Working with one student on functions this morning, I was reminded of how much there is that students can fail to understand.
I was trying to explain to this student that the x-value is the input, and that the f(x) value is the output. But because of the repetition of x in both of these terms, he got confused.
I finally solved the problem by telling my student to view the parentheses around the x in (x) as like the slot that takes the coin in a vending machine. It sort of looks like a slot, too, right? So what goes inside it must be the “in”put. Now he at least understands clearly that what goes in the ( ) slot is the input.
Staying with the vending machine analogy, I told him that the f(x) is what the function machine (like a vending machine) gives you after you put the x-value in the slot.
I did need to clarify that when you’re working out the value of inputs and outputs, you must insert the x-value twice: once inside the ( ) slot, and secondly on the other side of the equation, in where the ‘x’ stands.
Obviously this student has a lot of trouble with processing the visual symbols of math. But working with him reminds me of something important. It shows how much students can get confused by math concepts and math notation. I feel that it’s important for us educators to keep this in mind as we teach. There’s so much that we take for granted in our understanding of math. But for students who struggle with notation and with the visual aspect of math, notation can be confusing.
One thing I try to do when I work with students comes from something I saw in a Great Courses class by Bruce Edwards, an excellent teacher of higher math. Mr. Edwards likes to say things like, “Now this is next part is a little bit tricky …” Just by saying this, Mr. Edwards shows that he understands that not everyone will get the concept, and that, I believe, helps students relax.
Ever since I saw Mr. Edwards use this way of talking, I’ve been using it in my tutoring work, too. And I find that it helps students. It makes them feel like no one will think badly of them for not understanding, since I, the teacher, have acknowledged that the concept is “tricky.” As a result, students relax, and that helps them be more relaxed in taking in what you’re going to tell them. A nice thing to learn from a master teacher, and another lesson in the importance of the way in which we talk to students to help them learn. There’s so much more to being a good math teacher than just being thorough and clear. The affective aspects of communicating, such as showing empathy, are very important as well.