## The Calculus of Teaching

A few weeks ago I made a short trip down to Lincroft, New Jersey, to deliver a Maple training course to a group of math professors at Brookdale Community College. I’d been to Manhattan before, but never New Jersey, and didn’t really know what to expect. I was pleasantly surprised to fly in over a lush and verdant landscape, with temperatures hovering in the mid-70s. My host, Barbara, had graciously invited me to her home for dinner, and we had a great conversation about teaching math, the Verrazano Bridge, dealing with deer in the backyard (Barbara was originally a Brooklyn girl, used to the concrete jungle!), and of course, what to expect the next day at the college.

The group of professors that showed up for the training was exactly the one I’d have picked, given the choice. Most of them had never seen Maple before, but they were very excited to find out what they could do with it, and specifically, what Maple could do in the context of teaching Calculus, from pre-calculus all the way through to level III.

Using Maple’s point-and-click tools, I started by teaching the basics: how do I enter math in Maple? How do I create a plot? Can I add descriptive text? And what about matrices? From there, we were able to move into working through specific problems, based on material from the college’s curriculum I had received in advance.

A common comment that we hear (and that I heard at Brookdale) is that “it seems like Maple takes away the need for students to do any work.” At first glance, this seems to be true. For instance, I can enter a mathematical expression into Maple and easily find its derivative by right-clicking and accessing the context-sensitive menu. A student could indeed do this, without needing to learn any of the steps involved in calculating derivatives. However, this is not really the way Maple should be used in the classroom. It’s important that the instructor use it as any other tool, to support – and not replace! - the teaching process.

In fact, using Maple to teach mathematics can turn the standard textbook teaching formula on its head. The usual process is to first teach the mechanical steps by rote, and then to explain the concept behind them. Unfortunately, the students can get caught up in the mechanics and miss the point. In contrast, Maple lets you first demonstrate the meaning of a solution: the why of a particular method. As well as finding quick answers, it can be used to derive solutions from first principles and illustrate concepts graphically. It can illuminate theory, clarify abstract concepts, and give form and substance to general principles. Of course, once you have demonstrated a concept with Maple, it is still necessary to teach the mechanical steps, only now, the students should have a better understanding of why they are doing it.

There was a lot of material to go through, and Maple is so exhaustive, I knew there wasn’t a chance we’d get through it all in one day, but I was able to leave behind a list of resources available, to help them continue from where they’d started. I often get questions from customers along the lines of “How do I do … in Maple?”, so to conclude, I’d like to share some of those resources with you:

Where can I get help with Maple?

1. Inside Maple, type ?portal to open the Maple Portal, an effective place for any new Maple user to begin their journey with Maple. It also includes links to portals specifically designed for engineers, students, and math educators.
2. All of Maplesoft’s instruction manuals and Maple’s help system are available online.
3. Also online is a collection of Maplesoft training material, including tutorials, videos, and much more.
4. Anyone can sign up for our live web-based demonstrations of Maplesoft products. If you missed a session, recordings are of course available.
5. The Maple Application Center is a repository of Maplesoft applications created for and by our users. You get free access to applications for Maple, MapleSim, and other products over a wide range of topics from high school math to robotics and beyond.
6. MaplePrimes is Maplesoft’s online user forum. Membership is free, and lets you post questions about any of our products, as well as search the database for past user questions that may be helpful.
7. For information on products and solutions for Academic Instruction, click here. In particular, the Teacher Resource Center, designed to support math educators using Maple, links to course material, sample applications, training, and more. Two other great resources you can point students to for help are the Student Adoption Center and The Student Resource Center.
8. A series of recorded videos on Clickable Calculus is available; Dr. Robert Lopez “solves a spectrum of standard (and not-so-standard) problems drawn from differential equations, linear algebra, and vector calculus.”