The biggest breakthroughs in my own teaching occurred when I learned to spot the hidden complexities in mathematical topics. It might be obvious to say that topics such as the fundamental theorem of calculus or integration in three dimensions using cylindrical coordinates are difficult, contain many “moving parts”, and require sophisticated mental images to fully understand. But when was the last time you really thought about the potential for confusion and the underlying complexity in the most basic concepts you teach?

Think about one of the most basic topics in mathematics: arithmetic. I’m sure we all remember the timed tests and the months spent learning “carrying” and “borrowing” so that we could perform the standard algorithms without mistakes. But think back a little further. Do you remember working with manipulatives, such as the wooden blocks that are broken into singles, tens “sticks”, and “hundreds squares”? I have talked to many elementary school teachers who take it for granted that working with such manipulatives prepare students for working with the standard algorithms for addition and subtraction. In fact, most believe that the connection is so obvious that it hardly needs to be mentioned. But is it so simple?

I was watching a video one day of an elementary student working through the problem “70 – 23 = ?” using manipulatives and using the standard algorithm. When working with the standard algorithm, she kept getting the answer “53” (forgetting to “borrow” and saying that she first subtracts 0 from 3, then subtracts 2 from 7). To verify her answer, she was given blocks. She proceeded to count out 7 “tens sticks”, first removing 2 of the sticks and then counting off three ones from a stick to get an answer of 47. She was understandably confused that the answers didn’t match. She tried again with the same result. Then, looking at a hundreds chart, it appeared that her first instinct was to go up two rows (subtracting 20) and then left 3 (subtracting 3). While watching this video, I had a revelation. The standard algorithm had nothing to do with her mental image of subtraction gleaned from repeated interactions with the manipulatives! In each use of the manipulatives, the student first subtracted some number of tens, then subtracted any ones left over. In the standard algorithm, she was told to do the opposite, and nowhere in her explanation of how she solved the problem using the standard algorithm did she acknowledge that the 7 and the 2 actually represented 70 and 20. It was obvious to me that the gulf between the use of manipulatives and the use of the standard algorithm was much wider than I had ever considered. If this is true for one of the most basic ideas in mathematics, how many topics in my algebra, precalculus, or calculus courses was I assuming connections between topics and ideas that were anything but obvious to my students?

Here’s an example from algebra: systems of equations. If you’re like me, you’ve been frustrated with students’ ability to solve systems when the techniques seem so obvious, especially the substitution method. Yet I found time and again that students had a lot of trouble making sense of systems and solving them accurately. Putting aside difficulties with algebra and calculation that result in student mistakes, have you ever considered that the technique of taking two equations, such as *y* = 3*x* + 1 and *y* = -2*x* + 11, and creating the statement 3*x* + 1 = -2*x* + 11 is a very sophisticated idea? I’ve heard teachers say statements such as “since *y* is equal to both 3*x* + 1 and -2*x* + 11, then it must be true that 3*x* + 1 = -2*x* + 11.” In fact, I’m sure I’ve said this exact statement myself in the past. But this statement is totally and utterly false. It makes no sense. The “*y*” used in both equations are the output values of two different functions, and there is nothing inherent to the problem that says they must have the same value. In fact, if these functions represents real-world quantities, such as costs and revenue, or distances from school for Joe and Sally, or something similar, then each “*y*” represents a different quantity. Talk about confusing – the same variable used to represent two different quantities? Not only that, but for any input *x*, the two *y* values are almost always different values. Further still, we use “*x*” in both equations to stand for the input. But is there anything inherent in the problem that says we have to plug the same numbers in for *x* in both equations? For example, if the two functions model Joe and Sally’s distance from school as a functions of hours since noon, might I ask how far from school Joe is at 12:01 and how far Sally is from school at 12:02? There’s nothing stopping me!

Even if we say, for example, that the *y* value must be the same, isn’t it true that for most *y* values we could choose the two equations have the same value for different values of *x*? For example, 1 = 3(0) + 1 and 1 = -2(5) + 11, and we can see a graphical representation of this below.

Also, for the same value of *x*, the values of *y* are almost always different. For example, 13 = 3(4) + 1 and 3 = -2(4) + 11. The point of our solution methods is that we are assuming that the two *x* values must be the same and the two *y* values must be the same. In fact, our solution method forces this to happen, but why it works is generally not discussed. To really unpack what’s going on, let’s stop the double use of *x* and *y* for a moment and redefine our functions to be *y1* = 3*x1* + 1 and *y2* = -2*x2* + 11. Now we must define what we’re searching for. The solution to a system of equations is an ordered pair that makes both equations true. So to begin, we know that for the solution of the system, the output values must be the same. Therefore, it must be that *y1* = *y2*. If *y1* = *y2*, then the expressions 3*x1* + 11 and -2*x2* + 11 must calculate the same value, so 3*x1* + 11 = -2*x2* + 11. But right now there is nothing to say that the inputs must be the same. As mentioned before, there are plenty of different values of *x1* and *x2* that produce the same output. The key is that, for the solution to the system, not only must the two expressions calculate the same value, they must do so at the same input value. Therefore, we let *x1* = *x2*. Now that we have forced the input values and output values to be the same, the equation will tell us what input value meets this requirement.

Only after fully unpacking the complexities of a concept can I begin to imagine how to teach it and how to help students develop powerful, meaningful understandings.

The day I transformed as a teacher was the day I stopped taking the even the most basic, seemingly-straightforward topics at face value and focused on unpacking each idea, examining solution methods and calculations I took for granted as obvious, and really tried to put myself in the shoes of a novice learner – someone who didn’t understand the ideas and was trying to develop the formulas and algorithms for myself. The more I did this, the more I understood that each math course could be boiled down to a handful of key ideas, and that by focusing teaching on key ideas and teaching the myriad topics in the course as various representations of a small set of ideas instead of hundreds of disconnected procedures I could give my courses a coherence they previously lacked. More importantly, however, this approach empowered my students. Their confidence and problem solving skills swelled and my students actually began looking forward to my class. (Imagine that! Students enjoying math!) From their point of view, they finally had a teacher whose goal was to help them make sense of what they were learning. In future posts, I will provide some specific examples of how I unpacked the key ideas in my courses and was able to better focus my teaching.

Alan O’Bryan