Introduction
Imagine you have a Rubik's Cube in your hand. In a moment of carelessness, it slips and drops into a bucket of paint. You are devastated and quickly pull it out to dry before smashing it on the ground. The Rubik's Cube breaks apart into smaller cubes. As you stare at the ground, you notice that some cubes have paint on three faces, others have paint on two or one face, and some have no paint at all. A natural question emerges: How many cubes fall into each category?
Through vivid storytelling, this scenario introduces an adaptation of the Painted Cube Task (Driscoll, 1999), used in a professional learning setting for secondary mathematics educators. When MTEs position educators as doers of mathematics, they model the very vision of HQEI that NCTM (2014) calls for in every classroom. After the professional learning experience, Diana and Jose emphasized that “being a student for the math task was the most important part of the learning,” because it modeled what their “classroom would look like.”
Drawing on three educator narratives, this article illustrates how integrating NCTM’s (2014) MTPs with Liljedahl’s (2021) Building Thinking Classrooms (BTC) framework created conditions for mathematical agency, multi-representational reasoning, and connected discourse. While the Painted Cube task is well-documented as a vehicle for student learning (e.g., Driscoll, 1999; Suazo-Flores & Roetker, 2024), its use in professional learning and the instructional design decisions that make that experience powerful have received less attention.
Instructional Design for Teacher Learning
The professional learning was structured around four interconnected components: setting the stage, task launch, just-in-time facilitation, and orchestrated debrief.
Setting the stage
Visible random groupings (Liljedahl, 2021) disrupted status hierarchies and promoted equitable participation (NCTM, 2014). Groups of three were assigned roles, including scribe, includer/inquirer, and speaker, to ensure all voices shaped the group’s reasoning. Educators worked standing at vertical whiteboards with access to physical materials (Rubik’s Cubes, snap cubes) available but not prescribed, preserving strategy choice.
Task launch
Rather than a written prompt, the facilitator used storytelling to launch the task (Liljedahl, 2021), narrating the scenario. Teachers were asked: for a 3×3×3 cube, how many unit cubes have three, two, one, or no painted faces? This launch preserved cognitive demand (Stein & Smith, 1998) while fostering agency through multiple entry points, such as physical models, drawings, and mental imagery.
Just-in-time facilitation
To sustain productive struggle (MTP 7), the facilitator monitored group work and posed questions (MTP 5) based on observation rather than predetermined pacing. Assessing questions (e.g., “How did your group determine the total?”) checked understanding without narrowing the choice of strategy. Advancing questions (e.g., “What if the cube were 5×5×5?”) extended thinking across cases from 3×3×3 through n×n×n.
Orchestrated debrief
The debrief was designed using the Five Practices (Smith & Stein, 2011): the facilitator anticipated strategies, monitored group work, and purposefully selected and sequenced sharing to move from concrete to abstract reasoning. The facilitator invited peers to revoice strategies and explicitly name connections across representations. Linking Natalie’s concrete cube-counting to Diana’s layer decomposition and Jose’s symbolic generalization revealed a key insight: each strategy is a different way of representing the same algebraic structure, where 9 corner cubes always have three painted faces, 
edge cubes have two, 
face cubes have one, and 
interior cubes have none.
Three Approaches, One Mathematical Structure
As educators reflected on their experience, they repeatedly noted that “we all solved it differently,” highlighting the task’s richness and breadth of mathematical content. These varied approaches surfaced interconnected mathematical ideas that cut across strategies, including spatial reasoning, connected representations, pattern recognition, and functional thinking. Rather than converging on a single approach, educators exercised mathematical agency (NCTM, 2020) by selecting strategies that aligned with their own ways of thinking. Three narratives illustrate how different strategies converged on shared mathematical ideas during the debrief.
Natalie, a middle school teacher, immediately reached for a Rubik’s Cube, explaining, “This is how I need to think about it.” Her group developed a coding system, numbering faces and using check marks, to avoid double-counting, which became a model for spatial reasoning. As dimensions grew, Natalie’s insight to leave the interior hollow when building with snap cubes revealed a concrete understanding of the structure. During the debrief, Natalie’s concrete representation served as a foundational reference point for the broader discussion. Other educators connected her spatial reasoning to visual and symbolic strategies, collectively constructing meaning across representations, an illustration of mathematical agency emerging through strategy choice.
Diana, a middle school instructional coach, began with a conceptual question: Was the interior of the Rubik’s Cube a unit cube or a mechanism? This attention to hidden spatial structure was foundational, and the group ultimately took the position that a cube was inside. Resolving this advanced their reasoning, leading them to adopt a layer-by-layer decomposition, analyzing the top, bottom, and middle as distinct categories. This visual chunking served as a bridge in the debrief, translating Natalie’s physical coding into a visual representation, making multi-representational reasoning visible.
Jose, a high school teacher, was captivated by the diversity of strategies within his own group: “One person started to draw visual representations; another just visualized in their head.” He oriented his group toward pattern recognition by repeatedly asking, “Is there a pattern within it?” which shaped the group’s reasoning and prompted conjectures that would be tested and refined. By the 5×5×5 case, these conjectures had become generalizations grounded in the earlier concrete and visual work. In the debrief, Jose’s functional representations gave the whole group a shared symbolic language, building on Natalie and Diana’s conceptual foundation and producing the connected discourse the debrief was designed to create.
Across these three narratives, mathematical agency, multi-representational reasoning, and connected discourse were not incidental outcomes; they emerged because the instructional design preserved strategy choice, made representations visible across groups, and sequenced sharing to build toward connected understanding.
Implications for Mathematics Teacher Educators
This work extends prior research on the Painted Cube task (Driscoll, 1999; Suazo-Flores & Roetker, 2024) by foregrounding facilitation in teacher learning contexts. Existing scholarship documents what students do mathematically with this task; less attention has been paid to how MTEs can design conditions under which teachers experience mathematics in personally meaningful and pedagogically instructive ways. Four insights emerge.
- Position educators as mathematicians. Designing professional learning in which educators wrestle with non-routine problems, make conjectures, and construct understanding positions educators collectively as active learners rather than passive consumers of pedagogy. Jose’s reflection that he returned to his classroom understanding how to “create an environment for students to be fearless” illustrates how being positioned as a doer of mathematics shapes not just what educators learn, but how they see themselves.
- Task launch through storytelling. How a task is introduced shapes who feels invited to engage, a narrative entry point activates prior knowledge and positions all learners as mathematicians, while maintaining cognitive demand. MTEs can make this visible by debriefing the launch alongside the mathematics, asking educators what the storytelling made possible and how they might apply those insights in their own classrooms.
- Coherent framework integration makes discourse productive, not just active. Pairing BTC structures (vertical whiteboards and visible random grouping) with the Five Practices framework (Smith & Stein, 2011) ensured that educators see pedagogical frameworks as complementary and not separate entities, fostering coherence. In practice, this meant that BTC structures created the conditions for the Five Practices to operate. Visible random groupings minimized status barriers, vertical whiteboards made reasoning visible for sequencing, and the debrief moved from Natalie’s concrete through Diana’s visual to Jose’s symbolic representations, precisely the connected discourse the Five Practices are designed to facilitate.
- Design for transferability. The same launch, facilitation, and orchestrated debrief structure applies across professional learning contexts and tasks. Several educators implemented the Painted Cube task in their own classrooms after the session. Natalie’s observation that the task had “an extremely high ceiling but also a low floor” reflects a feature of the design as much as the task itself.
Conclusion
When MTEs design professional learning that positions educators as doers of mathematics, they accomplish two things simultaneously: they develop teachers’ mathematical understanding and model HQEI. The Painted Cube task, embedded in a coherent instructional design, created conditions for exactly this kind of dual learning. The challenge for MTEs is not finding the right task; it is designing the professional learning experience around it so that mathematical agency, connected representations, and productive discourse become the norm rather than the exception.
References
Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10. Heinemann.
Liljedahl, P. (2021). Building thinking classrooms in mathematics, grades K-12: 14 teaching practices for enhancing learning. Corwin.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. NCTM.
National Council of Teachers of Mathematics. (2020). Catalyzing change in middle school mathematics: Initiating critical conversations. NCTM.
Smith, M. S., & Stein, M. K. (2011). Five Practices for Orchestrating Productive Mathematics Discussions. National Council of Teachers of Mathematics.
Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: from research to practice. Mathematics Teaching in the Middle School, 3(4), 268–275.
Suazo-Flores, E., & Roetker, L. (2024). Construct it! Building painted cubes task: Serena’s case. Mathematics Teacher Learning and Teaching PK-12, 117(1), 36–40. https://doi.org/10.5951/mtlt.2023.0069