Guiding Students to Share Math Knowledge With Groupworthy Tasks

Why is knowledge not flowing freely from group to group despite my prompting and encouragement? How do I convince my students that collaborating with other groups will help to inspire deeper understanding? To further grow my students’ level of collaboration, disrupt the culture of noncollective thinking between groups, and promote more equal status, I needed to investigate my role in planning for and supporting intergroup knowledge mobility while working in random groups of three.

In Building Thinking Classrooms, Peter Liljedahl describes how to form collaborative groups in a thinking classroom. According to Liljedahl, a natural consequence of implementing engaging tasks in random groups of three is increased knowledge mobility among groups. Knowledge will flow from one group to another, and students will begin to rely less on the teacher as the authority of knowledge.

This creates a significant shift in the teacher’s role. In addition to supporting students’ self-reliance, teachers must create tasks that are challenging. In her book Strength in Numbers: Collaborative Learning in Secondary Mathematics, author Ilana Seidel Horn states that in order for knowledge to flow in a classroom, you must “ensure that the task actually requires the input of multiple students.” She argues that tasks have to be “groupworthy.” 

CHECKING IN WITH OUR STUDENTS

Prior to my role as math specialist, I had used the Building Thinking Classrooms model in my own classroom. Now, I have a dedicated colleague interested in doing this work with me. To measure the degree of this problem in his classroom, we gave our fifth graders a Knowledge Mobility Survey. The survey included questions related to the students’ comfort level with giving and receiving ideas from other groups and the degree to which they valued intergroup collaboration.

As I expected, our fifth graders were significantly uncomfortable with the idea of intergroup collaboration and didn’t value it. We needed to model the benefits of intergroup knowledge mobility by designing a task that required this degree of collaboration. By crafting and launching a groupworthy task, we gave our students the opportunity to experience firsthand what dynamic group work looks like, sounds like, and feels like to act as a springboard to future tasks.

THE PRIME FACTORIZATION CHALLENGE

The results of the initial survey indicated that students didn’t feel comfortable borrowing ideas, comparing answers, and discussing alternative solutions. Therefore, to kick off our intergroup collaboration campaign, we created a task called The Fundamental Theorem of Arithmetic Challenge. Instead of merely mentioning Euclid’s famous theorem to our students while practicing traditional prime factorization problems, we transformed this theorem into a rich groupworthy task.

As students worked on this challenge, they realized that many of their prime factorizations relied on other students’ work. For example, when Group A was testing 468, they discovered that a factorization of 468 was 2² x 117. Group A was unsure if 117 was prime, so they walked around the room to find Group B’s work for 117. Group B explained the divisibility rule of 3 to Group A. Group A was now confident that 117 was not prime, so they had to further factor 468 as 2² x 3 x 39 = 2² x 3² x 13. The room was buzzing with excitement. This task allowed students to experience what intergroup collaboration looked like, felt like, and sounded like. It was amazing to witness.

In the spirit of Horn’s work, this task successfully modeled the benefits of intergroup knowledge mobility to our students by ensuring that the task required collaboration. Each group had their own set of numbers to test, and one group’s work relied on another group’s work. They needed each other to achieve their common goal: to test this theorem with as many numbers as possible in one class period.

REFLECTING ON OUR GROUPWORTHY TASK

To check their understanding, we asked students to reflect on this experience. Here are some of their reflections:

• “I liked the feeling of our class working as a team.”
• “It’s fun to work together because you can make more friends and strengthen bonds.”
• “Everyone felt connected.”
• “It was awesome that my group could help other groups.”

The next day, we had  a whole class discussion about their firsthand experience with knowledge mobility. Students thought the process was an “exciting, meaningful activity.” Their words are a testimony to the power of this experience and to the value of reflection.

THE POWER OF A GROUPWORTHY TASK

In the weeks that followed, with this challenge fresh on everyone’s mind, we noticed a significant uptick in intergroup collaboration. Knowledge was flowing naturally.

Many weeks later, we redistributed their original Knowledge Mobility Survey. The results showed that on average, our students went from being uncomfortable to being comfortable borrowing and receiving ideas. “When students tap into ideas created by their peers, it enriches the classroom with diverse perspectives, creativity, and critical thinking, benefiting everyone involved,” my colleague shared. The realization that change can occur when one carefully plans with a goal in mind (getting students to grow beyond their comfort zone) has been so rewarding for me.

The activity exposed more students to multiple solution strategies and diverse ideas, which led our students to go deeper with mathematics. Learning to rely on each other has strengthened their understanding. In addition, increasing the flow of knowledge allowed more students to discuss and debate with their classmates—challenging spaces of marginality in our classroom.

Because more groups now naturally join together to share ideas, it allows for more students to contribute to the work. This has raised the status of many students and has helped build stronger math identities, as declared in Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin’s book The Impact of Identity in K–8 Mathematics Learning and Teaching: Rethinking Equity-Based Practices.

The key to the success in our knowledge mobility investigation was introducing a groupworthy task. The task ignited my students by fully immersing them in an experience that helped them know what knowledge mobility looked, sounded, and felt like. Intergroup collaboration played a major part in that. Our students have been inspired, and the spark of a culture of collective thinking has been ignited.