Tag Archives: pedagogy

David Mumford says we should replace plane geometry with programming and I’m not sure he’s wrong

MAA Mathfest is in Madison this week — lots of interesting talks about mathematics, lots of interesting talks about mathematics education.  Yesterday morning, David Mumford gave a plenary lecture in which he decried the lack of engagement between pure and applied mathematics — lots of nodding heads — and then suggested that two-column proofs in plane geometry should be replaced with basic programming — lots of muttering and concerned looks.

But there’s something to what he’s saying.  The task of writing a good two-column proof has a lot in common with the task of writing a correct program.  Both ask you to construct a sequence that accomplishes a complicated goal from a relatively small set of simple elements.  Both have the useful feature that there’s not a unique “correct answer” — there are different proofs of the same proposition, and different programs delivering the same output.  Both quite difficult for novices and both are difficult to teach.  Both have the “weakest link” feature:  one wrong step means the whole thing is wrong.

Most importantly:  both provide the training in formalization of mental process that we mathematicians mostly consider a non-negotiable part of general education.

But teaching coding instead of two-column proofs has certain advantages.  I am not, in general, of the opinion that everything in school has to lead to employable skills.  But I can’t deny that “can’t write five lines of code” closes more doors to a kid than “can’t write or identify a correct proof.”  People say that really understanding what it means to prove a theorem helps you assess people’s deductive reasoning in domains outside mathematics, and I think that’s true; but really understanding what it means to write a five-line program helps you understand and construct deterministic processes in domains outside a terminal window, and that’s surely just as important!

Computer programs are easier to check, for the teacher and more importantly the student — you can tell whether the program is correct by running it, which means that the student can iterate the try-check-fail-try-again process many times without the need for intervention.

And then there’s this:  a computer program does something.  When you ask a kid to prove that a right triangle is similar to the triangle cut off by an altitude to the hypotenuse, she may well say “but that’s obvious, I can just see that it’s true.”  And she’s not exactly wrong!  “I know you know this, but you don’t really know this, despite the fact that it’s completely clear” is a hard sell, it devalues the geometric intuition we should be working to encourage.







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Split-screen blackboard

From Andrew Gelman, an interesting pedagogical suggestion:

The split screen. One of the instructors was using the board in a clean and organized way, and this got me thinking of a new idea (not really new, but new to me) of using the blackboard as a split screen. Divide the board in half with a vertical line. 2 sticks of chalk: the instructor works on the left side of the board, the student on the right. On the top of each half of the split screen is a problem to work out. The two problems are similar but not identical. The instructor works out the solution on the left side while the student uses this as a template to solve the problem on the right.

Has anyone tried anything like this?  It sounds rather elegant to me.


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March math link dump

  • If blood found at a crime scene contains a series of genetic markers found in about 1 in a million people, and if you search a database of genetic material from 300,000 people and find just one match, person X, for the blood at the scene, what is the probability that person X is innocent of the crime?  If you said “1 in a million” you might be a prosecutor.  If you said “1 in a million, and I’m barring any expert testimony that says otherwise” you might be a judge.
  • Good article in the New York Times about the challenge of teaching teachers to teach.  Deborah Ball of Michigan talks about what math teachers need:
  • Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less. This was neither pure content knowledge nor what educators call pedagogical knowledge, a set of facts independent of subject matter, like Lemov’s techniques. It was a different animal altogether. Ball named it Mathematical Knowledge for Teaching, or M.K.T. She theorized that it included everything from the “common” math understood by most adults to math that only teachers need to know, like which visual tools to use to represent fractions (sticks? blocks? a picture of a pizza?) or a sense of the everyday errors students tend to make when they start learning about negative numbers. At the heart of M.K.T., she thought, was an ability to step outside of your own head. “Teaching depends on what other people think,” Ball told me, “not what you think.”

    The idea that just knowing math was not enough to teach it seemed legitimate, but Ball wanted to test her theory. Working with Hill, the Harvard professor, and another colleague, she developed a multiple-choice test for teachers. The test included questions about common math, like whether zero is odd or even (it’s even), as well as questions evaluating the part of M.K.T. that is special to teachers. Hill then cross-referenced teachers’ results with their students’ test scores. The results were impressive: students whose teacher got an above-average M.K.T. score learned about three more weeks of material over the course of a year than those whose teacher had an average score, a boost equivalent to that of coming from a middle-class family rather than a working-class one. The finding is especially powerful given how few properties of teachers can be shown to directly affect student learning. Looking at data from New York City teachers in 2006 and 2007, a team of economists found many factors that did not predict whether their students learned successfully. One of two that were more promising: the teacher’s score on the M.K.T. test, which they took as part of a survey compiled for the study. (Another, slightly less powerful factor was the selectivity of the college a teacher attended as an undergraduate.)

    Ball also administered a similar test to a group of mathematicians, 60 percent of whom bombed on the same few key questions.

  • Thurston teams up with the House of Miyake for a Paris runway show loosely based on the fundamental 3-manifold geometries.  Thurston talks fashion:
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