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…

# Category: Teaching

## 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…

## Thoughts on teaching induction

I don’t plan on doing this very often, but I thought I’d re-post one of my earlier blog posts—one that I wrote a year ago, when I had many fewer readers. Now is an appropriate time for me to re-post it because I am currently teaching induction in my Discrete Mathematics course. Enjoy. In their…

## Kindergarten Mathematics (part 2): a report

Last week I wrote a blog post asking for suggestions for math to present to my son’s kindergarten class. My readers posted many great comments. Thank you all. Today was the big day,… and it was a great success! I began by talking about what I do. My son introduced me as a math teacher….

## Kindergarten mathematics

This is a call for help. My son’s kindergarten teacher has invited parents to come in and talk about their careers. I’d like to go in and talk about math. I’d like to have some interactive hands-on mathematics activities for the kids to do. I also want them to be activities outside the typical kindergarten…

## Tennenbaum’s proof of the irrationality of the square root of 2

Yesterday I came a across a new (new to me, that is) proof of the irrationality of . I found it in the paper “Irrationality From The Book,” by Steven J. Miller, David Montague, which was recently posted to arXiv.org. Apparently the proof was discovered by Stanley Tennenbaum in the 1950’s but was made widely known…

## A new way to collaborate: DropBox

I have a long-time collaborator who lives in Georgia (I’m in Pennsylvania). I’ve had good luck collaborating with him via email, but it is a pain. As soon as one of us edits a file he sends it to the other person as an email attachment. We haven’t had any “forked” files, but we do…

## Is or an inclusive or or an exclusive or?

(That was a fun title to write!) At the start of our discrete mathematics course we talk about symbolic logic. Students are often confused by the logical operator “OR.” If p and q are statements then p OR q is true if either p is true or q is true or if both p and…

## An applet for teaching the limit of a sequence

I’m currently teaching real analysis. Right now we’re discussing limits of sequences. The definition is: The limit of a sequence is (or converges to ) if, given any , there exists a natural number such that for all . I used GeoGebra to create the following applet, which illustrates the definition of a limit. (Clicking…

## The nuts and bolts of writing mathematics

This was a handout that I made for my Discrete Mathematics class. At our college this course is the gateway to the mathematics major and is the students’ introduction to writing mathematical arguments. Here is a pdf version of the text shown below. The nuts and bolts of writing mathematics “Mathematics must be written so that…

## Look who majored in mathematics

Here are some people who studied mathematics, but are more well known for other things. Ralph Abernathy ~ civil rights leader and close associate of Martin Luther King, Jr., was a mathematics professor at Alabama State University Corazon Aquino ~ former president of the Philippines, was a mathematics major at the College of Mt. St….