The Reflective Educator

Education ∪ Math ∪ Technology

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Day: April 25, 2012

First the work, and then the theory

I love this quote shared by Gary Stager via the Daily Papert.

“They first learn engineering, then from there they progress to learning the ideas behind it, and then they learn the mathematics. This would be inventing, it’s a little probe toward inventing a different kind of content. It’s not a different way of teaching; it’s not pedagogy. It’s different knowledge. It’s a good example of turning knowledge – turning learning – upside-down. Instead of starting with this abstract stuff we had from the nineteenth century, let’s start with stuff that’s really engaging for the children, out of which the deeper ideas can develop.” Seymour Papert, 2004

Instead of students learning a bunch of theory, and then being able to apply it to practice, they would engage in building and in creating, which would provide a need and a motivation for the knowledge behind the thing that they are building. One thing that is missing in great degrees from our school is motivation. Why do I need to learn this math when I can see no use for it?

It’s not to say that there aren’t fascinating ideas which are worth learning without something practical to support them, but this is not the general tendency of school. We do not usually teach (with obvious exceptions being good teachers) that which is interesting. We teach practical knowledge. We tend to say, "Oh, we’d like kids to be able to be doctors. Well, what do they need to know? Let’s teach them that." with the result that students do not see the connection between what we teach, and what they can do with it.

Learning mathematical ideas through literature



Anno’s Mysterious Multiplying Jar



The Number Devil



The Fly on the Ceiling



A remainder of one



Sir Cumference and the Great Knight of Angleland



One Hundred Angry Ants



Among the Odds and Evens



One Grain of Rice



Two of Everything


The Phantom Tollbooth





On incentives in the teaching profession

So I read some interesting research on student incentive progams which has a couple of very important paragraphs. Here’s the abstract:

This paper describes a series of school-based randomized trials in over 250 urban schools designed to test the impact of ﬁnancial incentives on student achievement. In stark contrast to simple economic models, our results suggest that student incentives increase achievement when the rewards are given for inputs to the educational production function, but incentives tied to output are not eﬀective. Relative to popular education reforms of the past few decades, student incentives based on inputs produce similar gains in achievement at lower costs. Qualitative data suggest that incentives for inputs may be more eﬀective because students do not know the educational production function, and thus have little clue how to turn their excitement about rewards into achievement. Several other models, including lack of self-control, complementary inputs in production, or the unpredictability of outputs, are also consistent with the experimental data.

Notice the sentence in bold. Incentives for outputs (like let’s say student test scores) does not improve performance. Incentive for inputs (like increased collaboration, more training, working in high needs schools) does. Of course, the benefits gained by the incentives are not terribly strong (with the exception in the study of students being paid to read more) and so any benefit from paying teachers for changing the inputs to education may be minimal.

The merit pay for teachers movement has it all wrong, and the very people in promoting merit pay for teachers have access to this research. In fact, the list of acknowledgements at the beginning of the paper is a veritable who’s who of the leading education reformers in the US.

Converting degrees to radians

One of my students came up (with some help) this procedure for converting between degrees and radians.

  1. Memorize the fact that 60° is π/3 and that 30° is π/6.
  2. Note that 10° is therefore π/18 and that similarly 1° is π/180.
  3. You can then take any degree measure and convert it by converting the number of degrees into sums of degrees where you know the conversions. For example, 70° is equal to 60° + 10° = π/3 + π/18 = 6π/18 + π/18 = 7π/18.

Obviously this procedure is not by any means the most efficient way to convert between radians and degrees. Although I showed a much more efficient algorithm for converting between degrees and radians, it didn’t make sense for this student, and so he and I came up with this procedure (which I drew out of him by asking him questions about the angles), which he does understand.

In general, I’d prefer students use inefficient techniques that they understand completely than highly efficient techniques that they do not understand. Hopefully this student will continue to work on his procedure to make it more efficient as he has to use it over and over again, but if not, at least he will be thinking with something that makes more sense in his head.