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AMATYC defines “mathematics intensive” courses as those taught in the first two years of collegiate study that lead to advanced work in science, technology, engineering, and mathematics. Our mathematics intensive network focuses on courses such as college algebra, precalculus, calculus, differential equations, and linear algebra. These require development of abilities in problem solving, modeling, reasoning, connecting to other disciplines, communicating, and using technology, which are the goals student learning outcomes as defined in AMATYC’s (1995) Crossroads in Mathematics. Similarly, the National Research Council (2001) defines proficiency as reflecting conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive student disposition. These are wonderful goals, but how does a teacher help a student develop these skills and attitudes? How can teachers, curricula, peers, and the individual student initiate real understanding and positive student dispositions?

Some people learn mathematical concepts and others do not. Why? Piaget (Beth & Piaget, 1966) claims that students construct knowledge through the processes of assimilation and accommodation. Assimilation is the active process of constructing a new cognitive structure and accommodation is the active process of revising that structure so that it fits coherently with existing ones. A “memorized” or “mimicked” procedure that a student “learns” cannot be defined as a new cognitive structure that must be a coherent part of a student’s knowledge base. That procedure has not been assimilated or accommodated, and it will be likely forgotten. So, the goal is the development of lasting cognitive structures, and Piaget and Dubinsky claim that reflective abstraction, a type of individual reflection, is the key.

Cooley (2002) clarifies this in the following passage.

“Reflective abstraction is a mechanism for the isolation of particular attributes of a mathematical structure that allows the subject to construct or reconstruct knowledge that is new, that is, knowledge not previously known. A feature of reflective abstraction is that it clarifies and organizes logico-mathematical experiences in such a way as to recognize both nuances and broad generalizations among them. Any new constructions will be associated with knowledge the subject already has. The subject orders or re-orders a class of situations with the characteristics of the current object so that the new knowledge fits with previous schemas, or the previous schema has been reconstructed. The new generalization occurs precisely because of a mental construction or reconstruction.” (Cooley, 2002 p. 255)

It is necessary to identify the constructs of reflective abstraction so an instructor can act as an agent of change. Dubinsky (1991) refines Piaget’s original constructs of reflective abstraction. They are interiorization, coordination, encapsulation, generalization, and reversal.

Working definitions of the terms follow. (Cappetta, 2007)

- Interiorization: A student performs the steps in a procedure. The student reflects on the procedure and begins to define a concept.
- Coordination: A student examines two different processes and integrates them into a coordinated process that is used to analyze a mathematical concept.
- Encapsulation: A student encapsulates a concept by constructing individual meaning. Encapsulation is the act of personifying a concept. An abstract notion or a collection of abstract notions becomes meaningful to an individual.
- Generalization: After an individual has encapsulated a notion, it is extended and applied to a wider collection of mathematical problems.
- Reversal: A student constructs a new mathematical notion by reversing the steps of the original notion.

These are internal processes that an individual must perform. Certainly, good students may do these things naturally, but what can be done to encourage more people to do so?

Cobb, Boufi, McClain and Whitenack (1997) recognize that teachers can prompt shifts in the discussion that encourage students to reflect on what they are learning. Several calculus reform curricula were designed to get students to reflect. Notable efforts include the Harvard Project, Project CALC, and Calculus and Mathematica. Certainly, effective social discourse among students can promote individual reflection (Cobb, Jaworski, and Presmeg, 1996). As stated earlier, successful students regularly reflect. To summarize, the teacher, the curriculum, peers, and the individual student can promote reflection, therefore, how does this influence planning and assessment?

A teacher could design a lesson to help initiate interiorization, coordination, encapsulation, generalization, and reversal. For example, use the difference quotient to evaluate the derivative of a quadratic polynomial (Interiorization). Explain why without using a memorized rule. (Encapsulation). Explain the relationship between limits and continuity (Coordination). What would a derivative mean for a function of two variables? (Generalization). During first semester calculus before learning integration techniques, ask students to construct a function whose derivative is (Reversal).

Collaboration is an essential part of active student learning. Laursen and Rasmussen (2019) argue that inquiry-based mathematics education should require student engagement in meaningful mathematics, student collaboration designed to foster sensemaking, instructor inquiry into student thinking, and equitable instructional practice. The meaningful mathematics may reflect a curriculum that initiates reflective abstraction, student collaborative sensemaking appears to be a peer-initiate of reflective abstraction, and instructor inquiry is likely a teacher initiate. This approach and several others encourage students to reflect deeply, and hopefully it results in conceptual understanding.

The NRC (2001) concludes its description of proficiency as developing a productive disposition towards mathematics. An interesting research question would be whether a curriculum designed to promote reflective abstraction would improve students’ dispositions towards mathematics. Unfortunately, Sonnert, Sadler, and Bressoud (2015) found that “ambitious teaching” that included group work, word problems, “flipped” reading, and student explanation of thinking was found to have no impact on student attitudes overall. Even worse, these practices were negatively associated with student attitudes when graduate students implemented these methods.

If assignments typically reflect rote skills, it is little wonder that students do not reflect. A good task should give students opportunities to demonstrate each of the five constructs. However, it is somewhat troubling that students may not appreciate these. Phil Hill wrote on November 30, 2015, “the routine hard work required of students in active learning courses can lead to poor student evaluations . . . [this is] a major barrier to improving teaching practices.” eliterate.us/student-course-evaluations. Retrieved November 4, 2021.

In a previous study (Cappetta & Zollman, 2013) showed that a curriculum designed to initiate reflective abstraction was correlated with better student performance on mathematical concepts and better performance on mathematical communication. Many others have found success with similar strategies. Dagley, Gill, et al. (2018), McNichol, Frank, et al. (2021), Mazur (2009).

I invite AMATYC members to comment on the challenges of an approach like this and whether they think it could be effective. Analyze a recent exam for a math intensive course. Does that task initiate reflective abstraction? Are all five constructs present? The goal of the mathematics intensive community is to encourage collaboration among education professionals to improve teaching and learning. Hopefully this brief essay will help that happen.

References

American Mathematical Association of Two-Year Colleges (1995). *Crossroads in mathematics: Standards for introductory college mathematics before calculus*. Memphis, TN: Author

Beth, E. & Piaget, J. (1966). *Mathematical epistemology and psychology*. Dordrecht, The Netherlands: D. Reidel.

Cappetta, R. W. (2007). Reflective abstraction and the concept of limit: A quasi -experimental study to improve student performance in college calculus by promoting reflective abstraction through individual, peer, instructor, and curriculum initiates. (Doctoral Dissertation, Northern Illinois University, 2007)

Cappetta, R.W., and Zollman, A. (2013). Agents of Change in Promoting Reflective Abstraction: A Quasi-Experimental Study on Limits in College Calculus. *REDIMAT – Journal of Research in Mathematics Education*, 2(3), 343-357. doi: 10.4471/redimat.2013.35

Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. *Journal for Research in Mathematics Education*, 28(3), 258-277.

Cobb, P., Jaworski, B., & Presmeg, N. (1996). Emergent and sociocultural views of mathematical activity. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), *Theories of mathematical learning* (pp. 3-19). Mahwah, N.J.: Lawrence Erlbaum.

Cooley, L. (2002). Reflective abstraction and writing in calculus. *Journal of Mathematical Behavior*, 21(3), 255-282.

Dagley, Melissa A.; Gill, Michele; Saitta, Erin; Moore, Brian; Chini, Jacquelyn; and Li, Xin (2018) "Using Active Learning Strategies in Calculus to Improve Student Learning and Influence Mathematics Department Cultural Change," Proceedings of the Interdisciplinary STEM Teaching and Learning Conference: Vol. 2 , Article 8. DOI: 10.20429/stem.2018.020108 Available at: https://digitalcommons.georgiasouthern.edu/stem_proceedings/vol2/iss1/8

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), *Advanced mathematical thinking* (pp. 95-126). Boston: Kluwer.

Laursen, L & Rasmussen, C (2019). I on the Prize: Inquiry Approaches in Undergraduate Mathematics. *International Journal of Research in Undergraduate Mathematics Education*, v5 n1 p129-146 Apr 2019

Mazur, E. (2009). Farewell, Lecture? *Science*, 323(5910), 50-51. http://www.jstor.org/stable/20177113

- H. McNicholl, K. Frank, K. Hogenson, J. Roat & M. P. Carlson (2021) Improving Student Success and Supporting Student Meaning-Making in Large-Lecture Precalculus Classes,
*PRIMUS*, 31:7, 792-810, DOI: 10.1080/10511970.2020.1737850

National Research Council (NRC). (2001). *Adding it up: Helping children learn mathematics*. J. Kilpatrick, J. Swafford & B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Social Sciences and Education. Washington, DC: National Academy Press

Sonnert, G., Sadler, P., Sadler, S. & Bressoud, D. (2014) The impact of instructor pedagogy on college calculus students’ attitude toward mathematics. *International Journal of Mathematics, Science and Technology*, 46, 370–387.

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