Page 14 - Math 9-10 (2009)

THE MATHEMATICAL PROCESSES 13
integral part of the mathematics curriculum in Ontario, for the following reasons. Problem solving:
• is the primary focus and goal of mathematics in the real world;
• helps students become more confident mathematicians;
• allows students to use the knowledge they bring to school and helps them connect math- ematics with situations outside the classroom;
• helps students develop mathematical understanding and gives meaning to skills and concepts in all strands;
• allowsstudentstoreason,communicateideas,makeconnections,andapplyknowledgeand skills;
• offersexcellentopportunitiesforassessingstudents’understandingofconcepts,abilityto solve problems, ability to apply concepts and procedures, and ability to communicate ideas;
• promotesthecollaborativesharingofideasandstrategies,andpromotestalkingaboutmath- ematics;
• helps students find enjoyment in mathematics;
• increasesopportunitiesfortheuseofcritical-thinkingskills(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 stu- dents 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 successfully to solve problems of various types. Teachers who use relevant and meaningful problem-solving experiences as the focus of their mathematics class help students to 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 they become more mature in their problem- solving skills. By secondary school, students will have learned many problem-solving strategies that they can flexibly use and integrate when faced with new problem-solving situations, or to learn or reinforce mathematical concepts. Common problem-solving strategies include the following: making a model, picture, or diagram; looking for a pattern; guessing and checking; making assumptions; making an organized list; making a table or chart; making a simpler problem; working backwards; using logical reasoning.
Reasoning and Proving
An emphasis on reasoning helps students make sense of mathematics. Classroom instruction in mathematics should always foster critical thinking – that is, an organized, analytical, well- reasoned approach to learning mathematical concepts and processes and to solving problems.
As students investigate and make conjectures about mathematical concepts and relationships, they learn to employ inductive reasoning, making generalizations based on specific findings from their investigations. Students also learn to use counter-examples to disprove conjectures. Students can use deductive reasoning to assess the validity of conjectures and to formulate proofs.