Taking the answer out of the equation

In the quest to promote deep student thinking, sometimes the answer is the problem.

In the classroom, we can launch a beautiful, rich question only to see students reach the answer – and reach the end of their thinking. After all, why would they think beyond the answer? Isn’t the purpose of a question to lead to an answer? Isn’t the answer also the conclusion? Isn’t the answer the end of the journey of discovery?

No, it’s not.

The purpose of a question is not always to launch a journey toward a single answer. The purpose is often to give students an opportunity to think, to stretch, to learn strategies which they can apply to a wider range of scenarios. When students regard the answer as the end of the journey, they may miss those very growth opportunities. But how can we cause students to reach for deeper thinking when they are accustomed to ending the journey at the point of reaching an answer? A simple solution is to take the answer out of the equation. In other words, when you ask a question, give the students the answer to the question and change their task. Ask them to find as many connections as possible between the question and the answer. 

These two questions appear to be nearly identical, but their potential for leveraging student thinking is vastly different.

The first question, by itself, will lead to a journey-ending “28 cubic units.” An insightful instructional tactic would then be to ask the students how they know, to explain their thinking, to make connections. However, the reality is that the students will often do so from the vantage point of confirming and defending their original answer. All additional connections will be seen through the lens of the student’s original pathway.

Now consider the second question. The instructional architecture is much different in this case. If this question is presented with the answer showing, the students are released from finding and defending an original pathway. Instead the teacher is able to say, “Show me as many connections as possible between the question and the answer.”  

The answer has been taken out of the equation.

What remains is the pursuit of the thinking that connects the question to the answer. What are the multiple pathways that can be discovered? More importantly, how are those multiple pathways related to one another?

Expect your classroom to sound something like this:

“I see 4 towers of 4, and 6 towers of 2. (4 × 4) + (6 × 2) = 28”

“I see a prism on the bottom that measures 2 × 2 × 5. That’s 20. And there are 8 on top.”

“There are 14 in the front, and 14 in the back. 14 × 2 = 28.”

“I see a tall group with a volume of 16 and a short group with a volume of 12. 16 + 12 = 28.”

“I see it differently, like I’m slicing it with a butter knife. There are 2 groups of 8 and 3 groups of 4. (2 × 8) + (3 × 4) = 28.”

“Oh, I see that! What if we slice it the other way, horizontally? I see two groups of 4 and two groups of 10. 4 + 4 + 10 + 10 = 28.”

“I see 40 cubes. Imagine that there is not a section missing in the top right corner—that it is one solid rectangular prism. There would be 40 cubic units. But some are missing. The missing volume is 2 by 3 by 2. You see, 12 cubic units are missing—and 40 – 12 = 28.”

Then, after many connections have been established and recorded, including several connections you didn’t see yourself, you step in with the simple question, as you carefully and deliberately point to the algebraic representations of the students’ thinking. “I wonder how these equations are related?” You press your students toward an even deeper level of thinking.

You can do it because your instructional architecture has been well designed.

You can do it because you have taken the answer out of the equation.

Comments

It is typical for us to consider other areas of study, like History or English, as process/question-centered as opposed to answer centered. You hit the nail on the head--math should be approached in the same way. If it is, students will come to love the process.