AFFORDANCES AND GROUNDING WITHIN CONCRETENESS FADING WHEN LEARNING PROOF IN STEM’S GEOMETRY

Abstract

From a learning sciences perspective, this paper explores concreteness fading (Fyfe & Nathan, 2018) as a theory of instruction designed for facilitating cognitive grounding (Glenberg, de Vega, & Graesser, 2008) in academic domains. Originally devised by Bruner in 1966, concreteness fading is now considered an exemplary learning intervention for operationalizing the theoretical framework of Grounded and Embodied Cognition (Barsalou, 2008). The empirical study rendered for this paper examines concreteness fading in the STEM (Science Technology Engineering Math) domain of geometry, aiming to develop a conceptual understanding among postsecondary school learners (N = 8) of the Deductive Proof Scheme Class (Harel & Sowder, 2007) for the Centroid Theorem of Triangles (Clapham & Nicholson, 2014). In the study, concreteness fading is investigated according to four variations of instructional design, each implementing a different combination of what Bruner labels mode of knowledge representation (1966), which refers to the form that a representation can take for communicating information about a domain of knowledge. The enactive mode of knowledge representation utilizes tangible geometric object-shapes that learners physically manipulate. The iconic mode utilizes visual images of geometric object-shapes that learners physically draw. The symbolic mode utilizes numbers that learners physically write in formulas pertaining to geometric object-shapes. Each variation of instructional design combines the modes of knowledge representation differently, yet all designs follow the same concreteness fading (Fyfe & Nathan, 2018) delivery sequence, defined as presenting external knowledge representations to

learners that transition from less idealized instantiations toward more idealized instantiations. Specifically, the variations of instructional design are arranged as: condition 1 (n = 2) combines an enactive-iconic-symbolic delivery sequence; condition 2 (n = 2) combines an enactive- symbolic sequence; condition 3 (n = 2) combines an iconic-symbolic sequence; and condition 4 (n = 2) is a symbolic-only delivery. Unlike much of the current literature that compares concreteness fading to other presentations of knowledge representations, for example comparison to a concreteness introduction sequence or to a focus on concrete only and idealized only representations, this qualitative study is primarily interested in the influence engendered by each distinct mode of knowledge representation within the concreteness fading sequence. In this paper I ask the question, how do learners’ manipulation of less idealized objects, or drawing of more idealized icons, or writing of still more idealized symbols, uniquely contribute to cognitive grounding, within a concreteness fading instructional design in the STEM domain of geometry? And I answer that question through qualitatively coding participants’ formulation of mathematical proof, revealed in their written products and video recorded think- aloud protocols, on post training assessments of learning outcomes and of near transfer outcomes. The paper is organized into five main sections. In the first section, the motivation, I situate the study of mathematical proof as important for STEM domains, I spell out how the concept of cognitive grounding to real-world referents is crucial for learning proof, and I identify a formalisms first (Nathan, 2012) instructional approach as an impediment to students’ development of conceptual understanding. In the second section, the framing, I elucidate the theories of affordances (Gibson, 2015) and cognitive grounding (Barsalou, 1999) that underlie Grounded and Embodied Learning (Nathan, 2022), which is a framework used to guide my

incorporation of embodiment into the materials and activities created for the concreteness fading instructional design in this study. Section two also includes a detailed exposition of five prominent quantitative concreteness fading studies, to provide a brief overview of the existing literature and to point out the value that qualitative studies may potentially offer, such as the one presented in this paper. In addition, I explain the concept of cognitive offloading (Wilson, 2002), which in educational settings refers to students’ use of resources in the material environment to aid their cognitive processing during problem solving; and I explain two ways students can demonstrate offloading, specifically through pragmatic actions (Kirsh & Maglio, 1994) that are attempts to execute a planful goal to directly solve the academic problem, and through epistemic actions (Kirsh & Maglio, 1994) that are attempts to gain a better understanding of the academic problem. In the third section, the methods, I illustrate the unique way that embodiment has been designed into the materials and activities of this study, I share demographic information about participants, lay out the experimental design and procedures, and present the performance measures and qualitative coding scheme. In the fourth section, the results, I share: a) an analysis of mean performance by condition on mathematical proof for the centroid theorem of triangles, calculated by comparing percent-correct-scores on pre-test, post-test, and novel test; b) an analysis of individual performance by participant on conceptual understanding of the deductive proof scheme class, calculated by comparing the evidence utilized for “ascertaining and persuading” (Harel & Sowder, 2007, p. 6) on pre-test, post-test, and novel test; c) an analysis of two case studies that reveal the fascinating way participants employed pragmatic and epistemic actions to access affordances provided by particular modes of knowledge representation in their cognitive offloading while formulating a proof, which they demonstrated in four meaning making strategies described as an exploration phase, a guided search phase, a mooring phase,

and a problem-solving phase. In the fifth section, the discussion, I revisit the findings in relation to my initial hypotheses established prior to beginning this exploratory study, and I highlight the way participants utilized the mooring phase to cognitively ground their conceptual understanding of the deductive proof scheme class for the centroid theorem of triangles when presented with a novel geometric shape. The substance of the paper starts in the following section where I begin the story of my inquiry inspired by a belief in the importance for students to learn proof as preparation for the challenges of our rapidly evolving 21st century.