I have not tried ISEL, but teaching mathematics by teaching computer programming is an interesting idea. There have been various utopian proposals to standardize or invent languages in the hope that this would lead to harmony and world peace. None of them worked out. However, in the field of mathematics, something along the lines of a standard language does happen. I think it happens when intuition, as expressed in ordinary language, fails to settle questions. More precise terminology is invented to find a way around the difficulty. It isn't likely that students will be disposed to adapt precise terminology in their thinking any more than ordinary persons adapt the terminology used in legal documents in ordinary speech. (Rigorous mathematics and "legalism" have much in common.)
It isn't realistic to think of most students as some sort of abstract conceptual processing apparatus that likes to struggle with ideas. They have the ordinary motivations of human beings. Most are somewhat motivated to get through the class in some way. Programming tends to present itself as a problem of telling a "stupid" computer exactly what to do. So programmers must put aside whatever natural distaste they have for precise communication and write the directions carefully I think programming gives students motivation to use proper terminology. And when they fail to do so, the fact that the program works badly is a more objective and acceptable correction than for a teacher to tell them that they aren't speaking properly.
Ideally, a harmonious group of people would replicate the invention of mathematical terminology if they undertook the same tasks that have been done by the (not-necessarily harmonious) community of mathematicians. However, I think it would be rare to find such groups. People do not like to make concessions in the way they speak for the sake of communicating to others.
I don't think there is one language to which one can translate mathematics ideas. And I'm not sure what exactly these people are doing with the ISETL programming language, but I was referring to some other portions of the text, like:
...in classrooms, ...we go through the motions of saying for the record what we think the students 'ought' to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models. ...We assume the problem is with the students rather than with communication...
That passage doesn't resonate with me. I can't guess what the author's language means or what mental model he has!
My mental model of students is that they are focused on the question of what tasks are expected of them. The problem of dealing with concepts is purely a utilitarian necessity in order to accomplish a task, e.g. do the homework, pass a test, answer the teachers question correctly. Yes, there are a few “students” who also want to be “learners”. However, a typical case is that students don't like tasks that instructors assign “to make you think”. They regard these tasks as slightly unfair since they being requested to do something that they have not been precisely instructed how to do. The guessing about the instructor's mental processes is focused on the question: “What does [he/she] expect me to do?”, not so much on “How does [he/she] conceptualize the subject matter?”. In some classes the tasks must be accomplished by attending to objective facts and talking in peculiar ways. In other classes the instructor asks “What was Hamlet thinking when ....” and the task must be accomplished by an exercise in make-believe where the student pretends that a fictional character can think like real person. Going between Math class and English class means dealing with one kind of mentally disturbed person and then going to deal with a different kind of mentally disturbed person. Students are not focused on understanding the mental processes of these disturbed people. They are focused on the problem of “What must I do to make this instructor happy and keep him from going off the deep end?”.
I recall seeing a television program about “intelligence” or some such topic and it showed footage of a dog. It pointed out that when being trained by the owner, the dog, at first, did not understand what he was supposed to do, so he executed little snippets of different behaviors until he found one that was rewarded. The same process often occurs in classrooms
So I speculate that if the task set before students is, say, to do set computations by interacting with a computer, then they may find this more relaxing than talking about set computations with a human instructor (who may utter statements like “if x is in S” when he means “if x is an element of S”). The computer is more consistent and can't be accused of not liking Johnny, but thinking that Jimmy can do no wrong.
Perhaps my mind is in the gutter of secondary and undergraduate education. Perhaps Thurston is taking about some idealistic, earnest student or abstract “curious mind”.
The best I can make of the phrase “we go through the motions of saying what students ought to learn” is that math instructors give formally correct definitions and proofs but don't reveal their own intuitions and primitive ways of visualizing the subject matter. That's worth discussing.
I think I see what you're saying. But you also lose something when working with a computer - it forces you to think the way the computer "thinks" (the specific axiomatic system which is in place) whereas my view of mathematics is that each person has his own understanding of things, and we do not need to necessarily agree on axioms before we have a discussion.
Couldn't teachers just do a better job at making the kids enjoy trying to answer questions? Maybe by showing them that they can actually do it instead of having them consider the actual "thinking" questions as "unfair" ones?
And, by the way, do you live in NYC? Would you like to come to one of our workshops at the school?
I live in Las Cruces NM, so unfortunately I can't attend the school. ( Stephen Tashiro, the retired civil servant, not to be confused with Steven Tashiro, the California chiropractor).
Yes, as I imagine it, working with a computer would enforce conformity. For example, the students form a mental model of how the computer thinks about sets and that conveys the axiomatics of set theory. The activity is a puzzle (guess how the computer works) as opposed to a form of self-expression....well, I suppose the kids could exchange their opinions about what the machine is doing.
I don't have a clear picture of the teaching method that you propose. However, like the dog in the TV show, I can produce random snippets of ideas.
Students do like to express themselves. However, their joy in that task goes down as the “degrees of freedom” diminishes. They would rather have the assignment: “What do you think about...” than the assignment “What was Hamlet was thinking...”. And they would rather do that than answer “What are the known biographical facts of Shakespeare's life ?”.
You did say “make them enjoy themselves” instead of “let them enjoy themselves”. That gives me the courage to suppose that your school's method is to begin with a collection of ideas from the students and work toward the conventional ideas. How can this be done? You could have period of self-expression by the students and then the teacher could then lay down the law, so to speak. You could offer intuitions (like the negative numbers taking their place on the “number line”) that are so powerful that they win the day for the conventional way of doing things.
The practical way lead such a class gracefully probably depends on psychological skills - things I'm not interested in thinking about at the moment!
An idealistic method would be have a group discussion that began with intuitions and progressed by reasoning to the standard ideas. My favorite undergraduate course approximated that. It was point set topology taught by “The Moore method”. The students are handed pages with definitions and “statements” that the class struggles to prove or disprove. The teacher only criticizes the logic of the proofs. (By that era, R. L. Moore had been discredited because of his arbitrariness and racism, so the teacher preferred to call the technique “the Socratic method”.)
The big problem with collective reasoning in the classroom is that most students do not understand the nature of logic. They do not make any distinction between the assertion that something is “logical” and the assertion that it is “true”. They think of definitions as statements that are right or wrong. They don't see the arbitrary nature of definitions and they don't think of them as logical equivalences. (They want to say “An open interval is....”, not “An interval L is open if and only if...”.) To me, the mathematical curriculum would make more sense if it began with teaching logic. I'm not a believer in some sort of Utopian scheme where all children are indoctrinated in logic and society proceeds happily onward under the stewardship of wise technocrats. I merely think that it would make the job of math teachers easier.
I'm going to take your advice in the other thread and start a topic about an “advanced” mathematical topic. But first I must collect my thoughts – no! , intuitions about it.
My understanding of the Moore Method is that it starts with the axioms of topology and derives more advanced results in topology. But this means the students need to agree to these axioms to begin with - so it's already not their own discovery. In our workshops at the school we always start with a question, not with a specific definition or axiom system.
Yes, in the Moore method (as I experienced it) the instructor supplies the definitions and axioms. This is not particularly coercive since the subject matter doesn't involve many definitions and axioms. The statements that the class must prove or disprove are really the heart of the course. Since dis-proof plays a prominent role, discussions of specific counterexamples take place. It isn't exclusively abstract logical reasoning.
I read the example discussion from your link. Will I extract the correct generalities from it? Let's see:
I conclude that a necessary condition for a creative group discussion of math is that the participants can express themselves without making any uncomfortable adjustments in the way the speak. The way that people speak has some connection with the way they think, so that affects things also. But the way they speak is the observable part of the process. I'll claim that it is unimportant whether the group leader stipulates definitions and axioms or whether the leader poses questions. To me, having a question posed is just as restrictive and onerous as having a definition or axiom imposed. You must know how it is; your non-mathematical friends know that your are “interested in math” and so they bring up all sorts of puzzles or expect you to be interested in doing arithmetic. I feel revulsion for such things. (I also confess that I'm not particularly interested in combinatorics. The way the typical curriculum goes, people are taught combinatorics in order to insure that they never understand the idea of probability.)
Every college math department picks some random field of mathematics and uses it as the basis for the course that is supposed to teach math majors how to do proofs. Such classes are almost always awkward group experiences because people are having to work outside of their normal way of self-expression. A course on Point Set Topology that serves this function is destined to be awkward. But I don't think it's awkward because it stipulates unfamiliar definitions and axioms. It's awkward because using mathematical logic requires more precise speech. If you taught the same course to people who were accustomed to using precise speech and familiar with logic, it would be more pleasant.
So I get to repeat my opinion that it would good to teach logic in the secondary math curriculum. When I had such a job, I tried it. The kids liked logic better than “math”. It can be done without reference to math – little nonsensical word problems, “If Bob is happy then Bob both wears suspenders and chews gum. Bob does not chew gum. Therefore?” Take a logic book by Copi and remove all the philosophical baggage.
I find it difficult to continue this discussion through written medium. But let me try to state my points shortly: - Logic may help with thought processes, but it also might seem arbitrary and technical if just taught directly. I prefer studying logic naturally through discussion of mathematics and proofs (using simple examples like numbers or geometry) - It's not about choosing puzzles. It's about presenting a _significant_ question. For the participant, the question about combinatorics that we spoke of is significant. But in order to explain the significance of the axioms of point-set topology and make them not feel arbitrary is a very difficult task. If you have some free time take a look at this thing I wrote about motivating the axioms of Group Theory. http://sites.google.com/a/thewe.net/mathematics/Motivation/What-are-groups-
I read your article about teaching Group Theory and I have some thoughts about it. I'll put them in another thread. In this thread I want to talk about the teaching of logic.
The phrase "separation of concerns" is used by software development theorists and I have also seen it used in critiques of mathematical papers. The general idea is that you shouldn't get two different problems tangled up with each other when you don't have to. I think "separation of concerns" applies to teaching logic. Math teachers may think that math teachers own the franchise for teaching logical thinking. But when is math ever taught that way? In high school and undergraduate college courses, the material that is taught is based on foundations that are too advanced to be developed by rigorous logic.
For example: Negative numbers have no logarithms base 10, pi has a logarithm base 10, pi has only one logarithm base 10. All these are very plausible to students. Math teachers are glad to have them tacitly assume these things. No need to stir up doubts about them in the students minds because a rigorous demonstration would be a nightmare. Likewise, if you want the class to solve (2x-3)(x+1) < 0, better have them draw a graph and figure it out. Don't try to formally discuss all the reasoning behind that.
It always amuses me to hear an algebra teacher exhort a pupil to "Think logically!" when he is trying to work a problem like: "Bob can build a doghouse in 2 days. Working together, Bob and Ed can build 4 doghouses in 5.2 days. How many days will it take Ed to build 1 doghouse if he works alone?". What the student is being asked to do is learn a cultural convention (common only to algebra problems) that involves a fantasy about how people's "rates of work" combine. It has nothing to do with objective reality and not much to do with logic. Students are adaptable. They can learn the make-believe fantasies of math just as they can learn to make-believe that Hamlet was a real person and write an essay about what he was thinking. But this isn't learning logic.
As another example, how do students "prove" trigonometric identities? They traditionally begin by writing a statement that assumes the two sides of the identity are equal (the very thing they are supposedly trying to prove). Then they do manipulations an arrive at a tautology. Taken literally this shows: If the things that I was trying to prove is true then something known to be true follows. What kind of logic is that?
Perhaps high schools have reformed the way that geometry is taught, but it was once a course based on Euclid with the steps in one column and the justifications in another. At least that's approximately a proof. But geometry is exactly the wrong place to teach logic since it relies on pictures and tacit assumptions.
Logic should NOT be taught with mathematics because it would be hypocritical. It isn't practical to teach the traditional high school courses in a logically respectable manner. It isn't practical in most undergraduate courses either. What you get are sudden bursts of logic here and there (e.g. induction when summing finite series, quantifiers when you reach the definition of limit). It would be better to teach all the logic at the high school level, using mostly non-mathematical verbal examples. Don't raise the students expectations that math class is going to strictly follow its methods.
Finally, as I mentioned before, I've found that kids think logic is a lot more fun than math.