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,
Using a simple addition problem as a reader versus a speller.
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.
- Establish a starting point. What are students’ initial ideas regarding the equal sign?
- Mix it up. Don’t always write equations in the form a + b = c, rather c = b + a.
- See the equal sign as a relational symbol. Emphasize students proving that each side is the same as the other.
- 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.
How can we differentiate instruction without putting more work on our plate?
Teachers do not have time to prepare multiple tasks and lessons to meet the needs of each student in the classroom. The key is giving an anchor task that all students can enter and knowing how to adjust the task based on our observations.
In this class, students were guided to use “Katell’s” method (aka-left to right strategy) to solve the problem 65 – 12. Notice how the student recorded their thinking process.
How can we differentiate with one task?
During the guided structure component of the lesson, teachers are observing and questioning students. This student demonstrates and can explain the left to right strategy. What comes next for this student? During our planning phase, we need to ask ourselves, “What can I do for the student that already knows the answer”? Looking at this student’s work, how can we deepen the mathematical understanding?
After listening to the student explain his thinking, the teacher commented, I can understand your verbal explanation, but your written work is confusing to me. I wonder why I am confused? The teacher then walked away, and the student grappled with the idea of equality. Differentiation is meeting the student where they are at and extending the learning.
Do not try to create more problems for students to practice. Plan your task thinking of what mistakes students typically make and how you can help extend the thinking. Use your observations to go deeper and differentiate!
What are other ways you could take the task 65 – 12 and extending the task? Share your thought below.
How can we help student refocus on understanding versus just an answer?
Too often when we pose a problem and students shout out an answer. We need to ask ourselves do students understand the concept or are they obtaining the correct answer by fitting the symbols and numbers into a structure they know?
In this example, the anchor task was to subtract 34 from 87. The teacher wanted to screen the children first wondering, Do students know what it means to subtract one number from another? To find out more the teacher posted the following problem, removing the numbers.
Do students have the conceptual understanding or just fit the numbers into a given structure?
Students were asked to set up the expression that could represent the situation. The student A on the far right was the only student in the class that seemed to understand. When asked to explain his thinking, many observing teachers felt he understood the concept and that he gave us a platform to generate a discussion.
Do students understand the part-whole relationship here?
Following this analysis, the numbers were inserted into the problem. Subtract 34 from 87. It was interesting to see student A’s work. Much to our surprise, Student A who seemed to understand the concept had a hard time transferring that knowledge to another situation. Notice his equation is not correct but he gets the final answer.
Can student A transfer knowledge from one setting to the next? Look and listen to help guide students to conceptual understanding. Looking at answers does not tell us the whole story.
Less is more. Spend more time on conceptual understanding and listening and watching students versus solving more problems. The answer does not help assess student reasoning or how we can extend or guide the learning process.
Grade 1 Task- Not enough information
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?
Journal Entry: Given a number write four things you know about that number.
How can we use this to help assess where a student is before we even begin a formal lesson?
Can you follow Victoria’s thinking? Help me understand.
Take a look at this problem. What was she thinking? Do you agree or disagree? Explain.