What I Learned in Math School

Posted: April 21, 2013 in Uncategorized
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E.O Wilson recently published an article in the Wall Street Journal where he argues Scientists don’t need math skills. He feels that many potential scientists are being turned away from science because they find math hard. Wilson goes onto argue that scientists in many fields don’t really need much math. I’m not going to address Wilson’s argument directly but instead talk about what I learned about math in school.

There is a degree hanging on my mother’s wall with my name on it that says “Bachelor of Mathematics in Computer Science”. In high school I didn’t particularly enjoy math and I can’t say I was all that good at it. I wanted to go into computers and Waterloo had a reputation for having one of the best computer science programs in the country. At the time getting a computer science degree from Waterloo meant enrolling in the faculty of mathematics and getting a bachelor of math. I understand you can now do computer science at Waterloo without getting a B. Math but that wasn’t the case in my day.

Getting into any respected computer science program required good marks in high school math. High math marks weren’t coming naturally to me so I spent long hard hours working on getting acceptable math marks. Lesson one: If you get put in long hours learning something your not good at, and don’t enjoy, you can often get good enough at it. Other people learn this playing a musical instrument or through sports but the important point is learning how effort and determination can often make up for lack of talent. Even though you might never get as good as someone who is naturally talented you can often get good enough. Hard problems often require hard work and it is important to learn this.

The University of Waterloo Computer Science B.Math program gave students a broad exposure to most branches of mathematics in addition to a much deeper treatment of computer science. The math courses I took in my first two years were the same math courses any other (pure math, applied math etc..) math major took. As a first year math major I discovered that I didn’t know very much about math. First year mathies at Waterloo took a classical algebra course where we were taught how to prove things with mathematical rigour. By the end of the first week I was totally confused. High school math for me was about the mechanical manipulation of numbers we weren’t taught how to prove things. I would spend much of the next four years writing mathematical proofs. One of the early proofs in the classical algebra course was to show when a linear Diophantine equation had a solution. Until I dug-out my old textbook ago I had forgotten what a Diophantine equation was but looking at the textbook I can now clearly see what we were actually being taught in first year. The point of wasn’t to learn how to solve a Diophantine equation it was to learn how to go about proving things. Lesson Two: To show something mathematically you break the problem down into small pieces and solve each component. You build up your solution by first proving the parts (theorems and lemmas) that you need for your overall solution.

One of the reasons why teaching logical reasoning through mathematics works so well is because math is a very precise language. Lesson Three: When solving a problem, using precise language is important. Precise language allows me to be very specific about what I am talking about which allows me to say things that are verifiable. I can get away with saying almost anything using general non-specific language because I can phrase things with such generalizations such that I am not really saying anything at all. The language of generalizations works great for politicians and blog authors but doesn’t actually contribute much to coming up with real solutions to scientific problems.

A common type of question on my math assignments and exams was to show that a particular mathematical statement was true. In this type of question you need to provide an answer that walks the reader from a starting point to a known conclusion. I wasn’t being asked to find the answer to a question but to explain why the answer they have given me is true. Sometimes I couldn’t figure out all of the steps to go from statement A to statement B but I knew it was possible and required since I could work backwards from the result I was trying to show. One approach I tried (more than once) was to wave my hands and say A=B (as if by magic) and move on with the rest of the solution. The markers almost always called me out on this and put “Why” in big red letters next to a -1. I knew A was equal to B and the markers knew A was equal to B but knowing something is true doesn’t count for much in math if you can’t explain why it is true. Lesson four: If you don’t know the reasons behind something and you try to fake or hand-wave your way through you are going to be called out on it even if your conclusion is correct. This lesson is very important for scientists because science is about explaining the reasons behind the things we observe in the real world. Science isn’t about calculating how long a ball dropped from a 100m high building will take to hit the ground, science is about explaining why it takes 4.5 seconds.

These four lessons aren’t the only things I learned about math while getting my math degree but they are some of the most useful lessons I learned and are well worth the time and effort for anyone considering a career in science. Some scientific fields don’t require advanced calculus but many do and most require at least basic statistics. All scientific fields require the disciplined rigour that I learned in advanced math courses. Science isn’t about coming up with a creative story explaining the world that sounds good at cocktail parties it is about doing careful work to collect and analyze data that validates, or invalidates a theory. E.O Wilson’s students might be finding math hard but doing good science is also hard and our scientists of tomorrow shouldn’t be afraid of some hard work.

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