Every day in mathematics students are asked to make sense of problems and persevere in solving them. Problem solving opportunities require a different role for the teacher. Rather than directing a lesson, the teacher provides time for students to struggle with problems, search for strategies and solutions on their own, and learns to evaluate their own results. The focus in the class is on the students’ thinking processes.
As teachers, we model these attitudes in order to help students develop them. We let students know that we value problem solving. Teachers present themselves as problem solvers by being active learners willing to jump into new situations, not always knowing the outcome or answer
The problems students solve may not be based on what has just been studied. So while using mathematical and personal background knowledge, mathematical habits also provide students with a toolkit of ways to approach a problem needing to be solved.
Four steps to follow when solving a mathematical problem include:
1.Understand the problem. Based on the problem, teacher prompts could include:
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Can you think of a picture or a diagram that might help you understand the problem?
- Is there enough information to enable you to find a solution?
- Do you understand all the words used in stating the problem?
- Do you need to ask a question to get the answer?
2. Make a plan. Possible strategies for reasonable ways to solve a problem include:• Guess and check
- Make an orderly list
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
- Use a formula
- Be creative
- Use your head
3. Carry out the plan. Choose one. Persevere. If it continues not to work, discard it and choose another one.
4. Review/extend. Take the time to reflect and look back at what you’ve done, what worked and what didn’t. (Polya, 1945)
Teachers emphasize the importance of working on problems, not merely on getting the right answers. Errors can be opportunities for learning. Problem solving takes time and doesn’t end with a correct answer. We urge students to find ways to justify and explain their solutions to problems. These explanations allow us opportunities to observe students’ mathematical thinking and support their ability to analyze mathematical similarities and differences among strategies and representations of a mathematics problem or idea.