Page 20 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
The mathematical processes cannot be separated from the knowledge and skills that stu- dents acquire throughout the course. Students who problem solve, communicate, reason, reflect, and so on, as they learn mathematics, will develop the knowledge, the under- standing of concepts, and the skills required in the course in a more meaningful way.
PROBLEM SOLVING
Problem solving is central to learning mathematics. It forms the basis of effective mathe- matics programs and should be the mainstay of mathematical instruction. It is considered an essential process through which students are able to achieve the expectations in mathe- matics, and is an integral part of the mathematics curriculum in Ontario, for the following reasons. Problem solving:
helps students become more confident mathematicians;
allows students to use the knowledge they bring to school and helps them connect mathematics with situations outside the classroom;
helps students develop mathematical understanding and gives meaning to skills and concepts in all strands;
allows students to reason, communicate ideas, make connections, and apply knowledge and skills;
offers excellent opportunities for assessing students’ understanding of concepts, ability to solve problems, ability to apply concepts and procedures, and ability to communicate ideas;
promotes collaborative sharing of ideas and strategies, and promotes talking about mathematics;
helps students find enjoyment in mathematics;
increases opportunities for the use of critical-thinking skills (e.g., estimating, classifying, assuming, recognizing relationships, hypothesizing, offering opinions with reasons, evaluating results, and making judgements).
Not all mathematics instruction, however, can take place in a problem-solving context. Certain aspects of mathematics must be explicitly taught. Conventions, including the use of mathematical symbols and terms, are one such aspect, and they should be introduced to students as needed, to enable them to use the symbolic language of mathematics.
Selecting Problem-Solving Strategies
Problem-solving strategies are methods that can be used to solve various types of problems. Common problem-solving strategies include: making a model, picture, or diagram; look- ing for a pattern; guessing and checking; making assumptions; creating an organized list; making a table or chart; solving a simpler problem; working backwards; and using logical
reasoning.
Teachers who use problem solving as a focus of their mathematics teaching help students develop and extend a repertoire of strategies and methods that they can apply when solving various kinds of problems – instructional problems, routine problems, and non- routine problems. Students develop this repertoire over time, as their problem-solving skills mature. By secondary school, students will have learned many problem-solving strategies that they can flexibly use to investigate mathematical concepts or can apply when faced with unfamiliar problem-solving situations.
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