I started the year in Honors Precalculus a bit differently, looking at vectors first. With a colleague, we thought about how reflecting vectors could lead to a “bowtie” of sorts, and lead into circular functions in fruitful ways. It is now the end of October and I’m wrapping up a big part of the trigonometry unit. My students took a test which went well for many of them. The average was a 92%. The two students with the lowest scores earned 80%.

In some ways, those numbers represent what I want. An A- average seems great. Having two kids who still have some misconceptions about this topic still able to master 80% of the questions seems about right.

However, the numbers don’t tell the whole story. In looking at some of the student work on the test, here are a few samples that caught my eye:

This student circled this problem, and put a question mark at the top of his work. He included a clear calculation of the period of the function. Then, there are calculations all over the page (the student did not have a calculator during this assessment). What made this student doubt themselves on this problem? What part of the process felt “off” to them? I am eager to ask.

Another problem:

For the first time this year, I did not have my students complete a single-sided worksheet with about 40 problems like this on it. Clearly, this student’s solution is not a particularly efficient one. There are some lengthy calculations present because the student only wanted to think about the problem in degrees. Also, it shows me someone who likely uses their calculator briefly in questions like this, not hesitating to multiply 23 by 180, and then divide by 4, and also by 360. Because they haven’t needed to be efficient in this process, there is not an inclination to simplify the problem before multiplying and dividing. How can I better build up my student’s number sense to avoid spending time on something like this on a future test? Or should I instead provide a more basic calculator to each kid in case their working strategy involves computations like this?

Here’s one more solution that was interesting to me:

This student’s work is interesting because he had part of his answer incorrect. However, I think it was only off because of a miscalculation of the period (or of the coefficient of x in the equation).

Mostly, I’m ending this unit with some questions for my students about their thought process. I’m eager to see what they wrote in their math journals this weekend about the experience, and to have them reflect on a problem (on the test or from another moment) that did not make sense to them right away, as part of their ongoing math portfolio. I’m trying hard to see my students, and realize that as I look harder, I will only have more questions for them.