# The Easy Question at the End

I gave a test this week in my pre-calculus class and I’d like to reflect on one test question, the last one of seven. The question set up a realistic problem that modeled the measurement of light intensity using a rational function, the topic of last week’s study. The rational function modeled the intensity, in lux, that a measuring device would detect when placed meters from a low-intensity bulb and, correspondingly, meters from a high-intensity bulb—the bulbs were at a fixed distance apart—ten meters. The details of the function are not important, what I’d like to reflect on was the students’ response to the question.

• Identify the practical domain of the rational function for this application.
• Use your calculator to estimate the minimum and maximum distances from the low-intensity bulb that would register less than 5 lux on the measuring device.

First some background. The week before the test I presented, at the board, as a lecture, a very similar problem. So similar that it showed two light bulbs ten meters apart and a measuring device between them that was modeled by a very similar rational function. We (at least a few students and I) discussed the practical domain of the rational function, given the setup; how it only involved positive values, and in particular, values between zero and 10 meters. I thought the class understood. I then went through an analytic solution of the rational function, which eventually involved the solution of a quartic equation. The solution was long and messy (it was supposed to be). After that, we solved, as a class, on our calculators, the same problem by using a graph of the rational function and a graph of a horizontal line at to find the two points of intersection of the two graphs that would answer the second question of the problem (for a value of 4 lux). I displayed my graphs, in the practical domain, on a screen at the front of the room, and the students, supposedly, created the same graphs, and found the same points of intersection, on their own calculators. The day before we had gone through a different problem that was difficult to solve using a graphing calculator, but straightforward to solve analytically. The two problems were meant to stand as endpoints on a spectrum of problems that were variously easier or harder to solve analytically or graphically.

So how did my students do with a problem that I thought was a gift (and therefore assigned fewer assessment points)? Here’s a summary table of the results.

 Total Answers Correct Answer Incorrect Answer Identify Practical Domain 34 15 19 Analytic Solution Attempted 11 11 Calculator Solution Attempted 11 5 No Solution Attempted 7 7

I think I now understand why a well-formed lecture and a seemingly attentive class do not necessarily mean that the students are learning what I think I am teaching.

# Talk, Walk, Repeat

I’m an instructor at a community college in California and I teach mathematics. This quarter I’m teaching pre-calculus and calculus. As a part of a professional development program at the college, I have committed to reflecting on and writing about my teaching experience, once a week, for ten weeks. Here is my first reflection.

This quarter I’m also trying to use more active-learning techniques in my classroom. I’ve taught, up to now, in a conversational-lecture style, but comments from my colleagues and students during the last quarter have convinced me that I need to teach in a more student-engaging way—one that requires more active and less passive learning from the students. For right now, taking change slowly, I’ve altered the structure of my lesson plans to the following format: lecture, worksheet, lecture, worksheet, etc. The idea is to:

• Lecture on a topic for 15-20 minutes.
• Let students try to solve topical problems, alone or in groups, for 10-15 minutes.
• During which time I wander the classroom to observe and converse.
• Let students present solutions, with open conversation, for 5-10 minutes.

Not very original, but it gives me a way to start that is active-learning and lecture together—just a little outside my comfort zone.

This is week two of the quarter. Are my changed lesson plans working? Are my students more engaged? I am not sure. I need to find a way to evaluate what I’m changing, but since I’m in the middle of the change it’s hard to stand aside and just watch. And what should I be watching for?   I have seen concentration during problem-solving by students, and I have seen them working in groups of two-three, but I have also seen play-working by individuals and groups. None the less, without any way to justify it, I have an impression that the students are more engaged by the insertion of the activities in place of straight lecture. (And why shouldn’t they be? Who can concentrate on listening for more than 20 minutes?)

I’ll continue the structure next week and try to improve the continuum of challenge in the problem sessions. I’d like to engage all the students so none are too bored or too challenged. But, that too, I’m not sure how to measure.

Time to walk.