ASSESSING NUMBER SENSE: Equal Matters
Understanding the concept of equal and the role of the equal sign is one of the 4 foundational pillars upon which number sense builds. If our goal is to see students develop and master fluent, accurate and efficient strategies for solving addition and subtraction equations, we must include tasks that attend to developing a rich and robust understanding of what it means to be equal in math.
Why do I want this skill? Understanding equal is a skill that forms the foundation for future success in school math. The equal sign does not mean ‘here comes the answer’. We need the equal sign to trigger thinking about relationships so that when students meet equations like 0.200 = 0.2 or
2x + y = y + 2x they are not looking to complete an operation or get an answer.
There are 2 ideas embedded in the kind of thinking this young student is practicing. Automatic recognition of quantity: I just know the quantity each expression represents so I know this is equal.
Balance thinking: I see how the parts are representing the same quantity (but not thinking about what they add to).
The act of explaining out loud develops working memory & recall skills. She is practicing holding parts in her mind. She is developing confidence by explaining out loud.
This next clip shows fluency. He just knows what is missing. He was going to draw it out somehow but I interrupted by asking him if he just knew it. If we want fluency, we must prompt for it. But I am not sure what he understands about equal. He refers to balance but does he really mean balance on each side or does he mean just solve the missing piece?
The boys in this next clip demonstrate another aspect of understanding equal.
The boy talking “balances” the equation by adding to get to “friendlier” numbers. He knows that you need to change both numbers by the same amount and automatically knows what that same amount is (53). Watch
He says take- away is just a difference. I would prefer to hear subtraction. All subtractions are about differences. ‘Take away’ creates a faulty imagery. All subtractions are not taking away. But all subtractions are comparisons.
Does he understand how inverse and differences are connected? Does he actually demonstrate an understanding of difference on the number line? or is he more focused on the computation than on the relationship?
I like what I hear and see but is it robust enough? Does he understand the addition & subtraction relationship or is he only thinking in one direction. I ask because it will matter once we move to algebra. So I want to see and hear what he has to say if I ask him to prove with the inverse.
When I heard him explain I saw this in my mind
To understand the inverse I can move forward or back between the 2 numbers. If I shift everything to the right I have constant difference. My purple arrows show that. Here is what I thought when he said it was all about difference. Can students understand the connection between his explanation and my number line?
I would like to ask:
What are 2 more numbers that have a difference of 488?
How do you know?
Constant difference thinking relates to this type of ‘logic’ problem.
And these problems relate to what are classified as ‘systems of equations’ problems in high school and beyond.
The final check to know if students truly understand a concept:
Write a similar problem.
How are these problems similar? Are they similar to the original?
Are they similar to each other? Are they correct? Do they both work?
What do your students think?
CONSIDERING WAYS TO ASSESS NUMBER SENSE AND HOW TO USE THOSE ASSESSMENTS TO FURTHER STUDENT LEARNING is the goal of my work.
Find out more by participating in a Summer Numeracy Event. Studying student thinking will form a big part of our study.