On the topic of mathematical education

Yuqi Xiao, an undergraduate engineering student at UBC, has kindly allowed us to republish her article about mathematical education. You can visit her personal website at ukistatemachine.com!


“Why do we learn math if we only ever need it to count the number of apples?”

The morning after I started this blog, someone from my real analysis class shared the above Facebook post with me. It’s not uncommon to hear complaints about the utter “uselessness” of mathematics. Back when I was attending grade school in China, such a remark was quite popular: “you only need math when you are grocery shopping”. And our country was one that held elementary education at high prestige.

You go on to high school and then college, and still people who are taking math courses complain about the lack of real-life applications of abstract mathematics. I tend to think such complaints are not originated from ignorance or anti-intellectualism, rather a lack of perspective offered in current math curriculums. But is math really useless? Are mathematical concepts made up randomly just to make our school lives miserable? If the answer to these questions is no, yet the popular perception is yes, one may speculate our mathematical education has left out important contents that led to this misconception. While I am not an expert on the subject matter, I am extremely passionate about the foundation of mathematics, including how things like algebra were invented. And I may be able to provide an insight or two on the “usefulness” of mathematics. Eventually, we will identify what’s missing from our current math curriculums.

Understanding mathematics as a discipline

In order to make valid arguments, a writer and her audience must reach a consensus on the scope of discussion and relevant definitions. But math is such a huge topic, so why don’t we start by following the path that conventional mathematical education leads us on?

You may recall from your childhood that counting, addition, subtraction, multiplication and division were your first encounters with math. These topics were chosen as a starting point because they were the simplest concepts to understand. From then on, we can proceed with math from either of two directions. One direction is through the constructive approach. From counting natural numbers and basic arithmetic operations, we build integers, rational numbers, real numbers, and acquaint ourselves with more advanced topics such as calculus, topics that are typically taught in high school or college. The other direction is analytical, where we go back to visit the very beginning of logical deduction. Despite the absence of logical deduction from current math curriculums, this direction isn’t trivial. It can be highly intricate, and typically, it is not taught in university programs that are less math-intensive.

“The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle.”

BERTRAND RUSSELL, INTRODUCTION TO MATHEMATICAL PHILOSOPHY

Just as how in real life, the easiest thing to observe is neither galaxies from far away nor microorganisms resting on a surface right in front of us, the most obvious things to notice in math comes somewhere in the middle. Therefore, to gain a comprehensive understanding of the mathematical reality, both the constructive approach and the analytical approach are vital, much like how we need both a telescope and a microscope to observe the virtual reality.

This is where we see a gap in our mathematical education. Students are not formally taught analysis until upper years of university, which is when most students cease to interact with math. To these students, the foundations of mathematics and how we construct mathematical concepts remain unclear. The idea of proofs is missing. Math has always been plug and chug. So it is not surprising that students under this education system may convince themselves that concepts in math are completely made up.

Constructing mathematics

When presented in textbooks as lengthy equations and abstract theorems, mathematics seems unapproachable, one may assume it came out of nowhere. Whereas the development of mathematics is so closely related to human activities. Human physiology, agricultural needs, religion and even certain celebrities may introduce pivotal points to the construction of mathematical systems.

There are numerous examples I could name. For instance, why is it that we have adopted a base ten counting systems (i.e. nine special symbols and one “zero” symbol to denote empty space), instead of, say a base fourteen or base two counting system?

The number ten may seem like an arbitrary choice, but it happens to be the number of our fingers. As one of the most important and certainly most fundamental mathematical tools, counting was needed in early civilization to facilitate hunting, food distribution, and later on, trade. And fingers were the most accessible tool for counting at the time. Of course, there were other counting systems that were used historically. The ancient Mayan calendar had a base twenty. There is evidence that this system followed women’s menstrual cycles, which is 20-28 days, another indication of how human physiology impacted the birth of mathematics.

And it’s not just counting systems. To give a more advanced instance of mathematical formulation, let’s take a look at Georg Cantor and his idea of finitude. It may seem intuitive now to claim that the set of all natural numbers and the set of all real numbers are infinite. However, in the mid 19th century, infinity was considered to be an identity that’s preserved only by God. Cantor was the first to be able to prove that there are more real numbers than natural numbers, the cardinalities of both being infinite. The idea that there can be infinity outside of God, and that these infinities differ in magnitude, was considered unorthodox. Being a very religious man, Cantor was concerned with the orthodoxy of the relationship between God and mathematics. He attempted to define two kinds of infinity, one is transfinite, which we seek in mathematics, these infinities can have different cardinalities; and the other is absolute infinity, which he identified with God, this infinity is above all infinities, and is non-increasing. Despite framing his proposition in this manner, some Christian theologians considered transfinitude a challenge to the uniqueness of absolute infinity in the nature of God, and Cantor was accused of blasphemy. This is one of many instances in history where propositions in mathematics had to comply with religious beliefs.

Now to answer your question: how was algebra invented? You may be disappointed by the generality of the answer I am about to give: it is scattered all throughout history. For instance, how did someone randomly come up with the equation y=mx+b? This is a question that I can answer specifically. Someone named René Descartes invented the idea of representing an unknown variable with the symbol ‘x’. Before that, y=mx+b would be equivalent to something like: solve for the variable that induces a constant linear increase for a known variable with some constant overlay. I’m sure you can come up with real-world problems that call for such a definition. However, symbolic representations offer us a much more clear and concise way to generalize and unpack these situations.

To give a more concrete example, let’s look at y=x^2. This can be written in an approximate form of y=mx+b as y=1*x^2+0. How did this equation come about? For ancient Mesopotamians(modern day Iraq), this would mean adding the token x to itself for x number of times. For ancient Egyptians, it would mean the area of a square, each side with a length of x. As early as 1800 BC, humans have developed algorithms to calculate the square root of a known variable. Of course, at the time the question was posed in an approximate form of: what is the number when multiplied by itself gives us the number of which we are square-rooting? Nevertheless, ancient humans possessed a pool of knowledge in mathematics, which we’ve taken for granted today.

And that’s the story of how mathematics was constructed into what it is today. We’ve taken different routes, had setbacks due to societal conventions, and invented and discarded a lot of rules and ideas. But whenever there is a problem we want to generalize and to solve, we develop mathematics. Looking back in time, a huge portion of mathematics was heavily influenced by human needs and human activities. Mathematics is a human discipline, however it hasn’t been taught in a humanistic way. Maybe instead of shoving formulas in textbooks, modern education could lead its students down to the root of mathematics, explaining when it was needed, how it was developed. If we do a successful job in this, I’m sure the discipline would be much more appreciated.

Uselessness

Still, I cannot argue if someone claims that there are ideas in pure mathematics that just don’t have real-world applications, at least for the moment. Dedekind cuts (i.e. defining real numbers with the set of all rational numbers that are less than or equal to themselves) would be one of them. So much of mathematics exists only as formal theories and is studied only for its natural beauty. Should we then disregard these concepts, since they appear to be useless at this point?

While we may be tempted to answer yes because of all the pain we as students go through, having to learn abstract concepts that do not have apparent real-world applications, history tells us what seemed to be useless at the time may become tremendously useful when it’s called for its needs. Until very recent history, number theory has had trivial applications in reality. But today it is the foundation of cryptography, firewalls are built around it. Many mathematicians who specialize in number theory are being hired as consultants for internet security companies. Had we forfeited the field, modern day cryptography wouldn’t exist.

Even if we can’t find nontrivial real-world applications of some mathematical concepts after a prolonged period of time, would it do us any harm to have a more comprehensive knowledge system? Knowledge being acquired can always facilitate as tools to investigate more unsolved mysteries. As we refine and develop our mathematical system, we are only adding more substance into our toolbox. Hence, uselessness isn’t definite in mathematics, as it is not always essential.

The social sciences

To end this subject, I would like to describe in detail an example that shows us the importance of foundational mathematical education in the social sciences.

In a study conducted on social organization of sexuality (Laumann, E. O. (1994). The Social organization of sexuality: Sexual practices in the United States. Chicago: University of Chicago.), a group of researchers from Uchicago interviewed over 2,500 people, selected at random, over several years. One of the major claims they made was that men on average have 74% more opposite-gender sexual partners than women. Graph theory dosen’t agree with this claim. Now, without using graph theory terminologies, I will attempt to show you why such a claim is indeed far from being accurate.

Let’s look at a simple case. We denote women with Arabic numerals and men with letters, and let a line between them represent that these two people are opposite gender sexual partners. Taking five women and five men, we can generate the graph below.

This graph is made by my dear friend James after he blatantly bashed my drawing and hand-writing in my original graph. Thank you James :))

In this specific graph, woman 1 has five sexual partners — just because it’s her body and she can do whatever she wants with it^ today we celebrate that. Then, each of the other four women has one sexual partner. To compute the average number of sexual partners for women, simply count the number of lines associated with each woman, then divide that by the number of women we included. Do the same for men, and we will find that the digits are exactly the same. I claim that this is not an isolated incidence, because every time an opposite-gender couple become sexual partners, the total number of sexual partners for both women and men will increase by one. To put it in simpler terms, it’s impossible for a man to be sexual partners with a woman who is not a sexual partner with him, and vice versa.

With that in mind, we should be convinced by now that for a large enough sample size, the average number of sexual partners for men and women should be approximately equal to the ratio of female population to the male population, which is 100:105, according to the World Health Organization.

How do we account for the huge deviation from the expected value to the results in the survey? It turns out, men tend to round up and hence overestimate the number of their sexual partners. You can sort of tell since 80% percent of men responded with a number that’s divisible by five. However, this is a whole different topic. Point being made, foundational mathematics and the logic that it employs provide valuable insights for interpreting data in the social sciences, and should be placed higher precedence in this branch of education.