Although this document’s focus is teaching mathematics courses in large classes, the techniques presented are no less applicable to Physics, Chemistry, Engineering, and Computer Science.

**1. Encourage active practice**. “Math is not a spectator sport.” Provide notes to students with space to practice the problems you teach. Include essential terms and formulas, and ask students to fill in definitions and problems that are worked in class.

**2. Allow time for practice**. When working a problem in class, give your students time to work individually. Then ask them to compare their work with their neighbors. Display your own solution, or ask a student to come to the blackboard to present results. Discuss the student’s process and any errors. Ask about other modalities of solving the problem and discuss errors. One way to achieve this level of engagement is to distribute transparencies, and ask students to work in groups to solve the problem. When work is complete, display several transparencies, calling attention to correct methods and untangling any errors. Do not elide errors: Why was that error made? What was the thinking process in the error? Compare erroneous approaches to correct strategies. Discuss mistakes students make frequently in the type of problem under scrutiny.

**3. Begin class with a warm-up**. Display a problem to be solved as students are entering class. An immediate problem at-hand focuses attention on the day’s topic of study and subdues irrelevant startup commotion. Ask volunteers to put their solutions on the board. Or! Call on students at random. The prospect of hearing one’s name called from the lectern keeps everyone involved with the warm-up.

**4. Use iClicker**. Display a multiple choice question. The question should test understanding of a particular concept. iClicker polling helps you determine whether or not more discussion is needed on that particular concept. The distribution of choices is visible on top of the iClicker receiver. If the distribution is broad, a peer-teaching opportunity knocks: Before showing the distribution to the class, ask students to rationalize their choices to their neighbors.

**5. Promote practice and review**. Assign well-chosen practice problems for homework. To keep students from doing the exercise simply for a correct answer, ask questions that test understanding: “Explain what type of problem this is?” “Explain what type of concepts (or definitions or theorems or formulas) you need to use in order to solve this problem.” “Explain why that is the correct choice.” “Define variables that will help you to solve this problem.” “Describe what your first step will be.”

**6. Use an interactive lecture style**. When you ask a question (and that should be often!) pause long enough for students to think about the question and suggest an answer. After asking the question, call on random students. If the first student called upon does not know the correct answer, resist jumping to the next student,. Give prompts. Help. Ask the class to prompt (NOT shout out the correct answer). Put only the conditions of a theorem on the board and ask “What do you think we can conclude?”

### Strategies for developing problem-solving skills

**Let students see**. Instead of displaying a solved problem, solve it in front of your students. Talk through the steps, and ask yourself questions so your students see how an expert approaches solving a particular problem.

**When solving problems, ask questions**. Pose a question, and pause long enough for students to think about it. Some questions that fit any problem are:

- How do I start?
- What do I know?
- What formulas do I need to know here?
- Am I making sense?
- What are the possible approaches?
- What am I forgetting?
- Does this answer make sense?

**Model strategies**. Model your problem-solving thought process by talking through it out loud. Alternatively, present several problems of the same type, then ask students to look for patterns.

**Give credit for analysis**. Encourage students to make their thought process evident in exam problems, and give partial credit for being on the right track, even if the ultimate answer is incorrect. If the student solves half the problem and employs a correct strategy, award credit for sound thinking as far as it went.

**Ask for a narrative of thinking processes**. Before responding to a question in class or via e-mail, ask students to explain their thinking so far. Choosing an inappropriate approach or misunderstanding the entire problem up front is a common first step into the dark.

**Provide focus**. Ask focused questions. Give enough problems of the same type to allow students to familiarize themselves with one type of problem and develop necessary skills.

**Assign small group work**. Display problems in sequence, and ask students to work with their neighbors, in pairs, taking turns thinking out loud while working toward a solution. The silent partner monitors the process and, when completed, offers critical analysis.