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ray tracing using POV-ray

The Calculator of Tomorrow

The “of tomorrow” clause is a meme in itself. You might remember the Experimental Prototype Community of Tomorrow (EPCOT). EPCOT was later changed to Epcot so it wouldn’t mean anything, or so goes the story.

The House of Tomorrow is a movie, based on a book, about a youngster being groomed to save the world based on what some would call the “retro futurism” of Buckminster Fuller, who coined the term Spaceship Earth. That giant buckyball at Epcot is named Spaceship Earth, which seems fitting.

You’ll find, if you dig, a whole genre of retro futurism, which makes sense, as the year 2000 was pegged long before, in anticipation, as a kind of buoy or guidepost.

We would know, by the year 2000, what the future would hold. Would we be commuting to work wearing jet packs? Quaker Oats spoofed this vision in one of its breakfast cereal commercials. How about flying cars? What would the houses be like?

How smart would we be? Would nuclear energy be our primary energy source? Such questions haunted the 20th Century. I’ll get back to this thread.

The Calculator of Tomorrow plays off this “of tomorrow” meme, partly by accepting the calculator as already somewhat retro in flavor. The smartphone has calculator apps, however people still buy real calculators, for school.

The high school mathematics curriculum sometimes reluctantly allows calculators at least in higher grades, when we’re sure they’re not being used as a crutch, or something to that effect.

However, if you get to the college level and want to keep learning math, they want you to use a computer, especially for linear algebra which is all about multiplying matrices, finding determinants and so on.

The manual procedures are long and tedious, whereas entering a matrix into a calculator (if that’s even possible on a given model) is likewise rather impractical. Why use a cramped little calculator when the computer gives you Jupyter Notebooks and entire languages devoted to “operational mathematics”?

By “operational mathematics” I mean executable mathematics, the stuff we program and then “run” (often by hitting a “run button”). Printed textbook mathematics sits inertly on the page, suggesting a mental process (such as reading itself), should one wish to actually operate with the formula or equation in question.

In that case, should one wish to make the textbook formulae operational, one might well end up running some programs, especially if matrices are involved. Mathematica comes to mind, and Wolfram Language.

Speaking more autobiographically, I once worked for what had been a calculus teaching company that adopted Mathematica as its primary learning tool. That made sense, however new workflows were needed to couple students with their teachers. The company had solved the problem in software.

If you’re working at a desktop with a large screen (not a smartphone), then chances are you’re not crowded into a room with a lot of others doing the same thing.

You’re more likely in a cubicle or office, perhaps in a home office, perhaps a dorm (a home office with a bed). Your teacher is not there with you. Yet you need to turn work in.

These workflows had been added by the company (based in Champaign-Urbana) and could work for distance education more generally.

The company was purchased by a bigger one (based in Sebastopol) and repurposed to teaching coding skills more generally. I was their first Python teacher.

teaching Python (or any OO language)

A core challenge of learning STEM skills in the 21st Century is we need room to spread out. We need “maker spaces” with equipment. We still need chemistry and physics labs too. A lot of schools, including in the “richest nation on earth” have no budget for such equipment.

The charter school movement had a lot to do with families hoping private sector public schools (if that makes any sense) might fill the void. A political fight developed, over school inequality. An original ideal of public schools was these institutions would provide equal opportunity for all. Everyone would get quality time in the chemistry lab, in the gym.

The cramped circumstances of the contemporary classroom make “computer labs” somewhat unwieldy. We do have them, in many schools, with full desktop computers at each station.

However the mathematics teachers have not wanted “technology in the classroom” to mean something that big and bulky. Laptops have been the compromise. The schools buy these in bulk, complete with lockable charging cabinets. Chromebooks have flooded into many schools.

A textbook (possibly thick and heavy) and the scientific calculator, pens and notebooks, still comprise the necessary “props” (as in “theater props”) for most high school math. The “classroom of tomorrow” is much like the “classroom of yesterday” when it comes to mathematics.

Lets get back to Buckminster Fuller, Epcot, and The House of Tomorrow. The house itself (in the book and movie) is a geodesic dome. The EPCOT buckyball is a full sphere. The mathematics of geodesic domes and spheres is what Bucky Fuller would tour around and teach about, as a university faculty member. His students would typically have some kind of dome constructed, before the end of the course.

What fewer people know about Fuller was he concerned himself with ordinary polyhedrons, those shapes we call tetrahedron, cube, icosahedron and dodecahedron. These shapes have lived at the core of mathematics, architecture, chemistry, and of course philosophy, for a long time. We may learn about the Five Platonics as early as in elementary school. We might also learn of the Archimedean polyhedrons, or “Archimedeans” for short.

Constructing polyhedrons on the computer screen takes vectors and vector math, meaning linear algebra. Rotating polyhedrons around various axes, operationally, takes rotation matrices.

We’ve almost left the set of topics comfortably accommodated by the scientific calculator. As a consequence of the current economics, high school mathematics cannot afford to do too much with polyhedrons.

Computing their relative volumes and surface areas is still within reach, and has connections with calculus, especially if the surfaces are curved.

However, Buckminster Fuller did not leave the current pedagogy as he found it. He decided not to measure all polyhedron volumes relative to some unit cube’s volume (“cubic inch” “cubic meter”), but relative to a unit tetrahedron instead.

The tetrahedron has fewer faces and edges, topologically speaking (V + F = E + 2) and the triangle is architecturally more stable than a square, ditto the tetrahedron versus the cube.

Fuller came up with a new volumes table for polyhedrons, based around a unit volume tetrahedron, and a conversion constant for going back and forth between his table and the older, more conventional table — history few people know about because they’ve never heard of Synergetics, his two volume philosophy, mostly not shared in high schools (or in universities for that matter).

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concentric hierarchy of polyhedrons
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from the volumes table

As I hinted above, the Calculator of Tomorrow is simply a laptop computer (could be a Chromebook), or maybe a desktop computer.

We need to be able to visualize our polyhedrons, as we explore the volumes tables, talk about vectors, learn delta and lambda calculus and so on.

Does that mean public schooling means working from home offices or home studios a lot more? What happens to “equality of opportunity” when some students have the calculators of tomorrow and others do not?

Stay tuned. Watch my Youtube channel.

Lots online.

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