the more I'm finding that the differences don't always stand up to examination.
XYZ: So this provides a way of getting at certain assumptions about mathematics...
PDT: ...that are so ingrained that we aren't even aware of making them...
XYZ: ...but that may be false...
PDT: Or at least they tie us down to methods of teaching that don't work, that don't serve our students and don't serve the subject.
XYZ: So let's look at an example of how this works.
PDT: Well, I start looking through the grade 10 text book, trying to figure out what's wrong, and then all of a sudden it hits me that most of what I see there I find pointless and boring.
PDT: And my first reaction to that is, well, I'm a research mathematician-- it's only to be expected that I'd find grade 10 stuff a bit pointless, and not of great interest.
XYZ: I think that's how most people would respond.
PDT: Okay. But then I'm talking to a grade 10 math student I happen to know who's actually in danger of failing the course, and I ask him what's the matter, and he tells me he finds grade 10 math pointless and boring. So then I start to wonder just what's going on here.
XYZ: Okay -- so we turn to the analogy...
PDT: Right! So I went and got my daughter's grade 10 English book, and opened it and right away I am gripped. It's an anthology, a mixture of stories, poems, articles, reviews, etc. and most of it I found touching and compelling -- quite lovely pieces -- I'd happily read them for pleasure. In fact for a couple of weeks I kept the book by my bed. I'd never do that with the grade 10 math book. So what's the difference?
XYZ: Okay, this is interesting. First there is the idea we just mentioned that you're a math prof not an English prof, so we might expect that you'd find grade 10 math boring, but not necessarily grade 10 English.
PDT: Right, and that doesn't hold water because an English professor would also find the grade 10 English text touching and compelling.
XYZ: Right, so we're left with the second possible reason, that it's in the different nature of the two subjects. High school math just has to be boring -- it just comes with the territory -- but that's not true for English.
PDT: And that reason I totally reject. That's really the point I want to make here -- that math simply does not have to be this way, and there'd be more good learning if it wasn't.
XYZ: So what's the math text supposed to look like?
PDT: Well, I think the grade 10 English book is not a bad model: a mixture of "stories," "poems," "articles," "reviews," etc. That is, we want the mathematical analogues of these.
XYZ: And what are those? Good math problems?
PDT: Certainly. Good stories too. Commentary. Playing around with ideas and techniques. And what does "good" mean? I think we ought to use the same standards we use in English, and these are essentially artistic. The pieces of writing in the English book are works of art that have emerged from some deep creative place. And that's what the study of literature is based on -- such works of art -- and they are to be found in the classroom from the very beginning, they form the canon of the curriculum. But in mathematics this is not what happens.
XYZ: There are no such works of art?
PDT: Sure there are! I know lots of them -- wonderful problems, tantalizing, magical, wondrous, artistic in every sense of the word.
XYZ: But they are not in the grade 10 text book.
PDT: Even that's not true. There's quite a number in the book, but they appear in a fragmentary and peripheral way -- not at the centre. The focus of the book is elsewhere. The focus of the book is on the technical skills. It is these that provide the organizing structure for the entire book -- its heart and its soul.
XYZ: But why is this is not a problem in English? -- I guess because technical skills are not so important for that subject.
PDT: There may be some truth in that, but not nearly as much as is supposed. On the whole I feel that the importance of technical skills are overemphasized in math, especially at the elementary level.
XYZ: So we must de-emphasize technical skill in math?
PDT: I think that's true, though perhaps more should be said about that. But even that's just the beginning. A crucial difference between English and math curricula is that in English the books come first and the curriculum is then based on these, whereas in math, the curriculum comes first and then the books are based on that. In the first case, the books are essentially artistic; in the second case, the art is lost.
XYZ: And this leads to the alienation?
PDT: Exactly. The students never have a chance to be captured by mathematics -- in their universe, they give it little credit. And so I think that's where the alienation starts -- way back before things are perceived as being difficult in any sense, they are first perceived as irrelevant, essentially meaningless to the student's life. The students say "this isn't me," and shift their attention elsewhere. And then things are missed. And then finally, later, the anxiety comes.
XYZ: Let me recap'. You suggest that the reason we lose those two out of every three people is because the examples and the problems in the book are not works of art, and hence they have no meaning for the students.
PDT: Exactly. I could hardly have summarized it better myself.
XYZ: So you assert that meaning comes from art?
PDT: Yes I believe I do. What would you say?
XYZ: That meaning comes from life?
PDT: Well, I suppose that's true. In fact, that's certainly true. Whitehead said as much in his Aims: there is only one subject for education, and that's life-- in all its manifestations.
XYZ: So does meaning come from art or from life?
PDT: Well, let's pursue the idea that it comes from life. What would you conclude from that -- about the curriculum?
XYZ: That we should be doing more real life applications? Modeling, etc.?
PDT: That would seem to follow, but I want to be careful with that. I think at different times we have responded to precisely that question with more "applied" stuff, and it hasn't always worked, in fact it's often backfired. The students have found it too hard and often not really very relevant to their own lives.
XYZ: So the applications have to be "right."
PDT: And what does "right" mean? Well, let's look again at the grade 10 English book. Are the stories and poems about life?
XYZ: I would guess that they are.
PDT: For sure. Love, loss, caring, grieving, hopes, fears, frustrations, joys, triumphs. But does that qualify a poem for admission to the text book? No! -- I can imagine lots of poems written about these themes which were constructed without care, without integrity, without depth, which didn't really "work" as poems at all.
XYZ: To have meaning hey have to be works of art!
PDT: Exactly! Though there's an important interplay here, what Jerry King (1992 -- The Art of Mathematics. Fawcett Columbine) refers to as the paradox of the utility of beauty, and what Eugene Wigner called the unreasonable effectiveness of mathematics. Both of these bear witness to the possibly unexpected relevance of art to the concerns of what we call the real world. But notice that both of these epigrams are formulated in a way which leaves no doubt that it is art which occupies the primary position.
XYZ: That is certainly not the guiding principle behind the math book.
PDT: No indeed. Here, too often, the author presents a page of mathematical sludge and attempts to give it meaning by casting it in terms of building fences or rolling sheet metal. That does not fool the students. What they respond to are the artistic aspects of the work.
XYZ: So what is art?
PDT: Art is the ultimate source of meaning. This is such an important principle for education it can never be emphasized too much. Sorry, what did you say?
XYZ: What is art?
PDT: Ah, you certainly know how to ask a good question.
XYZ: It was, of course, sitting there, begging to be asked.
PDT: If it's beautiful, if it has integrity, if it hangs together, if it speaks with a clear voice, if it is able to care for the student and be cared for in turn by the student, then it is art. How's that?
XYZ: Not bad.
PDT: To be an artist is to care deeply about what you are doing -- to do it in a way that is "right," that has integrity, that speaks to your soul. I think that this happens far too rarely for students, either at home or at school, but it is the most important experience that anyone can have, the experience of being an artist -- it is golden -- nothing comes close to matching it. From it, you get your energy, your joy, your equilibrium. When I am fully immersed in an artistic endeavour, be it working on a math problem, or washing the dishes in hot soapy water, my personal anxiety disappears and I can "be there" for anyone. I am "in place." The sacred task of the educator is to guide the student towards that experience. But this is too often sabotaged by the scramble to cover the curriculum.
XYZ: Not only in math but in other subjects as well.
PDT: For sure. But this is particularly sad in math, because I think that mathematics, because of its nature -- its simplicity, its beauty, its directness, its precision -- can provide a stupendous canvas for the artist. Perhaps it shares this character with poetry. And music too.
XYZ: I think I agree with you, and I even think that many teachers would too, but as you say, they seem unable to respond to this because of their mandate to "cover" the curriculum.
PDT: Those who rebel often seek out alternative education programs.
XYZ: I've encountered some of these programs and most of them seem to be fabulous.
PDT: The teachers that lead them are real artists -- they care about their teaching, and they need to find a way to do it that has integrity.
XYZ: Perhaps the changes you have in mind for the math curriculum would allow more math teachers to operate with integrity without rebelling against the system?
PDT: I hope so.
XYZ: And you would use works of art as the basis of this curriculum.
XYZ: Can you give me an example of a "work of art" at the grade 10 math level?
PDT: Sure. Though I should warn you that it's no longer so easy to attach levels to different problems. Like a good poem, a good problem can work at many different levels and with many different abilities.
PDT: Here's an "extra" problem I pulled out of the grade 10 text book.
XYZ: Ah, the "extra" problems. I bet that's where most of the art ends up.
PDT: I'm afraid so. "Cola cans are sold in boxes of 40 packed as shown (a 5x8 array). Alicia discovers that the box can be made to hold 41 cans by simply rearranging them. How does she do it?
PDT: Yes it is. There's lots of possibilities for hands on exploration, with cola cans or pennies or whatever, and it's very satisfying to see the new array with that extra can. And then in checking that it really "fits" there is a bit of organizational strategy to work out and a small technical calculation, essentially right-angled triangle stuff. And if you're rusty on that, it's a good chance to revisit it. But most of all, what's needed is care in keeping track of bits and pieces. And even those simple technicalities are satisfying-probably because they are spawned by the art.
XYZ: And there are other questions which naturally arise. Like could it have worked with other sizes of array?
PDT: There sure are. For what m x n arrays would the same trick work? What arrays would allow us to fit two more cans? Or three? Are there ways of working with both dimensions at the same time? It's wonderfully open-ended. And here's an algebraic example:
Also wonderfully open-ended.
XYZ: But the kids would find these problems hard.
PDT: Of course they would. They are hard. But it's important to notice carefully what makes them hard. It's not the technical manipulations that we often think of as comprising mathematics. It's the organization and care needed to assemble and track the pieces of the problem. More and more, mathematics seems to me to be about those things: integrity and care in the service of art.
XYZ: I have a picture of a care package. Something you get from a parent or a friend when you're in an alien situation. You undo the string and open it up -- soul food!
PDT: That's a terrific image -- and it's the right one for a math problem. Our students should be sending little care packages to their solutions, their constructions, their manipulations. And receiving them from the problem too! Caring is a two-way street.
XYZ: I like that. I've always felt that to be true about writing -- that what makes it good prose or good poetry is not a lot of sophistication, but a real caring for the subject and a sense of what's appropriate and what works in terms of structure and technique.
PDT: That's well said. And that's what I want for math too.
XYZ: It means you can't afford to be in a hurry.
PDT: Certainly not. You have to be prepared to spend time exploring.
XYZ: Problems that appear hard when you're hard pressed are often easy when you insist on taking the right amount of time.
PDT: But when you've got a curriculum to cover, you don't give the problem a chance.
XYZ: So why is it that the English curriculum has come to be based on works of art but the math curriculum has not?
PDT: That's a good question. I think that math and science have been forced over the past many decades into a technical straight jacket in order to serve economic and technological purposes. And literature has escaped that because it's not perceived as being essential in the same way.
XYZ: And if English had been perceived as being economically essential...?
PDT: The same thing would have happened to English. Let's be fanciful for a moment. Suppose our great leaders had somehow decided that it was literature and not mathematics that held the key to our economic well-being. For example, imagine that, for some reason, Sputnik, instead of being a satellite, had been some triumph of literary technology, imagine that sonnets are found to heal heart disease and restoration comedy to annihilate tumors, and suppose that the best fuel for the latest breeder reactors is an epic tale. You can imagine that the word would come down that proficiency in reading and writing was now the number one priority of the education system. And reading and writing would quickly be reduced to a host of technical elements, and standard poems for the grade 10 curriculum would be written by a committee of educators. And most good poetry would suddenly be judged as too advanced.
XYZ: So that, as in math today, the English books would be written to fit the curriculum.
PDT: And the art would be lost. Mathematics, on the other hand, left to its own poetical devices, might well thrive. No one would think twice about throwing out all the boring stuff.
XYZ: This makes me want to return to the question of what you call the pointless boring stuff. Surely it is still true that the technical requirements for doing math at any level are more severe and less flexible than are the technical requirements for writing.
PDT: There's some truth in that. Math on the whole is less Dionysian -- more care is required. Certain manipulations have to be done right or you will end up with nonsense.
XYZ: And is it not also true in math, that a mastery of certain technical skills is typically a bigger prerequisite for "going on," than in English?
PDT: This is certainly an issue that needs to be carefully looked at.
XYZ: For example, with the current curriculum, a student who misses a week of math class can be terribly handicapped, but this seems not to be the case in English, not nearly so much.
PDT: Right! The sequence of works of art which embody the English curriculum tend to be fairly independent of one another.
XYZ: But that can't happen in math, or can it?
PDT: Well, actually, I think it can. We are all conditioned into thinking that math has to form some kind of tower, and if you're missing a piece at the bottom, the tower will collapse. But I don't buy that metaphor any more. Most of the mathematical thinking that gets done in the "real world" is not that far from the ground.
XYZ: But isn't it still true that a lot of the math in the curriculum really depends on an understanding of previous concepts?
PDT: I think you're right, and for me that signals the need for some fundamental shifts in the math curriculum. First of all, the examples/problems that we use should be simpler, less technical, more exploratory, and more discussion-oriented. In our efforts to "uncover" them in the classroom, different techniques will arise, some more elementary than others In my experience, what happens technically with these problems is often quite unexpected. Students find quite ingenious ways to get around sophisticated skills that they have not yet mastered, and don't think of using.
XYZ: And there's probably a lot of learning in that.
PDT: And excitement too!
XYZ: Will they learn the required skills eventually?
PDT: Well, the important ones will come up many times and the student who is interested in what's happening will finally get hold of them. And some students (probably many students at particular times) will even be hungry for them because they want the power that these skills will give them. And they will master them.
XYZ: But you'll have different students in the class with different technical backgrounds.
PDT: Yes you will. Mind you I think that we have that already, perhaps far more than we care to admit, but let's leave that one aside. We certainly have that situation in English, but it doesn't seem to hamper us so much.
XYZ: Not as much. It seems more of a problem in math.
PDT: Let me give you an example. Not long ago a senior economics professor called me up, dismayed that a number of students in his class, all of whom had first-year calculus, didn't know what Lagrange multipliers were. How could he teach his third-year whatever-it-was if his students hadn't met such a fundamental concept?
XYZ: I'm all ears.
PDT: Well, that's an important question. Lagrange multipliers, or at least the ideas behind them, turn out to be fundamental in methods of optimization, yet nevertheless, that topic will sometimes not be covered in the curriculum. So what I say to the economics prof is this: Is Lagrange multipliers actually an essential tool for your result? If so, then it's important, even essential, that you develop it yourself, for the class, from the ground up, in the context of your result -- that you recreate the ideas in the students' minds. If it's a central part of the economics model, it should be a central part of the economics class. Too often, mathematical results are terribly abused by teachers who take a cheap shortcut and simply refer to a result from the past, from another place, another context, totally underestimating the difficulty (and the importance) of transporting these ideas from one place to another. When that happens, the mathematics loses, the application loses, and most of all, the student loses.
XYZ: So the economics prof should spend more time on the technical result.
PDT: If it's essential to his model, yes. He might cover less ground as a result, but he will do more real modeling. What he has to realize is that in most of the real world, people don't use the standard cut and dried theoretical tools -- problems are usually too idiosyncratic to allow that. The tools almost always have to be refashioned to suit the circumstances, or even redesigned from the ground up, and I find that when I'm doing this, I'm often not aware that I'm redesigning a standard tool at all -- it looks more to me that I'm designing a tool from scratch, until I get to the end, and then I suddenly recognize that I've actually reinvented a special-purpose version of something quite familiar like Lagrange multipliers.
XYZ: And to be able to do that, that type of reinvention, I guess you would have to have studied those tools in a process-oriented manner.
PDT: Absolutely -- in a variety of situations, over many years. In fact, although the economics prof thought half his class hadn't seen Lagrange multipliers, they had almost certainly all played with, probably many times, the fundamental idea behind them -- that if you want to allocate a resource among several competing activities and maximize your overall return, it has to be true that you get the same marginal return through each activity. And I bet he could have presented his "model" by appealing directly to that basic idea.
XYZ: That makes sense.
PDT: I think that students encounter that idea many times in their education, but it gets lost in the technical shuffle that always goes on at the same time.
XYZ: That's a nice way to put it.
PDT: One thing I do in my calculus course is develop exactly that idea through a sequence of simple modeling exercises. My experience is that students who have encountered these may not be able to handle the formalism of Lagrange multipliers but can solve particular optimal allocation problems with their bare hands, so to speak, because they are able to think about the problem from the ground up.
XYZ: What are the prospects for these ideas in a highly technical program such as Engineering?
PDT: Well there are some interesting reform ideas in the air. But a number of my colleagues will certainly resist. They tend to freak out at the thought of a differential equations course that doesn't have this, or a complex variables course that doesn't have that. Why are you laughing?
XYZ: I'm picturing some of your colleagues freaking out.
PDT: Well maybe that was too strong. But honestly, sometimes I get convinced that they've never had a conversation with a practicing engineer ten years after graduation. But I certainly have, and it's the same story over and over: the topics we cover don't matter a bit -- almost none of it is explicitly used. What matters is the way of thinking, of solving problems, the methods of analysis. To be able to think and write clearly, and be comfortable with technical material.
XYZ: I believe it -- in fact I've heard that before.
PDT: Of course you have. Over coffee in the staff lounge, everybody agrees it's true. But somehow, when it comes down to the nitty gritty, of choosing a text, or setting a curriculum, or responding to the "demands" of some "user" department, we buckle under, we lay on the same tower of stuff.
XYZ: You're suggesting that, as a university teacher, you have quite a bit of choice about your curriculum.
PDT: I think we do. And therefore we have a lot of responsibility for the disgraceful state of most of our courses, particularly the basic ones.
XYZ: That's perhaps less true for high school teachers.
PDT: Certainly. They are told, principally by the universities, that close attention to the "curriculum" is necessary to prepare their students for the future, and for the most part they do not have the background and the freedom to counter that. And although there's lots of rhetoric about process and investigation, the text books give little idea about how to implement that in the context of the given curriculum. And school teachers certainly don't seem to have the time or the experience to develop alternative models.
XYZ: You're talking about math teachers?
PDT: For the most part, yes. It's worth pointing out that the situation is very different for English teachers. Everyone should look at a copy of the Ontario Secondary School Guidelines for math and for English, and compare them. The English guidelines are much more open, more liberating, more empowering. The teacher is given a lot of freedom over the choice of material -- the teacher is respected as a professional.
XYZ: Yes, I've seen both documents. The difference is striking.
PDT: The math teacher is hemmed in, treated like a lackey, like someone who really can't be trusted. If I were a math teacher I'd be mad as hell. Not only am I not valued as a professional, but the merchandise I've been given to "sell" is clearly second rate. It's hard to be an exciting teacher -- it's hard to generate energy in your class, it's hard to win the heart-felt allegiance of your students, when you're stuck with such material.
XYZ: And most high school teachers feel there's really no choice -- that they have to serve the university and the college and the workplace.
PDT: The next time a teacher asks me what it is that the universities want, I am going to reply that the universities are not the place to get that answer -- that they haven't thought carefully enough about that question. What I want to say to the high school teacher is: forget about the university, forget about the college, forget about the workplace -- tell them all to go get stuffed. Look instead inside your self, look inside your soul, you are an artist and you are a mathematician, you know what's good, you know what's right -- trust that, and give your students the very best you have to offer them.
XYZ: Holy cow.
PDT: Hey, we've all had the feeling of walking into a classroom knowing that the problem we've given ourselves to work with that day is absolutely golden -- that we don't have to worry about ourselves or our performance, that the material itself will illuminate the entire hour. There's a tremendous sense of energy and self-confidence which comes from that and radiates into the class. It's an incomparable feeling, and it's quite infectious.
XYZ: You're right. But it's rare.
PDT: And that's what's inexcusable. That it's so rare. One sure sign that the math curriculum has gone astray is the proliferation of talks at OAME meetings with titles like, How to take the numb out of number, and Making logarithms lively. As soon as you see a lot of that kind of talk, you can be sure the standard material is artistically poverty stricken. Here -- let me give you a poem from the math and poetry course:
The Kama Sutra of Kindness:
should I greet you
it is not love
the lake is cold
An enormous silence meets the initial reading of that poem. It simply rivets the class. There's no question of focus here -- there are a huge number of things to react to and take off from. I don't see anyone at the English teacher's conference giving talks on how to bring that poem to life. It carries its life with it like a flaming torch.
XYZ: And you want a math course with "poems" like that.
PDT: They exist! They're there. And we know them when we see them. We don't need anyone else to tell us what's good. That's very important to keep hold of. We know -- each of us inside ourselves -- we know what's good.
XYZ: Wow. Okay. What's the program? What do we have to do?
PDT: First of all, we have to develop some proper curriculum materials, and that means works of art -- good problems. They have to be the centre of the curriculum, around which everything revolves. Anything less than that is like constructing a poetry course with lots of stuff about rhyme and meter and form and symbol and metaphor without any real poems. A poetry course without poems!
XYZ: Okay so you find these poems.
PDT: Or construct them. The way poetry is written. From life.
XYZ: And then how do you present them?
PDT: In the first instance they'll stand alone. They'll declare themselves to be worthy of attention. And what will happen with them will vary a lot with different teachers and different levels of student.
XYZ: Lots of scope for professional activity.
PDT: Such a curriculum could give a real burst of life to the whole concept of professional activity.
XYZ: There's a lot of anarchy in the curriculum picture you're painting.
PDT: Anarchy is a great source of energy. More and more for me I look on my role of teacher as providing energy more than knowledge.
XYZ: I think there's a lot to be said for that attitude -- that if you can really can give them some energy, you've accomplished an enormous amount.
PDT: You see, knowledge is already embarrassingly plentiful -- with all this "web" stuff, there's a glut.
XYZ: Okay, I'm with you all the way here. But help me again-I'm still having problems figuring out how all the "stuff" that we do now is going to get done. I guess when you come right down to it, I'm having trouble believing we can get along without it.
PDT: Well, I'm not really saying that we can... Look, I think I really want to say something here -- several things in fact -- I think the technical side of mathematics is hugely misunderstood.
XYZ: Okay, shoot.
PDT: First of all, technical skills are important -- not only do you need to know them, you need to know them well. When you're solving a problem, you can't afford to be distracted by technical matters, you need your mind free to be creative. Also, problem-solving is not algorithmic -- we find our way uncertainly, by a hope, a hunch. But we are guided, deep down, perhaps without any explicit awareness, by a feeling of what's technically possible. So we really need a number of skills to be thoroughly internalized.
XYZ: And that's why drill is so important. It's the same with music and tennis: you can't play either of them unless a lot of stuff is automatic.
PDT: Okay! That analogy with music and tennis is just what we need. It holds to key to what we are missing in mathematics.
XYZ: Let me guess-in music and tennis the drill is inspired by art.
PDT: Precisely. It makes a huge difference.
XYZ:. And maybe that's one reason why many students are technically so weak. They simply do not put in the hours because they really have no inspiration to do so.
PDT: And you know there's something else here too. We all have a need to be grounded, to be protected, safe from the buffetings of an arbitrary universe, and careful precise routine activity plays an important role in this. There's an enormous amount of comfort and reassurance in it-but it has be "right", it has to be centred, it has to have integrity. I've already mentioned washing the dishes in warm soapy water. That soapy water is an important metaphor for me. When I leave in the morning, the dishes are warm and air-drying on the rack, the counters are spotless and bounce back the light which pours in from the window over the sink, calling to me, for I am now ready to meet the day.
XYZ: There's a lot of art in soap bubbles.
PDT: More on the same point! -- suppose a calculus student makes the following "simplification":
What do we do? -- throw up our hands in despair and blame poor technical preparation in the lower grades? And thanks for the soap bubble comment.
XYZ: I know some calculus instructors who'd do exactly that.
PDT: And so do I. But the fact of the matter is that this error may have nothing to do with technical preparation-but rather with focus and care. Perhaps the student who makes this error is simply not working carefully, is not checking things out, is not asking whether his answers are reasonable, is perhaps short of time, or short of patience, or short of caring about what he's doing -- anyway he's unable or unwilling to give this piece of the problem the time and attention it deserves.
XYZ: Not to mention that it might have been a problem that didn't really deserve very much in the first place!
PDT: Right on!
XYZ: So where are we? We've agreed that technical skills are important, and that they need to learned well.
PDT: I'd even say it more forcefully-to be really "there" for us, they need to occupy a place at the centre of our being, both intellectually and spiritually. And if you have trouble understanding what that means, think of tennis or music.
XYZ: Okay-but how many techniques do we need.
PDT:And that's the question of technical prerequisites. And I think that the answer is that to be able to do math, and certainly to succeed in the world, you don't need mastery of nearly as many techniques as we might suppose. Once you've mastered a number of skills of a certain type, you get the hang what can be done and what can't, of what's algorithmic and what's not. And that's the biggest part of what you need!
XYZ: It almost seem to me as if a lot of what we're doing now is not only wasted but damaging.
PDT: It's nothing short of shocking when you think about it. The politicians are all convinced that education is costing more than it should, but they simply have no idea of the enormity of the waste of time and energy in the science classroom. Ninety percent of the students spend ninety percent of their time operating at half speed, just trying to get through the hour. It's scandalous. If a significant number of politicians had the wit or the experience to know this, we'd be in serious trouble. The irony that saves our hide is that most of them have had such a poverty stricken exposure to school science themselves that they are unable to make reasonable judgments. So they are at the mercy of the educators who argue that we're all doing as best we can, and what's needed is just a little tinkering here and a dash of special ed there.
XYZ: Are you suggesting that we could get by...
PDT: In fact just look at the "solutions" that the educators have come up with-- sorry, I'm getting carried away here.
XYZ: Go right ahead.
PDT: ....just look at the "solutions" that we've come up with to save money-- curtail the music and the art and the drama and the crafts and the outdoor ed, and all the extras that breathe life into the curriculum. The school now seems to me to be an artistic desert. Where is the soul, the music and dancing, the colours and the bold outlines? Where is the energy?
XYZ: I have to say I agree with you.
PDT: What kind of integrity can educators have who propose solutions like this?
XYZ: Look, I really feel I agree with you. I do. But damn it all I'm still bothered by the feeling that somehow we need the math skills more than we need the English skills, that society depends on them more. So we necessarily have more freedom in designing the English curriculum.
PDT: Well I know "where that's coming from." In fact it's a unspoken and unexamined assumption in The Ontario Ministry's recent discussion paper on high school reform. But let's just stop for a moment and think about that. Suppose we moved to an artistically based curriculum, and simply let go of all the worry about the skills, and just let students who wanted them pick them up when they're ready. What terrible things do you think will happen? Exactly how will the world suffer?
XYZ: That's an interesting question.
PDT: I think we all conjure up a terrible picture of the consequences -- that bridges will start to fall down, economists will no longer be able to control the money supply, meteorologists will fail to predict the weather, nuclear power plants will become unsafe, video games will crash, coffee grinders will seize up, Canada will no longer be able to compete in the world economy and no one will be able to figure out what the world was like one trillionth of a second after the birth of the universe.
PDT: And I just see no evidence for that whatsoever. None. It's a totally unsupported assumption. I think things are more likely to go the other way -- that Canada might become a leader in creative technology -- video games that make sense, for example.
XYZ: It would certainly be interesting to see...
PDT: So when do we start?
XYZ: Well how exactly are we going to find the works of art?
PDT: Well, that's what I feel math educators ought to be doing now -- focusing their attention more narrowly on questions of subject matter and content. Because if that's wrong, you have to work incredibly hard to get the allegiance of the students, and only the most talented teachers will manage, but if the material is right, it will carry the class along like a great wave.