The following article is heavily influenced by Paul Lockhart’s brilliant article, ‘A mathematician’s lament’. I only hope to add my experiences as a Pakistani student to back his stance in the debate over Mathematics Education.
Throughout my life I have hated mathematics with a passion. I hated its rules and notations. I hated the fact that I had absolutely no say in whatever was going on in the class. I just had to sit there and listen to my math teacher go on and on about formulas, notations needed to write these formulas, practice questions which would help us memorize these formulas and eventually “practical problems” which were supposed to exhibit the relevance of these formulas in everyday life although even the eight year-old me could tell that these were merely the same practice questions loosely disguised in the most unlikely of social situations known to man. And frankly, I didn’t care. I didn’t care where x was, or how much older Mary was than her brother Mark or when train A would reach London. As far as I was concerned math was an obsolete science to which I didn’t want to contribute to and which, for the most part, didn’t really want me to contribute to it anyway.
Therefore it comes as a surprise to many people that I am currently a Computer Science major focusing on theoretical computer science, which is basically a branch of mathematics. I, who had once famously given a speech to my seventh-grade math class about the pointlessness of mathematics, am now the one trying to explain to other people the beauty of Erdos’ brilliant proofs. And it all started with the following beautiful proof of the infinity of prime numbers:
For any finite set {p1,p2…pr} of primes consider the number n= p1..p2..p3…pr +1. This n has a prime divisor p but this is not one of the {p1,p2…pr}, otherwise p would be a divisor of n and the product p1..p2..p3…pr , and thus also of the difference n-( p1..p2..p3…pr) =1, which is impossible. So a finite set {p1,p2…pr} cannot be the collection of all prime numbers.
I first heard of this proof in the first lecture of a discrete mathematics course I took during my sophomore year at university. The instructor didn’t even write the proof down, with all its messy set notation. He just told us about the idea of putting the prime numbers together in a group and showed us what goes wrong if we assume the group to be finite. At first I thought this was one of those introductory shenanigans professors deploy in the first class to get students interested. How could something so simple be counted as math? Where were the fancy symbols and the list of variables with their definitions? Where was the list of steps used to reach the conclusion? Where were the ten similar questions I needed to solve at home for practice? This was simply a clever idea used to solve a problem. Surely, this couldn’t be math! But, as I have learnt in the past year, this is basically what math is: a set of simple ideas used to solve problems. Sometimes the problems can be simplified to older problems for which people have already come up with solutions. Sometimes ideas which have been used to solve a certain problem can be used to solve an unrelated problem. But the simplicity of the process remains intact. It is the ‘idea’ which is at the heart of all mathematics, and to come up with ideas you just need creativity (and maybe a pencil and a notebook).
If a course can change the path of a person’s life, then this discrete math course changed mine. In the course of nine weeks, I was introduced to the kind of math I hadn’t even known existed. For the first time in my life I didn’t feel like a robot while doing math. I actually had to think about the problems and figure out strategies for solving them. While I was introduced to techniques like induction and graph theory, for the most part my assignments and exams required me to come up with my own strategies based on these techniques and my own logical arguments and common sense. Math was like an elaborate game and finally I felt like it actually wanted me to take part.
So, this brings us to the central question: why did I, and countless other students, hate elementary and high school math? What needs to be done to make mathematics more interesting to students? Although I do not have any experience teaching mathematics, I do remember the reasons why I hated it so much and know exactly what eventually made me realize that I wanted to study a branch of mathematics as my major. For the sake of this article, I am going to ignore factors which affect all subjects alike and focus on why math has become such a hated subject.
Looking back at my years of struggling with high school math the first word that comes to mind is boredom. And this was not caused by a lack of interest in school because I was generally a very enthusiastic kid. I loved studying languages, history, and science. It was just math that I dreaded. And looking back at the way math is taught it comes as no surprise. While all other subjects are taught as an amalgamation of the history, foundations, rules and applications of the subject, math is mainly limited to the rules of the subject. Take a typical sixth grade science class. I remember learning about the effect of different factors on the rate of evaporation by placing different shaped beakers filled with water all over the school campus. What followed was a memorable class in which we all had mock “evaporation races” as we timed the beakers to see which one would lose its water first.It was only once we had made our own conclusions about which factors affected evaporation, that our teacher explained Brownian motion to us. She also mentioned factors such as surface area and wind-speed, which most of us had been able to conclude for ourselves based on the observations we had made.
Now compare this to a typical sixth grade math class. Looking back, sixth grade was when some of the most wonderful mathematical concepts were introduced to us. It was in the sixth grade that we first encountered the idea of a variable and started to really analyze shapes. Statistics was introduced, and we started manipulating probabilities to get results which even now give me the feeling of being able to predict the future. But in the midst of all these amazing ideas, this is how a typical math class would go:
Teacher: An isosceles triangle is a triangle which has two sides of equal length. Okay?
Students: YES!
Teacher: So what is an isosceles triangle?
Students: A TRIANGLE WHICH HAS TWO SIDES OF EQUAL LENGTH !
And you can bet one of the questions on the progress test would be: “What is an isosceles triangle?”. In such a situation who would be interested in math? And these are not just two extreme examples I have mentioned to prove my point. Science that year continued to keep us hooked: we grew plants in inky water, caught insects in jars, experimented with mirrors and discovered the material we were supposed to learn, while in math we moved on to triangles which had no sides of equal length (I honestly don’t remember what they were called, though I think it begins with an s) and other lexical atrocities.
You may argue that science is an extreme example and that math just doesn’t have the exciting material needed to keep students hooked. While science teachers can use models, take their students outside or perform simple experiments to demonstrate their material, math teachers have nothing to interest a group of thirty kids. Not only do I disagree with this, I actually claim that it is the other way round and that it is the math teachers that have it good. While science teachers need extensive (and often non-available) funding to buy lab equipment and take their students out on field trips, all a math teacher needs are thirty pencils and notebooks. And how does he keep them interested? Well, he actually asks them to do some math. Do you remember the puzzle we probably all tried as kids in which we had to draw a house without lifting our pencils. That is just a simple example of a Eulerian path. And those complicated strategies for winning card games that our older siblings tried to explain to us were mostly simple applications of probability. The tower of rings of increasingly small diameters which we had to shift to another peg is the most common example given for recursive algorithms. The list of interesting mathematical problems which we solved willingly as kids is endless. Nim, Hex, magic tricks, and riddles in which we had to find loopholes in logical arguments are all example of the math we enjoyed as children and it is these problems which should be bought to the classroom to make math classes more interesting.
Another issue which I find with the way mathematics is taught, which is closely related to the first, is the extreme and almost exclusive emphasis on the utterly mundane aspects of mathematics. Take the isosceles triangle example above. Would it really have mattered if we had called the triangles, “triangles with two equal sides”? Maybe shortened to TWTES (pronounced tevtes). What’s important are the properties of these triangles. Instead of asking a child to spend time trying to memorize the pronunciation and spelling of this weird word, she should be asked to think about how they are made, and how the angles inside this triangle are related to each other. I am pretty sure if a child made a dozen different TWTES’ she would figure out most of their properties for herself and she would actually enjoy the mental excursion of discovering these properties instead of hastily be given a list of them in the last fifteen minutes of class.
Admittedly, there are some terms and jargon that a student of mathematics must learn in order for the classes to be held smoothly and for the students to eventually take part in the wider mathematical discourse. But no other subject puts even half of the emphasis that math places on its lexicon. Take the example of chemistry. If a subject has the right to focus on terminology it is chemistry, with it’s multitude of symbols, chemical formulas and specific reactions. But not once do I remember a chemistry teacher reciting the names of the elements along with their atomic symbols. Instead, we focused on the elements and their reactions and any time we needed help deciphering a symbol we could simply look it up on the huge periodic table taped to the classroom wall. Maybe that is what mathematics needs: a periodic table of shapes and functions which would be taped to the wall of every classroom. Then, children all over the world could forget about mathematical terminology and actually do some math.
And by ‘doing math’ I don’t mean the mindless repetition, or solving exercise problems at the end of every chapter. As a result of school mathematics, most people end up believing math is the application of known rules to problems that we know the rules can solve. That is the job of an accountant or a cashier or an insurance planner. A mathematicians job is much simpler. He must come up with the rules that other people are to use. When faced with a problem, he is not told that it can be solved using the second trigonometric identity; that is what he must figure out. And while this is harder than simply applying a set of rules, the result of coming up with a solution is infinitely more rewarding. You can compare the two as the difference between the joy a child feels in having an adult place him on a bike and push him along, and the joy he feels when he races through the park himself. It is hard to teach him how to ride and it might take him ages to learn but all parents understand that the end result is worth it. Math teachers should definitely do the same with their students.
And if difficulty was such a major barrier, why doesn’t it stop teachers of other subjects from trying to get their students to appreciate the beauty of their fields? By the end of high school most of us have faced the toughest aspects of most of the other subjects. We have read Iqbal’s poetry and critiqued it with our peers. We have a deep understanding of how the major systems of the body work. We have built electrical devices and have made original pieces of art in a range of different mediums. Then, why is it that most of us only experience the joy of coming up with a true mathematical proof well into our undergraduate programs? Surely there is something wrong going on here.
In Fall 2008, the Lahore University of Management Sciences (LUMS) opened its doors to 150 freshmen students to study science and engineering at its brand new 
