What we learn in school

Link: What we learn in school

I recently took on a remarkably talented tutee. Seeing the natural ease with which he, at 12 years old, sees deeply and sharply into all kinds of mathematical problems is a pleasure. For me, making sure this kind of student gets proper guidance is essential. Part of the reason, I’m sure, is that I was accelerated in math when I was younger… and I’m not sure anything in my education gave me any kind of idea what math was about. My first real exposure—and one of the reason I ended up as a mathematician today—came when I went to math camp summer after my ninth grade year. It’s hard to communicate how profoundly my relationship to the subject changed as a result of that program.

But what does school offer? Here’s an article from London that gives a sense of what school math is like for gifted students (and one in particular, named Joe). I’ll post the last half of the article here:

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“In the National Numeracy Project’s Framework for Teaching Mathematics (DfEE, 1999) the suggestion is that there are three ways of making the content of mathematics lessons more challenging:

  • by offering the same content within a shorter time (acceleration)
  • by offering additional content outside the regular curriculum (broadening)
  • or by offering increased cognitive challenge within the same curriculum area (deepening).

I would predict that if you talk to any group of maths teachers in any school you would find that acceleration is the predominant strategy to increase challenge. Broadening, after all, requires additional preparation by hard-pressed teachers so is unlikely to find universal favour. Non-specialist colleagues may lack the confidence (or competence) to ask deeper questions requiring higher-order levels of thinking, so acceleration by default becomes the most common strategy. Certainly for Joe it was the most frequent offer.

Of course, there are advantages for a teacher in accelerating pupils through the curriculum. There is no need to invent activities or acquire different resources as everything is already mapped out for the older pupils. It’s an easy option but can set up possible problems for the subsequent teacher and, as we saw from Joe’s experience, unless carefully planned can lead to repetition or, perhaps even worse, to gaps. For the student it may well offer increased self-esteem but it may also be lonely and seem like a random and disjointed set of experiences.

Thinking like a mathematician
For me, the saddest part of Joe’s story is that he is not inspired to continue his mathematical learning. He has no idea of what it is to behave mathematically because his perception is that maths is the acquisition of a set of skills, a set of exercises to be completed, a speed test. Essential though these skills are, they are just the starting point. I want able students to ‘think like mathematicians’ – to enquire, to generalise, to question and seek proof. I don’t want them to accept everything I offer or be able to regurgitate pre-digested algorithms. I want them to engage with the thinking and find some things difficult. I want them to know that mathematicians see things in lots of different ways: that maths is creative and although there may be accepted elegant methods of solving particular problems or calculations, there are other equally acceptable ways of doing the same. I want them to take time to deepen their understanding rather than skating through the curriculum at high speed.

Writers on the subject (Krutetskii, Kennard) say that these forms of behaviour (generalising, seeking proof and so on) are characteristic of able mathematicians. This is what they do naturally. The teacher’s role is to offer opportunities, activities, investigations for able students to display these characteristics, and to model such behaviour themselves and hence help students to refine their thinking. Spending time deepening understanding in this way is time well spent for able pupils, and can have knock-on effects on the rest of the class who are exposed to behaviour that they might otherwise never experience.

I want able students to think like mathematicians– to enquire, to generalise, to question and seek proof.”

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My new tutee is already accelerated. I’m going to try to offer the depth and breadth that, sadly, he may never get in school.

"I was told there would be no math"

Link: “I was told there would be no math”

When Sudoku came they said it didn’t involve any math, just logical thinking and pattern recognition. (No matter how we mathematicians claimed that logical thinking and pattern recognition are much closer to the heart of mathematics than arithmetic.)

And here comes the cousin of Sudoku: KenKen! Is it possible that everyone will start doing this one too, even though they can’t pretend they’re not doing math in this case?

I guess we’ll find out.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs.

Paul Halmos

Teaching with the Socratic Method

Link: Teaching with the Socratic Method

A professor of mine once said, speaking about his education, “If they had actually taught us anything, we’d all be geniuses!” And indeed, when I talk to people who hate/fear math, a bad teacher is sure to come up in the conversation somewhere.

This conversation with a group of third graders about base 2 is a lovely counterpoint to what many of us experienced in our own educations. Read the question and answer section (at least—the whole thing is good), and see if he doesn’t make you think differently about numbers, and education.

In teaching as in math, there are many options.

Hilbert’s Hotel

Link: Hilbert’s Hotel

I love this Waylay comic about Hilbert’s Hotel. She does a beautiful job of explaining the paradox succinctly, and of tapping into the sense of human (nonmathematician) frustration with the whole deal at the same time. It’s an old favorite of mine.

There’s more on Hilbert’s Hotel at wikipedia. The classic version is: a hotel has an infinite number of rooms, and a person is staying in each room. A new guest arrives and is given a room. How is this possible?

Here’s a harder version: the infinite hotel across the street has a power outage, and sends all their (infinite) guests over to Hilbert’s hotel for the night. How do you give all of them rooms as well?

Sketch of my math dance piece

It’s called “What I was working on today.”

I come out and aggressively challenge the audience:

“I’m Dan, and I’m a math student PhD, and tonight one of you is going to come up here and solve a math problem, because you’re not dumb. You—you look nervous. This is easy—everyone can do this. You—” etc.

Then I step back and relax.

“When I tell people I do math, sometimes I think that’s what they expect me to do. I’ve got news—doing math and being smart have nothing to do with each other. There are a lot of dumb mathematicians in the world and a lot of very smart people who don’t do math at all.

“I’m going to tell you a story… A teacher once gave his class an assignment to keep them busy: add up the numbers 1+2+3+4+5+6+7+8+9 and keep adding until you get to 100. The teacher figured this would be a good way to keep all the kids busy, and the kids set to work, adding 1+2+3+4 and it was a waste of time for all of them. Because they weren’t making their own choices, they were doing it the way the teacher told them. And you know what—it was boring for them, and they all they all got it wrong. But one student made a choice that was his own: instead of adding in the teacher’s order, he thought, what if I add the biggest and smallest numbers together? So he added the 1 to the 100, and got 101, and set it aside. Then he put the 2 and the 99 together and got 101, and set it aside; then he added 3 to 98 and got 101 and put it aside; and added 4 to 97 and got 101 and put it aside; and he realized that if he paired the numbers this way he would always get 101. How many pairs? 100 numbers means 50 pairs, so that’s fifty sets of 101. That’s just a multiplication problem! Fifty times 101. He could do that, and he did: the answer was 5050, and he was the only one to get it right, and in doing so he discovered a beautiful and powerful pattern.

This is what doing math feels like: you make choices and discover beautiful patterns. I can’t give you the full feeling of what it’s like to be a mathematician, but I think it starts with asking a question. Somewhere in there you all have questions about math that haven’t been answered. Today, I want to give you the opportunity to ask them, and if I can, I’ll answer them. My brain is open.”

And then I take questions. After I answer a few, I say “I’m going to show you what I was working on today.” Then I move into a dance where I try to relate physically what my current work looks like and feels like.

Notes from my group: I need to practice more on fielding the questions so I can make ANY question sound really interesting, and take it in cool directions. Second, they liked my dancing, and thought I could expand it from a brief taste to fuller arc.

It’s pretty exciting stuff. Afterwards people came up and talked to me about math in their lives. I need to start getting interviews and lectures on this blog.

A math op-ed, circa 1996

Link: A math op-ed, circa 1996

By Suzanne Sutton. Here’s a quote:

It is among the greatest ironies of education that a subject so graceful and elegant, so able to inspire and bolster confidence, and so useful for living a joyous and effective life, should be presented in a manner that strips it of its substance and glory, and leaves students feeling bludgeoned and inept, convinced they “stink at math”, unaware of its beauty, or their own precious abilities.

Another meeting with my advisor

First of all, the math dance piece went quite well (at least, I had some positive feedback). In part of it I ask for questions from the audience, and I’m not sure I answered them all as well as I could have, but I was able to spiral into and beyond the question “What is the quadratic formula” to explain that evidence of the quadratic formula has been unearthed dating back to 1000 B.C. and beyond, and all of the interesting questions it leads to. More on this particular subject later on.

I met with my advisor today, for the first time in a while. Meeting with him has been uniformly productive of late. He was pleased by my recent little result, and suggested some new avenues to explore. I’m starting to get into a better and better place with my question: I can draw analogies to previous work and have a sense of what should (or might) be true with what I’m working on.

More on this later too.