What is the cardinality of the Euclidean topology?

I’m teaching topology this semester. The students are looking at different topologies on the real number line. For homework I asked them to think about which topologies are “the same” (if any) and which are “different,” and why they thought that was the case. We haven’t yet talked about continuous maps or homeomorphisms, so I…

Three applets illustrating parametric curves

In my multivariable calculus we’re talking about parametric curves. I’m using this applet for displaying parametric curves. You can use predefined curves or enter your own. Although the applet is on my web page, it was created by Marc Renault, a friend who teaches down the road at Shippensburg University. I only tweaked it slightly when…

Why do mirrors reverse right and left but not up and down?

[I apologize to those of you who have been reading my blog for more than a year. I’m reposting something I wrote last year at this time. I was then, and am now, teaching Calculus III, and we just finished discussing the cross product. I ended the conversation by telling my classes how the cross…

Three applets for linear algebra or multivariable calculus

This semester I’m teaching two sections of Calculus III (multivariable calculus) and I happen to be teaching the first four weeks of Linear Algebra. The first couple weeks of both courses cover properties of vectors in Rn. (Of course, just to confuse the instructor and the students who happen to be in both classes, the…

Division by Zero listed as one of 50 Best Blogs for Education Leaders

This morning I was pleased to discover that my blog appears on OnlineUniversities.com‘s blog in their list of the 50 Best Blogs for Education Leaders; it was one of the five blogs listed in the Subject Specific category. What a nice surprise. I’m very honored for the recognition! This is the first I’d read their…

Steven Strogatz writes about the elements of mathematics in the NY Times

Yesterday the mathematician Steven Stragatz wrote the first article in a mulit-part series for the NY Times. In this first article, called From Fish to Infinity, he describes his intent. Crazy as it sounds, over the next several weeks… I’ll be writing about the elements of mathematics, from pre-school to grad school, for anyone out…

Showing two expressions are equal: stop the madness

I know that most you who read my blog teach mathematics at either the high school or college level. I’d like to ask you a question (at the end of the post). It is about how our schools teach students to show that two mathematical quantities are equal. This has bothered me ever since I…

An applet illustrating a continuous, nowhere differentiable function

Open any calculus book and you will find a discussion about how differentiability implies continuity, but continuity does not imply differentiability. The absolute value function is the standard example of a continuous function that is not differentiable (at ). The inquisitive student may ask: how bad can continuous, nondifferentiable functions get? Can we make a…

Applet for discrete dynamical systems

I’m teaching the last two weeks of one of my colleagues’ Differential Equations course. I’m leading the class through a chapter on discrete dynamical systems. In preparation for the first lecture I created a couple of java applets using Geogebra. I thought others might be interested in them, so I’m linking to them here. The…