Subsubsection 1.1 Types of Mathematical Understanding
For you, the student, I reiterate that this text (and class) should be different from what you may have encountered in prior mathematics, specifically Algebra, courses. Some of the specific differences will be explained below, but I want to explain the overarching one first:
This text was written with the intention that this will not be the last mathematics course you take. I assume you will take further courses (Calculus, Differential Equations, etc.) and gain all sorts of exotic mathematical skills. The goal now is for you to gain some deep understanding of symbolic algebra beyond a collection of procedures.
I highlighted the word "understanding" above for a reason. It turns out that there are two types of understanding in mathematics, described by Skemp in
[2]. They are as follows:
Relational Understanding: This is understanding at the level of knowing what to do, or how to arrive at an answer in a mathematical situation, and why.
Instrumental Understanding: This is understanding at the level of being able to apply rules without reasons to arrive at answers.
Let’s consider an example of each in the context of multiplying fractions (this one is from Skemp’s book). (Click on the example to show it and click on the solutions to diplay them, read them carefully. You’ll want to do this through the remainder of the text.)
Example 1.1.1.
Perform the multiplication \(\frac{2}{5}\cdot\frac{10}{13}.\)
Solution 1. Instrumental Version
\begin{align*}
\frac{2}{5}\cdot\frac{10}{13} = \amp\ \frac{2\cdot 10}{5\cdot 13} \\
= \amp\ \frac{20}{65} \\
= \amp\ \frac{4\cdot\cancel{5}}{13\cdot\cancel{5}} \\
= \amp\ \frac{4}{13}.
\end{align*}
Solution 2. Relational Version
Notice that \(\frac{10}{13}\) is \(10\) times \(\frac{1}{13}\text{.}\) Then \(\frac{1}{5}\) of that is \(\frac{2}{13}\) because \(\frac{1}{5}\) of \(10\) is \(2\text{.}\) Then we take \(2\) of those to get \(\frac{4}{13}\text{.}\)
In that example, which one is easier to perform and which one makes more sense? The first solution is probably easier, but that’s not the same thing as making sense. If you were given a different rule to follow for multiplying fractions, you couldn’t check that it was correct or incorrect without the second solution, which better utilizes an understanding of what multiplying and dividing really are.
"I prefer to understand mathematics instrumentally." This is something students have said when presented with the above distinction. The problem is that as you move further in your study of mathematics, instrumental understanding will greatly limit you; you can only memorize so many procedures and keep them straight. Relational understanding is the understanding that mathematicians seek to cultivate by relating new mathematical ideas to old ones and examining the properties of new mathematical objects and consequences of new mathematical ideas.
Be aware, cultivating your own relational understanding of mathematics, specifically algebra, will not always be easy. Sometimes you’ll find that applying procedures instrumentally is ok. For instance, you’ll probably multiply fractions as in the first solution to the example problem; formally justufying that both solutions are equivalent is hard. However, you should always try to understand the "whys" of the math you’re doing. Keep a growth mindset, and start your homework early, and you’ll get it.
