At a Glance...

Algebra is the mathematical field that arose from the need to understand and organize the real world, for example, the movement of stars, what light is, the shape of the Earth. Mathematicians have tried to answer these questions through observation and the invention of new calculation techniques.

Several authors (Driscoll, 1999; Squalli, 2002) have raised the importance of making connections between arithmetic and algebra. Arithmetic is generally seen as computational work that focuses on efficiency in finding the right answer. However, work in algebra aims to better understand numbering by allowing students to analyze the relationships between numbers. For this reason, it is important to develop analytical thinking (reasoning) skills in elementary school by laying the foundation for algebraic thinking.

Take, for example, the equation 2 + 3 = 3 + 2 for which students do not have to find an answer. In arithmetic, students could perform addition on either side of the equal sign to confirm that the equality is true. Instead, in algebra, the goal is to see that when the numbers are reversed on the other side of the equal sign in an addition, the result does not change.

The study of patterns is also part of the study of algebra; it is a more concrete way to get students to observe both change and order in the world around them. This introduces students to the observation of change and the analysis of relationships in those changes, as change is an important component of algebraic thinking.

Algebraic Thinking

In the search for a definition of what algebraic thinking is, many prioritize a perspective that is considered essential in algebra.

Here are three examples that reflect three different perspectives:

The development of algebraic thinking requires the intervention of several interacting factors, namely:

• fundamental processes to access higher levels of abstraction (abstracting, generalizing and operating on variables);
• mathematical skills developed from an algebraic perspective (problem solving, reasoning and communication);
• components of the learning environment (understanding relationships, representing with symbols, using models, and analyzing change);
• algebraic concepts grouped according to the big ideas (patterns and relationships, and expressions and equality).

The image below illustrates the interaction between these factors.

Algebraic Concepts Grouped According to Two Big Ideas

Grouping various algebraic concepts into big ideas is an important factor in the development of algebraic thinking.

To help teachers define and prioritize key concepts and implement strategies that provide effective and coherent instruction, two big ideas are presented, explored, and developed in the Algebra strand. While interrelated, these two big ideas each have a particular emphasis. They allow students to explore relationships in patterns and to understand relationships in equality situations.

Source: Guide d’enseignement efficace des mathématiques, de la maternelle à la 3^e année, p. 5-7.

Source: Guide d’enseignement efficace des mathématiques, de la maternelle à la 3^e année, p. 27.

In a mathematics class aimed at developing students' algebraic thinking, the traditional goal of teaching, learning to calculate, is not omitted; it is largely superseded. Developing algebraic thinking is a complex journey that relies on three fundamental processes: abstracting, generalizing, and operating on variables.

Abstract

It is to detach oneself from the sensory aspect of things in order to reason at a more general level (adapted from Raynal and Rieunier, 2003, p. 13), it is to mentally represent a concrete situation. Piaget considers abstraction to be one of the major processes that allow the construction of knowledge. For his part, Roegiers (2000, p. 77) explains that the appropriation of a concept generalizes reality (for example, a regularity that does not exist in reality, but which is observed in a pattern). The concept is thus situated on another plane than reality. This is the domain of abstraction.

Generalize

It is drawing valid conclusions, true in all cases, from the observation and analysis of a few examples (adapted from Squalli, 2002, p. 9). Generalization is at the heart of mathematical activity. Generalizing "...is particularly important because in humans it is the basis for the acquisition of concepts and the ability to abstract" (Raynal & Rieunier, 2003, p. 156).

In situations of equality, students can more easily formulate a generalization when it follows a process of proposing and verifying a conjecture.

In algebra, students can more easily generalize when making conjectures. In the primary division, the properties of numbers and operations are one of the subject of conjectures. For example, when students assume that changing the order of the terms in a multiplication has no effect on the product, this is a conjecture.

When students see a recursive phenomenon as they explore various equality situations, they can propose a conjecture. For example, students might say that if the number 0 is added to any number, the initial quantity does not change. Students should then check whether their conjecture is valid in other similar situations. For example, in the situation in the previous example, students could check it with various numbers as well as with concrete materials. When a conjecture seems to apply to all similar situations, students make a generalization with words or symbols.

In the primary and junior divisions, conjectures are usually expressed in words by students. They may also be represented by concrete or semi-concrete materials to illustrate their mathematical reasoning as clearly as possible.

Teachers need to expose students to a variety of problem-solving situations that challenge them to practice the skill of making and testing a conjecture; for example, they present the number sentence 50 + 6 - 6 = 50 and then offer them the following conjecture: "I wonder, when you add and subtract the same number in a number sentence, if it is the same as when you add or subtract zero." Teachers then invite students to discuss this conjecture with each other and determine if it is always true.

Students test this conjecture with other number sentences. They may not be convinced that it applies to any number sentence or to all numbers, especially large numbers. In the course of the discussion, they can propose their own conjectures as illustrated below.

After a check of various number sentences, students conclude that the conjecture is true and formulate a generalization.

Since students' vocabulary in elementary school is not yet very developed and precise, initial conjectures usually need to be rephrased or clarified. Ideally, therefore, the formulation of a conjecture should be practiced in a class setting, as shown in the example below. During the discussion, students can point out the limitations of a peer's conjecture and contribute to the formulation of a clearer and more relevant common conjecture. However, it is important for teachers to establish a learning environment in which students perceive each other's questions as positive interactions that can fuel the exchange.

Exemple

The teacher presents the number sentence 564 + 0 = 564 and asks students if it is true or false.

Student: "It is true."

Teacher: "How can you tell?"

Student: "When a zero is added to a number, it doesn't actually add anything. Therefore, we get the starting number."

The teacher presents other similar number sentences. After several such exchanges, the teacher then asks students to formulate a conjecture.

Student: "All numbers added with a zero remain the same."

Another student presents a counterexample: "No, because (100 + 300 = 400). The numbers 100 and 300 have zeros in them. When added together, they do not remain the same."

After further discussion, a student formulates another conjecture:

Student: "­When you join a zero to another number, you get the other number."

Other student: "That's not true."

Teacher: "So, are you referring to the number that is right next to a zero?"

Student: "No, added to another number."

After much discussion, the following formulation is adopted: Zero, added to another number, is equal to that number. When students see that this conjecture applies to all numbers, they can generalize.

Operating on Variables

It is to reason analytically, to think about operations, generalizations and not objects (adapted from Squalli & Theis, 2005). Research suggests that this is what distinguishes arithmetic from algebra (Squalli, 2002; Driscoll, 1999). The variable is usually represented by letter. However, in many situations, it can be represented by a symbol or concrete material, or it can be expressed verbally.

Algebra begins with the awareness of operations, operations in the broad sense of the word, that is, a series of intellectual acts involving reflection and combination of means in order to obtain a result or solve a problem. It is "[…] presented as a 'generalized arithmetic', as a tool for solving problems more powerful than arithmetic." (Squalli & Theis, 2005, p. 5)

Source : Guide d’enseignement efficace des mathématiques, de la maternelle à la 3^e année, p. 8-11.

In the Algebra strand, students' thought processes will evolve to the extent that the skills to solve a problem-solving situation, to reason and to communicate will be developed according to an algebraic perspective.

Ability to Solve a Problem Algebraically

A problem-solving situation refers to a problem that :

• is open;
• is large;
• can be solved in teams;
• is put into context;
• is a challenge for everyone;
• promotes the use of different strategies.

From grades 1 to 3, problem solving is designed to engage students in a process that requires the use of different strategies. Students who have developed strategies find it easier to initiate the resolution of a problem, to anticipate and predict outcomes, to reason and to find a solution.

One of the goals of problem solving in algebra is to acquire intellectual tools for reasoning (for example, looking for regularities, establishing relationships, using different representations). Problem solving in an algebraic perspective involves knowing models to attempt a solution.

In the primary grades, students have few of these. Teachers need to introduce and use them explicitly to help students make them their own. Models may appear to be more numerical in nature, but using them from an algebraic perspective will develop algebraic thinking.

For example, teachers introduce students to the use of an open or double number line to get them to think about the calculation, not do the calculation. The important thing is not to calculate, for example, the sum of 9 + 6 or 8 + 7, but to understand the equality relationship, namely, 9 + 6 = 8 + 7.

Example

Gradually, students appropriate models, integrate them into their strategies and use them spontaneously to solve a problem. A problem-solving situation that is contextualized and challenging arouses interest and motivates students to develop a solution. It involves a process that requires anticipation, backtracking and objectification, which promotes the development of algebraic thinking.

Ability to Reason Algebraically

The ability to reason algebraically allows students to organize their thinking.

Algebraic reasoning is about observing and acting differently depending on what one does in arithmetic, and using a set of analytical thought processes, such as generalizing, operating on variables, and expressing relationships.

The table below shows the main distinction between reasoning arithmetically and reasoning algebraically.

Source: L’@telier - Ressources pédagogiques en ligne (atelier.on.ca)

Through algebraic reasoning, students analyze numbers, symbols, quantities, operations, and then generalize them.

The intellectual process required for algebraic reasoning is not a simple and natural one. Teachers need to engage students in this process:

• by helping them to make their approach explicit;
• encouraging them to work backwards, that is, to reverse the process, from the answer to get to the starting point;
• by encouraging them to find patterns and organize information to represent the situation in a different way and to generalize;
• by having them observe the relationships between numbers or operations; by allowing them to objectify their approach.

In order to teach students to objectify, teachers should ask questions that emphasize algebraic concepts and that lead them to reflect.

Here are some examples:

• Does it work if I do the same with other numbers?
• What changes?
• What does not change?
• Can I predict the outcome based on the information gathered?
• Does the rule work in all cases?
• Do I always follow the same steps? What are they?

Ability to Communicate Algebraically

The ability to communicate algebraically, both orally and in writing, is developed through exchange. When students discuss their understanding of a situation or concept, it is done through two distinct elements: modes of representation and the use of mathematical arguments.

The mathematical exchange is the privileged moment to actualize the mathematical arguments.

Modes of Representation

To communicate effectively, students can use different modes of representation. Mathematical relationships can be represented using concrete or semi-concrete materials, symbols, or oral descriptions.

When representing an algebraic situation using one or two modes of representation, students use a variety of models such as tables, number grids, or number lines.

These models help them organize, record, and communicate their thinking as they explore relationships. The representation of a problem-solving situation using concrete, semi-concrete or symbolic models, along with an oral description, facilitates the observation of relationships and contributes to the development of algebraic thinking. The different representations allow students to appropriate algebraic concepts.

A significant mathematical step in the development of algebraic thinking is to understand that two patterns can be constructed with different material but have the same rule or that a situation of equality can be represented using different models.

Use of Mathematical Arguments

A mathematical argument is an oral or written justification of a reasoning in order to demonstrate or refute a mathematical idea.

The mathematical exchange is the privileged moment to actualize mathematical representations and arguments and thus to promote the development of algebraic thinking.

By taking ownership of an algebra problem, students suggest conjectures, present possible solutions, compare their ideas or justify their results using different representations.

Note: In the primary cycle, students may have difficulty developing a clear conjecture. The teacher must, through the relevance of their questioning, take advantage of mathematical exchanges with the whole class to encourage the formulation of conjectures.

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3^e année, p. 10-15.

A learning environment conducive to the development of algebraic thinking should incorporate the following four components: understanding patterns and relationships, representing problem-solving situations using symbols, using mathematical models to represent relationships between quantities, and analyzing change. Each of these components is discussed below.

Understanding Patterns and Relationships

Recognizing regularities is an important problem-solving skill; it allows for the appropriation of other concepts and the formulation of conjectures leading to generalizations. The concept of regularity is the cornerstone of algebraic reasoning.

By observing and analyzing the relationships between numbers within a pattern, within a number sentence, or within the base ten system, students discover patterns and can deepen their understanding of algebraic concepts.

From an early age, children become aware of regularities in the world around them (for example, the arrangement of ceramic tiles in the bathroom, the cycle of day and night, routines of the day).

In the early grades, algebraic reasoning can be developed through the exploration of patterns, presented in a variety of forms using various attributes such as movements, sound, color, geometric shapes, and numbers.

For example, the story Goldilocks and the Three Bears is an interesting tool to introduce students to the concepts of patterns and regularity, and to establish relationships between the terms in a pattern. Throughout the story, students discover that the pattern is always made up of three "big, medium, small" elements that repeat in the same order. This is a good opportunity to introduce them to patterns and relationships.

Recognizing, comparing, representing, describing, extending, and creating patterns in the context of problem solving leads students to establish relationships between the terms of a pattern.

Representation of problem-solving situations Using Symbols

Representing and analyzing problem-solving situations is a fundamental component of algebraic thinking. To be successful in algebra, one must be able to use symbolic representations of real or contextualized situations. Real situations represented initially with concrete or semi-concrete material will gradually be represented by symbols. In order to build a solid foundation for understanding algebraic concepts, it is important to move students gradually toward a more formal symbolic representation. The use of symbols facilitates the achievement of a higher level of abstraction, especially when representing regularities or numbers in situations of equality.

In Algebra, students need to develop the meaning of the symbol. The meaning of the symbol is the skill:

• to understand the value of symbols;
• to know how to make judicious and relevant use of it;
• to represent a context by symbols.

Some people draw a parallel between number sense and the meaning of symbol. Among others, Arcavi (1994) conceives of symbol meaning as a spontaneous and accurate appreciation or instinctive understanding of the symbol.

He states that the meaning of the symbol is manifested by the following actions:

• demonstrate relationships, generalizations and even justifications that would otherwise be difficult to grasp;
• choose a possible symbolic representation to represent and solve a problem;
• determine whether the situation requires a symbolic representation or another representation;
• manipulate the symbols, read them and visualize the results and possible regularities.

As early as the primary grades, students demonstrate an understanding of the properties of mathematical operations such as the commutative property of addition with specific objects or numbers. Students use concrete materials, drawings, words, or symbols to represent mathematical ideas and relationships, including the equality relationship. Students describe and represent quantities in a variety of ways, and improve their ability to use symbols to communicate their ideas.

Later, students become aware of the meaning of symbols (for example, squares, circles, letters) to express unknown quantities, and then also learn to use them to express relationships by writing equations.

The presentation of authentic problems promotes the understanding of the concept of unknown quantities while helping to demystify the use of symbols.

Example 1

A farmer has three pens: one for horses, one for cows and one for sheep. The three pens are connected by paths, as shown below. The numbers in the picture show the total number of animals in two pens connected by a path. How many animals can be in each pen?

Using small plastic animals, students determine the possible quantities of animals in each pen. For example, a student might place four cows (4c), four horses (4h) and eight sheep (8s) in the respective pens. Then, to justify their reasoning, the student writes a number sentence corresponding to their representation.

4 h + 4 c = 8 animals

4 h + 8 s = 12 animals

The use of manipulatives provides a concrete way to see the possible relationships between variables. In the primary grades, this strategy focuses on understanding rather than on the abstract use of symbols. It develops reasoning and algebraic thinking.

The ability to manipulate symbols allows students to analyze a problem, choose an appropriate representation, select an effective strategy for solving it, and evaluate whether their solution is plausible.

Use of Mathematical Models to Represent Relationships Between Quantities

Mathematical models are used to study relationships. Over time, mathematicians have created, used, and generalized certain ideas, strategies, and representations to make concepts easier to grasp. Through use, certain representations have become accepted models, for example, the number line and the ten frame. It is important that students use mathematical models in a variety of activities to understand relationships between quantities.

Faced with a problem to be solved, several representations are possible; some students use their bodies, manipulatives or drawings, while others represent the data more schematically. The way in which the data are appropriated and organized reflects the level of development of algebraic thinking. The models explored in the primary and junior divisions will differ depending on the level of abstraction of the students. A variety of models can be used to solve a problem. In this document, the word model refers to the various concrete, semi-concrete, and symbolic representations of the problem, but it also refers to how each student uses personal symbols to represent their reasoning.

Examples of Models

Problem 1 : A grocer bought 4 cases of lemons. In each case, there are 9 lemons. How many lemons did he buy altogether?

The Ten Frame

Problem 2: If he sells 6, how many lemons will he have left?

The Array

The Number Line

Problem 3 : How many lemons will he have if he buys 5 cases? 6 cases? 7 cases?....

Table of Values

Problem 4: On Monday, the grocer sold 9 lemons to one customer and 5 to another. On Tuesday, he sold 8 to Mr. Lauzon and 6 to Mrs. Qureshi. He says he sold the same number of lemons on Monday and Tuesday. Is this true or false?

To solve this problem, students can use the double open number line, the pan balance, or the math scale.

The Double Open Number Line

This is true because

The Pan Balance

The Mathematical Balance

Teachers should use these models and introduce students to using them to help them reason. In representing a problem, students analyze relationships using models, draw conclusions, and explain them using oral descriptions.

Models are tools that help students formalize their algebraic thinking. Models are primarily representations for exploring changes, illustrating relationships, and proposing conjectures in a variety of contexts in order to gain ownership of concepts. Fosnot and Dolk (2001, p. 77) describe them as mental maps used by mathematicians to organize and solve problems and to explore relationships.

Just as they do with symbols, students use models to make sense of the relationships between numbers and operations. By allowing them to analyze problems in a more abstract way, models facilitate understanding. And at the heart of these models, students have access to the meaning of number, the concept of relationships between numbers, and ultimately, the development of fundamental processes.

According to Fosnot and Dolk (2001), models, like big ideas and strategies, cannot be transmitted automatically; students must construct them themselves in order to appropriate them. It is therefore important to present them with problems that are conducive to modelling, so that students create their own symbols and models to represent situations, instead of systematically proposing the usual algorithms or learned strategies.

Thus, as students progress through school, they should come to conceive as a strategy the model that they first used as a tool; through this transfer, common and familiar mathematical relationships will be used to support less common situations presented in new contexts.

Note: Many algebra activities can be seen as number activities; in fact, it is mainly the algebraic orientation provided by the teacher that will allow students to enter the world of algebra.

Understanding Change

Students live in a changing world. Understanding that change is a part of life and that most things change over time (for example, each year their height and weight increases and their feet get longer) is the final component of developing algebraic thinking. Observed changes can be described qualitatively (for example, I am taller than last year; my hair is longer; the bucket filled with water quickly during the storm; it is colder than this morning) and quantitatively (for example, I grew 2 cm this year; the bucket of water filled with 50 mL in 30 minutes; the temperature dropped 6°C in 3 hours). Students need to learn to observe and understand changes in patterns and in situations of equality.

For example, to maintain equality in a given situation, students must understand and demonstrate how the change in one variable affects the change in the other variable. For example, in an exploratory situation where students are asked to find all possible combinations of two rods that, when placed end-to-end, are the same length as the reference rod, they quickly realize that changing one rod requires changing another rod.

When students understand this relationship, rods are no longer changed randomly, but are chosen more systematically, demonstrating the emergence of algebraic thinking. Understanding that most things undergo change, that these changes can be described mathematically, and that some of them can be predicted, is an important part of developing algebraic thinking.

Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 3^e année, p. 16-22.

Teachers remain the linchpin of the development of algebraic thinking in the primary classroom. The teacher's role is not defined solely by the choice of tasks, but also by their interventions, which aim to encourage students to go beyond arithmetic reasoning and to access a symbolic way of thinking. Doing mathematics thus takes on its full meaning.

Some authors (for example, Blanton & Kaput, 2003, pp. 70-77) believe that teachers need "algebraic eyes and ears" to identify and maximize connections to algebraic concepts in mathematical activities and to take advantage of opportunities to develop students' algebraic thinking.

To do this, the teacher can:

Vary the parameters of a problem

Use an existing problem and give it an algebraic perspective, promoting the search for relationships, patterns, conjectures and generalizations.

Example of a problem in the Numbers strand

In a team of four people, how many handshakes will there be if each person shakes hands with all the other people only once?

Starting from a number problem, the teacher asks questions that draw out connections to algebraic concepts, such as:

• How many handshakes will there be if one, two or three other people are added to the team?
• Organize the data in a table of values. Do you see a pattern?
• How many handshakes will there be if ten people are added to the team?

Ask questions to help students generalize

• What can be said about the relationship between the number of people in the team and the total number of handshakes?
• Is there a regularity? Explain your answer.

Ask questions with an emphasis on algebraic concepts

• Can you explain the problem in your own words?
• Can you solve the problem using another representation?
• How do we ensure that this solution is true?
• Would this approach work with other numbers? Always?
• What changes?
• What does not change?

These questions encourage students to make conjectures and explain them with mathematical arguments.

Develop and strengthen a symbol literacy

Teachers need to introduce students to symbol literacy and support them in its development. Too often, the application of many of the mathematical symbols is done automatically, with students perceiving these symbols simply as a command to perform a mathematical operation. These students then have difficulty solving a problem correctly and explaining what the number sentence they have written represents because they do not understand the symbols that make up the sentence. The teacher's job is to put in place strategies that allow students to :

• read the symbols and think about what they represent before acting;
• understand the correct meaning of mathematical symbols (for example, the "=" sign represents a relationship between the numerical expressions on either side of the sign and is not a precursor to the answer);
• recognize and use symbols as communication tools to interpret a number sentence and to express reasoning.

Creating an "algebraic" learning environment

An "algebraic" learning environment is one where the development of analytical thinking is emphasized. Teachers consciously identify times when reasoning is an integral part of their instruction. Arguing, abstracting, and generalizing become common practice during daily lessons in mathematics and even in other subjects, not an occasional enrichment.

Creating an "algebraic" learning environment means giving students the chance to experience the world around them with "algebraic" eyes and ears, that is, to be able to generalize explicitly.

Source: Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, p. 23-24.