In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Michael Dixon. His topic, from Precalculus: mathematical induction.
How can this topic be used in your students’future courses in mathematics or science?
The first time student are introduced to mathematical proofs is probably in high school geometry class, proving theorems using the axiomatic method. They work to prove things about Euclidean geometry with step by step deductive reasoning, as did Euclid himself in the Elements. They prove things about concrete objects that they can see and draw on paper, such as circles, angles, lines, and triangles. But then they move on to Algebra II where they are taught more abstract ways of dealing with numbers and expressions. Is there any way to prove things about numbers themselves? It’s not as easy to visualize, that’s for sure. What is a number? Is it something I can see and feel; is it the shape we write on the page? Or is it something beyond that? This aspect is one of the challenges that algebra students face as they are exposed to more and more mathematics. Mathematical Induction is one way to prove things about numbers using solid deductive reasoning that cannot be refuted. And not just about a few numbers; high school students would be more accepting of that. Mathematical induction is usually used to prove something about ALL of the natural numbers, starting from one and going on out past infinity. Induction can be used to prove what students might intuitively think about the natural numbers, such as that there are an infinite number of primes, or it can be used to prove less obvious things about numbers, such as 1 + 3 + 5 + 7 + …+ n = n2. We can prove these and more without having to compute billions and billions of cases. In just a few lines of mathematical logic, we can prove that something is true for every natural integer. This is more than just telling the students something and them accepting it, this technique PROVES that some statements are always true for any number we want to choose, no matter how large it is. That’s some powerful stuff.
How was this topic adopted by the mathematical community.
Mathematical induction has been around for thousands of years. While not in the same form as we see it today, induction can be seen all the way back to Euclid’s proof that there are an infinite number of primes, or in the writings of Aristotle. They used this logic to prove a lot of things, but it was not in the formal way of proving something about n and n + 1. This formal notation did not show up until around 1575, when Maurolycus that 1 + 3 + 5 + 7 + …+ n = n2, though he did not prove using n and n + 1, yet. Several mathematicians began using this formal method soon after, such as Pascal and , though no one had a name for it. Bernoulli then was one of the first to begin using the method of arguing from n to n + 1. Since then, mathematicians have been heavily using this method to prove countless things about the natural numbers. And eventually, around the 20th century the name itself, mathematical induction, finally became the standard term for the method starting over two thousand years ago.
How can technology be used to effectively engage students with this topic?
These videos cover mathematical induction in a way I hadn’t seen before, and cleared up a misconception that I had. I had always thought (because of the name) that mathematical induction was not the same kind of reasoning that is used in other axiomatic proofs. However, mathematical induction happens to actually be deductive reasoning, rather than inductive reasoning. The only similarity is that both mathematical induction and inductive reasoning deal with occurring patterns. The first video is more the engage part, while the second one goes a lithe further into the content. For the engage, showing the video at the beginning of the class is probably better, while the second might be given to the students as homework to watch on their own.