Introductory-level mathematics education is a festering wart on this country’s nose.
More locally, Rutgers—a university which touts some of the best researchers in applied mathematics, as well as a top-twenty graduate program—is doing nothing to heal the deep intellectual wounds incoming liberal arts freshman have sustained as part of their mandatory ‘mathematics’ education in public school. For those not familiar with Rutgers’ sorting process for matriculating students: incoming freshman have to take a math placement exam to determine if they will be eligible to take “Intermediate Algebra”, “Precalculus”, “Calculus for Business and Economics”, or “Calculus for Math and Physics Sciences”. If you place into Precalculus or higher, you have the option to take a course called “Math 103—Topics in Mathematics for The Liberal Arts”, to satisfy the Rutgers general ‘quantitative reasoning’ requirement. Many liberal arts majors enroll in Math 103 intending for it to be the last math class they ever take.
In this article I want to look at some of the selected topics for that class, and use them as a stepping stone to discuss the problems of math education today, as well as some possible fixer uppers.
Why do People Hate Math?
The Liberal Arts have been defined as “those subjects or skills that [are] considered essential for a free person (a citizen) to know” . Math still lives under the umbrella of the liberal arts—formally since the 5th century AD, in all likelihood since the time of the ancient Greeks. Why then, has it lost its connotation as a profoundly aesthetic, uplifting, and quintessentially human activity? Why do people not associate math with mental freedom and creativity? Take, for example, a clip from this Veritasium episode (1:24 ), where the young woman explains that she and her companion are “creatives, not intellectuals”.
Depressingly, this is the type of attitude many people have: “oh, I’m just not a technical person”, “oh, I HATE math,”, “oh, I hated science in highschool”, etc. You know the type.
So why do people think this way? Well, I could write about a book about the topic, but, fortunately, it’s already been done. A Mathematician’s Lament by Paul Lockhart is literally (yes, literally) the greatest exposition ever written on the failings of early mathematics education. Here’s the PDF. Read it. Anyway, long story short: public schools condition people to see math as this repulsive, unintelligible, and difficult thing, primarily focused with computation, long-winded definitions, and real-world application. This is compounded by the idiocy of requiring math teachers to have 5-year advanced degrees, effectively ensuring that no practicing mathematician or mathematically talented person will ever become a math teacher. There are countless more contributing factors (such as common core, standardization, etc) but again, read Mathematician’s Lament.
Rutgers Math 103 Topics
Returning now to our problem of letting 1st year liberal arts major see what mathematics feels like, let’s check out what they’re currently being taught. The Spring 2014 syllabus for Math 103 conveniently provides a list of 13 things students who ‘successfully complete the course will be able to do’. I’ve posted a representative few for you below—completely unaltered. The rest of the list is available in the link under “syllabus”.
- Determine winners of elections under different voting methods, and use these to rank the candidates
- Compute the future value of assets and value of a deferred annuity, and apply the notion of exponential growth to other contexts than financial ones
- Apply approximate algorithms to solve the traveling salesman problem
- Analyze the feasibility of performing certain brute force computations, in contexts of weighted voting systems, traveling salesman problems, and scheduling of tasks.
- Be able to articulate their understanding of the above items, in clear English
Now, let’s give incoming liberal arts majors the benefit of the doubt here and assume they have all devoted their lives to the pursuit of beauty, truth, and aesthetic gratification. They want to know what it means to be alive and better themselves both mentally and spiritually. Rutgers has a golden opportunity to show these starry-eyed romantics what math is all about—truth-with-a-capital-‘t’, internal necessity, elegance, mistakes, transcendence, fun, and logic. But instead, they teach them about ‘the feasibility of performing certain brute force computations’, ‘rank[ing] candidates’, and ‘scheduling of tasks’. I’m not saying these topics aren’t interesting, of course they are (resting on my bookshelf not five feet away from me is an entire book on the Traveling Salesman Problem). What I’m saying is that freshman liberal arts majors are being swindled! Bamboozled! They think they know what math is, but they don’t. How could they? It’s never been shown to them; they’ve never been allowed to poke and prod and explore inside that pearly world of Platonic idealism. They’re being deprived of humanity’s most enduring intellectual achievement, sans written language.
How do we fix this? How do we teach math–real math– to a bunch of bright beauty-seekers, when they’ve been conditioned their entire lives to approach it as this unattractive, arcane, and difficult thing?
Bring Back Euclid
Twenty-three centuries ago, in the great city of Alexandria, Euclid wrote the Elements—a collection of ‘465 propositions from plane and solid geometry and from number theory’. Through its effect on medieval philosophy and cosmogonic theory, to its impact on individual people, no book from Western Civilization (besides the Bible) has had more influence than the Elements. Abraham Lincoln was famously enamored of Euclid, as portrayed in this scene from Spielberg’s Lincoln:
Einstein called the Elements his ‘holy little geometry book’, and Bertrand Russell reminiscently said that: ‘At the age of eleven, I began Euclid…This was one the great events of my life, as dazzling as first love.’ (Dunham 31). The list goes on almost indefinitely.
What makes Elements so amazing? Could it just be the romantic allure of reading something written over two thousand years ago? A Western obsession with ancient Greece? Or is there something to Elements which made it the #1 geometry textbook for centuries? Something that the U.S education machine has forgotten or ignored? Mathematicians and historians will invariably tell you the following: Euclid was the first known person to take an axiomatic approach to math,—to start from first principles,or ‘common notions’, he found to be self-evident–and from them prove True geometric properties. But that’s not the full story. Let’s revisit ol’ Abe Lincoln and hear what he has to say on why one should study Euclid:
You never can make a lawyer if you do not understand what ‘demonstrate’ means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight.
Right on the money, Abe. Yes, Euclid teaches math; but, more importantly, he teaches reason. He shows that you can start from first principles and derive almost 500 True propositions! Using nothing but that gelatinous, pulsating organ under your forehead, you can find Truth in this world. And if your demonstration is air-tight, the jury will always vote in your favor. That is why Elements endured as a bridge across time,, linking students of the liberal arts, for so many generations. From Wikipedia:
For centuries…knowledge of at least part of Euclid’s Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.
Okay, let’s go back to our incoming Rutgers freshman and start wrapping up. Universities should bring back Euclid because it makes students feel like they’re partaking in something which connects them to their intellectual and philosophical heritage; something that brings them back to the dawn of mankind’s intellectual re-birth; something that places them right in the middle of The Library of Alexandria, surrounded by brown scrolls and bearded geniuses muttering in Egyptian and Greek. Euclid will let them explore a branch of math that is predominantly visual and uses just the most basic mathematical operations; and by doing so, put their general faculties of reason against the best whetstone there is. To conclude, let me give you an example of the type of math I’m talking about. The following example was taken from Mathematician’s Lament (book version).
Problem: Given a line and two dots randomly placed above it (as seen in figure 1), what’s the shortest path from one dot to the other which bounces off the line?
What you’ll need to know: that the shortest line you can draw connecting two points is a straight line from one to the other.
Solution: Choose your favorite dot. I like the right one, so I’ll call him Righty, and the other dot Lefty. Look at how far Righty is from the line. Call that distance x. Draw Righty on the opposite side of the line, with a distance of x separating him and the line (figure 2).
Notice that the shortest distance between the Lefty and the ‘reflected’ Righty is a straight line connecting them (figure three, purple dashed line). That’s obvious, now here’s the kicker: notice that the green dashed line in figure 4 is the same length as the purple dashed line in figure 3. Therefore, the green dashed line is the shortest line we could draw connecting Lefty and Righty while bouncing off the line.
That’s it. Do you see what I mean? No more words necessary. Of course we could sit and calculate the actual length and use the Pythagorean theorem and all that, but why? It’s not important. What’s important is that we solved the problem once and for all! Given these conditions, we can ALWAYS find the answer–our argument, our solution, they’re both true-with-a-capital-‘t’. So tell me, which is more interesting: the problem we just solved, or “comput[ing] the future value of assets and value of a deferred annuity”?
Leo Kozachkov (Staff Writer, Rutgers University) Leo Kozachkov is an undergraduate at Rutgers University, studying physics and mathematics. He is currently working as an Aresty Research Assistant under Professor Thomas V. Papathomas. He enjoys writing, drawing, creating/playing music, going on long walks with his beloved dog, and reading/hoarding books. His grandest hopes are to discover a new physical law, have a mathematical theorem feature his last name, and to write many books.