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 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.
counting activities for grades K and 1 from Mathodology fall institute 2017
Students using the think!Mathematics program reinforce the count on strategy. Games/activities can be used during the guided practice part of a lesson. Students are encouraged to count on from the larger number.
Addition Within 10
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?
Can you see the CPA progression? Modeling with actual items in making the lesson come to life first. Then move to pictorial representation with pictures of the objects. Last student model with cubes and record in a number bond.
Modeling with actual items in making the lesson come to life first. Then move to pictorial representation with pictures of the objects. Last student model with cubes and record in a number bond.
Then move to pictorial representation with pictures of the objects. Last student model with cubes and record in a number bond.
Lastly, student model with cubes and record in a number bond.
Can you picture spending several classroom periods on the number 5? Specifically, breaking it down into parts, and putting it back together again? Sounds like a lot of time, right? But, spending this amount of classroom time to decompose numbers using number bonds, allows students to gain a deeper, more flexible understanding of numbers. You might be wondering what those periods would actually look like, how you’d keep your students engaged and learning for that amount of time. In this blog, we will explore exactly that.
What are Number Bonds
Simply put, number bonds are the different ways we can break apart numbers. Number bonds are all about the relationship between numbers and quantities. The relationships of parts to a whole. Building the foundation for all mathematical operations. Building mental images of number relationships. While it sounds like a simple concept, it can be difficult for students to learn. So, it is important to dedicate time for them to learn the concept.
Use of color and representation helps to connect ideas.
Why Number Bonds
Who has the largest number? Students use cards to compare different representations of a number.
Number cards from mathodology coming soon!