Guided Learning Through Problems


I love new information. I read all the time (both fiction and non-fiction), and I like to learn new things about the world. I have interests in mathematics, science, and plenty of other niche topics like typography and running. As such, I’m frequently consuming information. I begin my day by reading, and I must spend at least an hour each day reading (particularly when I have time off between semesters, as is the case now). Put simply: I love learning new things.

To illustrate this, when I finished up with my exams for this semester (and felt pretty burnt out), I woke up the next day and started reading textbooks for classes I’ll be taking next semester and other topics I was interested in. I think that’s the best example that I can come up with to show just how much I enjoy the process of learning.

That being said, the observant reader may note that in the first paragraph, I used the word “consuming” instead of “absorbing”. That was not a mistake. The dirty little secret that I have is that I go through a bunch of information, but most of it isn’t retained as well as I would like. For some things, this doesn’t matter. But when I read a book about some scientific concept, it seems like a bit of a pointless journey if I don’t even recall the details a couple of weeks later. Likewise, I’ve found myself looking at textbooks with a cursory glance, interested in the new topics, but as soon as the problems and exercises came, I would stop reading. It was too much work to actually get into the book. Reading about a topic at a high level is fine, but digging into the details requires work that I was seldom giving.

This is a problem, because it means I’m only passively coming across new information. I’m not spending energy reflecting and really thinking about the content, which is where true learning and understanding comes from. That’s the difficult part of learning. Jumping into qualitative descriptions is relatively easy, but when is it time to sit down and really go through the details?

I bring this up because I’ve recently begun working through a textbook that is a bit different from the usual. Instead of being about theorems and proofs, the book is mainly a bunch of problems. The idea is to guide one through doing examples. This is much more difficult in the sense that working through a page goes from being a two-minute endeavour to a potential total-afternoon one. Progress is obviously much slower, since one has to think of the problems instead of simply reading through them. I’ve found that it’s very easy for me to skim through a worked example and say, “Oh, that makes sense.” Yet when it comes time to sit down and do a problem, it’s not so easy. You have to put pencil to paper and try things out. And in the end, that’s what helps us learn.


I think these kinds of books (or notes, or whatever you want to call them) are really the best idea if one wants to do some self-studying. It’s not that textbooks are bad. In fact, they contain a lot of useful information that actually does have a purpose. However, I think that this purpose is more of a reference than a tool to actively learn. For the learning part (within mathematics, at least), I think a lot of problem solving up front with the relevant details interspersed at the right moments can be much more effective than a traditional textbook. Do I think that this would work with every topic within mathematics? Of course not. For proof-based courses, one needs those definitions and previous theorems if there is ever a hope of doing more proofs. However, if I take as an example the particular topic I’m trying to learn at a more in-depth level (combinatorics), this is a great topic where working through problems can be more beneficial than reading through a textbook. In particular, I’m working through Kenneth P. Bogart’s Combinatorics Through Guided Discovery, which has been a great resource so far. The book forces me to think about the topics I’ve learned, as well as how an answer can be found in several ways. The writing is clear, and the problems flow together towards a total understanding of the material.

I really think that these kinds of resources are the next big wave within education. Instead of trying to learn from a textbook or video, the concepts will be illustrated directly through problems. Yes, it will take longer to get through the material, but if a student is motivated to learn, that shouldn’t be a large enough barrier. Sites like Brilliant do exactly this. They focus on problem solving over a bunch of details on the concepts, and I think this makes for an experience that stays with a student for longer.

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