**Mathematical induction-the champion of efficiency.**

*The beautiful idea:*

Induction is often used (and usually taught) to prove formulas that hold for all non-negative integers. For example, the sum of the first n non-negative powers of 2, is equal to one less than the next power of 2. If n=4 it looks like this:

1+2+4+8=16-1

Our next instructions are to add 16 to both sides:

1+2+4+8+16=(16+16)-1

and we see that the pattern continues!

Let’s do it one more time. Start with our newly discovered equation:

1+2+4+8+16=32-1

add 32 to both sides, and voila:

1+2+4+8+16+32=64-1.

It’s pretty neat, and also really clear from this kind of discussion…unless of course…

*Efficiency:*

Unless of course, our discussion only uses one equation, which corresponds to n=1, the so-called *base-case*, and you write:

1=2-1.

The base case is just too small to see what’s going on or to notice any patterns. Your students may even feel stupid writing this down, as if it’s a secret handshake and they really don’t know why it’s necessary.

You guessed it; this is the way it’s almost always taught. But wait!–it gets worse; the instructions for seeing how the “pattern” continues are too cryptic for many beginners to follow. They are hidden in the formal details of the so-called *inductive step*. So it’s a pedagogical lose-lose.

It’s done this way because the base case is the smallest example that will make the proof work both completely and rigorously. After all, a full proof would have to include the case where n=1, and a formal proof works just fine with that as our main example. So let’s be efficient and … actually let’s not.

*Focus on the beautiful idea*

The beautiful idea behind the principle of mathematical induction should be given priority. That’s the first part of this post, and the first thing to be communicated regarding mathematical induction. Formal proof-writing etiquette can come a little later at the university level, or not at all in elementary school.

(Yes—I’m suggesting that elementary teachers should introduce math induction!)

So let’s use n=4 and n=5 for our examples, then look at the inductive step, keeping in mind how they relate to these examples, and then rewrite the proof the next day.

In other words, let’s give up some of the efficiency requirement.

]]>*A little calculation *

“10 is the greatest common divisor, or greatest common factor, of the two numbers 130 and 50.” We may abbreviate this as:

gcd(130,50)=10.

We see this by inspection, by factoring our numbers. There is a harder way to find the answer, called the “Euclidean Algorithm.” It turns out that this method will usually be much easier for large numbers where factoring is difficult. Let us use this example to introduce the Euclidean Algorithm. Consider this little calculation:

130=2(50)+30 … gcd(130,50)=gcd(50,30) (why?)

50=(30)+20 … gcd(50,30)=gcd(30,20) (why?)

30=(20)+10 … gcd(30,20)=gcd(20,10) (why?)

and finally, gcd(20,10)=10, since 10 divides 20.

*What is the main pedagocial point?*

The question “why?” appears three times in the little calculation. If we’re clever, we can answer all three questions all at the same time, with a little lemma—and we should do this (if you don’t know the answer, try it!). The answer constitutes the central idea behind the Euclidean Algorithm. The pedagogical problem is that the required lemma is usually discussed first; so that the chain of equalities: gcd(130,50)=gcd(50,30)=gcd(30,20)=gcd(20,10) can be suppressed!

*Conclusion*

My suggestion to the teacher is to introduce the lemma only after a few examples have been worked by the student in this inefficient way. Thus, the lemma is an *eventual recognition of a pattern*—a celebration of just how useful it appears to be, rather than an efficient shortcut in order to avoid it.

Last semester I ran a mathematical problem solving course. I did a few specific things that are not usually done in a mathematics class. Several small assignments consisted of answering a single question, which the student chose from a list of five. Students therefore would read more than one question before choosing a question to answer. This was my first victory—thinking about several mathematical questions without necessarily writing about them or handing them in is already part of the math-loving process!

This brings us to the next trick—participation grade. When I was a student, any such notion was offensive. It usually meant the extroverts would get more points, or even worse, that teacher preference would play a role in the final grade. I corrected all of this by giving a mathematically precise definition of participation, with several options for obtaining the points. One option was to do five extra questions beyond the minimum at a convenient time during the course. So if students read questions and liked them, they could get a little credit. Another option for participation points came from short presentations of problems to the class. Since these points were for participation only, students lacking in confidence, who were normally too afraid to present, would still do so. Some of these presentations were admittedly not very good—but here’s the key—they got much better. The motivation for doing a good job changed from getting a good grade, to wanting to do a better job, and ultimately to sharing the enjoyment of having done the work. Classmates were quick to point out ways to improve these participation activities, since they knew that it was all about the process-the participation-and not about the presentation’s grade. I certainly don’t claim that all courses should be like this; but I am claiming that not enough courses are like this. Try it!

Extra Remarks: Congratulations to my Ph.D. supervisor, Dr. Jan Minac:

Fortunately there are some classes where love of mathematics can’t help but be taught, just because of the instructor. As a student, the only time I ever remember experiencing extensive choices for homework questions was at Western University, with Dr. Jan Minac as the instructor. So I was in no way surprised to learn that he just won a prestigious national teaching award! From the CMS media release: “Jan Minac shares personal stories of great mathematicians with his students, so that the students become so involved…” How involved? Click here for the full story. Congratulations Jan!

Here is a nice fact:

(1/2)+(1/3)+(1/6)=1

Let’s have some fun with this fact by making it the answer of a question. Think of it as a mystery equation, and it’s our job to point the problem-solver in the right direction to find it. Hmm…

The sum of these 3 denominators is 11. After a bit of thought we see that {2,3,6} is the only set of 3 positive integers with sum 11, such that the sum of reciprocals is equal to 1. So we will tell the solver that we have x numbers whose sum is f(x) and we’ll pick a function f so that f(3)=11. We’ll also mention that the sum of reciprocals is exactly 1.

Now we need a hint so that the solver can be directed towards x=3.

Consider the more general situation where the x numbers are only required to be positive real numbers instead of positive integers. If the sum of reciprocals is 1, then it is well known that the sum f(x) is minimized if the x numbers are equal to each other, and thus equal to x (think of famous inequalities!). Hence x^2<=f(x).

Let’s pick f(x)=5×-4. Then x^2<=5×-4 and this places x between 1 and 4. We are ready to state the problem. Here’s a version of it, close to the way it appeared on the 2005 William Lowell Putnam Mathematical Competition:

Find all positive integers x, k_1, k_2, …, k_x, such that the sum k_1+k_2+…+k_x=5×-4 and such that (1/k_1)+…+(1/k_x)=1.

Our mystery equation corresponds to the case where x=3. There are a couple of other solutions, corresponding to x=1 and x=4.

To be clear, I did not make up this question. I also have no knowledge of what was going through the mind of the person who did make up this question. I was merely having fun with this beautiful problem!

My point is that this question, while incredibly difficult, may not seem so bad from the point of view of the *problem-maker*. It’s only difficult for the problem-solver. More importantly, I believe that the problem-solver has a much better chance of answering this type of question after having experience as a problem-maker. As much as possible, we want to be proactive rather than reactive.

So I propose that we practice making up good questions.

Good luck on the Putnam!

(A version of this post will be included in an up-coming book I’m working on, about mathematical problem-solving).

(Click here for a pdf version of this post, with better equation formatting).

]]>I decided to cover the same material covered in university courses, realizing that I would have to do it in a much different way. Here’s an example:

University version: If a rectangle has perimeter p, what is its maximum possible area?

Here’s what happened in the math club:

I asked them if they knew what a rectangle was, and how to find its perimeter and area. We discussed examples. It turns out they knew quite a bit. Then I asked each of them to produce their own rectangle, with perimeter 12 units. They each had a favorite! We calculated the area of each one. By the end of the meeting, we were convinced that the square provided the maximum area. By the end of the next meeting, we *owned* the difference of squares formula. In the third meeting, we used this formula to understand the formula for the area of a circle, partly by considering an appropriate annulus—this meeting was only a couple of days after a solar eclipse.

So what does this have to do with superheroes?

Special transformations, accompanied by superpowers, can really capture the imagination. It might be Clark Kent removing his glasses; but probably it’s a tadpole transforming into a frog or a caterpillar becoming a butterfly! In this video (link below), we introduce Rover the rectangle, Squeaky the square and Ellise the ellipse. Rover can change his shape as long as he stays rectangular and keeps the same perimeter—he catches circles by flattening himself! The intuitive connection between superpowers and transformations illustrates how a mathematician may think and *feel* about the subject. It is a human activity. It is fun, it is personal—it is mathematics!

So far we’ve made three of these videos. Each one represents portions of a different, hour-long meeting. Perhaps the best way to use them is to watch them with children, pause to discuss certain points, and then go off on a tangent. For example, in our first club meeting we eventually changed the fixed perimeter from 12 units to 14 units, which led to a wonderful discussion about fractions.

I tell the math club members how our discussions fit into the context of a university course. They LOVE this! At the end of our first meeting, we wrote down the formula for the maximum area “in terms of p” and they hurriedly copied it down as if it were a treasure. This was one of many pleasant surprises. Perhaps you are already doing something like this; if not, please try it. Become a superhero!

http://www.youtube.com/watch?v=Ix03mrGrCwE

Note added: I was delighted to see that this video has been used by Bruce Ferrington, who keeps an excellent blog. Here is a recent post, “Same Perimeter, Different Area.”

]]>The problem is that fewer and fewer of our everyday tools have these open windows, and this really can’t be good for any of us. So when it comes to teaching, I prefer examples that come with an open window. I think this is a big part of what students want when they ask, “What is this good for?” After all, students master video games, without any direct application in mind; but they *can* see the entire picture.

I remember one day way back in the 1980’s walking through a field that probably no longer exists. I realized that I could calculate the height of the tree that was 30 feet from me, *without climbing it*. I did the calculation, even though no one asked me to do so. My enthusiasm was a result of understanding the whole point—I could gain information about a large triangle just by knowing things about right triangles with a hypotenuse of unit length. To me, this is trigonometry.

In this 3 minute video, I attempt to capture the excitement I had walking through the field. In the video, there is no field, but there *is* a clear story. For teachers, there will be a lot more to discuss after your students are done watching it. I hope you enjoy it!

http://www.youtube.com/watch?v=PSK858OJYt4

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In this 5 minute video, young Squeaky the Square discovers pi in his own uniquely square way! Please enjoy!

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Department of Mathematics

Norwich University

Northfield, VT 05663

802 485-2323

dmcquill [at] norwich.edu

There is a menu on the bottom of the above picture of Norwich University. Please explore various pages of this site by clicking on the menu tabs. In addition, several “blogroll” links are on the upper right side of this page, including links to papers in the journal *Discrete Mathematics*, as well as a link to a preprint on ArXiv.org on the Witt Ring and the Witt Cancellation Theorem.

Mathematics is a human activity; as such it is always changing and always exciting. It has a rich history and strong personalities. It is certainly a subject where we continuously learn more by re-examining what was done in the past. We are often looking for another perspective to deepen our understanding. I sincerely hope that you enjoy this article, partly on the Witt Cancellation Theorem, and partly a survey article, but also a tribute to many great mathematical personalities responsible for the birth of the algebraic theory of quadratic forms. We thank and honor a few of the founders, including Ernst Witt, Emmy Noether, and Leonard Dickson. We also celebrate the 75th birthday of a pioneering paper of Witt with a brief overview of recent spectacular work which is still building on his original creation of the the algebraic theory of quadratic forms.

The slides for my talk, “Vertex-magic Graphs,” given during the spring meeting of the Northeastern section of the Mathematical Association of America on June 11, 2011, can be found here. These slides have also been placed on the selected preprints page, found under the “publications” tab on the menu.

Click here for updates on our top performances in the Putnam competition!

A short 9 minute talk has been placed on the Talks on video page, where comments are welcome. I may produce more videos, or refine the current one. This video is about planarity and crossing numbers of graphs, and specifically about drawings of complete graphs, and it is intended as a companion to paper on a parity theorem for drawings of complete graphs and complete bipartite graphs. A direct link to the same talk on youtube is also provided on the blogroll. I’ve also added a link to another fun video by Ján Mináč and Leslie Hallock, entitled, *How Mathematics Could Save Romeo and Juliet*.