Essentials of Mathematics 11 -
Problem Solving

This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will use a variety of methods to solve real-life, practical, technical, and theoretical problems

It is expected that students will:

• solve problems that involve a specific content area
• solve problems that involve more than one content area
• solve problems that involve mathematics within other disciplines
• analyse problems and identify the significant elements
• develop specific skills in selecting and using an appropriate problem-solving strategy or combination of strategies chosen from, but not restricted to, the following:
  • guess and check
  • look for a pattern
  • make a systematic list
  • make and use a drawing or model
  • eliminate possibilities
  • work backward
  • simplify the original problem
  • develop alternative original approaches
  • analyse keywords
• demonstrate the ability to work individually and co-operatively to solve problems
• determine that their solutions are correct and reasonable
• clearly communicate a solution to a problem and the process used to solve it
• interpret their solutions by describing what the solution means within the context of the original problem
• use appropriate technology to assist in problem solving

SUGGESTED INSTRUCTIONAL STRATEGIES

Problem solving is a key aspect of any mathematics course. Working on problems involving estimation, analysis, and measurement can give students a sense of the excitement involved in creative and logical thinking. It can also help students develop transferable real-life skills and attitudes. Multi-strand and interdisciplinary problems should be included throughout Essentials of Mathematics 11

• Reinforce the concept that "problem solving" is more than just word problems and includes other aspects of mathematics than algebra.
• Introduce new types of problems directly to students (without demonstration) and play the role of facilitator as they attempt to solve such problems.
• Recognize when students use a variety of approaches; avoid becoming prescriptive about approaches to problem solving.
• Reiterate that problems might not be solved in one sitting and that "playing around" with the problem—revisiting it and trying again—is sometimes needed.
• Frequently engage small groups of students (three to five) in co-operative problem solving when introducing new types of problems.
• Have students or groups discuss their thought processes as they attempt a problem. Point out the strategies inherent in their thinking (e.g., guess and check, look for a pattern, make and use a drawing or model).
• Ask leading questions such as:
  • What are you being asked to find out?
  • What do you already know?
  • Do you need additional information?
  • Have you ever seen similar problems?
  • What else can you try?
• Once students have arrived at solutions to particular problems, encourage them to generalize or extend the problem situation.
• See Appendix G for examples of multi-strand and interdisciplinary problems that most students should be able to solve. These types of problems are indicated with an asterisk (*).

SUGGESTED ASSESSMENT STRATEGIES

Students analyse problems and solve them using a variety of approaches. Assessment of problem-solving skills is made over time, based on observations of many situations.

Observe

• Have students present solutions to the class individually, in pairs, or in small groups. Note the extent to which they clarify their problems and how succinctly they describe the processes used.

Question

• To check the approaches students use when solving problems, ask questions that prompt them to: - paraphrase or describe the problem in their own words - explain the processes used to derive an answer - describe alternative methods to solve a problem - relate the strategies used in new situations - link mathematics to other subjects and to the world of work

Collect

• On selected problems, have students annotate their work to describe the processes they used. Alternatively, have them provide brief descriptions of what worked and what did not work as they solved particular problems.

Self-Assessment

• Ask students to keep journals to describe the processes they used in dealing with problems. Have them include descriptions of strategies that worked and those that did not.
• Develop with students a set of criteria to self-assess problem-solving skills. The reference set Evaluating Problem Solving Across Curriculum may be helpful in identifying such criteria.

RECOMMENDED LEARNING RESOURCES

Comprehensive learning resources for this course are currently under development. As an interim measure, schools are encouraged to use the teacher-developed learning resources distributed to schools (student and teacher resources). Please note that the student materials require photocopying for student use.

Print Materials

• The Multicultural Math Classroom: Bringing in the World
• Problem Solving: What You Do When You Don’t Know What To Do

Please see the introduction to Appendix B for a list of suggested utility software that supports this course.

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 22, 2000

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