As a tutor at the secondary level, I get an inside look at the various issues that students face within mathematics. While the specific issues are different from person to person, I’ve come to see a pattern emerging. Anecdotally, I think that most of the issues students face could be summarized with one sentence.

Students aren’t sure what formula to use for a given situation.

This is a simple, yet profound issue. On the one hand, you can say that the issue can be easily remedied by telling the students to do a better job committing to memory the various equations that are of use. I don’t think this is really a good suggestion, since the secondary schools allow memory-aids (sheets with notes and equations on them) for many of the tests. As such, the student should have the equations themselves. This still leaves the problem of knowing when to use the equations, but a simple note on their memory-aid should suffice for this.

On the other hand, this issue is profound, since it indicates a lack of familiarity with the equations themselves. For example, if a student is trying to calculate the area of a square with length $s$, and they aren’t sure where to use $4s$ or $s^2$ to do this, I think there’s a fair chance that the student isn’t familiar enough with the concepts and equations for the perimeter and area of a square.

For the above situation, what’s the best way to tackle the uncertainty? Is it to tell them that $s^2$ is the right one to use since you’re multiplying? What about the fact that $4s$ is technically multiplication as well, even though we know that $4s$ represents a series of addition. This could be confusing to the student as well. Suddenly, a simple little problem might be pointing to a larger issue.

The approach that I use is something that I partially learned from one of my professors while taking a class in probability. When introducing a new concept and seeing that we are having a bit of difficulty answering his conceptual questions, his go-to strategy is to bring it back to the basics. He tells us to think about how this new concept boils back down to things we have seen before. By going back to the original definitions, it is easier to build back up and get to the new concepts with a solid foundation.

This philosophy for learning is something I now tell all of the students I work with. I condense it into two words: no magic. My goal as a tutor isn’t only to help a student solve their homework problems and prepare for a test. It’s to give them a sense of confidence in what they are doing. My goal is to move them away from asking, “Which of these two formulas do I use?” to “I know I have to use this specific formula, since we are talking about that concept.”

Related to this idea of making mathematics less magical and more grounded in logic is the issue of knowing where a formula comes from. Recently, I was working with a student on the concept of area for various shapes. In secondary school, the classic shapes that are introduced include a square/rectangle, a triangle, a parallelogram, a circle, and a trapezoid. The student learns about the area of all of these shapes. But really, what they learn are the formulas for the areas. Any time I ask students if they learn why these formulas are used, they just look at me as if I asked something strange of them. It’s as if the thought of why a certain formula corresponds to the area of a shape never occured to them.

To be clear, this isn’t their fault. It’s the fault of the objectives in mathematics classes. The objective is to have the technical skills within a given subject, not necessarily an overall awareness as to the purpose of the mathematics. However, in my experience, being adept at the technical side of doing mathematics is definitely helped by understanding why I need to use certain rules and techniques. Without that knowledge, it’s difficult to ground myself when looking at a problem and trying to think of a possible solution.

This precise scenario came up when I was working with a student on finding the area of a trapezoid. They told me something along the lines of, “I know it has a $b$, $B$, and an $h$ in it, but I can’t remember how it’s done.”

Instead of telling them that the area was $A = \frac{1}{2}h(B+b)$, I told them that we should work at finding out exactly what the area was, using only the tools we knew. I knew that the student understood how to calculate the area of a rectangle, so we should have been capable of finding the area of a trapezoid. I didn’t know the exact steps needed to get to it, but I did know that cutting the shape up into smaller pieces was probably the correct strategy.

This is exactly what I told the student. I didn’t want them to think that this area for a trapezoid was some mix of strange symbols arranged in just the right way to produce the correct area. I wanted to show them that this process jumps right out from the fact that we know how to calculate the area of a rectangle, which is a simple enough fact to get the student on board with. From there, it’s mostly about skillfully rearranging the shape to get new ones that are easier to calculate.

Once again, my mantra was no magic. I firmly believe that the student needs to see this process of obtaining certain results if they want to be more comfortable with using them. At the moment, I shake my head in disbelief every time a student tells me that their teacher gave them a formula, but didn’t explain where it came from. That’s doing the students a disservice, and frankly I wouldn’t be surprised if it resulted in lower performance.

At the same time, I realize that there is a fair amount of material in a given year for secondary mathematics. It’s not easy to go through it all and give derivations for each and every single fact. I can definitely sympathize with that. A rushed derivation is about as good as no derivation at all. However, I would recommend that teachers do their best to include more derivation during class, if they can. While it might be seen as boring (and yes, it sometimes can be!), it helps ground a student in their concepts, so that they are less likely to be unsure of what formula to use during homework and exams.

Remember, no magic. Mathematics can (and should) be explained along almost every step (as much as reasonably is needed). At the moment, students learn a lot of techniques, but they don’t have the experience of learning why a fact is true. In my humble opinion, this is where change needs to begin.