Wednesday, November 25, 2015

Not privileging the Symbol

"We don't want to privilege the word."

That was the justification that the director of a college writing program gave to a friend of mine for why he should allow his students the option to draw a picture rather than write an essay. Even back then, over two decades ago, our education experts were being careful not to privilege symbols.

But it has only recently occurred to me how this concern is playing out in today's Reform Math. Along with group work, group discovery, multiple solutions, and, of course, explaining answers to easy problems, there's doing math visually. This explains:

1. The large percentage of elementary school math problems today that involve classifying shapes.

2. The large percentage of K12 math problems that now involve reading charts and graphs.

3. The replacement of geometry proofs by visual "demonstrations," complete with spatial "translations," "reflections," and "dilations."

4. The growing reliance on graphical representations to solve algebra problems.

We see 4 at play, for example, in Brett Gilland's comment about the PARCC problem on my last Problems of the Week post. My inclination was to answer the question algebraically rather than appealing to geometric intuition. But, superficially, the problem really begs to be solved graphically, given how it's spelled. Consider the specific letters used to represent variables in the equation g(x) = mx + b. Because "m" is conventionally used to represent slop and "b," to represent the y intercept, many students will instantly recognize this as an easily graph-able line. The algebraic solution, via the Quadratic Formula, would prefer a different spelling: the standard spelling of polynomials (ax2 + bx + c), has the letter b representing the co-efficient of x rather than the constant.

While the specific spelling details of symbolic representations are superficial, symbolic representations themselves are anything but. They are what allow you to abstract away from two-dimensional space towards that which is increasingly difficult to represent graphically. Functions in 3-d are already tough; what about 4-d spacetime, or n-dimension space more generally? And consider other branches of mathematics like number theory and logic. Again, visual diagrams can represent the simple stuff (e.g., Venn diagrams for simple logical relations like if, only if, iff; or the number lines for simple numerical relations). But they do not take you very far at all. That's why, if you take a look at a math article, or even a physics article, you generally find many, many more lines of symbolic expressions than you find shapes, graphs, and diagrams.

But today's math teaching experts think otherwise. Their assumption seems to be that shapes, graphs, and diagrams are what make math meaningful, and that everything else is mere symbols. Thus, symbolic manipulation must be as mindless as "mere calculation."

The upshot is that what mathematicians and physicists use as tools for powerful abstractions are viewed by others as being as superficial as the decision to use the letter "m" to represent a slope and the letter "b" to represent an intercept.


owen thomas said...

/* "m" is conventionally used to represent slop */
you may be right about that.

Katharine Beals said...

Nice, vlorbik! I was thinking about fixing that but I think I'll leave it as is. Happy to have you back--it has been a while.