Concepts and Mechanics
As someone who teaches, I struggle with striking the right balance between explaining a concept to a student and giving them the steps to solve a problem (the mechanics). At first glance, it might seem like these two ideas are the same, but a student experiencing difficulties will often need one without the other.
If a student doesn’t understand a concept, then they will have a difficult time solving a problem. If they don’t understand that Newton’s third law talks about equal and opposite forces, then solving the problem is impossible. Maybe they forgot to include a force in their equations, giving the wrong answer. At the heart of the issue is that, conceptually, they’re missing the details that matter to solving a problem.
On the other hand, there are plenty of students who understand the concepts. Qualitatively, they can explain anything regarding the topic at hand. However, the quantitative aspect can be difficult, even when the concepts are mastered. This is because mathematics isn’t always easy! Even if a student has a rough idea of what they must do to solve a problem, this doesn’t mean they are comfortable with the tools to solve it. I’m familiar with a bunch of physics experiments, but as I’ve seen when going into the laboratory myself, I am not adept at carrying them out. Remember, there’s a big difference between understanding and being able to apply knowledge.
To address the former problem, we need to give more explanations. This is when we can review topics seen in class. The most important thing to do is get the student to explain the concept. This is at the heart of learning. If you can’t give an explanation of what you read or learned, you haven’t learned anything at all. Often, students are incentivized to put the majority of their attention on following predetermined recipes, while foregoing the explanations of concepts. I know this happens because I do the same thing. It’s difficult to spend your time studying for a test by explaining concepts when you know that the test will have little (if any) explanations required.
Despite that, while practicing and getting stuck on problems, it’s a good idea to try and get the student to explain the concept they are struggling with. This will either give you, the teacher, something to grab in order to guide the student to the correct explanation, or the student might even resolve their difficulty by talking about it.
If their difficulty lies in the mathematics themselves, the worst thing to do is start lecturing about the concepts. They know the concepts! Instead, a better move would be to work through a related example with them. That way, the student can see how the pieces move to solve a problem. It often only takes a worked example to give the student the tools they need to continue.
It’s important to note that we often think about addressing the former problem, but not the latter. We don’t want to give the students the answers, and instead we tell them to keep working at it. But that’s the wrong mentality. When first learning a new concept, a student needs the help of examples to ground them. It shows them how the new mathematics or concepts that were developed in theory apply to solving problems. It might seem like this should follow through from the theory, but often this doesn’t happen. By hammering home examples of how to use the tools, students can more easily solve problems later on.
Of course, we don’t want students to be just equation machines. So emphasize the concepts, but keep in mind that they don’t necessarily translate to the mechanics of solving problems. These are two different skills, and they need to be taught as such if we want students to be adept at both.