By Kristin Klingensmith

Institute for Learning

When most people think about creativity, the arts like music, painting, writing come to mind. Rarely do our minds wander to mathematics. Some might even argue that mathematics sits in direct opposition to the idea of creativity.

But, I am not that someone…mathematics is the medium that my creative, engineering, number-loving father uses in all of his tinkering. To me, he is the “MacGyver of Mathematics.” The way he reasons about, makes sense of, works with, and talks about numbers and quantities sparked my love of mathematics before I even knew there was such a thing.

Mathematics is all around us. It’s there in big ways, like when we put people on the moon, develop and distribute vaccinations, analyze data related to climate change. It’s there in small ways too, like taking inventory at a mom and pop shop, balancing a monthly budget, making decisions about when to refuel the vehicle. We use mathematics in our everyday lives, often times in ways that we don’t see. We are constantly taking in data, analyzing it, and making decisions based on patterns we see. So why don’t more of us recognize and honor that creativity is central to mathematics? “School math,” as I experienced it, may be part of the problem, but it doesn’t have to be.

Creativity: Smothered or Cultivated

In contrast to the love of number inspired by my father, I eventually encountered something I call “school math.” You may know this as the “I-do, We-do, You-do” approach to teaching mathematics. There is a lot of listening and following directions in school math. School math lays out a set of procedures and strategies that, if used at the right time in the right way, results in accurate answers. Because school math is about solutions, there is a lot (a lot) of drill in school math too. The focus is on computation and calculation. It was a common approach to mathematics teaching and learning in the past and is still pervasive in mathematics classrooms today.

School math does not honor different pathways. It is not concerned with the journey or the balancing act between what is known and what is being explored. Rarely with school math are learners given an opportunity to generate their own new idea.

Some of us were good at school math because we could follow the procedure we were told to use. Unfortunately, school math doesn’t work for the majority of learners. Some of us needed to reason about the “why” for school math to make sense. Some of us needed to explore and do our own thinking to hold on to the knowledge.

I was a graduate student before I experienced a mathematics class in which students were expected to be creative. We were expected to be and respected as doers of math. In this class, we did the figuring out and the sense making. We were responsible for representing our thinking on paper. We were given time and space to create solution pathways and collaborate with others. Then we connected and reasoned about our different solution paths to better understand the mathematic we were studying. I was blown away by the realization that we were the creators and authors of the material from which we were learning. There was creativity and brain tickling, and there was joy. The professor was a facilitator of learning, not the “sage on a stage” telling us what to do. Their choice of routines used in the class was deliberate and grounded in research that says mathematicians need space to be creative in their problem solving! The lesson routine that was used in this course finally gave me the chance to meld what I always loved about mathematics with school math.

A Lesson Routine Conducive to Creativity

The Lesson Routine has three phases: Set-Up Phase, Explore Phase, and the Share, Discuss, and Analyze Phase (see diagram[1]). You may have seen or experienced it or something similar to it in your own classroom experiences.

  • Set-Up Phase: Provides time for learners to begin making sense of the task. As necessary, teachers help learners access the context of the task without reducing the challenge by explicitly providing a solution path to follow.
  • Explore Phase: Provides an opportunity for learners to generate a solution path on their own before working collaboratively to make further sense of, represent, and discuss the mathematics in the task.
  • Share, Discuss, and Analyze Phase: Provides time and space for all learners to engage in a teacher orchestrated discussion of their work. The teacher intentionally selects and sequence the work based on the learning goals for the lesson.

The Lesson Routine assumes the use of a cognitively demanding task that learners find engaging and allows them to build on lived experiences and background knowledge. Tasks of high-level cognitive demand require learners to explore and to understand the nature of mathematical concepts, processes, or relationships.[1] The effective use of the routine also relies on the teacher’s understanding of the instructional triangle, the relationship of content knowledge, pedagogy, and the learners’ thinking.

[1] Klingensmith, K., Speranzo, L., Dostilio, J., & Bill, V. (2021). Accountable talk mathematics discussions: Teacher’s guide (p.11). Pittsburgh, PA: University of Pittsburgh, Learning Research and Development Center, Institute for Learning.

[1] Stein, M.K., Smith, M. Donut_Task_2. S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

Active Engagement of Learners by Phase
Learner Actions
  • Read and make connections between the task and their experience.
  • Mark what they consider to be important information.
  • Ask clarifying questions and/or share prior knowledge.
  • Self-select tools for solving the problem.
  • Work independently as they process and get started on the task.
  • Move together to work, share solution paths, and compare thinking with others.
  • Create and discuss different models and representations.
  • Engage in talk and debate with others in their group.
  • Make errors and revise their thinking.

Share, Discuss, and Analyze

  • Discuss the solution paths as a whole group.
  • Make connections between and among representations and solution paths.
  • Say back, add onto what their peers explain, ask for clarification, agree or disagree and say why.
  • Make claims and generalization about the mathematics being explored, question the claims made by others.
  • Revise their thinking, solidify their understanding.

Through the phases of the lesson routine, the learner is engaged in the complex process of sense-making that allows them to draw on their knowledge assets, all of the knowledge with which they enter the classroom.

Recognizing and Amplifying Learners’ Creativity and Assets

Now that the learner actions have been laid out, what role does the teacher play in providing opportunities that allow learners to be and see themselves as mathematicians? It starts with the belief system and is actualized in the pedagogical choices around facilitating the high-level task.

Throughout the Explore Phase, teachers listen as learners share their representations and how their work relates to the task being solved. Teachers ask questions to press learners to reason more deeply to add on to the work they have done so far or to make connections between two different pieces of learner-generated work. This is also the time that teachers look for overgeneralization and potential misconceptions with the intent of surfacing these for further discussion. From the insights gained while monitoring and facilitating discussion during the Explore Phase, the teacher selects several solution paths to be shared and the sequence by which they will be shared with the whole class. This action is critical not only to the way the discussion of the mathematics may go, but also whose work is recognized and held up for discussion.

Throughout the Share, Discuss, Analyze Phase, solution paths are shared and the teacher presses learners to make connections between and among mathematical representations and different ways of approaching the same task. Teachers listen as leaners contribute, looking for comments to “lift up” for further inquiry or to serve as a point of solidification about the mathematics being studied.

Teachers must actively and intentionally position the learners as the authors of the mathematics in the classroom, and work to make and keep learner voices the center of the discussion. It is not easy to support learners without taking over the thinking for them! But honoring that learners come to class with lived experiences that make them mathematicians is central to learner’s creativity in mathematics class. This also means honoring learners’ cultural identities and linguistic differences as assets that expand and enhance the collective creativity already in the classroom. Teachers are not creating mathematicians in their classrooms, they are accessing the creativity of the mathematicians already sitting in front of them!