One of the problems on the 6th grade NY State math exam looks like this:
What is the value of the expression below when r = 2 ?
9 – 3r
A 0
B 3
C 6
D 12
My first thought upon reading this was, more or less: “Nine minus six equals three”. So I looked at the four choices to see where the 3 was (in this case, next to B).
But then, afterward, I realized that what had gone through my mind was really more like this:
(1) Because I had just read `r = 2′, the `3r‘ looked to me just like a `6’. I know it doesn’t say `6′ — it says `3r‘ — but I saw a `6′.
(2) Because I saw something that looked to me like `9 – 6′, I just did the subtraction.
I think the take-away here was that I looked for a way to get rid of the variable r as fast as I could. In this case it happened so fast that it was done before I was consciously aware of what was going on. Something deep in my mind was apparently saying: “Danger, variable encountered! Must be eliminated!”
This reminds me of what we do all the time with natural language — something most people encounter far more often than they encounter algebraic expressions. When somebody says to us: “Wear that tie I like,” — and we happen to know that they really like the pink tie — we might actually end up thinking that they had said: “Wear the pink tie.”
In other words, we substitute the variable right away, as fast as we can, going from the abstract description (“that tie I like”) to the concrete result (“the pink tie”). Except in real life, with real objects, we make these sorts of substitutions so easily that we generally don’t even notice that we’re doing it.
Perhaps mathematical reasoning starts with something as simple as learning to repurpose the ways of thinking we all use for natural language (which is an innate ability), so we can apply those ways of thinking to numbers and variables.
I wonder if your typical 11 or 12 year old thinks like someone who (like you) is
1. exceptionally bright, and
2. has had years of formal mathematical training and practice?
I suspect that many of them don’t. I absolutely believe that we don’t give kids nearly enough credit for what they are actually capable of, but at the same time I don’t think we can assume they think like we do.
Typical 11 or 12 year-olds would probably be classified as ‘novices’ when it comes to doing problems like this. I’d suggest that you would qualify as an expert Ken, and we know from other work that experts and novices don’t look at problems the same way. They see things quite differently and pay attention to different details. Experts have an ability to filter out noise and fill in details in ways that novices can’t. It’s part of what makes them experts.
We should find the kids who struggle with these questions and see if we can figure out how they are attacking these.
Katrin, I completely agree with your point about the difference, and that is exactly where I am going with this. I do not suggest that kids should be expected to come to these problems thinking like experts. Rather, I’m constructing an evidential case for the opposite view.
In particular, I’m trying to identify the problematic assumptions that go into the creation of these tests. Instead of assuming that kids should just magically be expected to think like us experts when they come to these tests, I’m building up to proposing that we teach kids underlying skills which we take so much for granted that those skills — our own skills — are invisible to us.
The problem in effective teaching of math finds its root cause in that very lack of awareness, on the part of people who already know how do to math, as to the gulf between their own way of thinking — developed over many years — and that of 11-12 year old kids.
But in order to understand this problem, we first need to examine our own unexamined assumptions about what we take for granted.
Which is the reason that I’m going through this exercise.
Interesting.
Last year, I taught math ‘formally’ for the first time. It was in a general science course and it was for non-science majors. It confirmed my suspicion that, with very few exceptions, mathematicians should *not* teach math to anyone who isn’t already a mathematician (btw I’m not a mathematician and we had fun).
I also think that a big part of the problem is with the way we (fail to) teach teachers. It’s much easier to get excited about math when you have a teacher who is not afraid to play with numbers & problems.
Enzensberger’s book, The Number Devil is targeted at that age group. I think it’s a lovely approach.
OK, I think you’ve lost me. You said “with very few exceptions, mathematicians should *not* teach math to anyone who isn’t already a mathematician (btw I’m not a mathematician and we had fun).”
I don’t know what to make of that. Am I a mathematician? Am I a writer? Am I an inventor? Am I an artist? Am I a computer scientist? Am I musician? Am I a teacher?
The problem here seems to be the act of using categories to label people (as in “Mary is a mathematician”). I seem to fall into every one of the categories listed above. Yet of all of them, I would say I am first and foremost a teacher. I have never encountered a situation where my knowledge of a subject made it impossible for me to reach and connect with students who were encountering that subject for the first time.
I realize that I am fortunate, in that when I was young I had teachers who had a genuine love of math. Which gave me the most valuable lesson I’ve learned about teaching: That the most important thing a teacher can communicate is genuine enthusiasm and love for the subject. It is this enthusiasm and love in a teacher that triggers the student to become an integral and active part of the process, not just a passive receiver of knowledge.
I certainly agree with your last point — that it’s much easier to get excited about a subject when you have a teacher who is not afraid of that subject. Which might be the focus of Enzensberger’s book. I look forward to reading it!
You are all those things and more, Ken. But in a //good// way. 🙂
The ‘mathematicians’ of which I speak are the ones typically found in math departments (and sometimes in the theory isles of CS departments). These are the ones to which many of us were subjected in university math classes: they believe that knowing a subject automatically qualifies one to teach it and that anyone who struggles with math in ways they did not must be inferior (and therefore not worth their time).
An example: I was at a curriculum meeting where the discussion came around to how hard it is to find a good text for teaching discrete math to CS students. This is one of the courses seen as highly undesirable to teach by most of the faculty in math – which doesn’t bode well to begin with.
So, “It’s nearly impossible to find a good textbook for this course.”, they said.
“Why not create your own notes and exercises?” I said, and silenced the room. They looked at me as if I had just belched, or worse.
“You can’t teach a course without a TEXTBOOK!”, they said, and went back to trying to choose from among several ‘inadequate’ texts. Apparently there is only one way to teach a math course, and it demands the use of a textbook.
Many of these faculty have lost touch with the notion that math can be used in the process of solving other problems, and that the people solving those problems may have no interest in the math itself – just like most people don’t really want to use someone’s software – they just want to get their work done.
I am convinced that there is a reason why so many famous mathematicians did their best work before they turned 25: they had not yet been fully indoctrinated into the ways they were ‘supposed’ to think about math and so they tried unusual things. Many math professors are so thoroughly encultured to the Way that the only people they can teach effectively are people like themselves. There are exceptions of course, but the same is true in many disciplines.
p.s. Enzenberger was not trained as a mathematician. I read it to my son when he was in grade 4. He loved it. He hated grade 4.
Ah, ok, I see your point now.
I gave up using textbooks years ago — it always seems more productive to recreate the material myself each semester, so that it stays fresh and naturally evolves as student interest, appropriate platforms and key applications gradually change from year to year.
It turns out not to be so difficult to create your own course materials, if you’re teaching a subject you know and love. And if you’re not teaching a subject you know and love, I think that’s a bad scene all around.
Also, consistent with what you’re saying, I’m not sure I would know how teach without the context of driving applications — framing the math in the context of something the students already know they are enthusiastic to learn. That usefulness is actually a big part of what draws most learners to the math.
I totally agree.
I think though, that creating one’s own course materials is only not difficult if, like you say, you’re teaching something you both know and love, but ALSO: you need to be able to teach. Not everyone can. It’s a talent, just like being able to draw or play an instrument. Some people can learn (or be taught) to become virtuosos, and others, not so much. That’s even true of people in Education faculties – knowing //about// teaching does not necessarily make you a good teacher.
I think it also takes a good teacher to come up with good questions.