How to be successful in MAT240 / 247 at the University of Toronto.
Published August 17, 2019.
I received an email yesterday afternoon from an incoming freshman at the University of Toronto. You could tell that they were very excited to start and they couldn't wait to get their hands dirty with mathematics. They would be taking the advanced linear algebra course designed for the math specialist program and they wanted my opinion on how to prepare for it. My first reaction was to ignore the email. But then I remembered stories of Ragnar-Olaf Buchweitz told by Graham Leuschke and Michael Wemyss last month and decided to write to them. It turned into a blog post.
At the University of Toronto, I taught a lot of linear algebra courses. I TA'd MAT223 - your standard linear algebra/matrix theory course three times. The continuation of MAT223, that is MAT224 also three times. Then, I TA'd MAT240 - the advanced linear algebra course and its continuation MAT247 three times. On the other hand, I taught MAT223 once and I coordinated MAT224 twice. That is a lot of linear algebra.
The math major courses MAT223/224 are standard linear algebra courses. The material is the same in almost every university in the world. You could pick a test from an equivalent course in an arbitrary university in Canada and most of the questions will be suitable for exam questions in these courses. On the other hand, MAT240/247 are quite challenging courses, to be honest. But they also produce very good mathematicians.
I must say that I enjoyed TA'ing MAT240/247 most. Surrounded by very enthusiastic people, I was able to teach a lot of cool stuff in tutorials. Within the first month of their university life, not too many people see quiver representations or cohomology computations. I admit that I spent only a small portion of my tutorials on these and only a small portion of students actually understood what's going on, but somehow almost everyone was super excited about it every time I did this.
So, what can you do to prepare for MAT240/247 and to be successful in these courses?
Linear algebra is a very useful tool. Methods and concepts you learn in this course are going to be very useful in the future. However, what I value most about this course is that it is an introduction to how to be a mathematician. A linear algebra course is very suited for this because we understand linear algebra very well and the results are usually the best you could hope for. The proofs often follow a nice structure. The story is complete. Perfect landscape to teach how to do math.
Hence, the last thing I would recommend for preparing for this course is to look at a linear algebra book. First focus on fundamental skills.
You are going to be writing a lot of homework assignments. I can tell you that I probably marked over 5000 assignments for MAT20/247. I can guarantee that you are going to struggle at first, everyone does. The ones who survive get (surprisingly) better at the end of the semester.
Well, how do you practice writing?
Start writing about everything. Keep a diary. You are going to be expected to explain every step of your thinking in your assignment. You are going to be expected to justify each step of your work. So, start doing that. Write a recipe, write how to go from your house to the nearest grocery store, write about your morning routine, write a manual about how to pass a level in your favorite video game. As detailed as you can make it. Make sure that you use full sentences. When you write mathematics, your primary goal is to explain your work to the others. Be as clear as possible and use words as much as you can.
There are these weird three dots that some people use in assignments. One of them means "since" and one of them means "therefore" I think. I have never met a mathematician who uses these symbols, yet somehow some people teach these to our students. Don't use these symbols. Use words.
Make sure that what you write down is what you have in your mind. You have no idea how many times I have heard "I wrote this but obviously I meant that". This happens especially in the first weeks of September. So, if you really want to practice, practice that. Improve your writing skills. Just keeeeep writing.
Do you know who Alexander Grothendieck is? If not, you should look him up. Here is one of my favorite mathematical writings (warning: it is a pdf file). It is about Grothendieck's life and work. I keep reading it every now and then. When I first read it, I didn't really understand much. Every time I read it I see I understand one more thing. It is cool! And here is a Kimchi recipe by Grothendieck. Like I said, this is an exercise for you, too. Learn how to cook a meal, and then write a recipe of it on your own.
Mathematics is not something that you can do on your own. It is not easy, at least, and I can guarantee you will not enjoy fighting with those assignment questions alone. Form groups as early as you can. The first week, if possible. Just like writing skills, you need to improve your speaking skills.
The earlier you start, the better.
Meet regularly, but also just enjoy talking about math problems spontaneously while walking from one building to the other. Unfortunately, the UTSG campus is not very pedestrian friendly. You don't get to have those long walks focusing on math questions without being interrupted by cars. But still find cool places to talk math in campus before it gets covered by snow. Here is another idea: buy a white board, put it in your living room, invite people to your house, have a couple beers and just discuss mathematics. This is how AdIMOM started in my bachelor apartment and then became huge when I moved in with roommates. A decent size white board costs 50-60 CAD.
Speaking with your classmates is definitely encouraged. But you should be careful when it comes to writing solutions. Everything needs to be written by you. By this, I don't mean physically by you. After the discussions, you should be able to produce the solution on your own without speaking with someone else. If you are saying "Oh, I forgot this part. What was it again?" to your friend, it is not okay. It means, you didn't really grasp the solution. At that point, stop writing. You can go back to discussions if necessary. But stop writing. To make this easier, always write your solutions when you are alone.
If, for whatever reason, you are unable to speak with your classmates (you are too introvert, they are too sexist etc etc), just speak with an imaginary friend. Go for a walk and assume you have an imaginary audience listening to you. This is how I prepare for my talks in conferences. The night before, I walk around the city just presenting my research. Other people only hear a murmur and they don't care. The louder you can get, the better it becomes. The advantage of a real person is that they can say "I don't understand what you are saying" and then you might be learning even better when you attempt to explain better. There are more than 200 hundred students in these courses, though, you probably will find someone in your frequency.
You should also learn how to listen well, how to follow a math presentation, be it a lecture or a discussion among classmates. I love creating a classroom environment where students discuss. Every now and then, we have some real good discussions. But sometimes, we have low quality discussions and this is because we forget how to listen to each other. It is real funny because let's say Student A is saying something and Student B is responding. But they are too focused on what they are thinking about, they forget to listen. At this point, I interrupt and I write what A and B are saying on the board. Even better, I invite them to write their opinions on the board. When they stop for a second to listen to each other, they actually understand what they are saying and solve their problems.
When you are listening to somebody and you did not understand one thing, take a note. If they ask are there any questions, ask a question! Listen effectively.
Play with definitions and theorems. Try to change them. For instance, during the first weeks of MAT240, you will learn what a field is. There will be a lot of axioms. What happens if you change one of the axioms. Find examples where all other axioms are satisfied except that one. For instance, what happens if we don't assume that every non zero element has a multiplicative inverse? Instead of a field, call such a thing something else. Name it after yourself maybe. Or call it Ozgur Field.
When you learn about vector spaces over a field, go back to your example from above. Define a vector space over an Ozgur Field. Call it an Ozgur Space. Or name it after yourself or your best friend. Then, try to find examples of them. When you learn about subspaces, think about what an Ozgur Space Subspace should be. When you learn about linear maps, think about what an Ozgur space map should be. Always go back and play with definitions, try to see what happens.
Do the same with theorems. What happens if we change the definition of a vector space slightly? Which theorems still hold and which theorems fail and why? Like really just go out there and be curious. The more you bend things, the more you will appreciate vector spaces and how special they are.
You don't even need to be creative and make your own definitions. Take a theorem that you want to prove. Remove one of the hypothesis and try to prove it without the hypothesis. You will probably see that the proof in the book/from the lecture will not work. But ask yourself "Can I find a different proof which doesn't require this hypothesis"? Of course, at the same time, try to think a counter example to the statement of the theorem without said hypothesis. Maybe there is no other proofs, maybe you can't prove, the statement is wrong without the hypothesis. If so, keep that hypothesis and remove another hypothesis? The end game is not important here, just play with things as long as you have time. In the future, this will be an important skill.
Learn to Reflect
One thing I learned from Dr. Sarah Mayes-Tang has changed my habits. She asks her students every week: What is one thing you learned today? What are three things you learned this week? What was the most difficult concept from this week? What was one thing that you didn't fully understand? What did you to understand that confusing part? What are three things that you would do differently? Basically, every week some sort of course evaluations as she can collect these answers online. This is the instructor's side: We see these answers and we think about our course and maybe change it for the next class or next year. Useful. But it is actually much more useful for the student.
When you ask yourself these questions, you stop and think about what really happened that week. Yes, you were present in the lectures, you listened and took notes, but it is so easy for everything to slip away from you very quickly. So, constantly check on yourself. After each class, ask yourself: What three things did I learn today? Maybe also: Ask what did the instructor really want to teach me today? What was the most confusing part?
And not only daily, you should keep doing this reflection exercises regularly. At the end of each lecture, at the end of each week, at the end of every second week, at the end of each month, at the end of each semester. For instance, at the end of each semester you can ask yourself: If I were assigned to teach this course next semester, how would my syllabus be? In which order would I teach things? Which topics could be removed? Which topics should be emphasized more? Can I still fit everything to 12 weeks if I changed things a little bit? What were my top 5 assignment questions - top 5 I like, top 5 I got good marks, top 5 I hated. Things like this.
Now, if you also keep a diary or like a notebook where you keep all of your thoughts on this course, just add these questions and their answers to that book. This is also really good for your mental health. Eventually you will realize that no matter how much you learn, you know very little and almost nothing. That will make you sad. This will happen, I guarantee. For this reason, it is difficult to check how much you improved with time. If you keep writing your thoughts and your answers to above questions and save them, you can easily see your improvement. I still have my notebooks/diaries from undergrad courses and I look at them and I say "wow how stupid I was back then!" but you see, that wasn't the case. I faced a difficulty at that time, didn't understand one thing, but then I learned! Once I learned, past me became "stupid". Checking on yourself like this shows you how much you are actually improving. It is a good feeling! You need these good vibes!
I guess we came back to where we started. Write write write write. Speak, keep a diary, listen, write. (Read in Cardi B press press press press melody).
Take Care of Your Mental Health
Here is something unsurprising: If you only focus on your school work, you are going to get overwhelmed. Do something else. Check out the movies at the Royal, explore a new coffee shop in the city every week, hit the gym, explore the second hand bookshops in the Annex, find the best shawarma in Toronto, keep a blog on the best shawarma research in Toronto (I created 7-8 blogs during my undergraduate, what you write today will be embarrassing in a year, but who cares). When I first moved to Canada, I used Instagram to post photos of washrooms I used. You can definitely do better than that.
If you have the flu and you keep sneezing, you will not be able to do math. Mental health is not different. Take care of yourself.