## A Negative Minus A Positive

This week the students worked on solving simple linear equations. As an example:

`5x + 10 = -20, what is x?`

The ability to solve this problem varied wildly. Some students seemed to already be developing an intuitive feel for how to manipulate equations, and complained about showing every step. Most were able to follow the order of steps from the practice problems Mrs. Trauthwein did in front of the class. Some were totally lost.

But the stumbling points were not what one would expect, or at least not what I would have expected. For instance, most students correctly decided to subtract 10 from both sides of the equation above to combine the constant terms. However, the majority thought that (-20 – 10) was -10, instead of the correct answer: -30. I tried various means of explaining how to get the right answer, including drawing out a number line and parroting the old “same signs add and keep” mantra (which I personally think is confusing). I think the root of the problem is that they don’t see subtraction and negation as interchangeable. This was evident on the following problem:

`10 - 5x = 20, what is x?`

Many students did not seem to realize that 5x being subtracted from 10 means that the negative sign goes with the 5x. When they combined the constants, they would end up with:

`5x = 10`

erroneously dropping the minus sign completely.

Another issue I didn’t expect was with combining like terms. Students would often try to add a constant to a term that varied with x. The only explanation that comes to mind for why this doesn’t work involves showing that constants are really coefficients on an x raised to the zeroth power, which I’m guessing would only confuse them. So I stuck with: “you can’t combine things with an x with things that don’t have an x”.

This dovetails with the reading we are doing for our weekly Fellows meeting in that experts don’t remember what confuses novices. Experts already have the information organized and practiced, and have forgotten what tripped them up originally.

Still, the experts (myself and Mrs. Trauthwein) at least learned from the first block, and picked better (and more) examples during the following blocks to illustrate some of these points. That is one thing I have enjoyed so far about this teaching experience: the ability to adapt the lesson based on feedback. In my previous experiences teaching undergrads, I only ever gave a lesson once, since there was only one section of the class.