School administrators, it seems to me, are crazy about technology in the classroom. Technology is the answer in search of a question. In my school, there's a steady drumbeat of pressure to incorporate technology into the classroom. The message is rarely "Use this technology tool because it will help you do x better"; the impression I'm left with is that using technology makes the school look cutting edge to each group that comes in to take a tour. The push for I-pads in the classroom is such an example, yet there are many students caught up in games to the exclusion of school.

I'll be the first to admit that technology should be an important part of education provided it is used judiciously and appropriately. Graphing calculators should be a requirement for higher level math classes such as precalculus and calculus; after students have mastered the basics. The problem is it's never very simple at the public high school level; students get passed on despite having a poor mathematical background. Combine that with schools which are eager use technology and you're potentially creating all sorts of problems. It's no surprise to me that Shanghai led the world in math scores for high school students even without an I-pad in the classroom.

One of the reasons for using technology, I hear, is to save students from the drudgery of math so they can focus on the concepts. Arguments like these are often short sighted and made by people who don't have the math background needed to make good decisions on math policy. Let's look at a typical example: graphing calculators, we're told, can graph functions quicker and easier than humans. Obviously that's true but advocates seem to think that means a graphing calculator should be used to save students having to plot graphs by hand. But that's just wrong. Confused? Suppose, for example, you need to graph $latex x^4-4x^3-12x^2-6$. Plot that with your graphing calculator and maybe you get something like this:

The graph was created almost instantly; so are we done? Of course not, we're just getting started. The calculator hasn't solved the problem because the graph shown is lousy. Which brings us to today's post: good graphs. We need some criteria to decide if the graph created is a good portrayal of the function. I stressed the concept of a good graph by which I mean the graph should:

- be centered
- show all relative and absolute minimums and maximums
- contain all horizontal and vertical asymptotes
- display enough of the graph that the user can properly infer the information not on the screen (eg horizontal asymptotes or period of a sine function)

A good graph isn't always possible but for almost every function at the high school level it is. By the criteria listed, the graph above isn't very good. It takes some calculus for us to find the minimums and maximums and limits to find the behavior of the function "outside the screen". Knowing those basic skills allows us to plot the graph properly to highlight the essential features. I'll spare you the calculations but here's a good graph of the function $latex x^4-4x^3-12x^2-6$:

This graph is centered, shows 1 relative minimum, 1 relative maximum, and 1 global minimum. It has no asymptotes and the user can properly infer that $latex \lim_{x\to\infty}x^4-4x^3-12x^2-6=\infty$ and $latex \lim_{x\to-\infty}x^4-4x^3-12x^2-6=\infty$ from the picture. All the "action" is shown on the screen; that's a good graph.

To properly graph a function, I insisted that my students know the essential characteristics of a good graph are and be able to work (step by step) to establish:

- $latex \lim_{x\to\infty}f(x)$
- $latex \lim_{x\to-\infty}f(x)$
- vertical asymptotes
- minimums and maximums

Those who think that pushing some buttons on a calculator can replace having to master the basics shouldn't be setting math policy but (unfortunately) these just the people setting policy. If you teach calculus, the concept of a good graph can help you to push your students to master some basics.