Mathematics

Mental computation involves calculations done in the mind, with little or no use of paperand pencil. Students who have developed the ability to calculate mentally can select fromand use a variety of procedures that take advantage of their knowledge and understandingof numbers, the operations, and their properties. Using knowledge of the distributiveproperty, for example, students can mentally compute 70% of 22 by first considering 70%of 20 and then adding 70% of 2. Used effectively, mental computation can encouragestudents to think more deeply about numbers and number relationships.Knowing how to estimate and recognizing when it is useful to estimate and when it isnecessary to have an exact answer are important mathematical skills. Estimation is auseful tool for judging the reasonableness of a solution and for guiding students in theiruse of calculators. The ability to estimate depends on a well-developed sense of numberand an understanding of place value. It can be a complex skill that requires decomposingnumbers, compensating for errors, and perhaps even restructuring the problem. Estimationshould not be taught as an isolated skill or a set of isolated rules and techniques.Recognizing calculations that are easy to perform and developing fluency in performingbasic operations contribute to successful estimation.CONNECTINGExperiences that allow students to make more connections – to see, for example, howconcepts and skills from one strand of mathematics are related to those from another orhow a mathematical concept can be applied in the real world – will help them developdeeper mathematical understanding. As they continue to make such connections, studentsbegin to see mathematics more as a study of relationships rather than a series ofisolated skills and concepts. Making connections not only deepens understanding, butalso helps students develop the ability to use learning from one area of mathematics tounderstand another.Making connections between the mathematics being studied and its applications in thereal world helps convince students of the usefulness and relevance of mathematicsbeyond the classroom.REPRESENTINGIn the senior mathematics curriculum, representing mathematical ideas and modellingsituations generally involve concrete, numeric, graphical, and algebraic representations.Pictorial, geometric representations as well as representations using dynamic softwarecan also be very helpful. Students should be able to recognize the connections betweenrepresentations, translate one representation into another, and use the different representationsappropriately and as needed to solve problems. Knowing the different ways inwhich a mathematical idea can be represented helps students develop a better understandingof mathematical concepts and relationships; communicate their thinking andunderstanding; recognize connections among related mathematical concepts; and modeland interpret mathematical, physical, and social phenomena. When students are able torepresent concepts in various ways, they develop flexibility in their thinking about thoseconcepts. They are not inclined to perceive any single representation as “the math”; rather,they understand that it is just one of many representations that help them understanda concept.THE MATHEMATICAL PROCESSES21