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Students often have difficulty identifying and discriminating between shapes when reasoning solely with a concept image without a concept definition. This poster provides evidence that this reasoning is common in middle-school, that the reasoning is inconsistently applied, and that the reasoning might interfere with general geometry knowledge.
Students often have difficulty identifying and discriminating between shapes when their reasoning is based on a concept image (CI) without a concept definition. This poster session uses an extant dataset to explore how prevalent this type of reasoning is in middle-school, whether the reasoning is consistently applied, and how the reasoning relates to more general geometry knowledge.
Theoretical Framework
A CI is a mental image that students have of a shape, without a specified definition of the shape or its properties (Vinner & Hershkowitz, 1980). Students who reason solely with CIs can often identify some examples of shapes, but fail to identify other examples that aren't identical to their mental image of the shape, usually the shape prototype, i.e., the figure doesn't “look like” the shape. Orientation and proportions affect whether students recognize certain shapes, despite the fact that these are irrelevant to the defining properties of a shape (Carroll, 1998; Clements, 2003; Clements & Battista, 1992). Rather than rely solely on a mental or visual image, students should have a set of visual examples and a description of the necessary and sufficient properties that define a shape (Hershkowitz, 1989). Figure-1 shows examples of shape prototypes and non-prototypical shapes. The non-prototypical examples might not be recognized by students reasoning solely with CIs. For example, students might not recognize the third example of a non-prototypical parallelogram because it isn't “slanted or tilted” like most students' prototype.
Reasoning in this way is consistent with students operating at the first two van Hiele levels of geometric reasoning (Crowley, 1987; van Hiele & van Hiele-Geldof, 1959/1985). In Level-1, students recognize and compare figures by appearance and make mental comparisons to prototypes. Students don't consider individual properties but rely on visual perception. A student might only identify the prototypes in Figure-1 because of orientation and proportions and might explain that the others aren't examples because “they don't look like those shapes.” In Level-2, students are able to see figures as collections of properties, but they don't draw connections between properties or understand the necessary/sufficient attributes of properties. For example, a student might identify the parallelogram prototype in Figure-1 because it has “two pairs of parallel sides and is tilted to the right.” The student recognizes the necessary (and sufficient) property, but includes an additional property.
Reasoning solely with a CI is also influenced by students' ability to understand class inclusion and hierarchical relationships. Students in middle-school often relate classes in ways that preserve mutual exclusivity, while the development of shape definitions requires students to consider overlapping and interrelated classes (Mason, 1989; Serow, 2007).
Figure-1: Sample Prototypes and Non-Prototypes
Data/Methods
The cognitive theory underlying the type of reasoning discussed in this poster is thorough, but little research has focused on how common this reasoning is in middle-school and on how this reasoning is applied. The poster begins to fill this gap by addressing three questions: 1) How prevalent is reasoning solely with CIs in middle-school? 2) Do students reasoning solely with CIs apply this reasoning consistently across shape categories? 3) Does this reasoning interfere with more general knowledge of geometry? To explore these questions, the researchers conducted secondary data analysis on extant data from the <Project-Withheld-for-Review>, which aims to apply cognitive research to develop a formative assessment system to identify and address geometric misconceptions.
Twenty items designed to identify students reasoning solely with CIs were administered online to 1,362 students in grades 6-8. The researchers don't claim a representative sample; demographic descriptions will be presented. Eleven items explicitly referenced the shape category of parallelograms. These items, NPP (non-prototypical-parallelogram) items, focused on whether students could identify non-prototypical parallelograms (e.g., prototypical rectangles, squares, rhombi) as parallelograms. The other nine items didn't reference parallelograms directly. These items, SR (square-rectangle) items, focused on whether students could identify non-prototypical rectangles (i.e., squares) as rectangles. The researchers also developed a measure of general geometry ability (GGA) comprising released standardized items. Sample items and descriptives will be presented.
The researchers examined how many students were identified as reasoning solely with CIs by the NPP and SR items and whether students identified with this reasoning were consistent across the items. The researchers compared the performance of students who did and didn't exhibit this reasoning on the GGA items.
Results
Twenty-two-percent of students were identified as reasoning solely with a CI by the NPP items; 53% of students were identified by SR items. Fifteen-percent of students were classified with this reasoning by both sets of items, 7% by only NPP items, and 37% by only SR items. The differences in classification of students by the two groups was statistically significant (chi-square analysis, c2=0.37.755, df=1, p<0.01). Thus, reasoning wasn't consistent across shape categories. Student identified with this reasoning by either set of items showed statistically significantly lower performance on the GGA items (independent-samples t-test, p<0.01). Thus, this reasoning might interfere with more general geometry knowledge.
Significance
This poster begins to build a base of empirical work to support the cognitive foundations of reasoning with CIs and begins to explore the consistency of the application of this type of cognitive reasoning. Future work should explore the basis for inconsistencies, such as, are some shape categories “more general” and easier about which to develop sophisticated reasoning? Are some characteristics more persistent as defining characteristics (e.g., side length vs. side orientation)? Are there instructional approaches that encourage this reasoning (e.g., statements such as “a square isn't a rectangle”)?
From a practical perspective, this research indicates that reasoning with a
CI might be prevalent in middle-school and might interfere with more general
geometry knowledge. Therefore, it's critical for teachers to be aware of this
of reasoning, be able to identify it, and have methods to help students develop
more sophisticated reasoning. Shape properties is a topic that is often
considered appropriate for lower grades, but the current work shows that
middle-school teachers might need to include instruction or remediation on this
topic. Instruction should include a wide variety of shape categories, as
students might apply this reasoning inconsistently.
References
Carroll, W. M. (1998). Geometric knowledge of middle school students in a reform-based mathematics curriculum. School Science and Mathematics, 98, 188-197.
Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 151-178). Reston, VA: NCTM.
Clements, D. H. & Battista, M. T. (1992). Geometric and spatial reasoning. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics (pp. 420-464). New York, NY: Macmillan.
Crowley, M.L. (1987). The van Hiele model of the development of geometric thought. In M.M. Lindquist (Ed.), Learning and Teaching Geometry, K-12 (pp. 1-16). Reston, VA: National Council of Teachers of Mathematics.
Hershkowitz, R. (1989). Visualization in geometry: two sides of the coin. Focus on Learning Problems in Mathematics, 11, 61-76.
Mason, M. M. (1989, March). Geometric understanding and misconceptions among gifted fourth-eighth graders. Paper presented at the Annual Meeting of the American Educational Research Association. San Francisco, CA.
Serow, P. (2007). Utilising the rasch model to gain insight into students' understandings of class inclusion concepts in geometry. In J. Watson & K. Beswick (Eds), Mathematics: Essential research, essential practice. Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australiasia. Mathematics Education Research Group of Australasia Inc: Adelaide, pp. 651–660.
Vinner, S., & Hershkowitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometrical concepts. In R. Karplis (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 177-184), Berkeley, CA: University of California, Lawrence Hall of Science.
van Hiele, P. M. & van Hiele-Geldof, D. (1959/1985). D. Fuys, D. Geddes, & R. Tischler (Eds.), English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre van Hiele, (pp. 243-242). Brooklyn, NY: Brooklyn College, School of Education. ERIC Document Reproduction Service No. ED287697.
Students often have difficulty identifying and discriminating between shapes when reasoning with a concept image but without a concept definition. This session gives evidence that this reasoning is common in middle school, is inconsistently applied, and might interfere with general geometry knowledge.
Session Type: Poster Session