As a "freshman" to the field of developmental mathematics my viewpoint has been broadened by the rich discussion regarding the philosophy and pedagogy of teaching mathematics. Thank you.
I want to expand upon the mathematics eduator's "toolbox" by mentioning discourse theory. I am not saying it is a 'one size fits all' option, but it might help lead students to a broader, more abstract level of thinking by first presenting mathematical topics using real life analogies.
As a brief introduction, discourse theory expands upon the knowledge students already have. It is not a lecture but a discussion, lead by the instructor, and sustained by the students.
Our Algebra with Arithmetic course begins with the order of operations. The order of operations, as we know, is an algorithm that must be followed exactly (however, there is some, yet little room for working in a different order). The discourse can begin by asking what real life activities use a specific process. A student may answer: Making gumbo must start with a roux (some may disagree-but a discourse on the culinary arts can precede at a different time :) before the seasonings can be added, before the vegetables can be added, etc. Since mathematics is a very discipined field, any analogy breaks down at some point, but it can help them to start thinking about the rules and characteristics about that particular mathematical topic.
>From Schremmer, >Isn't there something between the two? Like what most of us did when we learned mathematics?
Again using the anology idea, students can be exposed to a topic by first 'exploring' a real life example through a discourse led by the instructor or (to allow it to be done at home) through a written series of socratic questions. The exposure to graphs can be shown by giving students a graph of (you name it in the science field) and asking them what the intercepts mean, what a particular point represents, what the up/down ward slope represents. Is the up/down ward slope practically a good or bad thing? Students know more about math than they know.
Students often think that mathematics is rigid, like an overcooked square biscuit. However, mathematics also takes creativity, which means thinking outside the box. Discourse theory can help us as well as our students think outside that box.
-Chris Elder Math Specialist Title III/Developmental Education Florida Gateway College Chris.Elder@fgc.edu<mailto:Chris.Elder@fgc.edu>