“So, so you think you can tell
Heaven from Hell,
blue skies from pain.
Can you tell a green field from a cold steel rail?
A smile from a veil?
Do you think you can tell?”
– from “Wish You Were Here” by Pink Floyd
I’ve taken some time to get back into the subject of mathematics, and secondarily math education. This is partly because I’ve had this niggling feeling that math is pertinent to my study of computing. I was always told that math was important to computer science, but in my work I didn’t feel it had relevance beyond algebra and discrete structures. The thing is, with the exception of a couple math classes I had in high school I didn’t find the presentation of math that interesting, nor did I feel like I was really grasping it. Nevertheless there was always some part of me that was interested in math generally for some reason.
Alan Kay helped steer me towards some interesting things about mathematics last year, and I’m grateful for that. This started when I watched a video of a presentation he did called ”What is Squeak?” and I became intrigued when he said that students aren’t learning what real mathematics is in most American schools. I went to good public schools, but I figured since he talked about most schools I probably wasn’t taught this either. I asked him about this last year, and I realized quickly that no, I wasn’t taught what real mathematics was.
He believes that students should be taught to think of mathematics as mathematicians do, even if they don’t go into it professionally. He’s seen examples of how this can be done. The reason is that math is not about the numbers, arithmetic, symbols, and algebra that we learned in school. It’s a way of thinking. It’s just a matter of translating this mode of thought into forms that children can understand and work with.
For the rest of this post I’ll be talking about a book called The Art of Mathematics, written by Jerry King, who is a professor of mathematics at Lehigh University. This isn’t a book report. He talks about a bunch of things in greater detail than I’ll cover here. So please do read it. I’ll just be talking about what really grabbed me.
It’s a very interesting book. In it King lays bare what is rarely acknowledged, that typically schools and universities do not teach the full depth, the beauty of mathematics. In a typical scenario a student is not exposed to real mathematics until their junior or senior year in college, if they take math courses for that long. Up until then they are fed a steady diet of procedures and techniques for solving math problems, but they don’t get to the essence of what mathematics is. The end result has been generations of students who for the most part fall into two categories (à la C. P. Snow).
There are those who became skilled in some aspects of mathematics for its utility in science and engineering, but have missed its beauty. Then there are those he calls “humanists” who have felt like math isn’t for them. They’ve felt forced to do it in school. They left it by the side of the road as soon as they had the chance, hoping to never deal with it again. They are educated but they are so distant from what mathematics is they don’t recognize it, much less its beauty. We see them out in society all the time. They say, “I’m not good at math,” and they say it with no stigma at all. Most of the rest of society would say, “Yeah, me neither. What was the point?”
None of us would say, “I can’t read” without feeling a profound sense of shame and deprivation, which goes to show that mathematics is not seen as essential in order for people to function in our society. But is that view correct? I think there’s reason to believe it’s not. There are some common everyday reasons I can imagine where having “a taste for mathematics”, as Alan Kay calls it, would be advantageous to all sorts of people in ordinary life (again, I’m not talking about algebra, but mathematical thinking). But more importantly, Kay has also said that in order for our civilization to realize the potential of the computer as a new medium it is essential that we understand (real) mathematics and science. He’s also said for years that “computers form a new kind of math”, a kind that doesn’t fit into the confines of classical math categories.
Jerry King wishes that mathematics would be upgraded to the status of a subject that students should be literate in to be considered educated. He explains why mathematics education at universities has not been good for decades, and touches on some essential ideas that every mathematician knows, which the typical math student is not shown.
Mathematics at its essence is a way of thinking about relationships of abstractions. It’s also a creative/discovery process, something that most math pedagogies don’t teach.
From this one can see right away that math is valuable to science, since deeply embedded in it is the study of the relationships between things. It also has value in computer programming, since in order to get our programs to work we have to understand the concept of abstraction, and understand something about the relationships between things in our code, and entities that are created in a running program.
Bertrand Russell said that mathematics is “p implies q“, and as King explains in his book this point of view allows us to take assumptions and reason about their implications. For example, “If p is true then q is also true”.
Beyond the reasoning skills it engenders, King says there’s also beauty in it which you can experience once you get to the real essence of mathematics. For mathematicians this is the reason they do it. I would say the same is true of computing. It’s the reason I got into it, and still pursue it, though as a field I would not be so bold as to say we know yet what its true essence is.
What King’s book reveals is that mathematicians do not merely have a more advanced knowledge of math in terms of how most of us understand it. They have a very different perspective on it than we do. Mathematicians believe The conventional wisdom among mathematicians is that this perspective is impossible to teach. You either “get it” innately, or you don’t, no matter how you’ve been taught. King is the exception. He contends that everyone has the potential to understand this perspective, and that it would do us good to do so. If only mathematicians could find it in themselves to have a passion for teaching what they really know.
What follows are summaries from my notes, and quotes from his book. I’ll try to make it clear when I’m talking about my own thoughts.
In my estimation the most important chapter in his book is called “Aesthetics”. King says that mathematicians do mathematics for aesthetic reasons, but the aesthetics of mathematics are not defined anywhere, either in the field or outside it. For mathematicians it’s a personal experience.
The crux of his book is explaining why most people are never exposed to the beauty of mathematics, and it’s a tough problem to solve:
Deeply embedded in our culture lies the notion that mathematics can be truly comprehended only by a gifted minority. Because so few members of the otherwise educated public possess even the rudiments of mathematical knowledge, mathematics has been assigned a special status. Unique among the collection of disciplines–such as philosophy, history, and literature–which in times past formed the basis of both the concepts of liberal education and the core curriculum, mathematics may be set aside under ordinary conditions without social or intellectual consequence.
Behind all this stands the concept–widely held and deeply believed–that there exists in a certain few members of the human race a type of “mathematical mind” which allows them to understand the logical complexities of mathematics. It is believed that just as there are only a few people capable of running 100 meters in less than ten seconds there are only an analogous few capable of understanding mathematics. And just as the inability to sprint at world-class level carries with it no social stigma, neither does the inability to understand mathematics.
Mathematical talent comes to you exactly as does sprinting talent: God either gives it to you or he does not. Or so it is believed.
Such beliefs provide comfort. Through them, members of the public can justify their often awesome mathematical ignorance. And because of them, mathematicians can rationalize their failure to teach–in a way such that the knowledge sticks–even the basics of the glorious discipline which occupies every moment of their conscious thought and almost every ounce of their energy. You cannot be expected to understand mathematics–so goes the myth–unless nature has provided to you the kind of mind necessary for the subject’s comprehension. Nor can you be expected to teach mathematics to people who lack the basic mental equipment for it, just as Minnesota Fats lacks the physical equipment to run the Olympic Trials.
Comforting they may be, but these beliefs have no more validity than astrology.
He speculates that a sense of aesthetics may really make mathematicians who they are, as opposed to the developed senses of logic, precision, the ability to manipulate symbols, or the ability to deal with layers of abstraction. Jules-Henri Poincaré, a French mathematician, believed…
that an “aesthetic sensibility” for mathematics defined the very soul of the mathematician. It acted as a “delicate sieve” without which no one in mathematics can become a “real creator”.
Poincaré is right as rain about beauty and every mathematician knows he is right. You have inside you an aesthetic sensibility toward mathematics which acts on your intuitive mind as a delicate sieve sorting out the elegant and harmonious ideas from those which are merely useless combinations of other ideas. You have it, that is, or you are not a mathematician.
Poincaré, however, did not extend the notion of the aesthetics of mathematics to teaching. He did not believe that students could understand the beauty of mathematics through the educational process, but King is hopeful.
Perhaps Poincaré was wrong. Maybe the notion extends. Perhaps you can bring students to mathematics early on by emphasizing its aesthetic value rather than its utility or its applicability. … I don’t know. But one thing is certain. We will do no harm by trying. For what we do now has failed. Mathematics as understood by mathematicians remains unknown to everyone else. … To the humanist–and to everyone else in the other culture–the subject is something to be shunned.
I refuse to believe that this is the nature of things, that mathematics must remain forever beyond all but a tiny minority of our citizens. The notion that there exists a large subset of the populace who are capable of appreciating and understanding music, art, and literature but are somehow innate mathematical cripples seems to be simultaneously arrogant, apologetic, and just plain wrong. … We can do ourselves no harm … by presenting to our students early on those characteristics of mathematics which, in Poincaré’s words, contain “this character of beauty and elegance, and which are capable of developing in us a sort of aesthetic emotion.”
Instead of aesthetics, the schools give us drill and tedium–and immediately forgettable techniques aimed at unwanted and unwelcome applications.
The other side of this problem is we need to understand…
why people are driven to do important and difficult things for the sake of beauty. We need to understand what Mr. Keats had in mind when he wrote:
“Beauty is truth, and truth beauty–that is all
Ye know on Earth, and all ye need to know.”
In the book King notes an interesting correlation between the beauty of mathematics and scientific and engineering ideas, derived from mathematics, that have been recognized as being very useful. He suggests that perhaps in the case of mathematics beauty brings more than just pleasure, but also advancements in our ability to make new discoveries and build things that work. That would be pretty cool if it was true. King also says that mathematics IS a kind of art.
He says that unfortunately there has been no formal study of aesthetics (he suggested philosophy is the appropriate venue for this) which might help the academic community better understand what mathematics is, because philosophers don’t consider art and aesthetics as serious subjects, since they provide amusement and enjoyment. To them serious subjects like truth, justice, and reality are the things worth spending time on.
[S]erious people–most of them–do not deal professionally with matters of amusement and enjoyment. But they ought. Because there is more at stake than understanding what kind of art appeals to whom and why it does. … We need to understand why people are driven to do important and difficult things for the sake of beauty and for its own sake alone.”
My own sense of the reason King says this is it would help educators understand how to teach mathematics so that people understand it, and convey why it’s important.
Quoting Seymour Papert, King says:
[W]hen mathematics is taught in the schools, students are asked to “forget the natural experience of mathematics in order to learn a new set of rules.” Moreover, as we all know, the existing “rule learning process” does not work, has not worked, and–in my view–cannot work.
What is needed is a real understanding of the mathematician’s “personal experience” with his subject. At the highest levels, there can be no doubt that this experience is largely aesthetic. What we must learn first are the characteristics of mathematical aesthetics so that we can talk about mathematical elegance in more than a descriptive manner.
He says this will require rethinking classical aesthetics to include theories (to be developed) that apply to mathematics. It will require bringing teachers into math classrooms in primary and secondary schools who have been touched deeply by their own personal sense of mathematical beauty, so they can communicate that to their students. The problem here is:
We are not … headed that way. The future seems to hold, for mathematics instruction, an increased dependence on technology in the form of computers, hand-held calculators, and video presentation. Surely, I cannot be the only person to notice the clear correlation between the declining mathematical abilities of American students and the aggressive introduction of these technologies into the mathematics classroom. The decline began as the technology came in.
At the university level it will require bringing research mathematicians back to teaching. The problem here, he says, is that mathematicians think their only job in life that’s worth doing is mathematics. If a mathematician talks about mathematics (rather than doing it) they are considered “on their way out”, a former mathematician. In the profession it’s seen as beneath one’s stature, even shameful, to talk about it in descriptive terms. Thankfully there are a few like King who don’t care about that.
Technology and math education
Alan Bradley: Some programs will be thinking soon.
Dr. Walter Gibbs: Won’t that be grand? All the computers and the programs will start thinking and the people will stop.
— Tron (1982)
I want to take a detour here to address King’s point about technology entering the math curriculum in primary and secondary schools. His aversion to technology in the educational setting is understandable, but I think he’s narrowing the focus of his blame too much. In much the same way that college mathematics is taught as a set of rules and techniques, so it is in primary and secondary school. The quality of instruction would be improved somewhat if the technology were taken away, but not a great deal.
I had a wonderful conversation with an engineering professor from Duke University in March. One of the topics we discussed was how math is taught in secondary schools, from her own experience either observing it, or hearing second-hand accounts. It gave me a sinking feeling in my stomach. She said that students are required to buy a specific TI calculator model, and teachers use it as an integral part of their “math” instruction. She used the example of inverting a matrix. Rather than teaching what one is really doing when one inverts a matrix, students are taught to enter the matrix into the calculator and to hit the “invert” button. That’s it. They are not taught where the principle comes from, why they would want to do this, or what it represents. It’s a continuation of the procedural mindset that King criticizes. From this perspective I can see its folly now.
I said to the professor, “That’s not teaching math. That’s teaching how to use the calculator.” Yet the educational system has deluded itself into thinking they are teaching mathematics by doing this. Obviously one needs to know about arithmetic operations, but that’s not all we need to know.
Later I remembered that some academics used to say, “Someday we won’t have to teach math. We will have calculators and computers that will do the math for us.” First of all, as I’m increasingly becoming aware, this displays a fundamental misunderstanding of what math is, and its importance. It’s also widespread. What it reveals is the mistaken notion that mathematics has nothing to contribute to human reasoning, and is only useful for science and engineering where math is used to express ideas and figure things out. And why would it be useful to the general population to understand mathematics? It’s just symbol manipulation and calculation after all, isn’t it? Something that machines are fully capable of doing by themselves.
We can see how the educational establishment, and the “humanists” as King calls them, view mathematics. I’m not saying that calculation and symbol manipulation are unnecessary skills, but let’s be clear (for once!). Calculation is a product of arithmetic, not mathematics. Arithmetic is an outgrowth of mathematics, but they are not the same thing. Symbol manipulation is a part of mathematics, but it’s only a skill within it. It’s like knowing how to write in that realm. Having it does not mean that one understands mathematics.
I disagree with King that technology is at fault for the decline in math education. What he and the professor from Duke are seeing is an extension of the way math has been taught in our school system and in our universities all along! Now, I’ve heard the claims that the current standard in public education in the U.S. (No Child Left Behind) requires teachers to “teach to the test”, and other things. That’s not my point. Procedural thinking in math education has been around for decades.
The perceived role of technology also has something to do with this. If teachers and those who create curriculum believe that progress means machines do more of the thinking for us, then this is what you get.
There’s nothing wrong with a computer or calculator making some tasks easier. It can enhance our work. What we should not do is use it as a crutch, making it automate as much as possible so we no longer have to think, because then we’re really robbing ourselves. Technology should be brought in consciously with an eye towards where it helps us think.
King gets into some foundational concepts, theory and forms, which he uses later to illustrate what mathematical aesthetics is like.
He says a theory is a framework or set of rules which prescribes the collection of allowable sentences one can make about whatever objects are under consideration.
A theory, therefore, gives you a framework for talking about the objects of the theory. More precisely, it gives you rules of inference from which you can prove theorems about the objects. Truth about the objects consists of sentences about them which are either assumed in the theory or else are provable within the framework of the theory.
Just an aside, but this almost sounds like a description of a programming language on a computer, minus the stuff about theorems and proofs. Could programming languages represent theories of computation? Hmm…something to consider. Anyway…
Extending this idea of mathematical theory a bit King says:
Plato, you will recall, asserted that what one has knowledge of are abstractions, which he called Forms. Roughly, what Plato did was replace real-world objects by their Forms in his epistemology. Each real-world object–say an apple–is a temporal thing which undergoes constant change. The Form of the apple, however, lives in an ideal realm and is eternal and unchanging. It is the Form of the apple and not the apple, said Plato, about which we can have knowledge. Moreover, because Plato’s truth could be known only about Forms and not about real-world objects the Forms were to him more real than the actual objects. … Plato argued, in fact, that Forms represented the only true reality since it was only Forms about which one could have knowledge.
An example of real mathematics
King takes a stab at defining aesthetics in mathematics, in formal terms. I’m not going to cover that here. Read the book. From my own sense of the aesthetics of mathematics I think he defined it well.
He presents a couple of proofs to demonstrate the aesthetics. I’m going to use one of them as a way to discuss what real mathematics is like. It also demonstrates a foundational principle of mathematics: p implies q (also expressed as p=>q, using “=>” as a double-lined arrow). The symbol p represents a hypothesis, and q represents a conclusion.
He puts forward the problem: Without leaving the room let’s set out to prove that there are at least two trees in the world with the same number of leaves.
To make a proof we start with a hypothesis. “The hypothesis is what we assume, the conclusion is what we deduce.” The hypothesis needs to be minimal. This is part of the aesthetics.
Whatever we mean by “trees” and “leaves” it seems obvious that there are more trees in the world than there are leaves on any single tree (probably many more). … So, for our hypothesis, we will assume that the number of trees exceeds the number of leaves on any single tree by some positive number. It is sufficient to assume that the excess is only 1.
If this were written as a typical math problem I had in school it would have been rendered as: “Given t trees in the world and a maximum of m leaves on any one tree, prove that there are at least two trees with the same number of leaves,” and I would’ve felt stuck, because my math training taught that we should only draw knowledge from prior math material (formulas and proofs), and whatever the problem statement gave us.
What I find interesting is that King approaches the problem differently than I would have, and probably most other math students. He uses the ideas about forms and theory to get an idea about how to approach the problem, and he thinks about the relationships of “trees” to “leaves”. What I find interesting and pleasant about the mathematician’s approach is that our imagination, intuition, common sense, and experience can play a part in solving math problems.
We can imagine forms of different trees. We’ve seen hundreds of them in our life. So we have a fair idea of their structure and what they look like. The theory is that there are t trees in the world and a maximum of m leaves on any tree. Using the forms of “trees” and “leaves”, and this theory, we can explore its ideas.
He starts with an idea that fits easily into our notions of the relationships. When I looked at King’s hypothesis I thought, “Of course. I can imagine that,” but I wouldn’t have come up with it, because I wasn’t taught to think of math in these terms.
If I had understood the notions of theory and forms, and their use in mathematics, I might’ve come up with the same assumption right away. Or I could’ve experimented with other ones, and through a process of discarding the theories that didn’t serve the objective of the proof maybe I would’ve eventually hit upon the one King used. I remember Alan Kay saying that children who are learning to think like mathematicians go through a kind of scientific process, trying out ideas and seeing which ones succeed and which ones don’t. Research mathematicians go through the same process.
I believe it was King who also said something I have heard from those in the math reform movement for years, that it’s the process that matters, not the result, and I think I’m beginning to understand this. Math education has been results-oriented for as long as I can remember, creating a system where the only acceptable goal is to get from Point A to Point B as efficiently as possible. What King indicates is that this mindset is part of the problem.
(Update 7-7-2009: I modified the two paragraphs below because I realized I was making some pedagogical points that were perhaps inappropriate. What I wanted to emphasize in the first paragraph was the importance of criticism in the education of math students. In the second, I did not want to diminish the importance of precision. It’s just a matter of where it’s applied.)
I think, however, there’s been a misunderstanding of “it’s the process that matters” when the math reform agenda has been implemented. Logic matters in mathematics, as does the concept of relationships between things. Depending on where students are pedagogically, one subject or both might be good to emphasize. This implies that there needs to be room for criticism and correction of flaws in thinking, something that reform-minded teachers have discarded in the name of protecting self-esteem. This is not a small omission. I think the idea needs to be that how you get to something that’s logically consistent need not be a straight A-to-B process. “The right answer” is the end result of creating something that’s logically consistent. It’s an afterthought. The goal is to reason about the ideas under consideration, and to wonder about the implications of those ideas (not necessarily in that order).
It’s become apparent to me that in addition there are circumstances in the realm of mathematics where a precise answer is not desirable (though precision within proofs is essential), because it gets in the way of exploring and thinking about ideas and how those explored ideas fit into a logical structure.
This brings to mind a story I heard years ago about a female executive interviewing for a position at Microsoft with Bill Gates. It’s been a while, so I may not have this word for word, but it basically went like this. Gates asked her point blank during the interview, “How many gas stations are there in the United States?” The woman responded, “I don’t know.” He pressed her further about why she didn’t know, and she said, “I would need to collect data to find an answer to that question.” Gates got frustrated and asked her, “Are you stupid?”
I heard commentary about this incident from others who had been associated with Microsoft, and what they said was Gates wasn’t expecting her to recall a fact, an exact figure on the number of gas stations. He was looking for a reasonable estimate, and he would’ve looked for a justification of it–a proof, perhaps–without her leaving the room. What was said about this and other “Microsoft questions” is “they really want to see how you think.” My guess is in this instance he was testing the candidate’s understanding of mathematics. I’m also guessing that she had a math education that was similar to mine. I would’ve given the same response.
I talked to Sanjoy Mahajan, a theoretical physicist at MIT who was a panelist at this year’s Conference on World Affairs at the University of Colorado at Boulder. We were talking about math education and he spoke about the ability to estimate, and that this is an important skill in mathematics. He used an example that a math teacher used with students (I can’t remember the name). He said the teacher would ask, “How many trees could fit in this room?” without bringing a single physical tree into it. He wanted the students to visualize, based on the trees they’d seen, about how many would fit, and they’d discuss it.
The setup for the proof on “two trees with the same number of leaves” is similar. In terms of mathematics we don’t have to go out and look at every tree and count their leaves and come up with a maximum. We just make a reasonable estimate, based on our experience with how many trees are, say, in a single forest, and how many leaves we’ve seen on a single tree, and our knowledge that there are many forests in the world. We estimate that there are more trees than leaves on a tree. It seems to work, and that’s our assumption.
Here’s the theorem statement from the book: Let t denote the number of trees in the world and let m denote the maximum number of leaves on any single tree. If t exceeds m + 1, then there exist at least two trees with the same number of leaves.
Just to make the mapping to “p implies q” more obvious I’ll break it down: p = “t exceeds m + 1″, implies = “then”, q = “there exist at least two trees with the same number of leaves”.
Here’s the proof from the book: Since m denotes the maximum number of leaves on any one tree, each tree will possess either 0, 1, 2, 3, … or m leaves. Imagine m + 1 boxes sitting in a row on your floor, each tagged, in order, with the numbers 0, 1, 2, …, m. Now imagine bringing the world’s trees one by one into your room (suspend disbelief). Place each tree in the box which bears the label equal to that tree’s number of leaves (thus, a bare tree goes into box number 0, a tree with one leaf goes into box number 1, etc.).
We have only m + 1 boxes and, by our hypothesis, we have more than this number of trees. Hence, some box must contain at least two trees when trees have been brought into the room. Thus, at least two trees must have the same number of leaves.
The reason we were able to prove this was because of our knowledge of combinatorial math, but it’s so simple it even works intuitively.
He introduced this topic of “aesthetic distance” from a philosopher’s perspective on how audiences appreciate art. He then put mathematics in the situation of being the object of appreciation. He said there are people who are too close to mathematics, and those who are too far away. The ones who are too close are scientists and engineers. They have quite a bit of knowledge around mathematics, but “their nose is pressed up against it”. They work with it all the time, but they don’t have sufficient distance from it to understand it and appreciate its beauty. They understand the rules and formulas they were taught, but they don’t really understand what mathematics is. Most others are so far away from mathematics that they have no comprehension of it at all and therefor cannot appreciate its beauty either.
In “the middle” between these two groups are the mathematicians, who are at a sufficient distance to understand what they are looking at, and to appreciate its beauty, and they are a small group compared to the others. With the way things are set up, only mathematicians can experience its beauty, and they aren’t doing anything to change the situation. They’ve locked themselves into their own “room”, so to speak, and they aren’t letting anyone else in.
King discusses at length the reasons why this has happened. To sum it up, it wasn’t always like this. There was a time when mathematics departments actually taught what they knew about math, rather than just procedures and techniques. King says that after 1957, our “Sputnik moment”, math departments at universities were seen as essential and no longer had to compete for money and students. They were even encouraged to engage in more research, for Cold War purposes, rather than teach. This fit in well with what mathematicians really wanted to do anyway–research. Math education, however, deteriorated. The change made mathematicians feel privileged, even superior. The privilege that mathematicians enjoy is in essence the cause of the problem with math education at universities to this day.
This book was an eye opener for me, mainly because it revealed that what I thought was math was not real mathematics, but rather aspects of math (deduction, reduction, symbol manipulation) and techniques derived from math (arithmetic, procedural tricks for solving problems, pattern matching). My sense is my math education in public school was better in terms of quality than what I got in college, but that isn’t necessarily saying much. College math was like how King described it: mostly techniques taught by rote, conveying no understanding. At least in the public schools I attended I got some sense of the aesthetics of mathematics.
I expected to get a sense of “the art of mathematics” from this book. Instead it talks about the subjects I discuss above, and more. I didn’t get a sense of the beauty of mathematics, but then King did say the purpose of the book was to talk about the subject, not delve into it. Nevertheless it is a good introduction for the uninitiated. He came out with a new book just a few months ago called Mathematics in 10 Lessons: The Grand Tour. I have it on my reading list. My sense of it is he gets into the actual subject of mathematics in that book.
—Mark Miller, http://tekkie.wordpress.com
Hand With Reflecting Sphere, M. C. Escher, 1935
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