# The purpose of mathematics in a liberal education when it is not a prerequisite to other subjects

Suppose a calculus classroom is full of students majoring in Classical Greek or music or literature or sociology or pre-medical studies or any of many subjects that do not require the course as a prerequisite, and not one of them intends to take a subject requiring as a prerequisite any of the technical skills (finding derivatives, limits, integrals, etc.) that could reasonably be taught in calculus course, and every one of them has said so.

Should a syllabus contain something like the following statement?

• The purpose of this course is neither to prepare you to use your knowledge of calculus nor to provide you with an opportunity to show you can work hard and overcome difficulties and thereby earn a grade to impress future employers, graduate schools, etc. Those may be side-effects, but neither of them is the intended purpose. The purpose is to contribute to your education by acquainting you with one of the great intellectual, scientific, cultural, and aesthetic achievements of humankind and with the kind of thinking that is used in this field.

• Most students have never suspected that there are other purposes than the two that this statement says are the purpose of the course. (In a reasonable system, they would find that out long before reaching calculus. The main thing that is unreasonable that differs from such a system is our current practice of pressuring or even coercing students known to be unqualified to take calculus to do so, anesthetizing ourselves by telling ourselves that the systems for screening out the unqualified actually work.)
• The usual syllabi say one must cover the following technical topics (computing certain limits, derivatives, integrals, etc.) within the allotted time: [insert list here]. The perceived imperative to cover everything in the list precludes many things that would help students understand. When I mentioned one such thing, I was told "It would be nice to include that but there's not enough time." I said "So scrap ninety-percent of the material and then there's time." I don't know if that was taken seriously. I don't of its being implemented anywhere. And the usual textbooks are for the sorts of students for whom it is reasonable to insist on covering everything in a list of technical topics.
• I think including a statement like this would be a mistake. While I understand the basic sentiment that you're mulling over, I think this would be lost on most students at this level, and they would just see "this course is useless." I'd like to add that I think a valuable purpose of this course would be "to provide you with an opportunity to show you can work hard and overcome difficulties," though again, I see why you included it as debatable here. – Andrew Sanfratello May 22 '15 at 8:22
• Is this is a reasonable scenario for your class? Applied majors are growing and are strong majority of my students, the majors you talk about are static. – David E Speyer May 22 '15 at 14:10
• @DavidSpeyer : It's realistic for a substantial proportion of each class. – Michael Hardy May 22 '15 at 19:16
• I agree with your syllabus's statement up until the last sentence: "They think learning the material is only a price paid for a grade, and are quick to object to any attempt to over-charge them."--I'm not sure what that sentence even means. I think it's as important for a liberal arts major to become familiar that Calculus exists and that it's not impenetrable as I think a mathematics major should be aware of history and literature. – Jared May 23 '15 at 5:57
• @Jared : That sentence was never intended to be part of the syllabus. – Michael Hardy May 23 '15 at 14:36

St. John's College offers a four-year undergraduate program called the Great Books Program focusing on the study of western civilization's classics from primary sources. Regarding calculus, the program's curricula includes Newton's Principia Mathematica and selected writings by Leibniz, Taylor, Einstein and others.

I have only heard and read about the program and have no direct familiarity with it, so this might not be considered a full answer. But perhaps researching into this and asking someone who took that program would allow you to get better insight into such studies of calculus.

• The nature of mathematics is such that when one of the great geniuses first writes a theorem and a proof, often some lesser mathematician later discovers a simpler and more easily comprehensible version. This makes the question of just who should read the works of the geniuses perhaps somewhat problematic. I suspect those who'd rather not take any math course for any reason if it weren't required might be better off doing something other than reading Newton. But probably that's not who shows up as students at St. Johns. – Michael Hardy Jun 1 '15 at 19:47

I understand and sympathize with your point. Just, as a matter of emphasis, I think it is never so nice to present a course from the point of view of what you do not intend to obtain.

I'd rather turn it into positive and explain that it is mainly through understanding how to manipulate some math you can come to appreciate "acquainting you with one of the great intellectual, scientific, cultural, and aesthetic achievements of humankind".

Students majoring in Greek or music or literature are already committed to education aimed at appreciating the "aesthetic achievements of humankind," and may already be approaching the topic from that direction. Students majoring in sociology and pre-med might actually have to understand aspects of calculus in their studies. And students majoring in $X$ often end up pursuing careers in $Y$, regardless of their intents back in college.

To say that obtaining a useful "knowledge of calculus" may be a "side effect" diminishes that aspect. To me there are two purposes, roughly equally important:

(1) The appreciation of the "aesthetic achievements of humankind" you articulate.

(2) Obtaining an understanding of the concepts of calculus, both as generic intellectual equipment, and for possible use in other avenues of study.

Concerning (2), I would emphasize understanding over skill at calculation. Unfortunately, calculus courses often focus on the mechanics (integration by parts, etc.) and just hope that understanding follows. For me, understanding only followed long after I received an $A$ in calculus.

• One problem with "may already be approaching" is that most students have never suspected that that aspect of the subject even exists. And pre-med students are sometimes quite ruthless about getting grades and think you're distracting them from that if you tell them anything besides "Here are the algorithms for solving the problems on this assignment." – Michael Hardy May 22 '15 at 19:15
• @MichaelHardy: I acknowledge that many students are unaware of the aesthetic side of mathematics, but I would not emphasize that as the intended purpose in the syllabus. Rather, I try to convey those aesthetic aspects throughout the course. – Joseph O'Rourke May 22 '15 at 20:46
• Student who are fanatical about grades are often closed to that sort of thing. I at least think they should be told they shouldn't close themselves off to it. – Michael Hardy May 22 '15 at 21:00

I learned songs because I enjoyed singing them, and only much later did I learn songs to perform them for others. I learned history so that I could understand some of the cultural references involving history, and also in the vain hope that I could make newer, better, and more original mistakes. I learned some literature (but it didn't sink in until years later) to see why people are interested in some of the things they are interested in, as well as in how a story could be told well. I learned some debate because my father told me to, but also to practice the ability to think and produce arguments in real time, as well as prepare researched arguments.

I learned math because I was good at some of it, and good at explaining it. I became good at certain modes of reasoning, which helped me with economic and engineering problems as well as more abstract problems. I can now go back to some of the stories I've read and think "If this character had known calculus, they could avoid the problem in the next chapter if they just..." .

If your goal is to prepare a good syllabus entry, you should emphasize the difference in perspective that analytical reasoning and inductive reasoning can provide. Even going through what seem to be mindless computations trains one in mechanical reasoning, and can be useful in trying to debug problems they find with software systems. Not a software engineer? What's that thing in your back pocket? You're going to have to deal with that user interface on your phone-watch-pda somehow.

Gerhard "Ready To Turn The Page" Paseman, 2015.05.22

Some math is needed in a liberal arts degree program because of the way it helps develop thinking skills. An individual may not get a job that is directly lined up with their degree, but that background may help them land a job. In the March 17,2014 New York Times article "The Walls Close In", 80,000 people applied for 2,000 job openings at a VW plant that opened in Chatanooga, TN, in 2011. How many had a degree is not given, but 78,000 people didn't get better jobs because of a lack of basic math skills. This information was verified by economics professor Matt Murray. Allen Seay

• The article online is now titled "Low-Wage Workers Are Finding Poverty Harder to Escape". – Rory Daulton May 31 '15 at 12:23
• Math courses for students not interested in math usually degenerate into teaching students algorithms for doing assigned exercises, and students graduate thinking that's what math is. That doesn't develop thinking. One big obstacle to honesty in math teaching is a list of topics that supposedly must be covered by the end of the semester because they are technical prerequisites to courses that the students will never take. – Michael Hardy May 31 '15 at 13:20
• Saying that math helps develop thinking skills in a liberal arts program, I want to share the following comment from "Conceptual physics", Twelfth Edition, by Paul Hewitt. On page 181, he says "Why study this material, especially since you'll forget most of it? My answer is that whether or not you use this knowledge, in the act of learning how to connect concepts and solve problems you're establishing connections in your brain that didn't exist before. It's the wiring in your brain that makes you an educated person. That wiring will be useful in areas you can't dream of right now." A Seay – Allen Seay Jun 1 '15 at 8:06

The purpose is to contribute to your education by acquainting you with one of the great intellectual, scientific, cultural, and aesthetic achievements of humankind and with the kind of thinking that is used in this field.

The whole thing sounds counterproductive - you can't make people have an aesthetic appreciation of something, and trying to do so by making it a required prerequisite of something else is just going to cause them to associate the subject with annoyance. Just as studying a "great book" in a mandatory class will leave half the audience with a distaste for it.

This goes double if the students are adults paying for the course, directly or through accumulated debt.

It sounds like what is really needed is to disguise the course in the language and structure of "History and Philosophy of Science". Teach people about the context, Newton and Leibnitz, and the problems it was developed to tackle, and have them actually do the worked examples, proofs and derivations almost as a side effect.

• Indeed, you cannot make someone become educated. In a classroom I try to enable students to do things, not make them do things. Only those (many, unfortunately) who perceive the earning of grades as being their goal and learning as being the means will think I'm trying to make them do something. If I could, I would abolish grades and degrees of that's what it takes to remedy this problem. – Michael Hardy Feb 10 '18 at 17:18