Thursday, December 23, 2010

functions and CCR

A recent paper on talks about the CCR Instrument, not John Fogerty's guitar, but rather the Calculus Concept Readiness Instrument - a mathematics placement test from Maplesoft. The paper's brief background discussion on what concepts students need to be "calculus ready" is interesting. A large literature is cited that states that the most important concept that students need to be familiar with (for Calculus and for general mathematics education) is the function concept. Unfortunately, the function concept and the concept of function composition have often been cited as weak points for teachers as well as students (see this study by David Meel, for example).

A really nice article to read if you would like to expand your understanding of the function concept and get a sense of how it has mutated over the last few hundred years is Israel Kleiner's Evolution of the Function Concept: A Brief Survey.

As mathematics has advanced, functions have become strange, generalized, and freed from conceptual limitations that threaten to tie them down. Kliener quotes Poincaré in offering an explanation as to why, in contrast, our education proceeds with with simple (antiquated, in Meel's assessment) and well behaved functions.

If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum.

For those of us who are lucky enough to wander through this bizarre museum, the concept of function gets stretched in interesting ways. I seem to remember that learning about functionals in Linear Algebra expanded my appreciation of  functions and their strangeness. Here's an example: consider two sets $A$ and $B$, and suppose that $a$ is an element of $A$. There is a function, call it $\hat{a}$, whose domain is the set of functions from $A$ to $B$, and whose co-domain is the set $B$. This function $\hat{a}$ is defined by the rule $\hat{a}(f) = f(a)$, where $f$ is any function from $A$ to $B$. The first time you see something like this it seems lovely and weird - functions become elements, elements become functions, and the rule that defines the element-become-function $\hat{a}$ looks like a bit of notational sleight of hand. Things can get even stranger of course, such as when arrows (function-like-things) are defined without elements at all.

Concerns about "Calculus Concept Readiness" reminds us how preparation for Calculus is often seen as the ultimate goal for K-12 education. Should this be the case? Discrete/finite mathematics may provide a better goal (convincingly and quickly argued by Arthur Benjamin in his TED talk). It can also be argued that discrete math provides a better setting for learning the fundamentals of the function concept than pre-calculus courses that focus more on functions that are well behaved for the purposes of elementary Real analysis.

Tuesday, December 14, 2010

skeptics vs. constructivists

A short while ago, noted skeptic Michael Shermer wrote an interesting article about Steven Hawking's new book "The Grand Design" (another brief review from the Washington Post is here). In Shermer's analysis, Hawking's philosophy of model-dependent realism sounds a lot like radical constructivisim. Skeptical of the extent to which this philosophy should be pushed, Schermer asserts that Hawking's model-dependence has its limits, and that science provides an unprecedented means of overcomming the relativism that model-dependent realism and constructivism seem to lead to.

Coincidentally, there are a number of articles in the most recent issue of Constructivist Foundations (volume 6, number 1) that discuss the same issues that Shermer brings up in his assessment of Hawking's model-dependent realism.

What Shermer and the authors in Constructivist Foundations wrote about, and what seems to trouble people most about model-dependence and radical constructivisim vis-a-vis science, is whether or not science tells us anything about "reality." Shermer answers affirmatively:

"Yes, even though there is no Archimedean point outside of our brains, I believe there is a real reality, and that we can come close to knowing it through the lens of science — despite the indelible imperfection of our brains, our models, and our theories."

Radical constructivists, like Andreas Quale in his contribution to the Constructivist Foundations issue noted above ("Objections to Radical Constructivisim"), doubt that there is any true reality described by theories, even if the theories seem to be getting better:

"...a scientific theory is regarded as a model, constructed to address certain questions that we want to ask, and then imposed on natural phenomena. If the model is successful, fine – but this is then better seen as due to the capabilities of the constructors (scientists) [rather than being closer to reality]."

What does this have to do with math? The debate between scientific realists and relativists maps partially (but perhaps not exactly?) onto the debates between neo-platonists and constructivist/intuitionists and formalists in the philosophy of mathematics (see a few quotes here). Also, after reading a few articles by radical constructivists, I am surprised how many of them are mathematicians and/or mathematics educators (a notable example is Ernst von Glasserfeld, another is Andreas Quale, quoted above).

Few doubt the constructed nature of scientific theories. As Shermer states, the issue is how far you go with constructivist arguments (is there an ultimate reality, or is it turtles all the way down?), and whether ultimately you are a realist, or a relativist. Perhaps asserting the existence of reality requires a Kierkegaardian leap, not of faith, but of rationality - and also marks the reasonable limits of skepticism.

Tuesday, December 7, 2010

multiplication table with stars

Here is a printable (2 page pdf) for exploring multiplication patterns with star polygons. Not for memorizing the multiplication table, for exploring.