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On Mathematical Intuition

Issue | Tue, 02/16/2016 - 23:41

To the Curriculum Committee, and in particular, the Fundamental Skills sub-committee,

I am writing this letter, the first in a series, in response to the solicitation of feedback by the committee.

Mathematics has had an unfortunate history, particularly in the United States, of being literature’s unloved sibling, not just among students but also often among those who educate them. It starts early. When a 14-year-old impetuously questions the necessity of Shakespeare, she will probably be told that literature is intrinsically worthwhile, that his plays are amongst mankind’s great accomplishments, that beauty is meaningful and important or that literary works can profoundly transform the individual. When the same question is asked about the value of trigonometry, the typical responses are less flattering: “better” jobs, foundational for “practical skills” and applicable in day to day transactions. Mathematics today has been relegated from the realm of the important to that of the useful. Although it is seemingly paid more attention than ever, when politicians call for better math programs or parents ask that their children take more math, it is generally motivated by some sort of practicality rather than philosophy.

The problem is that the abundance of useful applications for mathematics obscures a great part of its value. First, we forget the true breadth and scope of mathematics’ ambition. There’s logic, algebra, analysis, geometry and topology, probability and statistics. Although they are interconnected, each of these areas is unique in its methods, interests and fields of influence. Most importantly, they are each able to offer to students a distinct type of intuition — logic gives you an intuition for reasoning, algebra for generalizing, analysis for change, geometry and topology for the spatial, probability for extrapolation and statistics for data. That physics, computer science, architecture, engineering, economics and analytical philosophy all have roots in mathematics does not diminish the subject to simply serving those fields but instead points to its ability to plant seeds out of which important and influential ways of thinking often grow. It would seem that in a college whose mission is to bring students together to “seek, value, and advance knowledge, engage the world around them, and lead principled lives of consequence”, it is critical to make the various areas of mathematics readily accessible to students.

This is basically impossible with the way things are set up in Amherst College at the moment. It comes down simply to a lack of people. There is a remarkable shortage of tenure lines in the math department. When you take into account the demands that several other departments make by way of having math classes be pre-requisites, you have a department that is scrambling to fulfill its role as a service department and to offer as many calculus classes as possible (last semester, we had 13 calculus sections). This doesn’t leave much room to create electives in the various areas of math I mentioned that are accessible to a wide range of people and could offer them any kind of intuition about what they are studying. Since there are so few professors, they are forced to use the “efficient” solution of requiring that students have gone through a large number of core classes first before they can get to the non-computational classes. Even if you do go through the whole major, it’s entirely possible to graduate with very little mathematical intuition in any of the areas, since this requires precisely the type of small class setting and experienced professor that Amherst prides itself on, but is unable to offer many of its math majors. As a math major in her last semester, I am astonished that all my professors, but one, have been visiting professors or lecturers, and that I have only been in one math class under 30 people.

Without question, Amherst College has reneged on its promise of student-focused, small classes taught by experienced professors when it comes to math. However, as frustrated as I am as a math major, I’m even more frustrated on behalf of those who are not. Part of what has been so wonderful about my education here is being able to take upper-level seminars in a wide range of departments without having to major in them. It’s unfair that students from other departments do not have this opportunity when it comes to mathematics. This is particularly tragic given that all the students here have already trudged through some of the most tedious aspects of mathematics to get to the college in the first place but do not have the opportunity to enjoy the fruits of their labor. It is my belief that having acquired some form of mathematical intuition ought to be considered a fundamental requirement of students here. However, even a weakened argument that we ought to commit to making such intuition accessible to all our students would be a great improvement on what we have today. As a note of caution, what would definitely be a step in the wrong direction is asking that all students take a math class. Instead, if the department had the people it so desperately needs, it would be free to thoughtfully design seminars that were accessible to a range of backgrounds, and advisers could point students to these instead of telling them that they would “probably find a class in calculus helpful.”

This piece was pretty surprising for me to read. I've felt that the math department is one of the departments through which the benefits of a small college shines most. I guess we've had different experiences, and I'm sure we had different expectations for the department. I've had very few visiting professors as teachers (I can think of only one but I'm likely forgetting someone). More importantly, though, almost every single math professor I've had has had generous office hours, has offered to schedule additional appointments, and has in general been a great mathematician AND teacher. Most of the math professors, and in particular the tenured ones, beg students to eat lunch with them once a week in Val. Student input is always solicited in the hiring of new professors. Parties and talks are offered endlessly. Professors provide students candies and invitations to their homes. I do take your point about the relative dearth of upper-level electives, and I don't doubt that you're right about the lack of tenured positions for professors, but I do take issue with your assertion that "Without question, Amherst College has reneged on its promise of student-focused, small classes taught by experienced professors when it comes to math."

I agree completely. There is definitely a shortage of professors in the math department. Math is something beautiful, and it deserves to be taught in a way that respects that.

I believe it is crucial here to point out that quite honestly, none of the intro math courses actually offer any insight into mathematical intuition and way of thinking: Calculus I, II and Multivariable Calculus are merely the analogues of arithmetics and high school algebra that teach you how to add numbers, but don't teach you how to learn to prove the properties behind addition and subtraction that make higher mathematics tick.

TL;DR: Provide analysis, abstract algebra, and topology without having to take calculus, because calculus courses don't actually offer any preparation for those.

The math department's curriculum and structure do not reflect Amherst's mission. Hence the immediate priority should be to realign the department with the mission statement.

1. "Without question, Amherst College has reneged on its promise of student-focused, small classes taught by experienced professors when it comes to math." This is one example of the above; in contrast see this quote from the mission statement, "Amherst College is committed to learning through close colloquy". 2. Similarly when Shruthi says "Mathematics today has been relegated from the realm of the important to that of the useful", she points out where the math department lies at odds with the liberal arts priority placed on reasoning, critical thinking, and breadth over application. 3. Furthermore the math curriculum at Amherst does NOT hand "substantial responsibility for undertaking inquiry" to the student. Encouraging independent student inquiry into the material (at least in math) requires a certain pedagogy, not the one practiced by most Amherst professors.

To explain 2. I should add calculus and linear algebra (of the 'applications' flavor) do not teach critical thinking, reasoning or even what the stuff of mathematics is in the way the upper level math courses do (and perhaps also the discrete math/introduction to proofs class). Calculus and linear algebra (with applications) are not introductory classes; in the sense that they do not introduce you to the stuff of mathematics. The Amherst math department needs an introductory class along the lines of philosophical questions. This class should be most students' entry point to the math department. This is not a new idea; to see what could be, simply google 'How to Solve It' or 'What is Mathematics'.

I'm not sure if I can explain 3. briefly. Instead let me anticipate a counter-argument. If many students struggle with the material, how can one justify encouraging every student to go beyond the material in their own thinking? This line of questioning is clearly mistaken, and in exposing it one sees the value in "undertaking inquiry".

To summarize the above, I believe the issue is that the mathematics department is limiting itself to only seeing small changes. Reading student feedback, and responding to those; hiring more faculty, and offering more electives. This process buries the larger curriculum, and department wide issues: what is being taught, why it is being taught, and how it is being taught. These issues need not be addressed, and indeed at even the most prestigious of schools they are not. Perhaps this is for good reason; the current system works, and it works well. That said, I would have preferred to attend an Amherst where such issues were addressed.