Concrete-Representational-Abstract Pedagogical Progression

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Riemann sums are often utilized to approximate the area accumulated under a section of a function; a small quantity of rectangles results in relatively larger rectangles that may not be entirely representative of the accumulated area. As the quantity of rectangles increase, the rectangles decrease in size, resulting in the increase in the approximation's accuracy. Similarly, the Concrete-Representational-Abstract (CRA) model seeks to depict a student's pedagogical progression from tangible examples to abstractions.

The concrete phase of the model pertains to tangible constructions. Often, manipulatives, such as counters or base-ten rods, are used to illustrate concepts. Students are able to construct mathematical models, serving as the preliminary step towards formulating a logical framework. An instance of the concrete phase would be illustrated by a student constructing an angled ramp and measuring an object's displacement of the vertical direction-- referred to as the "rise"--and displacement of the horizontal direction-- referred to as the "run". Additionally, the student may modify the displacement in the vertical or horizontal of the original ramp as a basis for comparing the ramp to other ramps with dissimilar slopes. At this epistemic juncture, students may be cognizant of the concept of slope, yet may not be aware of its definition.

Subsequently, the representational phase serves as a liminal state between the concrete and abstract stages. Students are given proxies of the concept--namely charts, graphs, and figures--rather than a physical model. To illustrate, through the slope example, a student may practice finding the slope after receiving a table of x and y values; implementing a table of values allows the student to associate, respectively, the horizontal displacement with changes in x-values and the vertical displacement with changes in y-values, and learn to associate a quantification of the line's incline with the quotient of the change in y-values and the change in x-values. In the course of the step, formulas, such as the average rate of change formula, may also be introduced as they allow the student to form a conceptual lattice through connecting slope with other concepts.

As a culminating stage, abstraction provides a pattern for conceptual understanding. Frequently, students may prove or derive formulas to augment their knowledge. Utilizing an extension of the slope example, students may be provided with the average rate of change formula, and be tasked with ideating a scenario that describes the limit of the average rate of change at a given point. The inquiry, regarding the affect of a decreasing interval for the average rate of change, allows students to formulate a definition of a derivative--the slope of the tangent line at a given point without merely memorizing an equation; the idea of the derivative is, in toto, abstract rather than representable using tangible objects.

Learning is a manifestation of integration, a compiling of knowledge that accumulates under a student's function of intellectual comprehension; demonstrating proficiency in each subtopic can be likened to appending an individual rectangle to the approximation.Metaphorically akin to how the value for the Riemann sum is situated between that of the lower Darboux and upper Darboux sums, the acquisition of knowledge occurs between utterly concrete and utterly abstract states.