Page 21 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

REASONING AND PROVING
Reasoning helps students make sense of mathematics. Classroom instruction in mathe- matics should 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 relation- ships, 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.
REFLECTING
Good problem-solvers regularly and consciously reflect on and monitor their own thought processes. By doing so, they are able to recognize when the technique they are using is not fruitful, and to make a conscious decision to switch to a different strategy, rethink the problem, search for related content knowledge that may be helpful, and so forth. Students’ problem-solving skills are enhanced when they reflect on alternative ways to perform a task even if they have successfully completed it. Reflecting on the reasonableness of an answer by considering the original question or problem is another way in which students can improve their ability to make sense of problems.
SELECTING TOOLS AND COMPUTATIONAL STRATEGIES
The primary role of learning tools such as calculators, manipulatives, graphing technolo- gies, computer algebra systems, dynamic geometry software, and dynamic statistical soft- ware is to help students develop a deeper understanding of mathematics through the use of a variety of tools and strategies. Students need to develop the ability to select the appropriate learning tools and computational strategies to perform particular mathe- matical tasks, to investigate mathematical ideas, and to solve problems.
Calculators, Computers, Communications Technology
Various types of technology are useful in learning and doing mathematics. Students can use calculators and computers to extend their capacity to investigate and analyse mathe- matical concepts and to reduce the time they might otherwise spend on purely mechani- cal activities.
Technology helps students perform operations, make graphs, manipulate algebraic expressions, and organize and display data that are lengthier or more complex than those addressed in curriculum expectations suited to a paper-and-pencil approach. It can be used to investigate number and graphing patterns, geometric relationships, and different representations; to simulate situations; and to extend problem solving. Students also need to recognize when it is appropriate to apply their mental computation, reasoning, and estimation skills to predict results and check answers.
THE MATHEMATICAL PROCESSES
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