I had the interesting realization recently while tutoring one of my students. As he worked on his solution for solving a system of equations, I suggested that his work would be much more clear if he outlined what he was doing and showed steps to his process instead of simply writing a bunch of equations and having the answer pop out at the end.
He bemoaned having to be so complete with the work, not seeing the point of having to add these small elements to his solution, but I held my viewpoint. I knew it was important for him to write the entire solution in a clear manner, but he didn’t seem to care about achieving this level of perfection. He thought the elements I suggested were more or less superfluous, since the core of his solution was correct.
So what was my realization?
It was that, ironically, I do the same thing as he does in my own mathematics classes.
The quest for one hundred
As I’ve mentioned a few times in my writing, I’ve had the same professor for all of my mathematics courses in CÉGEP. During this time, I had a bit of a personal goal: get a perfect score on one of my tests.
Unfortunately, I never met this goal, though I did get close. My best mark was 69.5/70, which is over 99%. Other than that, my tests usually hovered around the lower nineties.
What becomes clear though as you look at my exams is that, apart from silly errors, most of my work is correct, but is lacking a small clarification or a point of completion. As you can probably see (and like I said, I only just made the connection), my situation is remarkably close to that of my student who I tutor.
Every time I saw this kind of error on my exam, I’d shake my head, not really even knowing how I could have been expected to write that anyway. I took solace in the fact that almost no one in my classes ever included the little things, and so I was just like the rest of the class.
In regards to my own performance, I didn’t fully appreciate what these little clarifications for completeness would do to my solution. Instead, I told myself to be happy that I got the core part of the problem and just forgot a little thing of almost no consequence. This is obviously the wrong view to take, but I didn’t make the connection.
The reality is that mathematics is a sort of logic. Therefore, all parts of a solution to a problem are important if the final answer is to be believed. Whether or not the final answer works is entirely dependent on the small and big details of the solution. Omitting a detail means the foundation isn’t rock solid. Sure, it might happen to be the correct answer now, but it is only in spite of the lack of detail, not because of it.
For example, I lost points on one of my tests because I failed to identify the curve I was using as a cylinder. I was dealing with the classic $x^2+y^2=1$ representation of the cylinder, but I failed to mention that the previous equation was for all z values. I thought this was an implicit assumption from the equation, but it’s actually important to explicitly declare it. This is because the equation I wrote is actually a circle in two dimensions, yet is a sphere or solid in three dimensions.
Therefore, what I thought was a harmless extra point of completeness was actually very relevant to the solution. It was the difference between a circle and a cylinder. Consequently, I now fully understand why my professor took away points in the question. Of course she knew I was talking about a cylinder, but the equation I wrote was really that of a circle, so further clarification was needed,
What I’ve learned from this is that there is no small detail that you should leave out from a mathematics problem. Simply put, it’s much more advantageous to be explicit about the work you’re doing than to make a bunch of implicit assumptions and hope everyone gets it. You will rarely lose points for being complete, but you definitely will on the other side.
I’ve found that this occurs for me just as it does others. Even with over a decade of doing mathematics in class, I struggle to be perfectly complete. However, it does give me a bit of inspiration to include all the important points of a solution as they come up.
Remember: the little frustrations are annoying, yet they will instruct one to be more complete in their mathematical pursuits.