Early Algebra: Generalized Arithmetic in the Classroom

Embedded in the number and operations units at each grade is the early algebra work focused on generalizations that arise in the course of students’ study of counting, numbers and operations. In mathematics a generalization is a general claim you can make about the way numbers or operations work. An example of a generalization is: in addition you can change the order of the addends and the sum remains the same.

Implicit Generalizations

What does it look like for students to work on generalizations? Generalizations are implicit in the strategies students use to solve computation problems. Students may not be able to articulate these generalizations, but they use their understanding of number relationships and the operations as they solve problems.

For example, in the following video, students share strategies for solving a story problem about 6 cars plus 7 cars.

All Gr. 1 videos.

In order for Elizabeth to use 6 + 6 to help her solve 6 + 7, she needed to know that 6 + 6 plus 1 more equals 6 + 7 and that if she added 6 + (6+1), the sum would go up by 1. A generalization implicit in her solution is: if you add an amount to an addend in an addition problem, the sum will increase by that same amount.

Derek used a different generalization. What did Derek do? What did he understand? How would you put the generalization implicit in his solution into words? You might start with: “In addition, if you…”

Articulating and Justifying Generalizations

At each grade level there are explicit opportunities for students to articulate, investigate and justify some of the generalizations that are often implicit in their computational strategies. Teachers help students articulate and justify these generalizations by asking questions such as: Would that work with other numbers?  Would it always work? How do you know? They ask students to use representations, models and contexts to investigate and justify their claims.

In grade 2 students make generalizations about what happens when you add two even numbers, two odd numbers, or an even and an odd number. Students are asked to justify their ideas.

Consider the representations students use to show why an even plus an even number is an even number.

They use similar representations to show why an odd number plus an odd number is an even number:

How do these representations justify the generalizations students make about adding even and odd numbers? What do the representations show about what these students understand about even and odd numbers? about addition?

Applying Generalizations

Once students have articulated and justified a generalization, they can use it to help them solve computation problems. In the following video 3rd grade students use a generalization they have already worked on, which they call the “plus/minus rule”, to solve 623 + 249.

All Gr. 3 videos.

How might you put the “plus/minus rule” into words?  You might start with: “In addition, if you…” How does Kiara show what she understands about this generalization?  How does Rose apply this generalization to the problem: 623 + 249?  How does Kiara apply it?