[tags]Calculus, concept map, math education, teaching, learning[/tags]
Today I am trying to plan out the first three 20-minute episodes for the summer calculus class. Lesson 1 is about the concept of the function, basic terminology, and how to compute function values using graphs, tables, and formulas. Lesson 2 will be about using symbolic input with functions (e.g. computing f(x+h)), and Lesson 3 will bat cleanup with end-of-the-section topics like the Vertical Line Test and piecewise functions.
can’t leave well enough alone am trying to find ways to make this material more accessible to students, and especially because I want students to chart their course as they learn this stuff and not just see calculus as a discrete set of algebraic tricks, I have spent the morning playing around with learning about an idea that I’d heard about before: concept maps.
The page at the link above is an outstanding brief tutorial/reference on the idea of a concept map in general. But basically a concept map consists of a focus question — some kind of question that we want to answer using a heirarchical structure of knowledge — followed by what looks like a flowchart of concepts connected by cross-links that show relationships between concepts. In reading this article I was struck by how appropriate this model of knowledge was to learning mathematics, whose power in general derives from the interrelatedness of its concepts — and whose main difficulty for students consists in seeing those interrelations. Just ask my linear algebra students; the hardest thing about linear algebra is keeping all the different concepts well-defined and clearly connected in your mind.
I’m certainly not the first person to try to use concept maps in a math class, and others are much better-versed at this than I am. But to try this idea out, I sat down with what was originally a point-by-point outline for Lesson 1 and turned it into a concept map. Here’s what i came up with (click to enlarge):
The focus question was: How do we represent precisely the relationship between two quantities? The map was made using the CMap Tools software, which is available — for free — here (click on the little icon where it says “Downloaded”). This map is the result of exactly 90 minutes of reading and experimenting with the software, prior to which I had no experience with concept maps; so this is probably not the best concept map one could make. In particular, I was wanting to add some concepts in the map about how you calculate function values for the various representations of functions. I’ll just save those for examples at the board, unless I get a bright idea (from the comments?) on how to build that in.
Although this probably has flaws (and I take comfort from the fact that “no concept map is ever finished”, quoted from the documentation) I am struck again by the degree to which the map “looks like” how I personally organize the information about this material. I am hopeful that these concept maps will help students get it organized too.
And concept maps present interesting opportunities for students if they are allowed, or required as an assignment, to create them on their own. Creating a concept map seems like a great out-of-class assignment to review for a test, for instance; or for a working group in one of my upper-division classes to do when gathering information to solve a new problem (especially in those classes where I won’t be using a book, like Geometry). The latter group of students might benefit particularly from such an exercise because they tend to be primarily education majors.
I’ve started up a folder in my del.icio.us links here for anybody interested in keeping track of what I find on concept mapping, particularly as it applies to math instruction and especially-particularly calculus instruction. And you’ve got links of your own to suggest, send ’em my way.