Introduction
As our world is advancing in technology, teachers need to prepare students to be able to make sense of complex, real world problems. This requires more than simply memorizing facts or procedures. Students must have the ability to adapt, apply, and transfer knowledge in flexible ways. Cognitive Flexibility Theory (CFT), developed by Spiro and colleagues, helps teachers to design instruction that mirrors the complexity of real-world thinking. Evaluating and improving our lessons with emphasis on the Integration Model, will promote learning through multiple representations, case-based reasoning, and cross-contextual thinking.
Lesson Summary
This lesson featured in this case study was teacher created using resources from the Illustrative Math and Baltimore County Public School Curriculum materials for Middle School Mathematics. It is designed to help students learn absolute value as a distance a number is from zero on horizontal and vertical number lines. Since students received little to no instruction on absolute value in grade 6 math, the lesson starts with a “notice and wonder” activity to assess students’ prior knowledge. Students share thoughts verbally or written on a Schoology discussion board. This engagement activity is revisited at the conclusion of the lesson.
Students work in breakout groups via Google Meet (for online students) with Desmos slides. They are exploring absolute value and how it relates to subtracting positive and negative integers. There is an example problem involving a balloon and anchor above and below sea level that students work through. The teacher monitors progress and provides support. Also, the teacher gathers data for differentiating instruction. After breakout groups, the class joins together to discuss learning and misconceptions.
The students engage in another breakout group session to practice the concepts with differentiated instruction. Some groups (2-4) work at a faster pace and require extension materials. The teacher provides additional support to group 1, who works at a slower pace. After circulating and monitoring progress in breakout groups, the teacher wraps up the lesson. Students apply their understanding utilizing a real-world application problem involving the distance between the top of a volcano and the ocean floor. Students submit their responses to the Schoology discussion board. This lesson builds foundational skills involving absolute value and integer operations and connects the mathematical concepts to real-world applications.
Integration Model Summary
Spiro’s Cognitive Flexibility Theory (CFT) was developed to address the challenges of teaching and learning in complex, ill-structured domains where knowledge cannot be reduced to fixed rules or linear sequences. CFT emphasizes the need for learners to develop the ability to adapt and reconfigure their understanding as new situations arise. The Integration Model, a core element of Spiro’s Cognitive Flexibility Theory (CFT), emphasizes the importance of learners interconnecting knowledge from various sources and perspectives to navigate complex, ill-structured domains.
Since ill-structured domains lack a single correct framework for organizing or applying knowledge, learners need to adapt and apply what they know in flexible and often unpredictable ways (Spiro, Coulson, Feltovich, & Anderson, 1988). Rather than approaching concepts in isolation, the Integration Model encourages blending multiple cases, examples, and viewpoints to develop a more nuanced and interconnected understanding.
This approach is especially crucial because real-world problems rarely follow clear-cut rules. Experts in ill-structured fields routinely draw on overlapping and even contradictory information to make informed decisions. By integrating diverse knowledge, learners are better equipped to avoid oversimplifying complex topics, recognize meaningful connections among ideas, and apply prior learning in novel situations. The Integration Model supports adaptive thinking that is required for deep understanding and application to real-life situations.
The link below is a great overview of this theory.
Lesson Strengths
In reviewing the math lesson in this case study, several elements align closely with principles from Cognitive Flexibility Theory. The lesson’s self-paced structure allows students to explore content at their own speed, time to revisit instructional videos, and make second attempts at the work. This supports the revisiting and restructuring of knowledge, which is essential to Cognitive Flexibility Theory’s emphasis on building flexible, interconnected understanding. Students can engage with the material in non-linear ways, which will deepen their understanding of absolute value and its applications to the real world.
Another key detail that aligns with the model is the real-time monitoring of student progress through Desmos. The instant feedback and redirection provide opportunities for students to refine their work. This continuous formative assessment helps students adapt their approaches and fix misconceptions, which promotes cognitive flexibility.
Finally, collaborative dialogue through breakout rooms and Schoology discussions gave students the opportunity to explain their thinking to peers. This is an effective way to reinforce and construct multiple interconnected schemas. Students are encouraged to explain their thinking, listen to alternative perspectives, and collaborate with peers. Social interactions are critical for understanding ill-structured problems from different perspectives.
Lesson Weaknesses
While the lesson effectively uses a real-world example of the volcano and ocean floor distances to teach absolute value, it presents a simple, well-structured task with a single correct response. According to Cognitive Flexibility Theory, learning is most powerful in ill-structured domains where problems are complex. This requires learners to consider multiple perspectives and adapt their thinking. A math task with a single right answer may limit opportunities for exploring contradictions and ambiguities.
Cognitive Flexibility Theory recommends cross-contextual thinking. This lesson stays narrowly focused on the math domain. Incorporating contexts from science, geography, or economics – such as exploring volcanic activity, oceanography, or economic impacts of natural disasters – could enrich the lesson. This would occur by connecting absolute value and distance concepts to more real-world applications. This cross-contextual thinking would encourage students to build more complex knowledge networks.
While the use of Desmos provides an interactive experience for student exploration and differentiation, heavy reliance on a single technological tool may limit accessibility for some learners who benefit from alternative or varied learning modalities. Offering alternative methods of learning the content could increase flexibility and support for diverse learners. Overall, the lesson provides foundational groundwork, but could be expanded to embrace more complexity, interdisciplinary connections, and varied learning approaches aligned with the principles of Cognitive Flexibility Theory.
Integration Model Application
Spiro’s Cognitive Flexibility Theory (CFT) highlights the importance of helping learners develop adaptable knowledge. A key component of this theory, the Integration Model, focuses on using multiple perspectives, examples, and cases to build a flexible understanding that can be transferred to new situations. In middle school math instruction, this means going beyond teaching isolated procedures and instead designing lessons that encourage students to make connections across different problem types, representations, and real-world contexts. By revisiting concepts from various angles and using multiple representations, teachers can support students in developing deeper, more adaptable mathematical thinking.
Click here to learn more about applications to this instructional model.
Suggestions for Improvement Based on Integration Model
There are several improvements that can be made to better align the lesson with Cognitive Flexibility Theory (CFT). One improvement is to incorporate short, case-based scenarios drawn from real-world contexts. For example, students could work with sea level data or sports statistics which require interpreting signed values and absolute values. By comparing cases side by side, students can see mathematical patterns which would lead to better understanding of how absolute value applies in diverse settings. In addition, the lesson could include slightly ill-structured problems with incomplete or conflicting information. This would challenge students to make sense of ambiguity and revise their thinking, reflecting the adaptive reasoning required in real-life situations. Finally, another suggestion would be to allow students time to work together to generate their own content to promote deeper engagement, further supporting flexible knowledge construction. This content could include creating teaching slides or real-world problems for peers to solve.
References
Spiro, R. J., Coulson, R. L., Feltovich, P. J., & Anderson, D. K. (1988). Cognitive flexibility theory: Advanced knowledge acquisition in ill-structured domains. In V. Patel (Ed.), Proceedings of the Tenth Annual Conference of the Cognitive Science Society (pp. 640–653). Hillsdale, NJ: Erlbaum.
Spiro, R. J., Feltovich, P. J., Jacobson, M. J., & Coulson, R. L. (1992). Cognitive flexibility, constructivism, and hypertext: Random access instruction for advanced knowledge acquisition in ill-structured domains. Educational Technology, 31(5), 24-33.
Spiro, R. J., Feltovich, P. J., Jacobson, M. J., Coulson, R. L., & Anderson, D. K. (1991). Cognitive flexibility, constructivism, and hypertext: implications for the design and use of electronic hypertext. Educational Technology, 31(7), 24-33.