First Principles

When I was younger and first going through the “jump” between secondary mathematics and physics to that of CÉGEP and university, I always got frustrated when teachers would just shrug their shoulders when we grumbled about having too many things to remember for the test. Their advice was to simply remember the fundamentals, and rederive any result that was needed afterward.

I still think that this isn’t the most useful advice for exam that only last for fifty minutes. During this time, the tests are usually a mad scramble to make sure one can answer all the questions in the allotted time. If you have to pause and waste five to ten minutes on a question, it can be difficult to finish the rest of the test.

However, I’m come to really understand my teachers’ advice for learning in general. As my mathematics professors keep on telling me, “Mathematicians are lazy. We try to remember as little as possible, while secure in the knowledge that we can recover the result if we want to.”

Does that mean I forget what the real number line is, or work through a delta-epsilon limit proof every time I run into a new problem? Of course not. For most applications and situations, this would be overkill. But what it means is that I try to remember the fundamentals of concepts, and I avoid trying to remember special scenarios. This is effective because, if you deeply understand how the concept works, it’s not too difficult to extend to certain special cases. The converse is not true. If you are able to remember the special cases, it doesn’t mean that you understand what’s going on, it means you know what the formula is for that situation.

In my experience with tutoring students in secondary school, this is most manifest in the manipulating of algebraic equations. This is arguably the basis of most of the mathematics that the average student will encounter for most of their lives. Being able to solve and manipulate equations is important for statistics, probability, calculus, and linear algebra (not to mention any physics or most other science courses). Not being able to manipulate equations introduces a huge crutch into one’s mathematical ability (at least, at the level of secondary school and introductory classes as I mentioned above). It’s therefore extremely important that students find themselves comfortable with the manipulation of equations.

Unfortunately, this doesn’t seem to be a skill that is easily acquired, and it seems to stem from a core issue: students don’t seem to be taught that manipulating equations requires that you do the same action on both sides of an equation.

It’s a simple enough concept, but a lot of a student’s mathematical education rests on the shoulders of understanding this concept, so it needs to be understood. Therefore, if there is one fundamental idea about solving equations that needs to be remembered, it’s this.

In a similar vein, when students solve equations, they learn of multiple methods: comparison, substitution, and elimination. These are presented as “different” ways to solve equations, but what seems to often be missed on the students is that they are all essentially the same, albeit in special scenarios. Substitution and comparison are basically identical, and elimination is simply adding terms on both sides of an equation. As such, instead of remembering all of the methods, you simply need to remember two ideas:

  • You have to apply an operation to both sides of an equation (as I said above).
  • When you have an equality between two expressions, you can swap one for the other.

Of course, these two ideas aren’t necessarily obvious to the student who is first learning algebra. However, with a good number of examples and practice, I’m confident that a student armed with these two principles can begin to understand equations on a deeper level, without thinking about “bringing X over to the other side of an equation”.

This is the kind of thinking I try to apply when I learn new concepts. As I go further down the mathematical and physical roads, I’ve learned so many things that it’s difficult to remember everything. Therefore, I keep note of important concepts. This lets me retain the crux of many concepts, without necessary needing the details. Then, if I find myself in a situation where the details become more important, I can always refresh my knowledge by looking at my notes or in textbooks. I think this method is easier on the mind and allows one to search for the deeper connections that various subjects have with each other, because you’re not focusing as much on the details.

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