My pre-college mathematics education was probably different from most others. Instead of adopting the standard approach of teaching Algebra I and II, Geometry, and Trigonometry, my school district took up the Interactive Mathematics Program (IMP), a problem-centric approach to learning the essential material from these subjects. The program was split into 4 courses, each the equivalent of a middle school year or high school semester, and each course was split into approximately 5 units. Each unit introduced a mathematically-oriented story, which gave rise to a guiding question that we sought to be able to answer over the course of the unit. For example, in the second year unit “Cookies”, we are introduced to the Woo’s, a family of bakers who wish to optimize their cookie baking choices to maximize their profits. The challenge though is that they have several constraints on their available ingredients and sales capabilities. Over the course of the unit, we learned several ideas related to systems of equations and inequalities that helped us answer this question.
Although I didn’t appreciate it at the time, IMP was attempting to do something that is rarely seen in education these days but is extremely important: motivating the concepts. At the beginning of the “Cookies” unit, I remember that class immediately started off with an attempt to solve the unit problem—no instruction on the best mathematical techniques, just a lot of pain, backtracking, and guesswork. I remember being overwhelmed just after organizing the information provided on the family’s ingredient and sales constraints. And guessing potential solutions was memorably laborious: does it work to make 100 chocolate chip and 170 oatmeal raisin? Rats! That violates constraint 4! What about 150 oatmeal raisin? Nice! That’s allowed! Aw shoot, that makes less money than my 150 chocolate chip, 130 oatmeal raisin combo! Fifty minutes of this and I was pulling my hair out. There must be an easier way to do this! And there was! Which we came to learn over the course of the next month or so that we spent on the unit. This particular unit sticks out in my memory primarily because it did such a good job at motivating the need for the main mathematical tools that we were learning during the unit. Had I taken a standard algebra course, I know I would not have truly internalized the utility of what we learned during that unit. For the most part, IMP did a decent job at contextualizing math concepts, and I think that this is a quality that is lacking in higher education in general. In designing curricula, we are not doing enough to motivate the content of the course, and I plan to explore this idea much further in a future post relating to introductory statistics curricula.
Another aspect of IMP that is particularly well suited to higher education is its emphasis on summarizing and organizing knowledge gained over the course of a unit. At the end of every unit, students are required to create a portfolio of reflections, class notes, and assignments that were instrumental in their comprehension of the main topics. So for example, in my “Cookies” portfolio, I included the first day of class activity that had us take a stab at the unit problem. I also included my class notes on feasible regions and solving systems of linear inequalities. I also wrote a cover letter summarizing the goals of the unit, what I learned from the assignments that I included in the portfolio, and my impressions of how these concepts were useful in everyday life. Portfolio time was always a laborious and dreadful experience for me during middle and high school, but I recognize now how useful a practice it is for being serious about retaining knowledge. As a graduate student, I have been exposed to several fundamental concepts time and time again in different courses, and I have found myself consciously wishing that I had put together portfolios for many of the classes that I took during college. Not only is the act of putting together a portfolio an invaluable synthesis activity, but the portfolio also serves as a one-of-a-kind reference manual. Because it is a collection of notes, homeworks, etc. created, curated, and edited by you, it can be infinitely more readable then a textbook. If you want it to, it can contain all of the steps in the proofs of key theorems, all of the exhaustive explanation that you worked through on your own, all of the fine details that finally made a concept click. A portfolio is a mini-textbook written in the language of your mind and can thus potentially serve as a reference for a very long time. It is a wonderful idea for IMP to introduce this concept to younger students, and I feel that it would be very useful for many high school and college classes to adopt.
So essentially the Interactive Mathematics Program, though by no means flawless, takes steps beyond traditional education practices that definitely have merit. Its case-based structuring of learning and its emphasis on creating concept portfolios can really impact the depth of learning and are ideas that nearly all higher education courses can benefit from.