I’ve been teaching MAM1000W for around 9 years now, and I am learning all the time. I learn both about new ways to think about old subjects (and how to try and best explain them), and I learn about the way students study, about what works and what doesn’t, and what are some of the habits of students who succeed. Not all of these ideas will be perfect for everyone, but I hope that they will help.
Passive versus active learning
Trying to teach as clearly as possible is a double-edged sword. Of course I want students to come away feeling like they have understood the subject, but if they come away with too much confidence, then they won’t do the one thing which they have to do to actually understand it…and that is practice, but practice of a very particular kind. There is a balance that we should all be thinking about when trying to improve on something (be it sports, music, languages, or maths), and that is finding the right questions to practice on which are hard enough to make us have to sweat a little, but not so hard so as to make us give up completely.
This balance is something that takes time to find, not only in terms of finding the right questions (which often means the right order of questions), but more importantly about finding how far you can actually push your brain. This is a skill which mathematics at university will absolutely give you. What is your brain capable of? I can promise you, it’s much more than you think! However, the only way to get there is to DO maths, not watch maths, or read maths, or listen to maths (although all of these are important). You actually have to do it.
One thing that I have seen recently is students coming to 8am lectures, then watching the videos as well, and possibly even going to both the 10am lectures as well. The world is full of resources for you to take in passively, you could spend all day every day consuming them.
I can promise you: This is NOT a good use of your time. You will not truly learn the material in class. You will learn it by taking the hints and general intuition that you gained in class and getting a pad of paper and a pen out and trying it yourself…and getting stuck…and more stuck…and finally figuring it out. This is active learning.
That step of finally figuring it out for yourself is key, and it doesn’t matter how many lectures you attend, there won’t be some sudden insight in the lectures which brings it all into perfect clarity.
Given three hours in the day to go to lectures and watch the videos, think about how much time you are then not spending DOING maths which you could be. I would say that 2/3 of that is not a useful way to try and get good at this stuff. 30-60 minutes a day doing maths will get you really really good at it, but that DOING is what you have to figure out.
Spend time writing maths clearly
I see this over and over again in tutorials, and I understand it…I did it myself. I see students going through the tutorial questions, scribbling down the working all over the paper, getting to the answer (which may well be correct) and then moving on.
Just imagine going into an exam and being asked to do something that you have not practiced at all. Wouldn’t that seem crazy? Well, that is what you are doing! In the exam, we want to see that you understand the mathematics, not just that you can get the right final answer. We want to see you explain, clearly, coherently, and in an unambiguous way that you know what you are doing. If you have only ever practiced maths by scribbling your answers in order to get the final result, you will really struggle to do what you are being asked to do in the exam!
My recommendation is to have two notebooks. Have one notebook where you are doing the quick scribbles – it’s an important process – but have a second notebook where you then write everything out neatly, with explanations saying what you have done. This will be both good in terms of the practice of writing mathematics, but secondly, it will be infinitely easier to revise from later if you are explaining to your future self why you are doing the steps that you are doing. Write it down. Write down why you did what you did and why you were allowed to do so. “Because the function is continuous on… we can use …”.
I have said it before, but I think that it’s really important to say again. The maths which you see in a lecture has all of that middle stuff in, but it’s often only spoken, and not written down explicitly. You need to take notes which include the things which are said as well as written. This is not always easy to do during the lecture, so go through it after and try and figure out precisely why we did what we did, and if you can’t get it, then take a look in the textbook, or the notes, or go back to the place in the video that it doesn’t make sense. This use of the videos is fine, and I absolutely condone it. Just watching the whole thing again to try and feel like you understand it more the second time is however not useful. It is like watching someone paint, or sing, or play sports and feel like you could do it.
Spend time discussing maths with others
I find that tutorials these days seem mostly pretty quiet, though there are people who are discussing together. You can do quiet work when you are on your own, and absolutely should. Take the opportunity during tutorials to explain the mathematics to someone else, and have them explain something to you. Talking through maths, teaching others and learning from others who have just battled through can be far more useful than listening to some guy stand at the front of the class who has understood this stuff for years.
As always, there is a balance to be found, and this is key. Spend time talking through maths, explaining, arguing. Spend time scribbling maths, spending time writing it neatly. Spend time reading it in a way that is as active as possible (how did they get this line? Can I get it?), but most of all spend time in that slightly uncomfortable space between understanding something easily and not understanding it at all. That is where you will truly grow!