Carefully sequenced problem strings support sense-making, reduce cognitive overload, and build conceptual understanding over time.
Learn how to design problem strings or use an example problem string from Algebra and Precalculus courses.
Thin slicing is a similar strategy to lead mathematical thinking through a series of carefully crafted problems. Chapter 9 in Building Thinking Classrooms discusses this process (Liljedahl, 2020)*
Watch Peter Liljedahl explain how he facilitated a thinking lesson for solving irrational equations using problem strings.
Incremental small improvements to math teaching take time and planning, worked problems coupled with intentional teaching practices support continual growth.
Structured discourse supports sense-making, clarifies understanding, and refines thinking through shared reasoning.
Try increasing the level of student discourse with simple moves that work. Increasing "Math Talk" is a goal for many classes that build student engagement and ownership that leads to deeper understanding.
Try building student accountability to learning by shifting mathematical authority to your students. Processes like classroom norms and RUME classroom authority change this dynamic.
5 Practices for Orchestrating Discussions are easy to apply strategies to increase discourse.
Evidence-based practice strategies support consolidation, flexible application, and long-term retention.
Resources from cognitive science for facilitating practice (retrieval practice, spacing, interleaving): Research summary
Retrieval Practice: Spaced retrieval practice
Interleaving: Interleaving mathematics practice
Reviewing prerequisite skills supports comprehension, reduces gaps in understanding, and builds confidence with complex content.
Emphasizing patterns and key structures supports sense-making, deepens understanding, and organizes knowledge meaningfully.
Calculational and Conceptual Orientations in Teaching Mathematics (Thompson et al., 1994.)*
Mathematical questioning can create student to student discussion with strengthens engagement and understanding. These 5 Talk Moves can support that process.
Use "Explanatory Questioning" (cognitive science practice) by asking "Why?" questions and encouraging students to engage in self-explanation think alouds or discussing with others (Star & Verschaffel, 2017)*
IB Mathematics Mind Maps provide models for these visual connections.
Comparing examples and non-examples supports conceptual clarity, reduces misconceptions, and clarifies essential features.
Worked Examples (cognitive science practice): Lighten cognitive load by asking students to examine multiple worked examples before moving into problem solving practice (Star & Verschaffel, 2017)*
Compare & Discuss / Contrasting Cases draw on cognitive science research to highlight big ideas:
Develop consistent routines for students analyzing multiple solutions using:
Contrasting Cases for Geometry provides multiple worked examples with contrasting strategies for students to analyze and build understanding.
*Liljedahl, P. (2020). Building Thinking Classrooms in Mathematics: 14 Teaching Practices for Enhancing Learning, Grades K-12. Corwin.
*Thompson, P. W., Philipp, R., & Boyd, B. (1994). Calculational and conceptual orientations in teaching mathematics. In 1994 Yearbook of the NCTM
*Star, J., Verschaffel, L. (2017). Providing Support for Student Learning: Recommendations from Cognitive Science for the Teaching of Mathematics. In J. Cai (Ed). Compendium for Research in Mathematics Education. NCTM.