At a Glance...

Introduction

The Algebra strand brings together concepts essential to mathematics for representing and analyzing relationships found in many everyday situations.

The study of algebra developed from the need to understand and represent the real world, for example, the position of planets, the movement of tides, the movement of falling objects. Mathematicians tried to solve these questions by observing regularities and modelling the phenomena with equations and graphical representations. This work has contributed to the development of mathematical symbols and methods of calculation. Now, "in this new millennium, algebra is no longer a discipline that dwells on the manipulation of symbols. Algebra becomes a way of thinking, a way of seeing and expressing relationships." (Ontario Ministry of Education, 2000, p. 26)

In the junior grades, students observe changes in the world around them and describe and represent them first in concrete and semi-concrete ways and then in symbolic ways.

For example, the growing pattern below is used to represent a situation. Students learn to describe the regularity that can be seen from one term to the next, to express the relationship between the position and the number of cubes in the term, and to represent this relationship by a table of values and by an equation.

Students observe that there is one constant (+1), the blue cube that does not change from one term to the next.

Students also observe the relationship between the term number and the term value to determine the pattern rule: number of cubes = term number × 2 + 1. In this case, it is a pattern rule with a multiplier (the part that makes the pattern grow) and a constant. This rule allows them to make far predictions.

Algebraic Thinking

In the primary and junior divisions, the main goal of the Algebra strand is to develop students' algebraic thinking. In the search for a definition of algebraic thinking, several authors emphasize a perspective that is considered essential in algebra.

Here are three examples that reflect three different perspectives:

Of the many elements that contribute to effective teaching in modelling and algebra, some have a greater impact on the development of algebraic thinking. Thus, it is important to recognize particularly the following:

• 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 a big idea (relationships).

The image below illustrates the interaction between these factors.

Algebraic Concepts Grouped According to A Big Idea

Overall and specific expectations in Algebra involve a large number of concepts. Grouping these concepts under big ideas helps to plan more effective instructional programming. In doing so, teachers are able to develop coherent learning situations that allow students to:

• explore concepts in depth;
• make connections between the different concepts;
• to develop algebraic thinking.

A big idea is a statement of an idea that is fundamental to learning mathematics, an idea that ties together a lot of mathematical knowledge into a coherent whole.

The big idea of Relationships forms the basis of the expectations for the Algebra strand in grades 4-6. This big idea places the understanding of relationships between quantities or numbers at the center of all algebra learning. Its development in terms of the two underlying key points, exploring relationships and symbol meaning, helps teachers identify and prioritize key concepts and implement effective and consistent instructional strategies.

Understanding the relationships between quantities or numbers in mathematical situations with or without context is the basis of algebraic thinking.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 30.

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.

Abstracting

To abstract is "to dismiss by thought, to disregard" (Le nouveau Petit Robert, 2006, p. 11). Abstraction is one of the characteristics of algebraic thinking. In the primary grades, students learn that no matter what objects are involved, two objects plus two objects is four objects. To understand this concept, students do not consider the objects per se, but focus on their number. To abstract is to detach oneself from the sensory aspect of things in order to reason at a more general level (adapted from Raynal & Rieunier, 2003, p. 13), to mentally represent a concrete situation, to move to a deeper level of conceptualization. Piaget considers abstraction to be one of the basic processes in the construction of knowledge. For his part, Roegiers (2000, p. 77) explains that the appropriation of a concept generalizes reality. The concept is thus located on another level than reality. This is the domain of abstraction. In algebra, abstraction is mostly related to generalization.

Generalizing

To generalize is to draw valid conclusions, true in all cases, from the observation and analysis of a few examples (adapted from Squalli, 2002, p. 9). It is to reason by generalization, going from the particular to the general. Generalizing "[…] is particularly important, because in humans it is the basis for the acquisition of concepts and the possibility of abstraction" (Raynal and Rieunier, 2003, p. 156). Generalization is therefore at the heart of mathematical activity. In algebra, it allows the development of the student's algebraic thinking.

To make a generalization, students observe and analyze situations and then propose conjectures. When proposing a conjecture, students must be able to express their reasoning in their own words. By supporting their conjectures with concrete and semi-concrete representations and mathematical arguments, students must then verify whether their conjecture is valid in other situations. This process, sometimes informal, helps students learn to formulate their generalizations more clearly.

A conjecture is the expression of an idea that is perceived to be true in any similar situation.

Example

Students analyze the relationship between the term number in the pattern below and the term value. One student proposes a conjecture that the number of squares in each term is always two more than in the previous one. Another student mentions that the number of squares is always twice the position term.

To verify the conjecture, the first student can check that in position 2 the term has 2 more squares than the term in position 1 and that the term in position 3 has 2 more squares than the term in position 2. This conjecture allows the student to predict that the 4th term will have 2 more squares than 6, that is to say 8 squares, and the student can verify this by building the term in position 4. In this case, the reasoning is additive.

The second student can verify their conjecture by checking that at the term in position 1 there is 1 group of 2 squares, in position 2 there are 2 groups of 2 squares and in position 3 there are 3 groups of 2 squares. This conjecture/relation allows the student to predict that in position 4, there will be 4 groups of 2 squares (that is, 8 squares) and that the term in position 20 will have 20 groups of 2 squares, (that is, 40 squares). The construction of the term in position 4 and other subsequent terms support this conjecture. Note that the conjecture must be revised if a counterexample is discovered.

Recognizing that their conjecture seems to apply to all similar situations in a given context, students formulate their generalization (the rule) using words or symbols. In the previous example, the first student formulates their generalization by saying, "The number of squares that make up a term is always 2 more than the number of squares in the previous position." The other student can formulate their generalization (the rule) orally or symbolically using the equation s = n × 2, where n reprensents the term number and s the number of squares that make it up.

Operating on Variables

Operating on variables is the action of analyzing and acting on what is unknown. "It is reasoning analytically, thinking about operations, generalizations, not objects (adapted from Squalli & Theis, 2005)." (Ontario Ministry of Education, 2008a, p. 11). According to several researchers, this is what distinguishes algebra from arithmetic (Driscoll, 1999, p. 1; Squalli, 2002, p. 8). Variables are usually represented symbolically by letters. However, in many situations, they can be represented by other symbols (for example, a square, a question mark, a fill-in-the-blank) or concrete materials. They can also 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 4^e à la 6^e année, p. 11.

Students' algebraic thinking develops in conjunction with the development of mathematical skills. Thus, students need to develop the ability to reason and solve problems algebraically and then communicate their algebraic reasoning.

Ability to Reason Algebraically

The ability to reason algebraically allows students to examine situations and organize their thinking. While arithmetic is generally perceived as a calculation on known quantities, focusing on finding the right answer, algebraic reasoning aims to better understand numeration by allowing the analysis of relationships between numbers to find an unknown quantity. This is why it is essential to develop, in elementary school, the ability to reason algebraically, especially in problem-solving situations.

According to Driscoll (1999, p. 1-19), algebraic reasoning includes the ability to “do and undo”, the ability to create rules to represent relationships between two changing quantities and the ability to formulate generalizations about the properties of arithmetic operations.

The ability to “do and undo” occurs when students succeed in working backwards. For example, students can make use of the connections between addition and subtraction and between multiplication and division. Given the equation Δ + 3 = 11, a student who can work backwards understands that 3 can be subtracted from the sum (11 – 3 = 8) to determine the unknown quantity, since we have added 3 to the variable to get a sum of 11. This is algebraic reasoning, since the action is performed based on thought and understanding, not a procedure for which the student cannot explain its meaning.

The ability to create rules to represent relationships involves the ability to generalize from rules. For example, in the growing pattern below, the student reasoning algebraically can first recognize that at position 1, the term is composed of 1 column of 2 dots, that at position 2, the term is composed of 2 columns of 3 dots… Thus, the student can generalize and represent the relationship between the number of dots (d) and the term number (n) by an equation [for example, d = n × (n + 1)].

The ability to generalize properties of arithmetic operations also demonstrates algebraic reasoning (for example, knowing that you can change the order of two numbers you want to add without affecting the sum or that the product of a number and 1 is always equal to the number).

Algebraic reasoning is also distinguished from arithmetic reasoning in the following way. Arithmetic reasoning is more concerned with static situations, whereas algebraic reasoning is more concerned with changing situations. Developing students' algebraic reasoning allows them to see and analyze more encompassing situations and to develop their repertoire of problem-solving strategies.

To stimulate students' algebraic reasoning, teachers can use an existing arithmetic problem and give it an algebraic perspective by adding a changing situation. These situations encourage the search for regularities and relationships, the use of variables, and the expression of justifications, conjectures and generalizations.

The table below highlights the difference between a problem that elicits arithmetic reasoning and a problem that elicits algebraic reasoning.

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

Here is another example where we can see the difference between algebraic and arithmetic reasoning.

In the picture below, there are the same number of squares on each side of the arrow. How many squares are there under the bell?

Click on the image to see the difference between the two reasonings.

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

The ability to reason algebraically does not develop in a simple and natural way. Therefore, teachers need to walk students through the process by encouraging them to:

• explain their reasoning;
• work backwards;
• analyze the relationships between quantities and organize information to represent a situation in another way;
• make conjectures and generalize.

In learning situations, teachers should ask questions that emphasize algebraic concepts and get students thinking.

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?
• Can the regularity be applied to any case?
• Do I always follow the same steps? What are they?

Ability to Solve a Problem Algebraically

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.

A problem-solving situation refers to a problem that :

• is in context;
• allows to use different strategies;
• is a challenge for the student.

Here is a problem that can be solved arithmetically or algebraically.

Sylvia has 15 red fish and 18 yellow-striped fish. Jacob has the same number of fish, but only 14 of his fish are red. How many yellow-striped fish does Jacob have?

Solving With Arithmetic Reasoning

Solving With Algebraic Reasoning

I know that Sylvia has 33 fish in total, because 15 plus 18 is 33. 15 + 18 = 33 Jacob has the same number of fish. So if he has 14 red fish, he has 19 fish with yellow stripes, because 33 minus 14 is 19. 33 - 14 = 19 We perform arithmetic operations to solve the problem.

I know that Sylvia and Jacob have the same number of fish. 15 + 18 = 14 + □ If Sylvia has 15 goldfish and Jacob has 14, then Jacob has 1 less goldfish than Sylvia. Since Jacob has the same number of fish as Sylvia, he must have 1 more yellow-striped fish than Sylvia. Therefore □ = 19. Instead of doing calculations, we interpret the problem and compare the quantities. We can represent the situation with an equation. To solve it, we compare the quantities on either side of the "=" sign.

In general, students tend to solve such a problem arithmetically. Teachers can then vary the parameters of the problem by giving it an algebraic perspective, for example, by asking students to represent the problem using an equation. To solve the equation, students can use a number line, which can help them think about the calculation, not do the calculation. The important thing is not to do the calculation, but to understand the relationship of equality between the two numerical expressions.

Ability to Communicate Algebraic Reasoning

The ability to communicate algebraic reasoning is developed when students express their understanding of a problem or concept and defend their ideas using different representations:

• the concrete mode, linked to exploration, manipulation and creation using concrete materials;
• the semi-concrete mode, linked to an illustration, a drawing or any other representation on paper;
• the symbolic mode, linked to any representation made from numbers or symbols;
• the " in words" mode linked to an explanation or a verbal or written description.

In order to develop a solid understanding, students need to have experiences in context by exploring problem-solving situations. Contextualization allows students to make connections between various representations and to develop an understanding of the algebraic concepts being explored. Teachers also use a variety of representations to help students take ownership of mathematical concepts, make connections between representations, facilitate transfer, and elicit cognitive flexibility in relation to the concepts.

Mathematical argument is an essential tool for communication in mathematics. Students need to be able to justify their representations, ideas, and understanding using mathematical arguments, using a vocabulary of causal relationships (for example, if… therefore, because, since). Mathematical arguments allow students to present their understanding in a much more accurate and thoughtful way. A mathematical argument is an oral or written justification of a reasoning to demonstrate or refute a mathematical idea.

The mathematical exchange is the ideal moment to communicate mathematical representations and arguments. When solving a problem in algebra, students formulate conjectures, present their possible solutions, compare their ideas or justify their results using different representations. In short, students communicate.

Source : Guide d’enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 18.

The components of algebra learning are the fundamental elements that learning in algebra should address, regardless of grade (National Council of Teachers of Mathematics, 2000, pp. 37-40). These components are part of the backdrop of the curriculum. Thus, in modelling and algebra, it should be recognized that the goal is to advance students' learning by having them understand relationships, use mathematical models, analyze change, and represent problems using symbols.

Each of these components is discussed below. Then, a problem is analyzed to illustrate the integration of these components in the learning of modelling and algebra.

Understanding Patterns and Relationships

Understanding relationships is an important problem-solving skill in algebra, as it allows for the acquisition of concepts and the formulation of conjectures leading to generalizations. When students perceive and describe patterns, compare, represent and create situations or patterns based on regularities, they are on their way to establishing relationships between quantities.

In the junior division, students explore relationships between changing quantities, particularly in situations involving growing patterns. To develop their sense of symbol, students observe, analyze and represent numbers in number sentences and equations.

Use of Mathematical Models

The ability to use models is an integral part of the mathematics curriculum. Mathematical models are used to represent situations in order to analyze the relationships between numbers and quantities. Over time, mathematicians have created, used, and generalized certain ideas, strategies, and representations to facilitate the understanding of concepts. In use, some representations have become accepted models, such as the number line and the array. It is important for students to use different models in a variety of activities so that they can move easily from one representation to another.

Example

The 3 × 4 multiplication is represented here in three different ways.

When faced with a problem, several representations are possible; a number of students use their bodies, manipulatives or drawings, while others represent the data more schematically. The way in which the data is appropriated and organized using models reflects the degree of development of algebraic thinking.

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.

To help students reason, teachers should offer a variety of models and representations, and encourage students to use them frequently.

In representing a problem, students analyze relationships using models, draw conclusions, and explain them using words. Models are tools that help recognize relationships and develop algebraic thinking. They promote analysis and introduce students to a level of abstraction that facilitates predictions and generalizations. Dialogue, mathematical exchange about the data in the problem represented by different models, and questions from the teacher are all ways to engage students in thinking.

Understanding Change

Students live in a changing world. Understanding that change is a part of life, that most things change over time (for example, each year students' height and weight increases, and their feet get longer), and that changes can be studied is a component of developing algebraic thinking.

Observed changes can be described qualitatively (for example, I'm taller than last year, my hair is longer, the bucket filled with water quickly during the storm, it's colder than this morning) or quantitatively (for example, I've grown 2 cm this year, the bucket of water filled with 50 mL in 30 minutes, the temperature dropped 6°C in 3 hours).

Students learn to observe and understand change in patterns, in equality situations and in problems.

Change and regularities are two inseparable concepts in exploring the relationships between two changing quantities. Students realize that the change in one quantity affects another. For example, by looking at the growing pattern below, students can recognize the change from one term to the next, that the number of squares increases from one term to the next. As they investigate the change further, students can recognize that the number of squares doubles with each position.

Students may even find that the number of squares that make up a term is equal to 2 × 2 × 2 × … × 2, where the number of multiplications equals the term number.

The study of a changing quantity is a major leap in the way we see things. For example, in a problem that involves the lengths 5 cm, 7 cm, 9 cm…, instead of seeing several quantities, we consider that it is a single length that is changing, hence the need to involve the concept of the variable.

Representation of Problem-Solving Situations Using Symbols

The representation of problems using symbols is a fundamental component of algebraic thinking. It is important to move students toward a more abstract and formal representation of situations using symbols; in this way, students develop their sense of symbolism.

Example of integrating the components in a problem-solving situation

The following example illustrates the integration of the components of algebra learning in a problem-solving situation.

After presenting the problem, teachers ask questions and invite students to complete various tasks to integrate the components of algebra learning.

Understanding Patterns and Relationships

Teachers highlight patterns and relationships by inviting students to analyze the problem-solving situation and represent it in a different way.

In general, the students note that at the Mega Fitness Center, the amount paid increases by $2 each time Jessie takes part in a class, while at the Extra Fitness Center, it increases by $5. By further analyzing the situation, students can recognize that there is a relationship between the amount paid and the number of classes taken at each Fitness Center.

Use of Mathematical Models

Here are some models that can be used to explore this problem.

Number Line

Manipulatives

Table of values Mega Fitness Center Number of Classes Taken 0 1 2 3 4 5 6 7 8 9 10 Amount Spent ($) 25 27 29 31 33 35 37 39 41 43 45 Extra Fitness Centre Number of Classes Taken 0 1 2 3 4 5 6 7 8 9 10 Amount Spent ($) 4 9 14 19 24 29 34 39 44 49 54

Understanding Change

Here are some questions that can be asked to analyze the change in this problem:

• What information allows you to predict (extrapolate) what will happen?
• What step or steps must be repeated to find the amount disbursed?
• When different numbers are used, what changes? What doesn't change?
• How does this representation (for example, with the manipulatives) resemble this one (for example, the table of values)?
• Would this approach still work with other numbers? How do you know?

Represent Problem-Solving Situations Using Symbols

By using strategic interventions, teachers can get students to represent the situation using symbols.

Examples

At the Mega Fitness Center: s = 2 × c + 25, where s is the amount spend and c is the number of classes.

At the Extra Fitness Center : s = 5 × c + 4, where s is the amount spent and c is the number of classes.

Source : Guide d’enseignement efficace des mathématiques, de la 4^e à la 6^e année, p. 20-27.

Teachers remain the linchpin of the development of algebraic thinking in the 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 pattern? 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 mathematical 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 3^e année, p. 23-24.