Peripheral Knowledge

Why is it that some students will pick up new subjects in mathematics and science relatively easily, while others will struggle for weeks on end getting just the simple concepts down? As a tutor, I see this all the time. In fact, I would almost wager to say that most students I work with aren’t actually having difficulty with the topic they say they don’t understand. So what’s going on?

If I ask them a question concerning this topic, they will often work slowly, but will get the answer correct in the end. The place where things get tricky is in the path to getting the answer. There always comes a step where the student, hesitates, unsure of how to proceed. It’s then my job to help them get past that hump and understand what is happening. However, those problems usually stem from past knowledge that has either been forgotten or poorly done in the past.

Here’s an example. Early on in secondary school, students begin extending the notion of a fraction. In particular, they learn that fractions are not unique, and that we can have (infinitely!) many fractions that all represent the same fraction. (Think of 1/2, 2/4, 3/6, and so on.) This is something the students will quickly understand. A few visual demonstrations also helps, and then they are sold on the idea. So far so good.

The next step is to introduce the concept of reducing fractions. If you have a large fraction, like 64/128, you want to rewrite it such that the numerator and denominator don’t have any more common factors. This lets us deal with “smaller” numbers in the numerator and denominator. Indeed, after a moment’s glance you will probably recognize that 64/128=1/2. The task is to get students to be able to go from these “larger” representations of a fraction to a smaller representation.

Once the mechanics of this process are hammered out, you would think that this would be a relatively simple thing for students to do. The actual “work” consists in factoring large numbers into their constituent parts. This is something that they have previously worked on, namely, while working on their multiplication tables in elementary school.

Yet this is where some hit a snag! It turns out that many student’s haven’t actually learned their multiplication tables, which makes reducing fractions go from being a straightforward task to one that is time-consuming and difficult. And yes, knowing one’s multiplication tables does essentially come down to memorization, but I would argue that is a very good use of memorization (unlike many other examples). Having a sense of how numbers “fit” together to produce other numbers helps in many other areas of mathematics, and it is very useful here.

For this specific example, the student will often have to work out each fraction individually, perhaps even making a factor tree for each number so that they are comfortable with how the number breaks up. Of course, there’s nothing wrong with this, but it makes it very difficult to finish a test within the allotted time when one has to work out the factors of a number one by one instead of knowing how they break up. This is only made worse with calculators, because it allows students to get by more easily, without knowing this basic concept. Then, when this need to be able to know the factors of a number comes up again, the student will have to pull out a calculator again, increasing their dependency on some outside source to help them.

This illustrates a broader point within mathematics in particular: most of the time, you can’t “escape” a concept by faking knowledge of it. Eventually, you will need to use that tool in some other problem, and you will be expected to know how it works. But, if the first time you learned it you did not take the time to understand and improve, that concept will come back to haunt you later on. This is simply a consequence of the fact that mathematics is cumulative, building on prieviously learned concepts. If a student didn’t understand something in the past, yet showed “enough” knowledge that they still continued on to new things, chances are they only got worse at the particular concept, since they never really worked at it again. This becomes a problem when it suddenly resurfaces and the student has to now learn both the new concept and the old one simultaneously.

This kind of knowledge (what I might call “peripheral” knowledge) is the kind that often causes the students I work with to struggle. It’s the simple reality that students start having difficulties because they’re trying to catch up on “old” material while also trying to get the hang of the new content. It’s no wonder that they struggle when this is happening!

There’s no easy way to fix this. The truth is that if the student wants to improve and really understand, they have to put in the work to catch up. If this means they have to work on their multiplication tables after school, then that’s what it takes. I try to make this very clear when I work with students and I see they are having these kinds of issues. I won’t be able to help them within a span of an hour or so every week, particularly when they bring work that addresses new topics, while I know that the underlying issue is the older topics. I do my best to suggest to them to do more practice on the underlying topic, because that’s where they will really see the largest returns.

Will it be easy? No. But it’s a lot better to be realistic about these kinds of issues than to simply “get a tutor” and hope that everything becomes better. In mathematics, the peripheral knowledge is what ends up separating those that struggle through every single problem and those that can work through them systematically. As such, I recommend to always be aware of your background within mathematics, and to acknowledge the areas in which you are weak and may need some work. Then, address those before automatically saying that you don’t understand the new material. Chances are, the new topics will make a lot more sense once you have worked through and are comfortable with the older topics.

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