Triad of Schema development for a calculus graphing Schema

The triad of Schema development can be used to describe Schema thematization, which is “a mental construction of a Schema so that it may be dissected, broken down, examined by its parts, reassembled, acted upon as an Object, and brought to bear in appropriate

situations” (Cooley et al., 2007, p. 371). In this way, Schema thematization can be thought of as a highly desirable and robust conceptual understanding of a particular idea. The triad of Schema development is then a description of the levels of understanding a student passes through as they develop understanding of a particular concept. Schema thematization of proof by contradiction would allow students to understand the proof method and know when to apply the method. However, only Cooley et al. (2007) have studied Schema thematization of any mathematical concept. Therefore, the rest of this subsection will be devoted to describing the methodology of Cooley et al.’s (2007) study and presenting the resulting triad of Schema development.

B. Baker et al. (2000) aimed to describe how students understand and solve non-routine calculus graphing problems. Forty-one students were interviewed, during which they were given a list of conditions and asked to graph a function that satisfied these conditions. After completing the interviews, the authors constructed a preliminary genetic decomposition for a calculus graphing Schema in order to establish specific criteria to categorize student thinking. The calculus graphing Schema that emerged was unique in that it involved the coordination of two Schemas: the condition-property Schema and domain-interval Schema. The condition-property Schema, shortened to property Schema, involves “understanding each analytic condition as it relates to a graphical property of the function and coordinating these conditions” (B. Baker et al., 2000, p. 565). The domain-interval Schema, shortened to interval Schema, involves “understanding the interval notation, connecting contiguous intervals, and coordinating the overlap of the intervals” (B. Baker et al., 2000, p. 566). The data was then analyzed in terms of the triad of Schema development to provide a framework for categorizing student understanding. As the calculus graphing Schema involved the coordination of two Schemas, nine categories were developed9. The progression from intra-property to trans-property and from intra-interval to trans-interval have been separated and are presented below:

9_{Each stage of the property Schema, such as intra-property, was coordinated to each stage of the interval}

• Property Schema

– Intra-property: Student interprets only one analytical condition at a time in terms of its graphical feature. There is awareness of other properties but student cannot coordinate them to produce a graph.

– Inter-property: The student begins to coordinate two or more conditions simul- taneously. This coordination is not applied throughout all connected intervals or across the entire domain.

– Trans-property: Student can demonstrate coordination of all the analytic condi- tions to the graphical properties of the function on an interval.

• Interval Schema

– Intra-interval: Work is on isolated intervals. The overlap of intervals or connection of continuous intervals causes confusion.

– Inter-interval: There is coordination of two or more contiguous intervals simulta- neously. This coordination, however, was not applied throughout all connected intervals or across the entire domain.

– Trans-interval: Describes the coordination of the intervals across the whole do- main. He or she is able to overlap intervals and connect contiguous intervals. (Cooley et al., 2007, p. 375)

Note how connected the mental constructions are at each stage. For the property Schema, the student begins by analyzing each condition separately at the intra-property level. Then, the student coordinates some of these conditions together at the inter-property level. Finally, the student can coordinate all of the properties together at the trans-property level. Similarly for the interval Schema, the student works on a single interval at a time at the intra-interval level. Then, the student can work on two or more intervals at a time at the inter-interval level. Finally, the student can work on all intervals at a time at the trans-interval level.

Cooley et al. (2007) built upon their previous study by focusing on Schema themati- zation of the calculus graphing Schema described previously. In order to examine whether students demonstrated a thematized Schema while solving questions related to the calculus graphing Schema, the authors interviewed 28 strong10 _{mathematics students. These inter-}

views involved participants completing a set of eight problems as well as explaining their thought processes and methods they used to assemble graphs.

After categorizing the student stages of Schema development, established by B. Baker et al. (2000), only one of the 6 students at the trans-property, trans-interval level of Schema development were determined to have thematized her Schema due to the student’s ability to “clearly explain what would change and what would remain invariant when the conditions of the problem were changed in the final question” (Cooley et al., 2007, p. 379). The authors concluded it is was thus possible for students to thematize their Schema, though the authors suggested that more research is needed to address how students thematize their Schema. This study will attempt to address how students thematize their Schema by conducting multiple sessions to capture students’ thinking over time.