Monday, April 23, 2018

Rewriting Equalities to Build Numeracy - Part 1

Recently, I was fortunate to attend a two-day seminar on teaching math to struggling learners in the middle and upper grades, led by Dr. Barbara Dougherty. Dr. Dougherty, the Director of the Curriculum Research and Development Group at the University of Hawaii, is an insightful and innovative educator with a passion for helping all students develop of deep appreciation for math. If I could sum up the most important lesson I learned during my two days with Dr. Dougherty, it would be with a statement she repeated many times over the course of the conference: Pay now, or pay later.

Dr. Dougherty was referring to the time it takes to build numeracy skills and operational sense. These two critical components of math education are often given a mere nod in the classroom, if not overlooked completely by teachers scrambling to cram as many procedures into students’ memories as possible before exam time. This rushed approach, while understandable, is a mistake, according to Dougherty. A solid sense of numeracy (being able to reason with numbers and other mathematical concepts) and operations (recognition of the relationships among addition, subtraction, multiplication, and division) underlies a deep understanding of math. Happily, these competencies are available even to students who struggle with numbers and even those with dyscalculia. But learners need time and the right platforms for exploring the relationships between numbers.

There are lots of ways to set the stage for this kind of inquiry, but one of the simplest is an exercise in which students rewrite equalities. This powerful strategy is deceptively simple: give students an equation and ask them rewrite it, keeping both sides equal. So,

6 + 4 = 10

might be rewritten as

10 = 6 + 4

9 + 1 = 10

15 – 5 = 10

5 x 2 = 10

7 + 3 = 20 – 10

etc.

My fifth graders have loved this activity, and even those who will swear they hate math cheer when I announce that we’ll be warming up for class with a new number sentence to rewrite. There are lots of reasons to try this in a classroom setting or with an individual student.

Rewriting equalities:
• It's fun! – My students, without any guidance from me, treat these challenges like riddles, and they are tickled when they come up with novel solutions.
• Builds operation sense – Crunching numbers with a real goal in mind makes practice relevant and motivating.
• Builds numeracy – With practice, students develop a sense for how to point the magnitude of a quantity in the direction they want; for example, they discover that most of the time, multiplying by a positive number will make a quantity much larger than adding by the same number.
• Accommodates a wide range of skill levels – More advanced students can use fractions, roots, exponents, etc., while those still building basic skills can gain comfort with simpler operations like subtraction or multiplication.
• Sets the stage for algebraic thinking – Many teachers take for granted that students who have only ever seen a mathematical expression with a single number to the right of the =, structured like this:  5 + 3 + 1 = 9 will be able to transition to algebra, when problems may have multiple numbers and variables!) to the right of the =, like this: 5 + 3 = 9 – x.
A misunderstanding of = is often behind the confusion because students haven’t been taught to think flexibly about an equality. (Check out Part 2 of this post for examples of how to implement this process in the classroom and for more about the “equals” sign.)