A Digital Mathematics Curriculum

I’m assuming a world in which curricula are as shells on the beach, if not grains of sand: frequent in the sense of plentiful, to be expected, statistically speaking, not a surprise. That goes for shells in general, but then far less frequently, a shell turns up that’s especially unique.

This curriculum is like one of those rather different shells.

Take the Martian component, for example. The point there is to encourage futurism and a willingness to tackle big picture challenges.

Don’t just throw up your hands and leave it all to elected officials to figure everything out. They’ll tell voters what voters want to hear. They have to stay popular, which takes work. So when does the planning happen?

“Plan for your planet’s future” advise the Martians, “or it might end up like ours.”

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a comic book version

Sometimes we’re looking forward, preparing for the time to come. We look backward too though, which is where the Neolithic component fits in, its lunar and solar calendars, the zodiac.

Also: navigation, timing migrations to match weather patterns, learning when to plant, how to store and distribute. Bookkeeping is involved. Borrowing and paying back. When were navigators finally able to gauge longitude, not just latitude?

I’m starting to describe Supermarket Math (storing and distributing goods and services), which is likewise “through the ages” in the sense that we use math in any age, as best we can, to embed our biospheric economic circuits within Motherboard Earth’s and elsewhere.

Last but not least (one might pick another order), we learn about risks, taking them, perhaps avoiding them when unnecessary, by modeling reality sufficiently precisely — or not. How strongly do we believe in our models? We’re sometimes quite tentative, for good reasons.

Pay attention to dangers, while realizing that just sitting around overthinking is potentially dangerous as well. That’s Casino Math. Games of chance, but wherein the exercise of skills makes a difference.

Those are the four arms of this Digital Mathematics curriculum: Martian, Neolithic, Supermarket, Casino.

I wrote all this up for Wikieducator awhile back, casting it out to the world as a Made in Cascadia product.

Past to Future is Neolithic to Martian, whereas at every time slice, in every age, we’re taking risks and getting along with our business, the Casino to Supermarket axis.

What I’m suggesting to teachers is to adopt these four directions, without feeling obligated to use the same labels.

Look forward, look back, and look side to side (to the wings), where you learn from your past and invest in your future.

There’s no time like the present for getting work done, where work might be daydreaming or study — or sleep, highly important.

The specifically digital piece has to do with the focus on discrete computations, with or without any assistance from machines. Some computer use is presumed. Coding languages get some focus, including what they do with electrons inside those microchips.

Is an algorithm a kind of machine? Following rules, with or without an abacus, will get us to the same results.

In my own case, I’m influenced by Wittgenstein’s linking mathematics to grammar. Working out some problem using language, expands to tool using more generally.

Pascal’s Triangle and Tetrahedron, the Binomial Distribution, Bell Curve, the shoptalks of statistics and machine learning, dominate in Casino World.

The stochastic techniques applied to thermodynamics started in sociological literature, not the other way around. People have always studied people.

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Marquee

Supermarket Math gets into rows and columns as data structures, then ups the dimensions, the number of axes we might need. The database and spreadsheet technologies have their place here. Graph theory does too, in the sense of networks, supply chains, critical path diagrams.

Websites, eCommerce, SQL and NoSQL… all part of Supermarket Math.

I’ve labeled my future “Martian” to leverage a specific narrative. Mathematics needs myths and stories to weave it together.

History, with timelines, serves that role marvelously until we start imagining the as yet indeterminate future. At that point, science fiction kicks in.

In my specific narrative, the Martians have come to Earth. We may call them ETs (extraterrestrials). The point is to have them use a somewhat different mathematics than we Earthlings are accustomed to using, and by this means branch to a geometry not many schools yet teach, but which is rich in polyhedrons and friendly to geometric metaphors, for just about anything.

Earthlings and Martians have agreed to collaborate on building a hydro-powered dam. The framework here is the Earth Energy Budget, with the fusion-powered oceans evaporation cycle driving the rain and rivers cycle, which makes our “water wheels” ever turn.

An electricity generator is one such type of water wheel and having it in the picture is our segue to lots of physics, starting with electrons and the difference between AC and DC current. We may study electrical networks and circuits quite a bit.

The Martians have a differently shaped unit of volume then we Earthlings do. Theirs is the regular tetrahedron, a different Platonic from our cube. Ambassadors from both sides had to come up with a conversion constant, and they did so, by golly, by using a canonical sphere both teams could accept, of radius R and diameter D.

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Canonical Sphere: Hand Drawing

The Martian tetrahedron-shaped unit volume T has edges D, six of them, while the Earthling hexahedron unit C has edges R, numbering twelve. C > T in terms of volume.

Picture a one-page triangular book, open flat, its front and back covers flat against the table at 180 degrees to one another. The single triangular page, hinged where the covers meet, along the spine, travels back and forth, its free tip tracing out a semi-circle.

The book cover edges, and page edges, are all D, the same D as above.

When the page forms a regular tetrahedron with either cover, we call that the Martian unit of volume.

When the page is vertical, its tip at maximum height above the table, that volume will equal the cube’s of edges R. You might need some trigonometry to prove that.

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TetraBook (paused Youtube)

Those of you schooled in 1900s New England Transcendentalism might recognize where I’m going with this. I’m interested in bridging to the so-called Concentric Hierarchy, the “Sesame Street” (core neighborhood or shoptalk) in the mytho-poetic writings of one Buckminster Fuller, RBF for short.

RBF uses a different model of 3rd powering to demonstrate the effects of scale. Double linear dimensions, and the tetrahedron’s surface area grows by a factor of four, its volume by a factor of eight. We learn from the Santa Fe Institute about the role of fractional powers, powers other than one, two or three, in expressions for natural “laws”.

Going in the direction of the past, rewinding, we watch world maps continuing to morph. We wonder what was burned in the Library of Alexandria. Cartography comes under the heading of world data displays, visualizations. Maps are about more than getting from point A to point B. We need them to study the state of our planet, in some previous time, now, in the future. We also invent planets, the better to think about ours. We think about Mars.

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thinking ahead

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