About 18 months ago in March, 2024, two black teenage girls from a New Orleans high school announced they found a new proof of the Pythagorean Theorem. What makes their proof especially impressive is that out of more than 400 unique proofs of the Pythagorean Theorem, theirs is the FIRST proof that relies entirely on trigonometry. For hundreds of years a trigonometric proof of the theorem had thought to be impossible, but Calcea Johnson and Ne’Kiya Jackson proved otherwise.

I share this story not only because it is an amazing mathematical story, but because it is also a great example of a math teacher who had created the classroom culture that allowed these two young mathematicians to flourish.

In this post I’ll share my own thoughts on the TWO steps for creating a classroom culture that embraces productive struggle such that these mathematical adventures can take place in YOUR classroom.

- Step 1: Teach wide math, not narrow math.
- Step 2: Explore-first instructional strategies

## Teach wide math, not narrow math

Narrow math is the type of mathematics that allows no room for flexibility, creativity, or alternate points of view. In narrow math, students can only be successful if they can use the ONE method that is offered by the textbook. In general, narrow math focuses on abstract algorithms and leaves to space for alternative thinking.

In problems like this, if the student does not already know the algorithm, then they are completely out of luck. The path to success is narrow: know the algorithm or fail.

Each math concept can be reframed as a wide math example.

For 24 x 15 we might tell students that one definition of multiplication is the area of the rectangle that is 24 units wide and 15 units tall. Students are then merely told to find the area of the rectangle.

Suddenly a narrow math task becomes WIDE because students have a plethora of strategies for finding the area of the rectangle, one of which is the standard algorithm for 24 x 15.

Students who are not yet familiar with the standard algorithm can build upon the area model to find partial products.

Strategy 1

Strategy 2

Students also can build upon the area model to find flexible factoring.

“Convert ⅝ into a percent” is often presented to students as a task in which students are forced to use long division to do the conversion.

Long division…or go home.

This can be turned into a wide task by introducing a tape diagram visual.

Students now have access to a variety of strategies for finding the percent represented by ⅝. Here are just two examples…

Strategy 1

Strategy 2

Wide math creates the mathematical space for MORE students to successfully access the math concept to be learned. Wide math also allows students to explore alternative strategies that make sense to them. The focus is now on understanding the math concept rather than just blind answer-getting.

Virtually all math concepts can be introduced to students in a WIDE way. Most often this involves some sort of visual representation.

## Explore-first instructional strategies

The second step to creating a culture that embraces productive struggle is to use explore-first instructional strategies. As I just hinted, WIDE math naturally opens the door to students exploring their own methods prior to the teacher formally teaching.

Your job as the teacher is to provide some time for students to explore and develop their own understanding of the mathematics prior to you formally teaching the concept.

From *Improving student achievement in mathematics* (International Bureau of Education, 2000) we know that when students are provided an opportunity to discover mathematical ideas and invent mathematical procedures, they have a stronger conceptual understanding of connections between mathematical ideas.

The amazing thing about inquiry-first teaching is that students do NOT need to successfully solve the problem during their moment of inquiry. It appears as though students benefit from inquiry-first instruction merely because of the opportunity to engage in authentic problem-solving itself, regardless of whether they successfully solved the problem.

Allowing learners to struggle will actually help them learn better, according to research on “productive failure” conducted by Manu Kapur, a researcher at the Learning Sciences Lab at the National Institute of Education of Singapore. In the study, they created two groups of students: the first were scaffolded carefully by the teacher and successfully solved the problems. The second group collaborated with one another without any prompts from the teacher. Even though the second group was unsuccessful in solving the problem, when the two groups were tested on what they’d learned, the second group “significantly outperformed” the first.

To convince students to become active problem-solvers, I use class norms that encourage discussion, questioning, and collaboration and minimizes the focus on answer-getting. I was recently introduced to these norms by my good friend Jamie Garner at Stanislaus County Office of Education.

The norms establish a culture in which students are comfortable working with one another and are not burdened by the fear of mistakes and unfinished thinking.

I provide **laminated discussion scaffold cards** to students to support the mathematical conversations that need to be commonplace.

One the other side of the laminated discussion card is a list of things students can try when they have no clue what to do with the tasks.

There are many “flavors” of explore-first instruction:

My personal preference is TTP because it best explains exactly how to implement inquiry in your classroom immediately. I have written about TTP here.

In summary:

- Teach wide math…not narrow math.
- Allow students to explore the math concept BEFORE you formally teach it.

Do not hesitate to contact me if you have any thoughts or questions about this!

Duane Habecker

.

.

.