*“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 3*x* – 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. 3 b. 3 c. 3
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:

- have multiple solutions or different ways to think about the task.
- allow access by a wide range of students.
- 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.

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

Post some of your favorite reversibility questions!