On Teaching Grade 4 Olympic Math

For several years now I have taught an Olympic Math interest course at the Grand River Chinese School. This is a weekly, one-hour course on topics in mathematics that students might see on contests, but probably not in their regular curriculum.

I have gradually begun to make available the lesson plans I have used for teaching this course. Indeed, the Number Theory unit is in a state that I am very happy with. I will hopefully soon upload the homework and quizzes for the classes as well to that web page.

During this post, I’d like to reflect on some of the things that I found useful, and some attempts that I found did not work well. Finally, I give a proposed new curriculum that I would love to try out next year.

What Worked Well

  • True or false questions. This seems a really minor point, but I have come to believe that this style of question is ideal for students of this age. True or false questions let me ask questions that involve mathematical reasoning, not just computation, without forcing the students to write down their justification. It also develops a student’s intuition.
  • Geometry (and measurement). I always start the year off with this unit; it’s very accessible, great for developing intuition, and is also helpful for students who write math contests. Next year I might not start with measurement, but that decision is hard for me to make because of the success this unit has had.

What Didn’t Work Well

  • I tried to teach classical logic to the students two years ago or so. It wasn’t a good idea. Students at this age don’t have the experience to understand the rigor involved in classical logic; they need something that is easier to intuit and construct. In the future I’d like to revisit logic, but teach a constructive, intuitionist logic instead.
  • Algebra. While a lot of teachers do cover this in their curricula, and many of them succeed, I have never really been able to motivate algebra in a way that students are engaged in. It doesn’t help that this subject is probably going to be covered in school for them soon. Instead, this year I’m taking an approach where I get them used to variables by using them on the homework, lessons, and quizzes.

The Next Step

My goal for next year is to start and teach just one unit for the four months I’ll be here: discrete mathematics. This will include discussion on sets, multisets, pairs, tuples, and other collections; functions and relations; the proposed revision of intuitionist logic; induction and recursion; algorithms and algorithm design; and finally, probability and combinatorics. This might seem like a lot, and indeed it is. But I think this particular way of ordering the subjects will allow me to cover them in the four months.

Why teach STEM?

One thing that I hear way too often is how high school is useless. The complaint is that the vast majority of people will never use fancy topics like al-gee-bra or will never care what the difference between a neutron and a proton is.

Furthermore, high school apparently does not teach important topics such as how to vote, how to find a job, how to give first aid, and other such topics.

I am not denying that there are valid complaints about the public schooling curriculum. However, this complaint is entirely bunk.

Firstly, at least here in Ontario, the so-called important practical topics are indeed taught. The Ontario high school curriculum requires three courses that are intended to address these practical topics of everyday life: Civics, Careers, and Healthy Active Living. I distinctly remember voting, jobseeking, and first aid covered in these three courses respectively.

Secondly, that science education should be made optional is entirely misguided. The subject of whether compulsory courses should exist at all is a touchy one, but if one accepts that there should be compulsory courses, science should certainly be one of them. The proportion of Canadian graduates in STEM is extraordinarily low among developed countries. Canada is falling behind a vast array of European countries. Just about everyone will accept that innovation comes foremost from STEM graduates. So if we’re falling behind in STEM graduates, we are falling behind in innovation. It is therefore absolutely critical that even if we just make a single course compulsory, it should be science.

The idea that science education is displacing important first aid and survival education that will save thousands of lives is comical, because it ignores the field that has saved millions of lives in Canada alone: medicine. When was the last time you saw someone saved by first aid? When was the last time you saw someone saved by modern medicine?

Thirdly, if we are to keep our modern standard of living, we have to teach ourselves to understand abstractions. Even if someone literally does not use algebra ever in his or her life (which, by the way, is exceedingly rare), the process of learning algebra develops the valuable ability to think with abstractions. James Flynn discusses how so many of the reasons that our society is so much better today than ever before is our ability to understand abstractions, rather than the rote memorization of the past.

While important, voting is far easier to learn through experience than something like abstract mathematics, and as Flynn stresses, the latter is equally important. So if time is really a concern, doesn’t it make more sense to teach someone the quadratic equation than to teach them how to use a ballot box?

Luckily, of course, it’s not a one-or-the-other, and we can teach both in Ontario high schools, which is of course the optimal solution. So once again, this argument is ridiculous.

If there is a topic today that is more productive to teach than mathematics or science, it’s computer science. But shockingly, the people complaining about compulsory STEM are not complaining that CS is optional.

Sampling bias in everyday life

Sampling bias occurs when a survey or series of observations deals with a sample that is, based on something inherent in the methodology, biased in some way. (The Wikipedia article does a much better job explaining it than I ever can.)

This can be seen in everyday life, without conducting systemic studies. I’ve compiled a few examples below.

  • I always seem to travel on the most congested lane. It turns out that this is not just bad luck—the most congested lane has more cars, so basic probability dictates that I’m more likely to be on that lane.
  • How long does the average relationship last? Not as long as you think—you’re more likely to witness a relationship if it lasts longer, so we all have a distorted view of the average relationship.
  • Consider the average number of friends each of your friends has. Because the people you are likely to make friends with are more sociable than the average person, this number is higher than the number of friends the average person has. In effect, this means that most people have fewer friends than their friends do. This is called the friendship paradox.

These fun examples illustrate how observations we make can be biased in subtle ways. Understanding sample bias is not just for the professional statistician, but is in fact important for everyday life.