If you’ve been around the Math Wars as long as I’ve been, going back to New Math post Sputnik, you might find yourself nodding along with some of my account.
Regarding New Math: my mother freaked out over how “owly” all this newfangled stuff was, and what might it be doing to her kid? Why were we all of a sudden learning about “number bases” for example, with base ten only one of many? This requirement we make room for other bases seemed suspect. Where were they going with all that? How would knowing about base two ever help in the job market? Many parents expressed skepticism.
We may forgive my mom some of her profound distrust of the establishment, which, in its infinite wisdom, had already spewed highly radioactive super-toxins into the pristine Pacific Northwest environment: Strontium-90 was in the milk.
If that’s what was in the milk, what might be lurking in the math? Were number bases the equivalent of Strontium-90?
Tom Lehrer wrote his mocking New Math song for the album That Was the Year That Was, so parents could realize they were all in the same boat, and laugh about it. The movie Incredibles 2 (2018) revisited those days of yore, with the superhero American dad feeling stumped by junior’s homework.
Those were also the days before “environmentalism” became a thing, with president Herbert Walker Bush only much later saying that he was one. By then, those concerned about the environment had grown and a president could run on those issues. Climate Change, on the other hand, was still waiting in the wings, well ahead of Peak Oil.
I remember both the New Math, otherwise known as SMSG by insiders, and the New New Math of decades later, branded “fuzzy math” by its detractors. Then came No Child Left Behind, then Common Core. The buzzwords keep coming and going. I came up with “GNU Math” as a pun on the GNU project (which encourages cleverness) and as homage to base-2 friendly New Math. In hindsight, it pays to think in binary, octal and hexadecimal. I’m glad for our time with truth table based Boolean logic, and all the set stuff.
However, the swirling controversies went so much deeper.
Some would harp on introducing Algebra even earlier, at a more tender age, claiming Cuisenaire Rods could work miracles. Caleb Gattegno helped anchor this Algebra First movement, but then he was a UK more than a US phenomenon.
Others championed Geometry First (I was in this camp) with an emphasis on figurate, and polyhedral numbers (1, 12, 42, 92…). Coordinate systems could come later.
The Platonics would come right away, with the concepts of vertex, face and edge, angle and frequency, transformations, such as taking the dual. Rotation. Translation. Scaling. Power Laws. Double the edge lengths and find volume increases eight fold, area four fold. What power laws do we find in nature besides these?
A somewhat parallel debate was meanwhile going on in Computer Science: how early do we introduced the mechanism of classes, in the Object Oriented sense?
In the years following Smalltalk, many languages would seek to evolve object-oriented versions of themselves. Most notably, the C language became C++.
The common wisdom was to introduce the procedural paradigm first, laying a foundation, and then switch to object oriented. “Ontogeny recapitulates Phylogeny” was the summarizing encapsulation of this approach.
On the other hand, why should newcomers to telephony go through the age of landlines first, before moving to cell phones. Why not start with the state of the art, if it’s at all intuitive.
My camp was arguing for “classes first” or at least for “types early”, meaning we did not want to make the object oriented stuff seem advanced and “back of the book” i.e. the kind of stuff most teachers would never get to.
My entry point to “classes first” (or “types early”) has been through the quasi-ubiquitous “dot notation” whereby the “dot” — literally the period on your keyboard — becomes our “containment” operator.
The Russian Doll contains a smaller Russian Doll, which in turn contains a smaller Russian Doll and so on.
The point is not to invoke the Specter of Infinite Recursion (SIR) so much as to suggest the subject matter of mereology: the logic of what’s inside of what.
A big debate on which I appear to have been on the losing side, was whether mathematics, as taught in the early grades, pre-college, might come to absorb dot notation with this particular meaning. I hoped it would.
The grammar in question is subject.behavior() and subject.status, allowing us to actively trigger events, dial in settings, while also passively reading the control panel, garnering feedback.
Dot notation might take its place along with subscripting and/or using square brackets, to supply some object with numerical indexing (most typically). A, A, A… an old standby, now to be augmented by A.verb().
What happened instead is the mathematics curriculum publishers circled the wagons, saying this material was not in their database and not in their job description to provide.
Having worked in the textbook industry, I understood that profitability came from updating the look and feel, while counting on the content to not need a deep overhaul.
Those like myself, saying we would soon need “dot notation” to infuse our math textbooks were not popular, though some maths did dabble in both BASIC and Logo.
What happened for the most part is administrators decided to mirror the four year college and make computer science remain its own thing. Learning to Code might have applications in mathematics, but imbuing mathematics with coding languages would represent too much of a departure.
The scientific calculator was the device most people expected would be required in a math class.
The Chromebooks were earmarked for other coursework.
My camp continued to register skepticism over this conclusion, noting how code schools were filling with refugees from higher tuition alternatives, roughly for the purpose of fleshing out their “three Rs”: reading, writing and arithmetic.
Dot notation gets to be there, almost because computers are ubiquitous and computer literacy is now just literacy.
Having given up on Mathematics, could we turn in any other direction?
Spatial geometry comes alive when computers are permitted.
Calculators don’t render Platonics, but laptops do.
HTML amounts to punctuation, an added grammar. Might the Communications Arts emphasize developing fluency not only in human languages, but in computer languages as well?
CSS includes knowing about color theory and design principles.
Programming moves us towards theater, with active agents cast in scripts.
That’s right: if a school wanted to bring in more computer programming, but didn’t feel like taking on computer science in quite the mainstream way, it could choose to re-galvanize the humanities curriculum instead, through visual arts, music making, crafting sets (with device controllers) and theater (making movies).
Polyhedrons crop up in literature all over in art, including in the works of Leonardo da Vinci, Albrecht Dürer and M.C. Escher, and indeed throughout the history of ideas.
For spatial geometry, we might draw upon New England Transcendentalism, updated with the latest information, about tensegrity and flextegrity, about geodesic domes, the macroscope (geoscope), World Game and so on.
What was Alexander Graham Bell up to with those “kites” anyway? What would attract crystollographers to ask this question?
Lets study dystopian versus utopian science fiction as a genre and cultural mirror. We might carve a path straight from futurism and whole earth modeling, to machine learning, through programming, without needing to wait for STEM disciplines to do the trailblazing. PATH might take the lead.