Leveling Transparency via Situated Intermediary Learning Objectives (SILOs)

When designers set out to create a mathematics learning activity, they have a fair sense of its objectives: students will understand a concept and master relevant procedural skills. In reform-oriented activities, students first engage in concrete situations, wherein they achieve situated, intermediary learning objectives (SILOs), and only then they rearticulate their solutions formally. We define SILOs as heuristics learners devise to accommodate contingencies in an evolving problem space, e.g., monitoring and repairing manipulable structures so that they model with fidelity a source situation. Students achieve SILOs through problem-solving with media, instructors orient toward SILOs via discursive solicitation, and designers articulate SILOs via analyzing implementation data. We describe the emergence of three SILOs in developing the activity Giant Steps for Algebra. Whereas the notion of SILOs emerged spontaneously as a framework to organize a system of practice, i.e. our collaborative design, it aligns with phenomenological theory of knowledge as instrumented action. When mathematics-education designers set out to create a new learning activity, they bear in mind the activity’s ultimate pedagogical objective. Reform-oriented designers, however, bear in mind intermediary objectives, too, for students’ immersive experiences in situated, multimodal, spatial–dynamical activities designed to foster grounded understanding of the ultimate target concepts. Broadly, reform-oriented activities unfold in two steps:  In Step 1, learners interact with media—physical or virtual materials and ready-made objects—to solve problems that require manipulating, organizing, and/or transforming these media with attention to quantitative relations as well as emerging patterns or principles pertaining to these relations.  In Step 2, learners are guided to reflect on, and rearticulate their insights using normative semiotic systems, including frames of reference, vocabulary, and symbolic notation and to reenact discovered processes as standard algorithms using the formal representations (Diénès, 1971; Freudenthal, 1983). This paper focuses primarily on Step 1. Step 1 is of immense importance to the construction of knowledge (Kamii & DeClark, 1985; Piaget & Inhelder, 1969; Thompson, 2013). And yet, we find, educational designers have little, if any, conventional forms, nomenclature, or methodology for articulating Step 1 learning objectives prior to the design process. Perhaps, we submit, this disconcerting lacuna in the design toolkit is related to the ultimate futility of attempting to articulate Step 1 learning objectives prior to building and refining activities and observing people engage with them. Namely, Step 1 objectives emerge only through the design process. Yet this emergent nature of a design’s Step 1 objectives, we further submit, should not deter us from eventually defining those objectives. This paper resulted from reflecting on an apparent omission in our own design process: Building a certain design, we kept referring nebulously to a set of latent, contextualized, mathematically oriented, informal ideas we wanted students to discover via engaging in its Step 1 activities. The objective of this paper is to name that unnamed class of ideas and define its role within the design process. We will name this class situated, intermediary learning objectives (SILOs) and demonstrate how this ontological innovation lends coherence to a comprehensive, complex, multi-stage process. We hope that, through this paper, fellow designers will join us in “learning and becoming in [design] practice” (the ICLS 2014 theme). In the remainder of this paper we: overview relevant educational-research literature (Section 1); present Giant Steps for Algebra (Chase & Abrahamson, 2013) (Section 2); explain how three SILOs emerged via developing the design materials and analyzing pilot implementation data and how these SILOs inform our technological redesign (Section 3); and offer implications for theories of knowing and learning (Section 4). Theoretical Background: Constructing Means for Constructing Meaning When we design concrete activities for mathematics learning, what are our learning objectives for these activities? These are not quite mathematical learning objectives per se, because they may not be articulated in formal register and might not even involve numerical values. And yet we do eventually form clear ideas for what the students should be discovering about the target concepts through engaging in the concrete activities. In so doing, we implicitly exercise a theoretical view on the relation between the manual and the mental. One such view is ascribed to John Dewey, who characterized conceptual learning as the individual’s process of ICLS 2014 Proceedings 23 © ISLS formalizing their reflection on experience—their guided passage from implicit know-how through to articulated know-that. Such characterizations of grounded understanding are not only vital for building theories of learning but also bear direct implications for the practice of designing effective learning environments that seek to guide children from informal experience to formal concept. This schematic conceptualization of grounded mathematics learning as an experience-to-concept two-step process cuts across multiple theories and frameworks, including our own. To begin with, our distinction between situated and general knowledge is a hallmark of Realistic Mathematics Education (RME). Freudenthal (1983), founder of RME, developed a pedagogical methodology based on the principle that children should create their own models of problematic realistic situations. Gravemeijer (1999) elaborates on the function of modeling activities in RME, emphasizing the imperative of letting students’ models emerge: “The premise here is that students who work with these models will be encouraged to (re)invent the more formal mathematics” (p. 159, original italics). This progress from explorative actions to consistent rules that generalize these actions is theorized more explicitly in RME via the formulation of two related constructs, “model of” and “model for.” A “model of” results from modeling a particular situation. A general “model for” eventually emerges from noticing homology across mathematically analogous “models of.” Our SILOs (situated, intermediary learning objectives) can be viewed as a checklist detailing structural properties and relations inherent to a “model of.” In turn, we deliberately articulate the SILOs in linguistic forms that would also capture general conceptual structures, just as in a “model for.” This ontological relation between actions, objects, and concepts has long fascinated theorists of human activity. For example, distributed cognition is a theory of human practice that elucidates relationships among participants to a collective human practice and the artifacts that mediate this collaboration (Clark, 2003). Broadly, the array of tools supporting our cognitive activities—pen and paper, calculator, computer, and so on—are cognitive artifacts, that is, artificial tools or devices that carry, elaborate, and report information during problem solving (Norman, 1991). As such, mathematical learning can be theorized as developing psychological structures for regulating the mental activity of distributing quantitative problems over available cognitive artifacts. This effect is dialectical: even as we learn to act and think in new ways as facilitated by these tools, they in turn bear the potential of reifying for our reflection what and how we act and think (Hutchins, 2010). It follows that different material instantiations of one and the same mathematical concept may bear different pedagogical affordances, because their uptake forges different cognitive routes, different neural residue. SILOs articulate this residue pragmatically in terms of the models’ structural properties that students learn to monitor. And yet, this emergence of cognitive structures from mediated actions with external media is not at all guaranteed. A novice might learn to problem-solve using a cognitive artifact that embodies a mathematical function yet without ever understanding this function or how the artifact embodies it. Is this cause for concern? We turn to discuss the psychological construct of transparency, which captures relations between, on the one hand, artifacts inherent to a cultural practice and, on the other hand, a social agent’s understanding of how features of these artifacts mediate the accomplishment of their objectives. Thus when we say that an artifact is transparent, we refer to the subjective relation between a particular agent and the artifact (Meira, 2002). For educational designers, the notion of transparency suggests a particular framing. Namely, the role of designers can be conceptualized as creating learning tools that learners can render subjectively transparent. In a word, the transparency perspective confers upon educators the role of enabling students to see and learn how mathematical artifacts do what they do. For example, in a study of physically distributed problem solving, Martin and Schwartz (2005) found that participants generated more salient and transferable conceptualizations of fractions when using “obdurate” square tiles as opposed to classical pie-shaped manipulatives. Why? From the theoretical lens of transparency, the pie pieces obscured the notion of “whole” precisely because the study participants did not need to assume agency in distributing onto those media their tacit sense of whole—the circle implicitly did that work for them. On the other hand, those students who worked with square tiles were obliged to construct the whole themselves, and that more challenging, agentive experience apparently endured. Whereas mathematical models per se are often static, such as those fraction squares, they are created through active engagement. Indeed, scholars of embodiment pay close attention to perceptuomotor routines as these relate to conceptual knowledge. In particular, when students operate physically within concretized conceptual domains, design-based researchers attend to how the students carry out spatial–dynamical analogs of formal operations (Antle, 2013). An application of embodiment theory to mathematics education is embodied design (Abrahamson, 2009), “a pedagogical framework that seeks to promote grounded learning by creating situations in which students can be guided to negotiate tacit and cultural perspectives on phenomena under inquiry” (Abrahamson, 2013, p. 224). When students participate in embodied-design activities, they solve problems that initially do not bear symbolical notation, do not require calculation, and do not call for quantitative solutions; they call only for qualitative judgments, informal inference, or naïve physical actions. Embodied designs clearly demarcate the two-step design framework that is thematic to this essay and, as such, underscore the informal nature of Step-1 situated, intermediary learning objectives (SILOs). That is, if we theorize perceptual judgment and motor action as bearing seeds of mathematical concepts, then we need ICLS 2014 Proceedings 24 © ISLS language for bridging actions and concepts. SILOs articulate subtle elements of learners’ informal inferential reasoning about perceptual judgments or motor-action solution strategies that they are to discover and refine. With the introduction of embodied design, our literature survey shifts from evaluating implications of learning theory for pedagogical design to educational research work dealing directly with the development of design frameworks for grounded mathematical learning. A profound contribution to the design of mathematics learning environments comes from Richard Noss and collaborators, whose learning theory and design frameworks co-emerged dialectically through empirical research studies (Noss, Healy, & Hoyles, 1997). Of particular relevance to our thesis is their set of design heuristics promoting students’ situated abstractions, “in which abstraction is conceived, not so much as pulling away from context [i.e. the particular features of a situated learning activity], but as a process of constructing mathematical meanings by drawing context into abstraction, populating abstraction with objects and relationships of the setting” (Pratt & Noss, 2010, p. 94, citing Noss & Hoyles, 1996). Pratt and Noss (2010) implicate the epistemological root of mathematical concepts in children’s purposeful construction of utility for new ideas that are instantiated into designed artifacts in the form of interaction potentialities. The SILOs framework differs from that of situated abstractions in terms of grain size, ontological and epistemological foci, and pedagogical underpinnings. In particular, SILOs articulate a set of initially unavailable interaction constraints that the learner determines, implicates, and wills as potentially conducive to more effective problem solving with a given cognitive artifact; in response, each of these willed constraints is then materialized into the artifact by the instructor who grants the learner’s will by enabling into functionality a pre-programmed “hidden” constraint. SILOs are thus functional concretizations of the user’s wish-list into working technological features of an interactive device. Yet SILOs are complementary to situated abstractions in the sense that SILOs can be conceptualized as articulating prerequisite structural conditions for enabling and appreciating utility. In summary, although scholars may differ acutely in their epistemological positions on the constitution of mathematical knowledge, they generally agree that models—forms or structures that learners use in organized activities to promote problem-solving processes—can serve instrumental roles in conceptual development. Having both situated and singled out our proposed heuristic construct of SILO in a legacy of educational theory, philosophy of knowledge, and design frameworks, we now turn to demonstrate this construct’s application in an actual case of design practice, namely Giant Steps for Algebra. The next section will explain the design problem that gave rise to this design, and then we explain the design itself. Setting the Context: Designing Giant Steps for Algebra (GS4A) The story of learning algebra in schools is often told as the challenge of progressing from arithmetic to algebra. A main character in this story is the “=” sign or, rather, students’ evolving meanings for this sign (Herscovics & Linchevski, 1996). When students first encounter algebraic propositions, such as “3x + 14 = 5x + 6”, their implicit framing of these symbols is operational, because the framing will have been fashioned by a history of solving arithmetic problems such as “3 + 14 = __”, where you operate on the left-hand expression and then fill in your solution on the right (Carpenter, Franke, & Levi, 2003). Yet algebraic conceptualization of the “=” sign should be relational, as this sense contributes to correct treatment of algebraic equations (Knuth, Stephens, McNeil, & Alibali, 2006). Given that the arithmetic visualization of “=” apparently impedes students’ transition to algebra, how might this visualization be countervailed? One way is to plant an alternative metaphor. The balance metaphor is undoubtedly the most common visualization of algebraic propositions. This metaphor is typically introduced to students by invoking interactions with relevant cultural artifacts such as the twin-pan balance scale (see Figure 1a). The equivalence-as-balance conceptual metaphor enables a relational, rather than operational, view of algebraic equations. In particular, it grounds the rationale of algebraic algorithms, such as “Remove 3x from both sides of the equation,” in interactions with a familiar artifact.