On Tue, Sep 25, 2012 at 8:06 AM, Paul A. Tanner III <firstname.lastname@example.org>wrote:
> This forum has been challenged: > > > In other words, I hypothesized that, in fact, we know everything > there is to know about teaching mathematics, and that the real debate on > education is about something else.... > > I challenged this forum to prove me wrong by either > > > > (a) pointing to any ongoing discussion on interesting open questions of > pedagogy, or > > > > (b) starting such a discussion, here. > > > > So far, not one person has risen to the challenge. > > > > To meet this challenge: > > How about this one, in the form of a question: Should we teach the adding > or subtracting of fractions based on the least common denominator as an > algorithm that can be stated as a single equation, as we already do with > respect to the teaching of the multiplying or dividing of fractions? > > ------- End of Forwarded Message > >
What's wide open, in terms of math pedagogy, is where if anywhere, some programming should fit in.
Programming doesn't just figure into some math track segments, it occurs in theater. They hand you a "programme" when you (a member of the audience) walk in, and the actors perform according to "a script".
Perl, a scripting language, was invented by a linguist, Larry Wall. The need for information handling in the humanities is at least as great as within some of the STEM projects.
Your school may or may not offer segments that cover:
(a) ASCII then Unicode (b) SQL.
The former story has to do with mapping permutations of binary strings to human language symbols, while the latter is about bookkeeping, tabulation, record keeping, storage and retrieval, filing.
Representation of glyphs, recording of numerals, form a basis for mathematical manipulation of same.
Storage in memory = persistence, the ability to set aside an operation or project and return to it later.
In other words, both (a) and (b) are pivotal stories / activities when it comes to core "how things work" explanations.
Adults with no knowledge of (a) or (b) are excluded from many STEM discussions for lack of vocabulary and concepts.
When it comes to fractions, we're likely talking about members of the set Q in our C > R > Q > Z > N discussion, one of the arcs in our story, solving equations a backbone theme.
We also bring the GCD in as a first look at a recursive process, taking Euclid's Method (as many texts know it) is uber-critical to our story of algorithms.
Yes, they had algorithms before Al Khwarizmi, from whose name the word "algorithm" derives (no, not from "Al Gore" and the word "charisma" has a different etymology).
How will we marry Euclid's Algorithm with programming and then move forward into relatively prime numbers (strangers), prime numbers, composites, Fermat's Little Theorem (not proved at first as 'what does it mean?' is more important on first pass) and Carmichael Numbers (to help explain the theorem)?
We do our Sieve of Eratosthenes at this point, but also get into totients and totatives, which the 1900s K-12 textbooks rarely did, not having RSA as a goal and not being informed by a computer-savvy age.
1900s texts derive from the 1800s in large degree, pre Von Neumann, pre Turing, pre Conway and Guy and 'The Book of Numbers', which we will often cite (have already in hypertext).
Knuth may be read on Euclid's Algorithm.
The extended version thereof will come later in the spiraling, as a tool in RSA (and in the solution of Diophantine Equations before that).
Should our visiting within Q include continued fractions? Of course, with a tie-back to phi through 1 = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + .... )))..)))).
Looking ahead to a standard course such as Litvin & Litvin (used at Phillips Academy etc.), we see a class of Q type, rational number, represented in a computer language. Operations + and * (add and multiply) will trigger methods, the operational guts of the class template.
Members of Q are instances of that template, which template may in turn inherit from a parent, and does in the Litvin text.
This brings us to a hallmark of 21st century STEM pedagogy, in which mathematics is immersed: numbers have been demoted to "another type of object" with subtypes i.e. there's a new family tree in town, of "objects" and numbers, represented as glyphs (numerals), are but one type of object among many, other types including character strings, boolean objects, vectors, quaternions, various collections, data structures (such as sets) and so on.
Learning the methods of string type objects, such as concatenation, capitalization, are as important as using the add and multiply methods associated with numbers. We have a unifying heuristic for all of them, as reinforced by the computer language.
So I've answered my question at the outset: where does programming enter math pedagogy.
As a STEM topic, it permeates everywhere, crossing over into the humanities. Closing it out of math teaching merely derails the math curriculum, though it's fine to have the old chalk 'n talk presentations persisting, perhaps in recorded media.
( Letting students do more of the presenting, in lightning talk format (per Ignite etc.) is more of a norm today, as we've outgrown the Prussian origins of many practices, but we can still play lectures asynchronously (after they happen) on our iStuff. Access to top grade lectures is likely not going to be a key challenge.)
Programming provides language fluency that (a) connects core topics and (b) organizes numbers and numeric operations within a larger tree of STEM-relevant object types.
With this more zoomed out kind of fluency, the idea of mathematics as mostly "just about numbers" begins to fade. More logic, grammar, rhetoric in general is restored, as we think more about the structure of cogent arguments, trains of reasoning, rational thought.
Divorcing out the numeric parts of information processing and naming that "math" as was done in some K12 curricula in the 1900s, is a huge mistake that only leads to dolt-hold in adulthood. Yes, the "a-dolts" and their crummy teaching ideas survive to this day, but not where there's been some concerted effort to upgrade the materials.
These poorly educated adults (many of them posing as teachers) are retrainable whereas today's students cannot afford to recalibrate to that low a standard of cultural illiteracy. Using the school system to provide livelihoods for relatively uneducated adults who don't know about Unicode, and don't know about SQL, was the norm in the late 1900s, but is proving less so today given the Internet and new generations who manage to avoid being slowed by over-exposure to these soon-to-be-retired institutions and practices.
Successfully escaping the antediluvian reflex conditioning of the past is the name of the game, as always, where pedagogy is concerned.