A Problem-Based Approach to Mathematics

As I mentioned in an earlier post, I wanted to go into greater detail about a course I took at the Phillips Exeter Math & Technology Conference this past week, specifically one called “A Problem-Based Approach to Trigonometry”. Kevin Bartkovich, math instructor at Phillips Exeter Academy, was the leader, and he did a phenomenal job recreating what a typical math class would be like at Exeter Academy.

The first important point to understand about Exeter math is that the curriculum is not the standard Algebra I/Geometry/Algebra II/Precalculus/Calculus sequence. They have written their own curriculum that is an integrated approach to math. From Year 1, topics such as vectors, geometry, algebra, matrices, functions, etc. are learned. As students work their way through the problem sets each year, the exercises become more involved and lead to even more interesting problems.

Second, there is continual spiraling throughout the curriculum, so topics are constantly being reinforced.

Third, all courses at Exeter are taught using the “Harkness Method“, which is basically Socratic. Course sections are limited to 12 students, and everyone sits around an oval table. The instructor encourages discussion and interaction between students, and through their conversation and collaboration they learn the material. There are lots of hands-on activities to supplement the exercises. Here’s a picture of one of my classes in action (I’m at the far end of the table):

If you teach math, you’re probably asking yourself, “How can you possibly teach something like trigonometry using Socratic techniques? Won’t you end up having to just tell your students the conclusion they’re supposed to get?” Well, here’s where the Exeter math curriculum plays a vital role. After working on a sampling of problems for a week, I began to think of the problem sets as something almost organic, in that each exercise serves a greater purpose, and each one is critical to understanding the entire curriculum.

Take, for example, this problem from the 3rd Year Math course:

The rectangle shown has been formed by fitting together four right triangles. As marked, the sizes of two of the angles are α and β, and the length of one segment is 1. Find the two unmarked angles whose sizes are α and α + β. By labeling all the segments of the diagram, discover formulas for sin(α + β) and cos(α + β), written in terms of sin α, cos α, sin β, and cos β.”

Try figuring it out for yourself – it’s actually a lot of fun as you see the pieces fall into place. If you get stuck, though, here’s the solution.

Once you label all the segments, you should be able to determine the formulas for sin(α + β) and cos(α + β)!

So, the point of this exercise (and it’s pretty typical of all the trig-related problems in the curriculum) is to enable students to discover on their own formulas that will be applied in future activities. They are not necessarily expected to solve the exercise alone, but through collaboration with their peers, they should be able to understand it.

I have one final remark: while I was at the conference, I was struck by how generous and eager everyone was to share their knowledge and expertise, especially the course leaders and the Exeter faculty. Their entire math curriculum is available for download here if you wish to explore it further. If more schools adopted it, I believe it would have a profoundly positive effect on math education in the U.S.

Boston, Massachusetts

Last night we took a bus into Boston, and walked around for a couple of hours. There aren’t many cities in America that played such an important role in our country’s founding. We walked along the Freedom Trail and saw the site of the Boston Massacre, the church where Ben Franklin was baptized, the cemetery where Paul Revere and Sam Adams are buried, the Old State House, and Boston Commons.

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The 27th Annual Anja S. Greer Conference on Mathematics, Science, & Technology

I am in Exeter, New Hampshire now, attending The Anja S. Greer Conference on Mathematics, Science, and Technology at Phillips Exeter Academy. It runs from Sunday through Friday, and it is a math teacher’s dream.

First, the Exeter Academy campus is absolutely beautiful, as you can see from the pictures below:

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Second, the courses offered are top-notch. Every participant signs up for two regular classes, and then you fill in the day’s open slots with one-time “Conference Within a Conference” seminars. The regular classes I am taking are “A Problem-Based Approach to Trigonometry”, and “The How-Tos of Project-Based Learning”. Both fit my goal for the 2011/2012 school year to increase the number of student-centered learning activities in my courses at Harpeth Hall. More on them in a later post.

Every evening there is a presentation by a mathematician on an interesting topic. Sunday’s presentation was by Frank Morgan of Williams College, about how one of his students’ projects was to calculate the optimal path a baserunner should take around a baseball diamond. It garnered lots of publicity and generated further research. He also showcased some of his current students, and their work in tilings.

Last night’s speaker was David Kung of St. Mary’s College of Maryland, on “How Math Made Modern Music Irrational”. He is a violinist as well as a math professor, and his talk delved into how tunings have evolved over the centuries.

The food is unbelievable good, but the best aspect for me is the exchange of ideas that occurs all day long between math teachers.

Update: Our final featured speaker was Nils Ahbel, whose topic was “Reflections on a 119 Year-Old Curriculum”. The gist of it was that 119 years ago, a group 10 prominent educators recommended that the U.S. math curriculum should be a sequence of courses that led up to calculus, which is pretty much what we have now. The problem is, a very small percentage of high school students end up taking calculus, so Nils argued for a more relevant math curriculum that incorporated much more statistics, and real-world data. You can view his Prezi for the presentation here (it’s very impressive).