The following is the text of a talk that I gave at this summer’s CUMC at McGill. It went pretty well, and ended with a good discussion on what’s going on in the world of math education at various universities across the country. Hopefully this dialogue will continue - perhaps through a wiki that Andrew Schaaf and I will be working on over the next little while.

Through the years great thinkers have considered the problem of how to teach effectively. Building on some of these ideas, the Math 121 Executive Class is an attempt to nurture a deeper understanding of mathematics through small group investigations of exploratory problems. In this talk, I’ll cover some of the ideas this project was based on, how well it worked, how it can be improved, and how to apply what we’ve learned in this project to other classes.

We’re all pretty familiar with the lecture approach to teaching at this point, given that it’s the dominant form of teaching at university and high school. The lecture method has its advantages: it’s efficient in that one professor can teach hundreds of students at once, following a set curriculum is easy, and finally, it’s a format most people - especially administrators and professors - are familiar with, and most university students deal pretty well with.

On the other hand, lecturing to hundreds of students makes it difficult for the professor to connect with the students, and leaves struggling students feeling unsupported. Although it is possible to make big lectures interesting and engaging, far too often material seems dry and subjects disconnected. This method of teaching makes it easy for students to drift into a routine of just memorizing what will be on the test instead of getting any deep understanding of the material covered, the latter arguably being the point of a university education. Many people do not learn best from lectures

Increasingly, faculty and students are trying to transform and supplement traditional lectures in ways that help students explore the material more deeply, and keep the joy of learning alive.

If you were at last year’s cumc, you may remember Professor Peter Taylor’s talk on math education. He spoke of a vision of education that follows the ideas that Alfred North Whitehead discussed in The Aims of Education. Us math types probably know Whitehead as the mathematician and philosopher who co-wrote the Principia Mathematica with Bertrand Russel in the early 1900s, but he was also a philosopher and a highly regarded thinker on education. In The Aims of Education, Whitehead proposed that learning occurrs in cycles of three stages.

The first stage is the stage of romance. I’m sure we all know the feeling of first learning about something and becoming captivated. The new idea seems so vivid and strange, that you just want to learn everything about it.

Next comes the stage of precision. In this stage, the student gets down to the real work. They learn the rules and practice until they’ve mastered the subject.

After this comes the stage of generalization. This is something of a return to the romance stage, but with a greater understanding of the material. It is here that the student can see the big picture and can apply what they have learned to other fields.

Whitehead proposed that these cycles occur both on a large scale, over many years of study, as well as on a small scale as one tackles an interesting problem.

Professor Taylor noted that in math education, all too often the stage of romance is bypassed in favour of the stage of precision. He advocated the use of interesting exploratory problems to allow all three stages to come to fruition, but especially to allow the romance to flourish.

This led to a number of projects including the creation of the Math 121 executive class which I helped TA last year and will be coordinating this fall. Math 121 is one of the largest math classes at Queen’s. There are over 1000 first year students registered in this class every year. This is the first year calculus class for students who aren’t in the math or engineering department, so the students vary widely in their aptitude and knowledge of mathematics. There are lifesci students, who got 98% in their advanced placement calculus classes, and there are students who just scraped by, or weren’t exposed to much calculus in high school. Since each lecture section has over 100 students in it, it’s impossible to teach the material so that everyone can keep up and no one is bored. Because the class is so large, there are no marked assignments and tutorials are often used for quizzes instead of review.

Dr Taylor (the head of the math dept) and Dr Ableson (the coordinator for math 121) came up with an idea for a separate class that would meet for an hour every week. This class emphasizes small group learning and one-on-one contact with TAs (something obviously missing from the lecture halls) through challenging exploratory problems. These problems are bigger and more complex than individual students could be expected to do on their own (even we TAs were challenged by many of them!) but they use the principles of calculus taught in the regular class and apply them in neat ways. The class is optional - if students choose to attend, it counts towards 10% of their mark, 5% of which is attendance, and 5% comes from the journals they do at the end of each term.

The problems usually take two or three classes to complete. The problem is introduced, and as a large group we brainstorm ideas on how to tackle it. Once we’ve found a few that hold promise, often with hints from the TAs, the class breaks up into small groups of about 4 people which work on the step we’ve come up with. Once most of the groups have completed this first step, we move back to a large group discussion, where a TA or a student explains what was just done. We then go back to brainstorming, and repeat the procedure until the problem is finished.

This worked with varying degrees of success. In order for it to work properly, the TAs need to know how much the class can handle on its own, and figure out how long the students can work independently before losing focus. I found that as time went on and I got to know the students better, it became much easier to figure this out. I also found that sometimes it’s tricky to put yourself in the heads of the students and give helpful hints that don’t entirely give the problem away.

The least successful sessions happened on days when we felt rushed to finish the problem quickly - it was far too easy to revert to the lecture format instead of the more Socratic method of giving hints in the right direction and getting them to figure it out on their own. On the whole though, the class seems to have fulfilled its purpose - encouraging students to enjoy math, and find hidden connections.

There ended up being three sections with around 15 people in each section. For those who came out, it was a great success. In their journals, many of them said that they were surprised by the many different applications of the math they were learning. For example, Tim Hou said:
“My experience in the Executive Class has both been challenging and rewarding. Before coming to university, I thought of math as a purely theoretical subject that had limited applications on life; however, the Executive Class has allowed me to see that many of life’s problems could be solved eloquently with calculus.”

Anahita Safavi-Naini:
“For me, the success of the Math Exec course lay in the feeling with which I left every class — intellectual curiosity, confidence in my reasoning ability, excitement, and a drive to apply myself not just in MATH 121, but especially in my other courses, providing me with the motivation to learn that I sometimes thought I had left behind in high school.”

In the future, we’d like to expand on what we accomplished in these sessions. Obviously, three times 15 is well below 1000… we’d like to encourage more people to attend. The majority of the people attending the class were those with a lot of confidence and aptitude in math. This was certainly beneficial to them, since otherwise they would have been extremely bored, but we’d also like to encourage those who have a bit more trouble with math to come out. I think that the approach taken would be really helpful to struggling students - both raising their confidence level, and exposing them to different ways of thinking about the material.

One of the ways we’d like to address this is very simple - changing the name. Perhaps calling it an “executive” class attracts a certain type of student, but it would seem likely that it would also scare off some people - probably many of those we would most like to help. Those attracted by the executive label, would probably still come out if the name were changed.

Otherwise, we think that word of mouth will help attract new students. Some of the people in the class will be orientation leaders and residence dons, and hopefully they’ll tell the frosh about the course. We put up exerpts from the journals on the website and point students towards them. The professors and TAs will also be spreading the word about this opportunity early in the year.

On a larger scale, we’d like to expand this program to other classes and maybe include some of this material in the course proper. There are many challenges ahead. The large enrollment in the class makes it difficult to foster the kind of learning environment we’ve attained in smaller sessions. There is some possibility of changing the regular tutorial sessions to include some of the “exec” material, and Professor Taylor is looking at introducing the problems into the lectures.

In expanding the program to other classes within the university, different problems arise - mostly in convincing those in power that this would be a good thing. The percieved risk in fiddling with Math 121, a first year program that won’t affect students registered in the faculty of math, is pretty low. The percieved risk in doing something like this in, say, an engineering course, seems much higher. Accreditation and “job training” are worries, as is the worry about the student reaction to unfamiliar changes. A first year course in engineering that was designed to incorporate different methods of learning experienced a significant amount of backlash, although I think a lot of this was due to initial organization problems than anything else. At Queen’s, we’re lucky in that Professor Taylor is the head of the math department, and the faculty of applied science seems to be moving in this direction, but change still comes slowly. Hopefully the success of the Math 121 project will lessen these fears in time, and make it easier to accept change.

Professor Taylor has been active in promoting more of this type of learning in high schools and elementary schools in Ontario as well. In the Kingston area he has run a high school outreach program with a similar format to the 121 exec course, and he’s been working with the government on modifying the curriculum. 75 years after Whitehead published his essay, maybe this will be the generation that finally gets the message.

Posted in Math, Learning |

You can follow any responses to this entry through the RSS 2.0 feed. Trackback from your own site.