At a Glance...

Numbers and operations play an important role in mathematics learning, as students rely on their understanding of them to master various mathematical concepts. In addition to being directly related to the other strands, numbers and operations are used daily by everyone. For this reason, the Numbers strand has historically been at the heart of mathematics learning.

However, learning about numbers and operations has evolved over time. The Numbers strand is more than :

• the application of algorithms and procedures;
• the search for the right answer;
• a series of arithmetic exercises.

The Intermediate Numbers strand is :

• understanding numbers and the quantities they represent;
• making connections between numerical concepts;
• the use of strategies that are understood and effective for computation in a variety of contexts.

Source: Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 5.

Big Ideas

“A Big Idea is the statement of a fundamental idea for the learning of mathematics, an idea that binds a lot of mathematical knowledge into a coherent whole." (Adapted from Charles, 2005, p. 10)

The expectations and learning content of the mathematics curriculum are based on many concepts. The big ideas allow teachers to see how these concepts can be grouped together to allow for more effective planning.

In doing so, teachers are able to develop coherent learning situations that allow students to:

• explore concepts in depth;
• establish links between the different concepts;
• to recognize that mathematics is a coherent whole, not a collection of disparate pieces of knowledge.

In the sections below, the two big ideas of the Numbers strand are each supported by three statements. These two big ideas represent the foundations of learning the Numbers strand and are approached in terms of whole numbers, fractions and decimals and percentages.

Source: A Guide to Effective Instruction in Mathematics, Grades 4to 6, pp. 21-22.

The two big ideas that form the basis of the Grade 7 and 8 Numbers strand expectations are Number Sense and Operations.

Big Idea 1: Number Sense

Number sense allows us to understand the numbers around us and to treat them with discernment.

The development of number sense in students should serve as a backdrop for teaching the Numbers strand. Number sense is a difficult concept to define, since it is not about specific knowledge, but rather an overview of numbers. Number sense can be seen as “a good intuition of numbers and their relationships that develops gradually, by exploring numbers, visualizing them in a variety of contexts, and connecting them in various ways. (Howden, 1989, p. 11)

In other words, number sense is the ability to recognize numbers, determine their relative values, and understand how they are used in a variety of contexts, including counting, measuring, estimating, and performing operations. It is therefore a deep relational understanding of numbers that involves many different ideas, relationships and skills.

Number sense is demonstrated or can be "observed" in mathematical situations. Students with strong number sense are aware of the importance of context in using numbers, can more easily estimate quantities and the results of calculations, make judgments about numbers as a result of calculations, and understand their use in context. Students are able to recognize a variety of relationships and represent numbers in order to use them in various contexts.

From an early age, children count, learn to determine quantities and recognize links between quantities and numbers, in a wide range of contexts. In the primary cycle, students explore whole numbers and progress to understanding the meaning of numbers below 1 000. In addition, students develop an intuition for the relative magnitude of numbers by comparing them and delving deeper into the meaning of place value. Students also have the opportunity to explore the meaning of fractions.

In the junior and intermediate grades, the development of number sense continues with the treatment of large numbers as well as various types of numbers in relation to each other. Students deepen their use of fractions and explore decimals and percentages. Number sense that has been built around natural numbers is then enriched with the use of various number notations.

Number sense is a way of thinking about and seeing numbers, of being able to "manipulate" them to grasp their meaning and use them effectively. It cannot be taught or demonstrated as such. However, in order for students to develop number sense, teachers must engage them in a variety of manipulative, exploratory, representational, constructional, visualization, communication, and problem-solving activities.

Below are the statements that define Big Idea 1 - Number Sense.

Statement 1 - Quantity Represented by a Number

Understanding quantity means developing a sense of "how many… " or "how many…" there are.

Statement 2 - Number Relationships

Establishing relationships means recognizing connections between numbers in order to better understand their meaning.

Statement 3 - Number Representations

Moving from one representation of a number to another allows for a better understanding of numbers.

Source: A Guide to Effective Instruction in Mathematics, Grades 4to 6, pp. 26-27.

Big Idea 2 - Operations

Operations enable us to choose the operations to be performed and execute them efficiently according to a given situation.

Operations combine the mastery of a multitude of mathematical skills and concepts related to numbers and operations. In a given situation, it allows for numbers and operations to be used with enough flexibility and versatility to be able to perform a calculation efficiently.

Students with a well-developed sense of operations (Small, 2005 a, p. 136) understand operations and the effect they have on numbers, make connections between the properties of operations, recognize that operations are interrelated, and develop computational strategies. In addition, students can adapt these strategies to different situations and express the relationship between the context of a problem and the calculations performed. For example, students are able to explain the reason(s) behind their choice of mental calculation and justify the effectiveness of their strategy.

In the primary grades, students developed a sense of operations by dealing with various types of problems. These experiences allowed them to grasp concepts related to various operations (for example, multiplication can be seen as repeated addition, addition is commutative) and to develop strategies for performing operations.

In the junior and intermediate grades, students continue to develop their sense of operations by dealing with numbers in more complex situations and gain a better understanding of the meaning of each operation and the relationships between them. Students become increasingly comfortable with a variety of computational and problem-solving strategies, allowing them to make more informed choices in different situations. In addition, their sense of operations extends to the application of basic operations on fractions and decimal numbers.

Below are the statements that define Big Idea 2 - Meaning of Operations.

Statement 1 - Quantity in Operations

Understanding operations allows one to recognize their effects on quantities.

Statement 2 - Relationships between operations

Understanding the relationships between operations allows them to be used more flexibly.

Statement 3 - Representations of Operations

Knowing a variety of strategies for performing operations allows them to be used effectively in different contexts.

Source: A Guide to Effective Instruction in Mathematics, Grades 4to 6, pp. 72-73.

The two great ideas are both complementary and interdependent - one cannot exist without the other. Having a sense of number means understanding numbers, what they represent. This understanding is essential to grasp what happens to numbers during operations. The goal of the Numbers strand is for students to use their number sense in conjunction with their understanding of operations to solve problems.

Each of the big ideas is explored in terms of statements with similar themes: quantity, relation and representation. The similarity of the statements is no accident. In fact, the statements allow for the recognition of essential notions while learning how to count (understanding quantity, or "how much"), how to recognize relationships between numbers and operations and finally, how to be flexible when representing and using numbers and operations.

The teaching of the Numbers strand, based on big ideas, aims to create connections and develop a more global view of numbers. Note that these big ideas and their statements are not limited to a set of numbers. For example, the fact that a number can be represented in different ways is not unique to decimal numbers, but applies to numbers in general.

Everyone needs to develop a strong number sense and operations sense in order to solve problems. To enable this development in students, teachers need to keep the importance of these big ideas in mind.

Source: A Guide to Effective Instruction in Mathematics^, Grades 4-6, pp. 24-25.

Structuring programs around big ideas and emphasizing problem solving provides students with coherent learning situations that allow them to explore concepts in depth.

Children are better able to make connections in mathematics and thereforelearn mathematics if the curriculum is structured coherently around the big ideas. Grouping expectations into big ideas facilitates student learning and teacher professional development in mathematics. Teachers will find that it is much more useful to discuss and determine the most effective instructional strategies for a big idea than to try to determine the specific strategies and approaches that would help students achieve particular expectations. The use of big ideas allows teachers to understand that the concepts presented in the curriculum are not to be taught separately, but rather as a set of interrelated concepts. In order to develop a curriculum, teachers must have a thorough knowledge of the key mathematical concepts of the grade they are teaching as well as an understanding of the connections between these concepts and the future learning of their students (Ma, 1999). This includes an understanding of "the conceptual structure and fundamental attitudes inherent in elementary mathematics" (Ma, 1999, p. xxiv) and how best to teach these concepts to children. Developing this knowledge makes teaching more effective.

Source: A Guide to Effective Instruction in Mathematics^, Grades 1-3, pp. 2-3.

Communication is an essential component of learning mathematics. It is a skill that goes beyond the appropriate use of mathematical terminology and symbols in verbal or written communication. It is also, more importantly, a vehicle through which students develop an understanding of mathematical concepts in contexts that involve mathematical reasoning and arguments. This is what Radford and Demers (2004) call the conceptual dimension of communication.

These researchers also emphasize the importance of taking into account the social dimension of communication. Indeed, "communication" means "exchange" between two or more people. The exchange will be beneficial for all those involved to the extent that there is a climate of commitment to dialogue and a culture of respect and listening to what others say.

To increase the effectiveness of numbers instruction, teachers need to foster a culture that values communication as a means of learning. This means creating opportunities for students to interact with each other to clarify their thinking and strategies. Communication is central to all learning situations.

The Role of Representations in Learning Mathematics

In mathematics, communication is not just about words. Ideas should be conveyed in different ways: concrete (for example, with rods), semi-concrete (for example, with a number line or diagram), symbolic (for example, using numbers and mathematical symbols in an equation) and, of course, verbally, using words, whether they are read, seen, said, written or heard.

These various modes of representation, shown in the diagram below, allow for multiple cognitive inputs, thereby making connections between ideas that are essential for learning. Teachers use models to represent mathematical concepts to students who, in turn, use them to solve problems and clarify their thinking.

Mathematical Models

Mathematicians have long constructed models to explain and represent discoveries and observations of the world, and to communicate them effectively. For example, when thinking about a number, some people visualize it in a mathematical model such as an open number line or a number grid. This helps to better identify the number and to recognize that it is more than… , less than…or close to…another number. Models are thus representations of mathematical ideas.

Young children visualize the world around them in their own way. To draw the tree in front of a house, a child will drawslines on paper and represent it in two dimensions - even though the child has touched the tree, walked around it and taken shelter under its branches (Fosnot & Dolk, 2001, p. 74). This representation is not a copy of the tree, but a construction of what the child knows. It is in fact a "model" of the tree. The same is true for students whose first models used to solve problems reflect their interpretation of reality.

Fosnot and Dolk (2001, p. 80) argue that models, like big ideas and strategies, cannot be automatically transmitted, but must be reappropriated and constructed by students. The value table is a good example: intuitively, students "organize" numerical data by placing numbers disparately on a sheet of paper, but the value table allows them to be ordered for processing and analysis.

However, a clarification about models and manipulatives is in order: the model is not the mathematical idea. Thus, the tree that the child drew in the example above is not a tree, but a representation of the situation that will be used to discuss it. The same is true for all the models used.

Base ten material is a model because it assumes that the person using it already has an understanding of the concept of grouping. However, presenting a tab from a kit of base ten materials and claiming that it is a ten is false. The object is not a "ten," but a concrete way to represent the "idea" of a ten. Here it represents a ten of small cubes, but it could represent a unit, a tree or even a beam. Depending on the intention, it could also represent a tenth, for example, by representing the number 2.5 with 2 planks (2 units) and 5 strips (5 tenths).

The number line is another model to which students should be exposed. The number line does not represent the quantity corresponding to the numbers that are placed on it; it allows us to "see" the numbers in relation to each other. For example, a number line with the numbers 44, 42, and 41.5 on it represents the order relationship between these three numbers.

In order to foster students' reasoning, teachers should use and encourage students to use a variety of models. Models do not have to be taught formally; they can be introduced in problem-solving situations. For example, the number line is an excellent model for exploring the addition of several numbers. However, most students do not "conceive" that it can be created without being scaled. Imagine then a mathematical exchange, in the context of an addition problem, where they present their problem-solving strategies. Teachers can use this opportunity to make a connection between the number line used by the student (Figure 1) to perform an operation and the possibility of using an open number line (Figure 2).

Example

\(\ 5 + 3 + 10\)

Similarly, in order to represent situations involving fractions, students often tend to use a surface model (such as a circle or a rectangle). However, this type of model does not accurately represent situations where the whole is a length or distance. Teachers can then take advantage of an opportunity where students are using a surface model to represent the fraction of a length and show them how a length model (for example, a number line) would be more appropriate.

Students need to be exposed to a variety of representations so that they can make connections between them and consolidate their learning. Over the course of their schooling, students must make a transition from using a model as a teaching tool in a particular situation to using a model as a strategy for generalizing mathematical ideas, solving problems, and applying the model to new contexts. This transition from a familiar context to a new context is a fundamental step in learning mathematics. It is found in the Achievement Chart under "Application".

Here are some models that students can use in the Numbers strand:

• a number line;
• a double open number line;
• an array
• a table of values;
• number grid (e.g, hundred chart);
• base ten material;
• equations;
• a surface model to represent fractions;
• a length model to represent fractions;
• a model set to represent fractions;
• a volume model to represent fractions;
• money to represent decimal numbers.

Source: A Guide to Effective Instruction in Mathematics^, Grades 4-6, pp. 8-12.

Teaching Problem Solving

To help students fully understand the concepts and processes of the Numbers strand, it's important to place them in problem-solving situations right from the start of a learning unit. When they work as a team to solve an engaging, non-routine problem, students become adept at formulating a hypothesis and a mathematical argument. They also learn to take risks, persevere and have confidence in their problem-solving abilities. This is where learning mathematics really comes into its own.

Problem-based instruction requires teachers to present learning situations that sustain student interest. The context or situation of the problem then becomes a determining factor. "Problems should be based on real contexts (situations that occur authentically in the classroom), real-life contexts (situations that are derived from experiences that could be had by students outside the classroom), and even fanciful contexts (situations that appeal to student imagination)" (Vézina et al., 2006, p. 4). Indeed, the context can be an eye-catcher for students and give them a reason to "do mathematics". Consequently, the context must be chosen, formulated and shaped judiciously in order to touch their sensibilities. Context, then, is an element of problem solving that can be used to engage students.

Problem-based teaching also requires that teachers present students with learning situations that are rich in mathematical content and that encourage them to think. Teachers must then allow students to develop their own problem-solving strategies without overly directing them. Finally, teachers need to clarify mathematical concepts as students present their strategies and solutions during the mathematical exchange. The mathematical exchange is a time of objectification in which students explain and defend their thinking and analyze the thinking of others. Learning and understanding are forged through this confrontation of ideas and effective questioning by the teacher. In addition, the mathematical exchange allows students to consolidate their learning and develop various skills such as problem solving, communication, reasoning, listening and analysis.

Teaching through problem solving is about understanding. In the Numbers strand, students will solve problems to develop a better sense of operations, which will result in the use of understood strategies and not in the use of memorized steps and blindly applied.

Source: A Guide to Effective Instruction in Mathematics, Grades 4to 6, pp. 13-14.

Students in the Numbers strand develop their knowledge and skills gradually. It is characterized by a gradual deepening of number sense and sense of operations, which takes place throughout the elementary grades.

The following tables propose a number sense development scale (Table 1) and an operations sense development scale (Table 2). Each scale describes a five-stage developmental continuum from initiation to versatility as illustrated in the diagram above.

This continuum, which reflects a path from the concrete to the abstract, is based on the following three premises.

• Students must go through all the stages for each new concept. If some of the stages are skipped, it will be difficult for them to fully develop number sense and operations sense to the multi-skill stage. However, with time and depending on their background of experiences, students will be able to move through the early stages more and more quickly.
• The journey through these stages is not exclusively unidirectional. Rather, depending on the learning situations presented, students may need to return to a previous stage to consolidate their learning.
• The stages are not disjointed. There is an intersection between consecutive stages, and student understanding can fall within this zone.

In each table, the stages are briefly defined and accompanied by a few examples of observable behaviours that serve to clarify their meaning. Teachers can use these tables as part of a diagnostic or formative assessment to determine the stage at which students are situated in relation to a particular concept. They can then plan learning situations that correspond to student development, enabling them to move on to the next stage. Progression from one stage to the next depends on the relevance of learning activities and mathematical exchanges in the classroom. In other words, the more meaningful experiences students have, the sharper and clearer their understanding will be.

Note: It is important to note that the five stages in the two tables are not related in any way to the grades or achievement levels in the mathematics curriculum rubric.

Table 1 below describes the stages in the development of number sense. It is important to remember that the word "number" in this table includes natural numbers, whole numbers, fractions and decimal numbers. When a set of numbers is the object of study for the first time, students are generally at stage 1. For example, when studying decimal numbers in Grade 4, students are at Stage 1 for this set of numbers. However, students may be at stage 3 for natural numbers.

Table 1 - Number Sense Development

The following Table 2 describes the stages of development of operations sense. It is important to link operations sense to the set of numbers with which students perform operations. For example, a student may be able to perform basic operations with fractions using concrete materials (stage 2) while being able to perform basic operations with natural numbers using personal strategies (stage 3). The progression from one stage to the next can also be done at a different pace depending on the operations. For example, the student may be at step 4 with the operations of addition and subtraction of natural numbers, but at step 3 with the operations of multiplication and division of these same numbers.

Table 2 - Number Sense Development