On Thu, Jun 20, 2013 at 6:51 AM, Robert Hansen <firstname.lastname@example.org> wrote:
> > Devlin's argument isn't against teaching any particular arithmetic > algorithm. He is against all of them. Sure, introduce an algorithm, > whatever it is, but dare not teach it long enough that the student develops > a personal fluency with arithmetic. Heavens no. That would be a total waste > of time because we have machines that can add and subtract. >
When Devlin came to Oregon for our ISEPP-organized "math summit" in 1997, his schtick was not against teaching basic arithmetic but against compartmentalizing the teaching of arithmetical algorithms such that only "math teachers" or "math classes" bore the brunt, while none of the other subject teachers had to spend much time on it, and just assumed "the math teachers" were doing their job. 
In point of fact, elementary schools are largely taught by all-subjects teachers which gives students more continuity. Or maybe they have "home room" teachers yet go from room to room more like in high school. Compartmentalization is in the architecture, but there's still lip service around inter-disciplinary integration (in some schools -- I'm sure that's a dead end in others).
But when it comes to basic arithmetic, Devlin's contention was basic numeric literacy is something all teachers should gang together and diffuse through the entire curriculum, not concentrate in one corner under the heading of "math" in particular, as if "math" were no more than "arithmetic" (that still confuses people).
Part of the point was to drive home that these are basic skills common to all adults and NOT stereotypically relegated to one walk of life or another. Arithmetic is our common heritage. Airline pilots, chefs, accountants, doctors, nurses, pharmacologists, bus and taxi cab drivers, farmers, business managers, disaster relief workers, environmental monitoring workers, zookeepers... all need to add, subtract, multiply, divide, and sometimes take a log, raise to powers.
Geography / geodesy should come with plenty of trig (doesn't have to be spherical -- when you zoom in enough it's locally planar, with triangles = 180 degrees). Geography includes demographics, ecology, rates of change. Geology includes exponential rates of decay, the concepts of half life etc.
My core heuristic (on MathFuture mostly), some may remember, is Geometry + Geography as the two basic subjects. Under Geometry comes all mathematics irrespective of empirical concepts and measures and under Geography comes all physical phenomena, from subatomic to astronomic, humans somewhere in the middle (scale-wise).
History could be taught with great attention to numbers systems and engineering. UCSC's Ralph Abraham (mathematician) was of the same mind (also at said summit).
For example, back to the "cooking / following a recipe" meme: if you're doing problems around home economics and budgeting, talking about how much rice to buy or the average rate of bean consumption, then you've got (a) units of measure (b) prices (c) time and rates of change.
Why not take this opportunity to dive into all the operations necessary to (a) compute the monetary cost of serving 30 people X, Y and Z (b) compute the calories, proteins etc. (c) develop a budget for the next six months, in terms of stores needed (d) compute the energy costs in terms of joules through the meter
(in South Africa, food energy values on the labels are in joules as well, not calories, which helps students make the connection between the energy they burn in their bodies from food, and the energy the import through the wall sockets).
You can do a lot with computer games and simulations at this point.
'Oregon Trail' is / was popular, because you needed to buy and sell stores, negotiate for supplies.
But this could just as well be "history" or "economics" as "math".
Fast forward: I think we're closer to Devlin's vision today, in that STEM is about countering over-specialization and over-compartmentalization. My own contention is once within the purview of STEAM (adding anthropology) it's not so important to decide "exactly which" subject is being studied at any given moment. Is this technology or mathematics? Is this engineering or science? These need not be considered useful questions.
I would be happy to see "math class" and "math teachers" disappear all together in some experimental curricula, in favor of STEM as a holistic network of topics and skills. The idea of "math class" as a completely divorced subject may be convenient for administrators, but it churns out adults with less integrated thinking and a weaker grasp on academic subjects in general (this would be my contention).
Of course in my overview synoptic comments here, I'm embracing more than just elementary school, during which the basics should be learned.
I was doing multi-digit addition in 3rd grade at Junior English (Rome) and having Narnia by C.S. Lewis read to me by the same teacher.
In Manila, our Introduction to Physical Sciences teacher did a lot with scientific notation and decimal points, logs and exponents. Looking back, I see that's how it should be: science teachers teaching math.
Physics needs vectors and should teach vectors.
In the smarter / better curricula of tomorrow, I'm guessing we'll see far less artificial fragmenting into "different subjects". We'll be content that it's STEM and not worry about the subcategory. Over-specialization was taken way too far in the 1900s and we paid an enormous price. Devlin's vision is not out of line with my own in this respect.