What do you think when you see … ?

I’ve fallen behind in the MTBoS blogging initiative, but my one day weekend turned into a four day weekend thanks to some well-timed snow, and it’s time to catch up!


This year, I have been preoccupied with how students forget so much, so quickly.

This year, I am teaching AP Calculus, which means that I am preparing my students for a cumulative final exam. This exam presents a daunting challenge: how do I get my students to remember the material that I taught in September when we get to May?

However, I have realized that Calculus presents a similar but even more daunting challenge: how do I teach my students Calculus when they do not remember their Algebra? It is a common refrain among Calculus teachers that “Calculus is easy, Algebra is hard,” but time constraints force me to be selective about when I explicitly teach the supposedly prerequisite Algebra content. Sometimes I can use the review to my benefit. For instance, I knew that none of my students remembered the “log laws” and by reteaching it, I was able to help students discover the derivative of \log_bx and a^x, and we explored how you can find the derivative of complicated logarithmic functions in more than one way. But other times, the inability to recall content has derailed my lessons. For example, when my students used the first derivative test to discover that x=-\frac{b}{2a} is the x-coordinate of the vertex of a parabola, they were unimpressed because they did not recall the formula.

In my Mathematical Foundations class, I have struggled with the same problems but on a different level. This course is meant to support students who are concurrently taking Algebra 2, so at the beginning of the year, I decided to start with a unit on linear functions. I wanted to deepen their understanding of linear functions by focusing on point-slope instead of slope-intercept and by doing a lot of modeling. However, I immediately hit a ton of roadblocks. I found that not only did students struggle to solve linear equations, most were still adding and subtracting using their fingers. How did these students get to Algebra 2 with such huge gaps in their mathematical understanding and skills? How can I best help students with Algebra 2 skills when they have yet to master elementary level content? 

I have posed these questions to coaches and administrators, and I believe that vertical alignment is likely an important part of the solution, but I can also start doing a better job on my own in my classroom. As a teacher, I play a huge role in what my students remember – I message what is worth remembering, I teach students how to study, and I create situations where they need to remember. I want my students to remember what I teach them, and I have found one question that I ask over and over again that has helped me toward that goal …

What do you think when you see … ?

In October, I went to a two-day workshop for AP math teachers, and one idea stuck with me. In his session, Jamil Siddiqui was going over the solutions to some problems by asking the “class” a series of questions. At first glance, I thought Jamil’s questions were just a more practiced version of the “what do I do next?” that I see first year teachers ask all the time. These questions are not open-ended and they do not encourage student-to-student discourse, yet I discovered that there is something subtle and clever about Jamil’s approach.

Instead of modeling how to solve the problem, Jamil was modeling how to analyze the problem. For example, if the problem was asking about speed, Jamil would ask, “If you see ‘speed,’ what do you think?” We were expected to respond with the “absolute value of velocity.” I got the sense that this call and response was rapid and habitual in Jamil’s classes; each question he asked was an exercise in recalling essential information, every time strengthening the memory of that content.

Then Jamil would ask, “If you see ‘absolute value,’ what do you think?” I thought: distance from the origin? A “V” shape? Jamil corrected us – were expected to respond with “piecewise.”  That answer to that last question fascinated me. Piecewise is not the definition of absolute value. However, in Calculus, piecewise is often the key to analyzing an absolute value function. Each key word is linked to the most important thing to remember, not the definition.

After the session, I followed up with Jamil and he shared his seven page Calculus “Key Words” glossary with me. I took it as a starting point and launched my own version in my class soon after.

How do I use this question?

In each of my classes, we created a glossary in the back of their notebooks. One column is labeled, “When you see/hear …” and the other column is labeled, “You think …” In the first column, I chose words that my students run across often. For calculus, this meant integral, derivative, maximum, etc, and in Foundations, this meant x-intercept, factor, product, area. For the second column, I brainstormed the most essential piece of information to associate with that word, and in most cases, this was not the definition. For example, when you hear derivative, you think instantaneous rate of change or slope of the tangent line. When you hear x-intercept, you think y=0. And so on. I included vocabulary that is introduced in my class and from previous years; I include whatever students need for my class.

Every day in class, I take two minutes to practice recall with these key words. I call the activity a “definition check,” and we do it everyday right after the warm up. I project a word, randomly cold call a student and ask, “what do you think when you see ___?” By letting students look at their glossary or pass if they don’t remember, I provide a low-stakes opportunity to practice recall. I ask the same question to 2-3 students, and then move on. And I often go back to words from much earlier in the year to make sure they continue to remember those words.

During class, I use this question like a hint. When I’m circulating and a student needs help, I point to the notation or the directions, and say, “what do you think when you see ___?” Sometimes they need their glossary to answer my question, but once they do, they’re usually off and running.


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