Wednesday / May 22

# Enriching a Math Task Using Reversibility Questions to Promote Robust Understanding

“How do I know that students have learned more than just an algorithm?”

“What are some tasks I can give students that push them to think critically?”

“What tasks can I use to create more opportunities for student discourse?”

These are questions we as math teachers sometimes ask ourselves as we are planning units and lessons or when we are evaluating our curriculum materials. We know that students need more than practice or computational-type exercises, but some curriculum materials may emphasize procedures or algorithms rather than concepts. For students to remember and retain content, it’s important for them to see the big ideas and connections within and across topics to build conceptual understanding. So how can we change those computational-type exercises to be more robust?

Let’s think about a problem such as:

Simplify 3x – 2(4 – x).

It’s a typical problem, one we often see in middle and high school algebra. It’s straightforward even though it could be simplified in different ways. The question, though, is how can we get more from a problem than just the answer?

One way is to change the problem from a straightforward question to one that forces students to think in a different way. We can use the RFG Framework–reversibility, flexibility, and generalization!

 3x – 2(4 – x) Reversibility (Reversing students’ thinking) Flexibility (Using one problem to solve another) Generalization (Noticing patterns that lead to big ideas) Find an algebraic expression that simplifies to 5x – 8 Simplify: a.      3x – 2(4 – x) b.      3x + (–2)(4 – x) c.       3y – 2(4 – y)   What do you notice? Find an algebraic expression with four terms that simplifies to a monomial. Find one that simplifies to a binomial. Find one that simplifies to a trinomial. What do you notice?

RFG tasks are useful in building students’ understanding because they:

1. have multiple solutions or different ways to think about the task.
2. allow access by a wide range of students.
3. lead to conceptual understanding that supports algorithm development.

You might think of other ways that they support student learning.

Try a reversibility problem where you give students the answer and they construct the problem. Here are some examples to help you get started.

1. Find an equation whose solution is –3.
2. Find a quadratic equation whose solution is –4 and 3.
3. Find a function whose graph is in Quadrants I, III, and IV.
4. Find the dimensions of a pyramid whose volume is 48 cubic centimeters.
5. Find a set of five data points that has a mean of 18.
6. Find a set of five data points that has a median of 25.

Post some of your favorite reversibility questions!

Barbara J. Dougherty is the past director of the Curriculum Research & Development Group and a professor in the College of Education at the University of Hawai‘i at Mãnoa. She is a former member of the board of directors of the National Council of Teachers of Mathematics and is the co-chair of the Mathematics/Special Education Workgroup, a partnership between the NCTM and the Council for Exceptional Children. She served on the author panel for the What Works Clearinghouse Practice Guide on assisting elementary school students who have difficulty learning mathematics for the U.S. Department of Education Institute of Education Sciences. She is the author or coauthor of approximately 22 book chapters, 29 articles, and 36 books, including Classroom-Ready Rich Algebra Tasks, Grades 6-12: Engaging Students in Doing Math (2023). Her research, funded by more than \$11.5 million in grants, emphasizes supporting students who struggle in middle and high school, with a focus on algebra. She holds teaching certifications in middle and high school mathematics and K–12 special education.

Linda C. Venenciano is a professor in the School of Learning and Teaching at Pacific University, where she teaches and supports early childhood through high school preservice teachers in mathematics education. She is a former classroom teacher and has taught mathematics to students from first grade through undergraduate. She previously served as the interim director of the Curriculum Research & Development Group and as an associate professor of mathematics education at the University of Hawai‘i at Mãnoa. She is the author or coauthor of 21 peer-reviewed publications and 12 books, including Classroom-Ready Rich Algebra Tasks, Grades 6-12: Engaging Students in Doing Math (2023). Linda is currently serving on the editorial board of Investigations in Mathematics Learning and has served as a program chair for the annual meeting of the Research Council on Mathematics Learning and as a guest editor for a special issue of Educational Studies in Mathematics.