Meaning of Equality

Often we teach the meaning of equality as the equal sign indicating “put the answer.”  To prepare students for future mathematics, we need to rethink the way we communicate this representation to students.  The equal sign is too important to attach such a limited meaning, especially when students are moving to abstraction.  When moving to abstraction, it is important to use this symbol as a “reader” versus a “speller.”  What do I mean by that? Let me explain.  First, a speller sounds foreign because the student is trying to make sense of abstract symbols that they don’t fully understand. A reader connects meaning that they can comprehend while connecting the story to the symbol.

For example, when teachers record the following equation,

A speller would say, “Four plus five equals nine”.

A reader would say, “Four apples and five apples is the same as saying ‘I have nine apples’” .

In the journal prompt below, we want students to see the statement as 5 and 4 is the same as 2 less than 11. If we focus too much as a speller, students see symbols as performing actions rather than relating ideas, which leads to misconceptions of the equal sign.

Less is more. Hesitate in thinking math becomes harder by creating larger numbers, rather provide depth early on using children’s intuitive understanding of basic number operations.   A simple journal entry can help assess what students understand about operations and the meaning of equality.  Once you find that a student understands the meaning of addition and subtraction, it is time to create tasks that dive deeper in understanding equality.  These task can be as simple as asking questions such as, 4 + 5 = ____ + 6.  Additional equality tasks can be found in the Algebraic Thinking and Reasoning with Numbers books (Groundwork Series) by Greenes, Carole, et al., (https://www.mheducation.com/prek-12/program/MKTSP-O8302M06.html).

In the book, “Thinking Mathematically,” Carpenter, Thomas P., et al. suggest some benchmarks to keep in mind while moving towards the conceptual understanding of equality.

1. Establish a starting point.  What are students’ initial ideas regarding the equal sign?
2. Mix it up.  Don’t always write equations in the form a + b = c, rather c = b + a.
3. See the equal sign as a relational symbol. Emphasize students proving that each side is the same as the other.
4. Compare sides without calculations. Encourage students to look for relationships without performing calculations.

Listen to students and be aware of where they are in the process. Encourage them to question if it is true.   If so, why is it true?  By listening and looking we may be pleasantly surprised at what we find out.

Below is student work demonstrating various levels of understanding within the concept of equality.

Spiky says that 5 + 4 = 11 – 2 is false.

Curly says that it is true.

Who do you agree with and why?

This student understands how to simplify expressions but lacks applying the meaning of equality.

This student is confused in understanding what it means to have expressions on each side of the equal sign.

This student understands the meaning of equality. What can we ask next?

This student demonstrates a shallow understanding of equality. What can we do next?

This student can solve for an answer but needs practicing in mixing up the form of an equation.

At 2017 Mathodology Fall Institute participants worked on a task from think!Mathematics which required them to use the given shapes to make a composite figure. Figure C posed a problem for many.  Is it possible?  Participants worked and worked and did not find a way to solve the problem with the given shapes.  The teacher allowed the task to move on without closure for a purpose.  When students work to become confident problem solvers we want them stating, “Figure C is not possible with the given shapes.  I would need an additional… to complete this shape.”  We have built perseverance and another way to assess them.  Do they know what to ask?  “I need two more pieces of one shape.”

When we do not provide immediate closure, we allow students to continue to explore.  Many participants emailed after the institute because they would not quit until they figured it out.  Below are a few of the pictures they sent.

Try giving a task where there is not enough information and see how your students handle it.  Will they be confident to state this is not possible, and know what to ask?

Great perseverance by participants who refused to give up?

A journal entry from a grade 1 student. This student can draw it, write an equation, and give a number bond for the given task.

How might you further assess this student?

Give two questions you can ask this student to extend the thinking that is recorded here?

Math War

Who has the largest number? Students use cards to compare different representations of a number.

Number cards from mathodology coming soon!