Mathematical Equivalence: Developing Conceptual Understanding of the Equal Sign

One of the best things about teaching on Outschool is the ability to create classes based on personal interests, passion and/or academic need. Many learners conceptually misunderstand the equal sign in mathematics; and often, traditional classrooms don't have time to dive deep into the topic. Here, learners analyze the equal sign and apply their knowledge in a friendly environment.

Mathematical equivalence is a Big Idea which refers to the relationship between two quantities that are equal and interchangeable. The equal sign is a relational symbol, not necessarily operational. Understanding mathematical equivalence is an important transition from knowing arithmetic; to mastering middle school pre-algebra and beyond.

Asked to define the equal sign, many children interpret it as an “operational signal” meaning “calculate the total,” or “put the answer’ instead of as a relational symbol of mathematical equivalence. Children struggle to solve mathematical equivalence problems featuring operations on both sides of an equation. Understanding the equal sign as an operation is ok for traditional arithmetic, however, it is conceptually deeper at the foundational level of pre-algebra.

Research shows that most elementary math textbooks mainly use operations equals answer format (i.e. 2+2=4) and do not show the interchangeable nature of the sides of an equation. Intellectually, children would benefit from exposure to and practice of various problem formats. For example, solving for the right side of an equation: ____ = 2+4, no operations: 4=4, and solving for both sides of an equation: 2+4  =   _____ + 3. Problem sets organized by equivalent sums promote relational thinking based on the Transitive Property of Equality. For example, 
5+ 4 =-___, 3+ 6 =___, and 1 + 8=___. Therefore, if 5+4= 9 and 3+6= 9, then 5+4= 3+6.

Children’s narrow experience with math often leads to difficulty in understanding mathematical equivalence procedurally, perceptually and conceptually. These challenges lead children to incorrectly define the equal sign when rewriting and solving equations. To succeed in pre-algebra and beyond, students must recognize that any number, numerical algebraic expression or equation can be represented in an infinite number of ways that have the same value.

Ideally, learners develop their conceptual understanding of mathematical equivalence, this three-week class:

* Week One

** Perform a short oral pre-assessment of grade-level math equations to determine baseline understanding of the equal sign, when presented on various sides of an equation.

**Introduce the equal sign outside the context of arithmetic. Learners first use the equal sign to represent the equivalence relation (equal or not equal) between two concrete sets or two cardinal numbers. For example, 9 ? 9, 11 ? 4, 2 ? 3. Arithmetic often prompts students to “add up all the numbers” even when the problems are not presented in the traditional left-to-right format.

** Learners practice solving arithmetic problems written in both the traditional format (for example, 5+ 7 = _____) and the nontraditional format (for example, _____= 5 + 7).

Week Two

** Learners practice concentrated fading exercises to help conceptually understand mathematical equivalence. Ideas first present in concrete form and detail, for example, picture form equations. Concepts then advance to abstract, symbolic forms, including numbers. The progression helps learners activate their informal knowledge of a relational framework and begin to understand more abstract relations. 

Week Three

**Provide opportunities for learners to compare and contrast different problem formats and problem-solving strategies. For example, explain why a correct strategy is correct and why adding all the numbers in a mathematical equivalence problem is incorrect. Discussion asks learners to think more deeply beyond simply solving problems.

** Leaners perform a short oral post-assessment of grade-level math equations to determine course understanding of the equal sign, when presented on various sides of an equation.

***Each lesson promotes precise math language, using the phrases, “is equal to” and “is the same amount;” instead of using “equals” or “2+2 makes 4.” Incorporating precise language helps children move away from the shallow interpretation of the equal sign as “put the answer.” Also, grade-level arithmetic problems present as “equations” or mathematical problems” rather than “number sentences” which can promote left-to-right interpretation of a problem.

***Ideally,  learners will develop a conceptual understanding of the equal sign and its symbol of equivalence, instead of an "operational signal" meaning "calculate the total," or "put the answer here."