With only so much amount of time in the day, I have felt the need to evaluate what my priorities have to be as I sit through my lectures. Although each lecture is only about 50 minutes in length, to dissect and engage with every 15 minutes of content takes me around an hour on a good day! So each lecture takes me about 3 to 6 hours. While I am okay with 3 hours to each lecture, the 6 hours per lecture tend to set me back and create scheduling issues for the future. I would like to find a way to break down 50 minutes of content consistently within 3 to 4 hours, but I do not know if its possible. However, I do know that if I’m going to be spending all of this time engaging with math lectures, I might as well learn deeply, and hopefully efficiently, by keeping some principles in mind. Thus, I sought out advice on how to deepen and better my experience with math from How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston.
Luckily for me and other readers out there, the chapter Reading Mathematics offered just what I was looking for. Before I demonstrate my list of principles from the advice offered in the book, I want to acknowledge that I struggle to be consistent with large lists. Therefore, I have chosen the four most important principles that I believe can keep me best engaged and efficient in my learning.
“Before reading decide what you want from the text. The goal may be as specific as learning a particular definition or how to solve a certain type of problem.” – page 16
Trudging through a lecture for hours with no end near in sight is incredibly intimidating. I have found it to be very helpful to skim through the lecture or notes (if provided), to seek out important definitions, equations, or ideas. At least by initially having a roadmap of the content, I can anticipate how long certain topics will be and if they will be related to other presented topics beforehand. Therefore, this principle allows me to consider the relationships of ideas as I learn about them. Also, if I notice that there are few topics to cover, I am not as discouraged when I am taking hours to get through one topic, knowing in advance that there are fewer topics to cover.
“The first reason for using pen and paper is that you should make notes from what you are reading – in particular, what it means, not what it says – and to record ideas as they occur to you.” – page 17
For me at least, I benefit from writing “what is says” and then “what it means.” By slowly writing down what I see in a lecture video or a textbook, I understand better what the author is trying to teach. As soon as I reach any hesitation or vague understanding, I make sure to comment my raw thoughts and confusion in my designated commenting area with a contrasting blue pen to my main black pen. This principle has helped me in recognizing and tracking and addressing my patterns of confusion.
“The second reason [for using pen and paper] is more important. You can explore theorems and formulas by applying them to examples, draw diagrams…Physicist and chemists have laboratory experiments, mathematicians have these explorations as experiments” – page 17
Following this principle, I make sure to not concern myself with fully understanding a theory or idea before looking at an example. Although I have a stubborn habit of rereading something and thinking about it until it makes sense, it has also been helpful to attempt related problems which can then reveal how and why the theory works. Although I would think that understanding the theory is vital to solving a problem about that theory, sometimes solving the the problem is vital to helping me understand the theory. I guess you could say that you can read about a subject as much as you want, but finally interacting with it is when you learn whether you truly understand it.
“Ask ‘what does this tell us or allow us to do that other work does not?'” – page 19
This principle pairs well with the first principle as it forces me to be in the driving seat of my learning. If I happen to get caught in the lull of simply memorizing without understanding, constantly asking this question throughout the lecture can help me stay engaged as I relate it to something I already understand. In other words, build your concept images!
Thanks for reading. (:
Question to the audience: How long does an “hour’s worth” of reading/lecture take you to understand on average, and why?