teaching machines

Why 12?

We looked previously at how an octave—or doubling, as we called it—is partitioned non-linearly into intermediate tones. But we didn’t ultimately decide how many intermediate tones there should be. The pioneers of Western music converged on a 12-partition. But why not 8? Or 10? Or 11? Or 13? Ultimately, we want our instruments to sound […]


A year ago some students and I started designing programming languages for generating 2D vector art. I called my language Twoville, which pays homage to Seymour Papert’s metaphor of learning math in Mathland just as we learn French in France. The language is a place to learn about and forge 2D shapes. After the semester […]

Five through Fifteen

On our walks home from school, my 8-year-old son has been telling me of a math problem that he has not been able to solve. Yesterday, after a week of not finding a solution, he wrote down the problem on paper for us to investigate together. Here is the task in my own words: Distribute […]

Twelve Steps

Music theory is a daunting subject, built out of centuries of mathematical analysis and aristocratic pride. Sometimes I think I’d be better off reinventing it than trying to learn it at this late stage of its maturity. Here’s how I might get started. One day I’d be absently fooling around with a bit of wire, […]

Parametric Puzzle

Check out these these parametric equations: $$\begin{array}{rll}x &=& \cos v \cdot \cos u \\y &=& \sin v \\z &=& \cos v \cdot \sin u\end{array}$$ Do you know what they do? They are most assuredly not magic. Here, let’s rename the variables, and you can try again: $$\begin{array}{rll}x &=& \cos \textit{latitude} \cdot \cos \textit{longitude} \\y &=& […]

Intervals in Deltaphone

For a few years now, schools and industry have been telling kids that they can code. They say the jobs are plentiful, and the salary is enviable—the workforce is waiting. The nobler agents of educational reform will also tell our kids that programming is a creative exercise that will make them better thinkers in other […]

Funeral of Trite Ideas

The art teacher at my son’s elementary school once held a special ceremony called the Funeral of Trite Ideas. My son and his classmates said goodbye to some friends that they had known for years in their drawings: stick figures, flat outlines of cars, suns with smiling faces. I am considering holding my own Funeral […]

Flatcaps in Libigl

Madeup’s dowel solidifier has one job: thicken a sequence of line segments into a solid. But what if the sequence isn’t a polyline, but rather a branching structure like a tree or a fork? One could model each branch as a separate dowel and hope that nobody looks too closely at the joints, but that’s […]

Debugging Secret Coders

My sons have thoroughly enjoyed Gene Luen Yang’s six Secret Coders books. They are graphic novels about three teens who fight evil with programming, and their weapon of choice is the Logo programming language. Last summer we were reading the fifth book in the series. It ends with this challenge to complete a code snippet […]

Prior Art

A year ago I made a little widget for visualizing the additive color mixing of RGB triplets. I had been teaching a course on front-end web development, and I wanted a tool to illustrate mixing that was easier to access than the traditional many-chambered Venn diagram. Just this week I discovered that my widget is […]

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