One of the most difficult things for me to do when I am learning is to make the conceptual leap from one idea to the next. Often, I’m not confused about how to do a problem. Rather, I’m stuck on an idea that preceded it which I am not fully on board with.

This is the reality that I’m frequently confronted with when I tutor students. Most of the time, the actual application of an equation to a problem isn’t an issue. Instead, the trouble comes from some conceptual piece of the puzzle that simply isn’t clicking into place. Usually, this comes from the teacher not fully explaining a concept, which then makes the student confused about how to jump from one idea to another. It’s also often a simple thing that manifests itself in the form of the student not being able to do any of the problem.

An example of a gap left in explanations to me is how students have to learn about various functions and what they look like. The list includes the following functions:

– Constant: $y = c$

– Linear or First-Degree: $y = mx + b$

– Quadratic or Second-Degree: $y = ax^2+bx+c$

– Inverse: $y = \frac{k}{x}$

– Exponential: $y = ab^x$

Ignoring the fact that the exponential function is thrown in there completely randomly, the other ones are given a bunch of different names. However, what is missing from this list is how the functions relate to one another. If we remover the random exponential function that was thrown in, a much more intuitive pattern can emerge from this list: each function is a successive power of $x$.

When I first showed this to the student I was tutoring, they didn’t even know what I was talking about when I used the idea of a negative exponent. But after walking through some examples with them, it was much clearer.

Here’s my modified list:

  • Inverse function (negative one degree): $y=kx^{-1}$
  • Constant function (zero degree): $y=cx^0=c$ (Ignoring the case of $x=0$)
  • Linear Function (first degree): $y=mx^1+b$
  • Quadratic function (second degree): $y=ax^2+bx+c$

If you look carefully, there are two important things I did here. First, I listed them in a logical order. They are ordered from the smallest power of $x$ to the largest power. This makes the functions easy to recognize. Count the powers and you’re good to go.

Second, I explicitly showed extra work in this list. I showed the powers of one and zero even though the final answer does not necessarily show them. The reason is simple: the students are learning and this is an easy foothold to take when learning about functions. When you’re suddenly bombarded with all these possibilities for functions, it can be difficult to keep them all straight in your mind. This way, you can easily see how the powers of $x$ are in each equation.

This is a simple point, but it’s something that is so important to get right in the beginning of teaching a concept. The longer this goes unnoticed, the more trouble the student will have since they won’t ever feel as if they have a comfortable foothold into the concept. Think about walking on ice. You’d feel a lot safer if you were wearing crampons or skates as opposed to summer shoes. In the same way, our goal is to give students these conceptual footholds, and the best way is usually through relating with something they already know.

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