### Teaching Repertoire: How I Teach Calculus (Part II: Problem Solving)

The dichotomy I've stated is a slight over-simplification. But with few exceptions the complement of lecture is problem solving. When I say "problem solving," I am not restricting myself to word problems: rather I mean any mathematical task that is not a routine computation, calculation, or exercise. Anything where I would expect many of my students would get frustrated if they were sent home to complete this task for homework. (I do assign frustrating homework from time to time -- just not very much of it.)

And I have the luxury to do this because my department has a delightfully leisurely syllabus for Calc I. This would not be possible in other departments.

Unlike lecture days, problem solving days are much less structured. I still start with the announcements. Sometimes, while I still have the attention of the whole group, I will give a five-minute mini-lecture or reminder or else I will model an example that illustrates key ideas, but then I will hand out the sheet of problems for the day. I will encourage my students to form groups (most choose to work in one-person groups, which is FINE), and they will start working on the problems. I will circulate around the room answering questions, giving hints, or just listening.

Almost always problem-solving day is announced ahead of time, and the problems are usually posted on the class webpage. (If they're problems I've written they are; if I use problems from the Big Binder that came with my textbook, then they don't go on the web.) If someone doesn't want to come on problem-solving day, that is FINE. This is college, and they need to learn how to make their own decisions. I do take attendance. They know that I know when they come to class.

As I circulate around the room, I ask the students how the problems are going and how class is going. After 15 minutes or so I will decide (based on my interactions with my students) if I need to present anything to the whole class. If so, I will talk for five or 10 minutes about something that will help them work on the task at hand. I may need to clarify an idea, present another example problem, or work through the solution to one or more of the problems on the sheet. Sometimes I will ask a student to present a problem at the board, but I don't do that very often. I will only select students who I'm sure can present things well and who can field questions accurately. While we are convened as a large group, I will field questions. Once everyone is done asking questions, we return to small group (individual) work.

I still haven't figured out the best way to wrap things up on problem-solving day. Often I will devote the last five minutes of class to going over at least one of the problems. Sometimes I will withhold answers until the end.

And I have the luxury to do this because my department has a delightfully leisurely syllabus for Calc I. This would not be possible in other departments.

Unlike lecture days, problem solving days are much less structured. I still start with the announcements. Sometimes, while I still have the attention of the whole group, I will give a five-minute mini-lecture or reminder or else I will model an example that illustrates key ideas, but then I will hand out the sheet of problems for the day. I will encourage my students to form groups (most choose to work in one-person groups, which is FINE), and they will start working on the problems. I will circulate around the room answering questions, giving hints, or just listening.

Almost always problem-solving day is announced ahead of time, and the problems are usually posted on the class webpage. (If they're problems I've written they are; if I use problems from the Big Binder that came with my textbook, then they don't go on the web.) If someone doesn't want to come on problem-solving day, that is FINE. This is college, and they need to learn how to make their own decisions. I do take attendance. They know that I know when they come to class.

As I circulate around the room, I ask the students how the problems are going and how class is going. After 15 minutes or so I will decide (based on my interactions with my students) if I need to present anything to the whole class. If so, I will talk for five or 10 minutes about something that will help them work on the task at hand. I may need to clarify an idea, present another example problem, or work through the solution to one or more of the problems on the sheet. Sometimes I will ask a student to present a problem at the board, but I don't do that very often. I will only select students who I'm sure can present things well and who can field questions accurately. While we are convened as a large group, I will field questions. Once everyone is done asking questions, we return to small group (individual) work.

I still haven't figured out the best way to wrap things up on problem-solving day. Often I will devote the last five minutes of class to going over at least one of the problems. Sometimes I will withhold answers until the end.