Of course “curriculum matters” may be meant in two ways: it matters, or I’m about to discuss matters pertaining to curriculum. Both meanings obtain.
My focus has been high school and below, not above, but when planning K-12, one needs overview of K-16 at least. I say “overview” because Merlin learns it backwards, with greater generality the younger he gets. Those who actually teach in upper levels may not be facing beginners. They’re often more interested in impressing their peers, usually by ascending to higher levels yet.
You may not know this myth of a wizard, at one point tutor to King Arthur. The point being, we get more specialized with age, whereas we’re born generalists, but without the experience. Merlin, in living backwards, remembers his previous lives. Something like that.
Conway’s Law says organizations reflect their own inner architecture when providing APIs (dashboards, control surfaces) to the public. Some organizations appear more chaotic, perhaps leaderless, although these are not necessarily features with a simple linear relationship.
Inscrutable high entropy black boxes may nevertheless beat us at Go, meaning they exhibit goal directed behavior even if we can’t figure out what their internals might be. Chaotic means we can’t crack the code, not that there’s no code to crack.
These days, a lot of companies have reimagined themselves as driven by big data to optimize profitability in a longer term sense, meaning there’s obvious dedication to empirical measures and feedback loops.
Stakeholders like the idea that management might be in touch with reality.
However — paradox — the optimizing operations may themselves be a source of black boxes, meaning some inscrutable product of machine learning algorithms gets to play a flagship role. Company profitability gets pinned to a secret sauce, and only some computer knows the recipe.
Is that smart? Do we have a choice? These may be the pressing questions in any given boardroom.
In other words, companies are nowadays finding themselves in a position to turn over some decision-making to AI, without having a way to second guess, only having a way to play the role of supervisor in supervised learning. You may improve the model, but dare you unplug the model? What does mere human intelligence have to offer instead?
Let’s ask about the error function (used to evaluate performance, cost of error, distance from optimal) and the whole process of computing gradient descent.
The client may find a new data point that’s worth its weight in gold, a new measure, a predictor.
A next iteration of the model makes use of this newly discovered measurement to outperform a previous generation’s models.
The vista may still seem Darwinian, and with the added wrinkle of whole new senses developing. In the rear view mirror, we see that as a realized possibility.
What has all this to do with curriculum matters?
The was all by way of distilling to K-12, to high school especially, some clues about where linear algebra might take them (us).
By the way, Elastic Interval Geometry (EIG) may be esoteric, but was a convolution of multiple weighting factors and vertex vector sums, creating a springy type network that’s always seeking to settle down, to minimize distance from some eventual equilibrium. These were animations, generated by EIG engines.
For example, check out Gerald de Jong’s Darwin at Home on Youtube. Visualizing EIG tensegrities might be a way to inuit machine learning?
I’m not suggesting all teachers everywhere need receive these same broadcasts or podcasts or Youtubes, when it comes to nudgings such as mine.
Clearly I’m not providing enough context to expect to recruit many followers or disciples. This might as well be a memorandum within some small faculty or choir, in terms of an audience ready to consider such suggestions, and even speed their adoption.
I’m geographically situated, like everyone else in this world. My circumstances affect my mode of communication. For example, I’ve already “come out” at the Linus Pauling House as some sort of Portland weirdo, at the intersection of several disciplines and even religions. They (the Wanderers) already know about how I’m always yakking about some “macroscope” or “geoscope” with Glenn Stockton, who speaks of his “global matrix”.
In our zip code, I don’t stand out that much, for all my eccentricities, and my interface to local schools is based on years of being a parent and participating in the public process. I learned a lot from observing the debate team culture, and not just around Portland.
My mode of relating to neighbors, coworkers, through meetups, discussion lists, tweets, social media or whatever, is peculiar to me, special case. We each have our fingerprints, our patterns of relating. There’s no way I’ll be standing behind a podium in your meeting hall or grange. My ads will not feature on TV. I used to scribble notes to E.J. Applewhite from my “Urnerbank of Bhutan” but that was just me being literary, taking advantage of my perch in Thimphu.
People finding this writing in Medium are likely to have seen other such writings of mine, or will seek them out. Par for the course, safe predictions.
“Gradient descent” is all about exercising those calculus concepts, such as when we seek local minima, perhaps roots, as we do with Newton’s Method. The matrices have gotten a lot bigger, so we let the computers do the heavy lifting. Back propagation is a process of fine tuning. The neural network learns to classify correctly. That’s one way among many that machines learn.
High school already embraces linear equations, including solving simultaneous ones, the beginnings of inverting a matrix.
I think we’re maybe intimidated by these bigger matrix operations, even though we have computers to do them, given a legacy of seeking to geometrize n-D arrays in terms of mutual orthogonals or some such inscrutable polytopes, whereas the idea of dot products, with relative weights, is intuitive without the hypercross.
We’re not even promising the features are all linearly independent. We may not have time to cross-check all possible correlations, let alone explain the ones that we find. Neural networks don’t begin with any guarantee of key features being strictly orthogonal. Two ears go with one nose, more often than not. Not a problem.
Statistics, nowadays data science, has a multi-lane highway to and from high school linear algebra, which doesn’t mean to not use them for rotation matrices in ordinary XYZ space (and/or IVM space in my world). Convolution is everywhere.
Go ahead with all the Euclidean constructions, and then some, just don’t insist on an approach to linear algebra heavy on vector spaces yet devoid of any mention of their role within neural networks and machine learning. Take advantage of where the adult world has been going. The dragon’s tail needs to follow the dragon’s head. Easier said than done.
We know from other teachers on Youtube that these data science topics reduce to accessible animations and substantive demonstrations. We don’t need them to “wait until college” before we supply all manner of context, in the sense of relevance. Non-vacuous mathematical workflows: the job of faculty is to provide these for students, and nurture their growth.
The justification for spinning maths in this direction (using the British plural for good reason), is not that schools are creating an army for just one profession. We don’t all intend to become data scientists at the end of the day, unless that simply means in touch with the information in experience.
The intelligent layman (layperson, ley person, public road user), or in the US, president, as in chief diplomat (lots of people skills), is what a public and democratic school system seeks to groom.
Private schools, or systems with a different political basis, may well shape a different type of graduate, nor would I argue that all who are diplomatic (skilled with people) are carbon copies of one another. They come in many complementary forms. Diversity, not uniformity, not conformity, is most consistent with the push and pull of a feedback-driven democracy, over the long haul. Is that a theorem? Monocultures lack adaptability.
A generalist layman with sufficient overview to participate intelligently in a democracy, needs a sense of how machine learning works as an antidote to superstition, as a door-opening possibility, and as a narrative any adult might share by way of explaining to others how an apparent intelligence may nevertheless be artificial, a result of silicon-based computations and not necessarily much (if any) conscious human connivance.
We need some ability to make these distinctions in order to be intelligent with our just allotment of paranoia, investing energy in our defenses where it most belongs. Some HBO viewers may think Westworld is just around the corner. Science fiction often comes with the feature of being highly believable.
My other big push of mine — outside the A, B, S, T & E modules that is (“building blocks” used in geometry puzzles) — is towards phasing in more cryptography.
Any high schooler should have teachers standing by, prepared to explain RSA, Diffie-Hellman, AES. Start with HTTPS in the web browser, if the internet is available.
At the time of this writing, chatter about cryptocurrencies is flooding in from every corner, attracting many schemers and dreamers.
Without trying to moralize or pass judgement on this whole idea — a phenomenon of the natural world at this point — teachers should be empowered to dissect the concepts. Do not deny high school teachers time to motivate discussion of blockchain concepts.
I push the “modules thing” (tetrahedron-shaped voxels) because we need examples of different approaches to perennial topics and shifting in and out of “tetravolumes” is a way to bring in some worthwhile history.
What was the “geodesic dome” and how was it intended to help humanity?
That’s not just some stupid politically motivated question. We’re opening a window into spherical geometry, carbon chemistry, architecture, urban and regional planning, even cartography.
Diving into these topics with a plunge into an exotic literature helps level the playing field and excite curiosity.
Said literature points us back to two generations ago, or more. Grandparents help close the circuit.
When humans started living a lot longer, say with agriculture, more generations started to overlap.
Or maybe agriculture made some of us weaker in some dimensions, that’s one more worthwhile investigation.
The point is: whenever we started getting lots of elders, available to look after and educate children, while parents pursued their jobs, the vector of civilization started really taking off. Less fell through the cracks. More knowledge got passed on. That’s one theory anyway.
Anthropology makes STEM into STEAM.