Page 16 - Math 9-10 (2009)

THE MATHEMATICAL PROCESSES 15
CD-ROM technology, and/or Internet websites to gain access to Statistics Canada, mathemat- ics organizations, and other valuable sources of mathematical information around the world.
Manipulatives.2 Students should be encouraged to select and use concrete learning tools to make models of mathematical ideas. Students need to understand that making their own mod- els is a powerful means of building understanding and explaining their thinking to others. Using manipulatives to construct representations helps students to:
• see patterns and relationships;
• make connections between the concrete and the abstract; • test,revise,andconfirmtheirreasoning;
• remember how they solved a problem;
• communicate their reasoning to others.
Computational Strategies. Problem solving often requires students to select an appropriate computational strategy. They may need to apply the standard algorithm or to use technology for computation. They may also need to select strategies related to mental computation and estimation. Developing the ability to perform mental computation and to estimate is conse- quently an important aspect of student learning in mathematics.
Mental computation involves calculations done in the mind, with little or no use of paper and pencil. Students who have developed the ability to calculate mentally can select from and use a variety of procedures that take advantage of their knowledge and understanding of numbers, the operations, and their properties. Using their knowledge of the distributive property, 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 encourage students to think more deeply about numbers and number relationships.
Knowing how to estimate, and knowing when it is useful to estimate and when it is necessary to have an exact answer, are important mathematical skills. Estimation is a useful tool for judg- ing the reasonableness of a solution and for guiding students in their use of calculators. The ability to estimate depends on a well-developed sense of number and an understanding of place value. It can be a complex skill that requires decomposing numbers, compensating for errors, and perhaps even restructuring the problem. Estimation should not be taught as an iso- lated skill or a set of isolated rules and techniques. Knowing about calculations that are easy to perform and developing fluency in performing basic operations contribute to successful estimation.
Connecting
Experiences that allow students to make connections – to see, for example, how concepts and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. As they continue to make such connections, students begin to see that mathematics is more than a series of isolated skills and concepts and that they can use their learning in one area of mathematics to understand another. Seeing the relation- ships among procedures and concepts also helps deepen students’ mathematical understanding.
2. See the Teaching Approaches section, on page 23 of this document, for additional information about the use of manipulatives in mathematics instruction.