Abstract
In this research article, I present evidence of the existence of visual templates in pattern generalization activity. Such templates initially emerged from a 3-week design-driven classroom teaching experiment on pattern generalization involving linear figural patterns and were assessed for existence in a clinical interview that was conducted four and a half months after the teaching experiment using three tasks (one ambiguous, two well defined). Drawing on the clinical interviews conducted with 11 seventh- and eighth-grade students, I discuss how their visual templates have spawned at least six types of algebraic generalizations. A visual template model is also presented that illustrates the distributed and a dynamically embedded nature of pattern generalization involving the following factors: pattern goodness effect; knowledge/action effects; and the triad of stage-driven grouping, structural unit, and analogy.
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Notes
I prefer to use the term figural pattern in order to convey what I assume to be the “simultaneously conceptual and figural” (Fischbein, 1993, p. 160) nature of mathematical patterns. The term “geometric patterns” is not appropriate due to a potential confusion with geometric sequences (as instances of exponential functions in discrete mathematics). Also, I was not keen in using the term “pictorial patterns” due to the (Peircean) fact that figural patterns are not mere pictures of objects but exhibit characteristics associated with diagrammatic representations. The term “ambiguous” shares Neisser’s (1976) general notion of ambiguous pictures as conveying the “possibility of alternative perceptions” (p. 50).
Especially in the case of patterning tasks that involve figural cues, among the most important perception types that matter involve visual perception. Visual perception involves the act of coming to see; it is further characterized to be of two types, namely, sensory perception and cognitive perception (Dretske, 1990). Sensory (or object) perception is when individuals see an object as being a mere object in itself. Cognitive perception goes beyond the sensory when individuals see or recognize a fact or a property in relation to the object. For example, students who see consecutive groups of figural stages in Fig. 7 as mere sets of objects exhibit sensory perception. However, when they recognize that the stages taken together actually form a pattern sequence of objects, they manifest cognitive perception. Cognitive perception necessitates the use of conceptual and other cognitive-related processes, enabling learners to articulate what they choose to recognize as being a fact or a property of a target object. It is mediated in some way through other types of visual knowledge that bear on the object, and such types could be either cognitive or sensory in nature.
I share Resnik’s (1997) view that the “primary subject-matter [of mathematics] is not the individual mathematical objects but rather the structures in which they are arranged” (p. 201). Pattern stages basically convey positions in some overall structural relationship and, so, do not have an identity or distinguishing features outside that relationship despite the fact that specializing on a particular stage, which is considered a useful generalizing strategy, may give the impression of a structure.
Intrinsic to classroom teaching experiments that employ design research are two objectives, that is, developing an instructional framework that allows specific types of learning to materialize and analyzing the nature and content of such learning types within the articulated framework. Thus, in every design study, theory and practice are viewed as being equally important, which includes rigorously developing and empirically justifying a domain-specific instructional theory relevant to a concept being investigated. Further, the content of the proposed instructional theory involves a well-investigated learning trajectory and appropriate instructional tools that enable student learning to take place in various phases of the trajectory. Finally, instruction in design studies is characterized as having the following elements: it is experientially real for students; it enables students to reinvent mathematics through, at least initially, their commonsense experiences; and it fosters the emergence, development, and progressive evolution of student-generated models.
I am aware of issues surrounding researcher-driven interviews. The X 3 interviews that I conducted with the 11 students were third in a series of clinical interviews conducted during the 10-month project. A colleague who was external to the project conducted both X 1 and X 2 interviews. Also, the X 3 interview would in fact be seventh in the case of the cohort 1 participants. Yanos and Hopper (2008) suggest the use of repeated interviews as a way of minimizing what Bourdieu (1999) refers to as “false, collusive objectification” whereby interviewees respond in ways that please their interviewers. Further, the findings that I have drawn from the X 3 clinical interviews and that are reported in this article were triangulated with two other longitudinal data sources (i.e., available student written work and earlier clinical interviews conducted by a colleague).
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Support for this research was funded by the National Science Foundation awarded to the author under DRL 0448649. The opinions expressed are solely of the author and do not necessarily reflect the views of the foundation.
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ESM 2
Graphical illustration of Emma’s direct formula \( s = n + n - 1 \) in relation to Fig. 9 (DOC 118 kb)
ESM 3
Additional examples of linear figural patterns that yield a CSG (DOC 5217 kb)
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Frank’s justification of the formula \( S = 2x - 1 \) in relation to the Fig. 9 pattern (DOC 68 kb)
ESM 9
Frank’s justification of the formula \( {\text{S}} = 2\left( {x - 1} \right) + 1 \) in relation to the Fig. 9 pattern (DOC 72 kb)
ESM 10
Cherrie’s justification of her formula \( x = 2n - 1 \) in relation to the Fig. 9 pattern (DOC 100 kb)
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Rivera, F.D. Visual templates in pattern generalization activity. Educ Stud Math 73, 297–328 (2010). https://doi.org/10.1007/s10649-009-9222-0
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DOI: https://doi.org/10.1007/s10649-009-9222-0