Extract from Ontario curriculum Mathematics:

1. Number Sense and Numeration

It is important for students to develop the mathematical competence that comes from understanding numbers, number systems, and their related operations.

Number is a complex and multifaceted concept. A well-developed understanding of number includes a grasp not only of counting and numeral recognition but also of a complex system of more-and-less relationships, part-whole relationships, the role of special numbers such as five and ten, connections between numbers and real quantities and measures in the environment, and much more. Experience suggests that students do not grasp these relationships automatically. It is the teacher's job to provide students with a broad range of activities that will help them develop many of these ideas about number.

Helping students to understand numbers must always include introducing them to procedures for accurately performing operations with numbers. Key facts and processes must be mastered. Students also need to develop "number sense". Number sense includes:

an appreciation of and ability to make quick order-of-magnitude approximations with emphasis on quick and accurate estimates in computation and measurement;

the ability to detect arithmetic errors;

knowledge of place value and the effects of arithmetic operations;

a grasp of ideas about the role of numbers and about their multiple relationships;

an appreciation of the need for numbers beyond whole numbers.

Mental Mathematics and Estimation

By the end of Grade 6, students should have consolidated their understanding of basic computational facts and be able to use computational strategies to do mental mathematics. This ability develops over time, supported by regular practice, as students identify relationships between numbers and learn when and how to use the various operations effectively. Techniques of mental mathematics should be introduced along with concepts of place value and the use of pencil-and-paper calculations. For example, the mental skill of adding numbers ending in zero, such as 20 + 40 + 70, can be learned by a student who understands place value and can add 2 + 4 + 7. Instruction in computational estimation should include not only applications involving whole numbers and decimals, but also those involving fractions and percent. Early experiences in estimating with percent and fractions can help students develop number sense.

Pencil-and-Paper Computation

Students (and adults) require facility with pencil-and-paper computations. It is important for pencil-and-paper computational procedures to be introduced through the use of concrete materials. Students should use these materials until they understand the concepts well enough to move from the manipulative stage to the semiconcrete medium of pictures and then to the abstract form of numbers on a page.

Many methods of pencil-and-paper calculation have been appropriately replaced by operations of a calculator or computer. For example, long division, operations with long lists of large numbers, and the calculation of square root can be done more efficiently using technology. Teachers should also provide students with a variety of experiences and investigations involving number.

Fractions and Rationals

Concepts and operations with fractions should be introduced using concrete materials. Models, tiles, manipulatives, and diagrams should be used to relate fractions to decimals, to find equivalent fractions, and to explore operations with fractions and decimals. Fraction symbols build on the understanding developed in these ways. Mathematics instruction should help students gain conceptual understanding as well as use fractions and rational numbers effectively and accurately.

It is recommended that, initially, simple denominators such as 2, 3, 4, 5, and 10 be used. As students gain more experience and skill in working with fractions, denominators such as 6, 8, and 12 can be included. Later, the focus shifts to using fractions in ratios, rates, and percent. As well, students will extend their fraction sense to include skill in operations with fractions. It should be remembered, however, that the use of fractions in real-life situations often involves estimating (e.g., "My friend lives half a block from here"), and this skill should be developed along with accurate calculation.

Calculators

The ability to use calculators intelligently is an integral part of number sense. It should be noted that the use of calculators does not do away with the necessity for students to master the fundamental mathematical operations. Students should use calculators in their schoolwork, just as adults use calculators for many purposes in the course of their daily lives. More importantly, students must learn when it is appropriate to use a calculator and when it is not. They must learn from experience with calculators when to estimate and when to seek an exact answer, and how to estimate answers to verify the plausibility of calculator results. Calculators allow teachers to engage students in meaningful mathematical investigations, such as solving science problems with large numbers, before their skill with pencil-and-paper computation is equal to the task. Proper calculator use stimulates the growth of number sense in students.

Computers

The computer is an important tool used by mathematicians to perform a wide variety of tasks; the ability to use computers effectively and appropriately is central to students' development of mathematical competence.

An important use of computer software is to engage students in the exploration of concepts. Computer programs should help students develop number sense and deal with large amounts of data in an organized way. Spreadsheets should be used by all students to manage and operate on long lists of numbers. Also, the computer can serve as an aid to students in clarifying operations rules that will help them develop concepts used in early algebra.

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Number Sense and Numeration: Grade 1

Overall Expectations

By the end of Grade 1, students will:

understand whole numbers by exploring number relationships using concrete materials (e.g., demonstrate with blocks that 7 is one less than 8 or two more than 5);

understand numerals, ordinals, and the corresponding words, and demonstrate the ability to print them;

understand the concept of order by sequencing events (e.g., the steps in washing a dog);

compare and order whole numbers using concrete materials and drawings to develop number meanings (e.g., to show place value, arrange 32 counters in groups of 3 tens and 2 ones);

represent fractions (halves as part of a whole) using concrete materials;

understand and explain basic operations (addition and subtraction) of whole numbers by modelling and discussing a variety of problem situations (e.g., show that addition involves joining);

develop proficiency in adding one-digit whole numbers;

solve simple problems involving counting, joining, and taking one group away from another (e.g., how many buttons are on the table?), and describe and explain the strategies used;

estimate quantity in everyday life (e.g., guess, then count how many beans are in the jar);

use a calculator to explore counting and to solve problems beyond the required pencil-and-paper skills.

Specific Expectations

Students will:

Understanding Number read and print numerals from 0 to 100; read and print number words to ten; demonstrate the conservation of number (e.g., 5 counters still represent the number 5 whether they are close together or far apart); demonstrate the one-to-one correspondence between number and objects when counting; count by 1's, 2's, 5's, and 10's to 100 using a variety of ways (e.g., counting board, abacus, rote); count backwards from 10; locate whole numbers to 10 on a number line; compare, order, and represent whole numbers to 50 using concrete materials and drawings; investigate number meanings (e.g., the concept of 5); use mathematical language to identify and describe numbers to 50 in real-life situations; discuss the use of number and arrangement in real-life situations (e.g., there are 21 children in my class, 11 girls and 10 boys); use a seriation line to display relationships of order (e.g., order of events in a story); model numbers grouped in 10's and 1's and use zero as a place holder; use a calculator to explore counting, to solve problems, and to operate with numbers larger than 10; use ordinal numbers to tenth;

represent and explain halves as part of a whole using concrete materials and drawings (e.g., colour one-half of a circle); estimate the number of objects and check the reasonableness of an estimate by counting; Computations demonstrate that addition involves joining and that subtraction involves taking one group away from another; demonstrate addition and subtraction facts to 20 using concrete materials; represent addition and subtraction sentences (e.g., 5 + 6 = 11) using concrete materials (e.g., counters); identify the effect of zero in addition and subtraction; mentally add one-digit numbers; add and subtract money amounts to 10¢ using concrete materials, drawings, and symbols; Applications pose and solve simple number problems orally (e.g., how many students wore boots today?); use concrete materials to help in solving simple number problems; describe their thinking as they solve problems.

Number Sense and Numeration: Grade 2

Overall Expectations

By the end of Grade 2, students will: