Prologue: A Cautionary Tale about Disembodied Design
A while ago I consulted for a large-scale, federally funded effort to develop educational media for young children to learn mathematics. The project was based out of Hollywood, where the studio was abuzz with highly creative animators, scriptwriters, songwriters, and joke experts. The studio was still agog from a recent international award for their flagship product, and they could not wait to brainstorm the âmerchâ that would surely emanate from the new project. It was all very flattering and alarmingly lucrative. As I was taxied and flown down and up California, wined and dined on cocktails and sushi, I began to affect a certain hero persona. Assistant professor in the Bay, big-time maven in LA. Abrahamson, Dor Abrahamson. And stir that latte, donât shake it. My job title was âEducation Expert.â Yet gradually, I began to suspect this title was little more than a euphemism for âImprimatur.â
On one memorable occasion, we were huddled in a glass room, with collaborating team players video-conferencing in from Juno, Tampa, and Providence, to consider how we might feature simple arithmetic expressions, such as 2 + 3 = 5, in video snippets and interactive tablet games. I witnessed a screening of a mock-up cartoon scene fresh from down the hall. A grinning Cyclops held up a bubble with a â2â in one hand and a bubble with a â3â in the other hand. Gurgling, he moved his two hands toward each other until the bubbles mashed and mergedâand of course the SFX were there, stars and ribbons galoreâonly to reveal a consolidated bubble with a â5â in it.
âBut,â I ventured brazenly, âhow might this help the children actually understand the meaning of addition?â Silence fell. Lowering his brow quizzically, clinically, the creative director proceeded to explain: âThe kids will see that adding is bringing two things together to get something else.â âWell, yes,â I persisted, âbut what are those things that are coming together so that we see how 3 and 2 make 5?â Now I was losing him. âSay,â I continued, âthat instead ofâor along withâthose symbols, the monster held up two fingers in one hand and three in the other and then brought all the fingers together? In fact the monster could offload the two fingers from the one hand onto the other hand, so that all five fingers are now unfurled and splayed in one manipulation. The monster could then use the free hand to count up the total of five fingers on the other hand. That is something the kids could make sense of, imitate with their own hands, and even show their parents and friendsâit lets them build on their counting skills to learn the meaning of adding rather than just trying to memorize all those sums.â âNo go,â was the peremptory reply. âAnd why not?â I rebutted. âItâs an old animation rule,â they explained. âCartoon characters can only have four fingers on each hand. Itâs just that they look really bad with five.â
I was aghast. We had apparently reached an irrevocable disagreement about a fundamental issue that was very important to me. In a matter of days, though, this design impasse was resolved brilliantly: I was fired.
*****
Fallen from grace, I bereaved my consultant identity, substituted taco for sushi, and then reinvented myself as a reflective practitioner seeking lessons learned. Never mind the octodigital ogreâmay he long simper in cartoon heaven, I muttered. Something deeper was at play that I could not quite put my fingers on. All ten of them.
My brief misadventure was a wake-up call. I had witnessed firsthand an apparent disconnect between research and industry and had begun to fathom the enormity of this disconnect: the lost opportunities, the questionable ethics of funding, the tragic inequity of kids ultimately missing out on big chances because us adults cannot talk to each other. How could my dearest pedagogical convictions be of so little interest to these engineers of commercial products whom we the people, via our governmental agencies and reviewing colleagues, had endowed with the means and mandate to educate our children?
Because guilt is the mother of all creativity, I opted for âItâs not you, itâs meâ therapy. The blame is on us, educational designers and design theorists, who are not getting our message through. Or perhaps our message is incoherent. Perhaps, even worse, we have no effable message at all, but only tacit skill that we bring to bear anew on each project. Perhaps we delude ourselves into believing that our creative process is deductive, explicable, and transferable. Perhaps design is a know-how that cannot be articulated in such forms and granularity as are necessary for delegation and replication. Perhaps designers have terminal expert blind spots (Nathan & Petrosino, 2003). So, is design unprincipled, more art than craft? Perhaps we should attemptâheroically if not quixoticallyâto demystify and essentialize what it is we do when we design for learning (Schön, 1983).
Where do we begin? We have âbigâ theories of learning that we often implicate as our cynosures, and yet we need mid-level guidelines for implementing these theories in the form of materials, activities, and other instructional resources that ultimately enable teachers to support students in developing target concepts (Ruthven, Laborde, Leach, & Tiberghien, 2009). The field has been taking measures to pool our resources in ameliorating this dearth of heuristic design frameworks (e.g., Abrahamson, 2009a, 2013; Barab et al., 2007; Ginsburg, Jamalian, & Creighan, 2013; Kali, Levin-Peled, Ronen-Fuhrmann, & Hans, 2009). However, a momentous implementation gap still remains between target concepts and learner actionsâbetween the content we want kids to learn and what we should have them do with the media so as to learn that content. This gap is exacerbated by recent technological development in industry coupled with inchoate messaging from academe with respect to reform-oriented constructivist pedagogy (Glenberg, 2006; Marley & Carbonneau, 2014; Sarama & Clements, 2009).
Good tools evolve at a glacial pace, but we live in impetuous times. In evolutionary scale, computers have sped up from 0 to 100 mph in a nanosecond. This accelerated evolution, however, has caused the historical precedent of pedagogy falling behind production: effective educational design principles are not evolving fast enough to catch up with each new sea change in software engineering, social media, and human-computer interaction (HCI). As a result, to paraphrase Seymour Papertâs (2004) assessment of educational technology, âwe are sailing on a ship that has no rudder on a journey to nowhere.â So what are the captains to do? What about us lowly deckhands?
Harboring optimism, still one might welcome these times as bearing great prospects; perhaps pedagogical designers must first witness the dire consequences of under-theorizing and under-explicating their practical acumen before they take action. Perhaps such communication breakdowns not only alert us to the need for design principles but also spur us to reflect on, diagnose, and respond to this need. This chapter is an attempt to do just so, with my consulting ignominy serving as both motivation for change and case study for analysis. I set off by thinking deeper about the grinning monster incident, because I view that particular encounter with the diligent animators as paradigmatic of the theory-to-practice hiatus that ultimately results in suboptimal learning materials that underserve our end-clients, the students. I will then draw on theories of learning to offer several design principles. These principles amount to what I call âembodied design,â a framework for creating activities that enable students to build meaning for the mathematical ideas they are learning (Abrahamson, 2009a, 2013; Abrahamson & Lindgren, in press). Carrying these principles forward, I then turn to present two exemplars of embodied design that are based, respectively, on learnersâ naĂŻve perceptual and motoric capacities.
Just before I begin, let me make a brief statement about technology, because this chapter appears in a book on learning technologies and the body. This polysemous semantic unit, âtechnology,â bears ancient etymology, and yet in modern times it primarily evokes fossil-fueled steam engines, hydraulic machinery, automatized industry, nuclear plants, electrical appliances, electronic devices, and so on. As a researcher of learning, however, I lean toward a broader definition of technology as techniqueâany technical procedure in which we deliberately apply available resourcesâcognitive, corporeal, natural, or artificialâin the pursuit of an objective we envision. From this theoretical perspective, in which fork, iPhone, and algorithm are all exemplars of technology, the colloquial differentiation of technology as âlowâ or âhighâ according to its material composition is unproductive to a discussion of learning, even if it is centrally germane to the practice of implementing and developing design in the form of products. In fact, I will be exemplifying one and the same design framework, embodied design, with both paper-based and computer-based artifacts. In so doing, I wish to focus our discussion of technology and the body on the learnerâs phenomenology through technology rather than on technical specifications of media and devices. I will discuss the conceptual change that may come about when technology extends the bodyâthat is, the embodied mind.1
Toward Grounded MathematicsâIssues and Principles
Let us step back to examine why I found that four-fingered friend so monstrous. This section draws on theories of mathematics education in an attempt to make progress in broaching the breach between educational researchers and commercial designers.
Meaning: Making Sense before Symbol
The mathematical proposition â2 + 3 = 5â is just about as simple as subject matter content gets in grade school curriculum. And yet, I maintain, the ostensible simplicity of single-digit arithmetic operations makes that generic mathematical proposition rhetorically useful. In particular, analyzing this proposition and how the creative director proposed to instantiate itâto wit, monstrouslyâI can mount my jeremiad against what Thompson (2013) calls âthe absence of meaningâ in mathematics education. Thompson submits that thinking about this mercurial thing called meaning is an effective analytic strategy for researchers to unpack and diagnose the ills of mathematics education at large. In particular,
attending to issues of meaning allows us to see problems of mathematics learning as emergent from fundamental cultural orientations as much as from epistemological problems of learning sophisticated ideas.
(Thompson, 2013, p. 57)
So what or where is the meaning of â2 + 3 = 5â? Kant (2007) thought that a mathematical proposition such as this is âsynthetic a priori.â It is synthetic, because it denotes a non-tautological assertion regarding a state of affairs, in this case the numerical equivalence of two mathematical expressions. In other words, the proposition is synthetic because its predicate (â5â) cannot be determined solely through analyzing elements of its subject (â2 + 3â)âsome supplementary knowledge and process must be brought to bear to generate or evaluate this proposition. On the other hand, Kant denied that the validity of this proposition is contingent upon humans reflecting on worldly interaction. Rather, the propositionâs validity is a priori to the agentâs psyche or dealings, based on universal laws of nature that transcend and anticipate experience. To evaluate this proposition, an individual enlists what Kant called a schemaâan intuitive psychological framing that imposes a specific type of structure on the sensory manifold.
Are mathematical schemas indeed part of our innate gear? If not, where do these intuitions come from? And what would that mean ultimately for mathematics pedagogy? We turn to a 20th-century giant, the cognitive-developmental psychologist Jean Piaget.
Piaget (1968) investigated the epigenesis of schemas, those would-be intuitive framings of perceptual input. For his research, Piaget used innovative experimental methodology, which included qualitative analysis of young childrenâs behavior as they participated in naturalistic cultural practices within domestic settings, as well as in task-based clinical interviews in laboratory settings. Based on his findings, Piaget implicated goal-oriented sensorimotor interaction as the experiential origin from whence cognitive schemas emerge. Knowledge is not some would-be mental archive of static pictures depicting the-world-as-we-find-it but, rather, dynamical mental activity drawing on what-we-learned-about-the-world-as-we-interacted-with-it. Piaget wrote, âKnowing does not really imply making a copy of reality but, rather, reacting to it and transforming it (either apparently or effectively) in such a way as to include it functionally in the transformation systems with which these acts are linkedâ (Piaget, 1971, p. 6). Adults, Piaget concluded, perceive and interpret the world in ways that are fundamentally different from children, and this adult capacity reflects radical cognitive reorganization of naĂŻve viewpoints, a reorganization that is achieved gradually, painstakingly, through reflexive generalization of functional regularities latent to myriad worldly interactions. Eventually, Piaget asserted, conceptual development gives rise to formal logical reasoning that is no longer manifest in external sensorimotor behavior.
Philosophical stances regarding Numberâs transcendent, a priori qualia notwithstanding, subjective numerical knowledge is thus hard earned via schematizing worldly interaction into formal operations. As such, mathematics âuses operations and transformations (âgroups,â âoperatorsâ) which are still actions although they are carried out mentallyâ (Piaget, 1971, p. 6). Even if we wish to philosophize mathematical laws as preexisting the cognizing agent, Piaget reasoned, still âwhat is involved [in the development of numerical knowledge] is the actual perceiving of correspondenceâ (p. 311, italics added). Piaget maintained that rudimentary correspondences are those of inclusion (perceiving the unit 1 within the compound 1+1 known as 2) and order (perceiving quantitative increase from 1 to 1+1 to 1+1+1, etc.âi.e., 1, 2, 3, etc.) (p. 310).
An agentâs knowledge that â2 + 3 = 5â emerges, therefore, not from moving cryptic symbols on paper, as in the proverbial âChinese Roomâ (Searle, 1980), but from interacting with objects. These symbol-grounding worldly interactions (Harnad, 1990) involve not the symbol string â2 + 3 = 5â per se but concrete instantiations of each of the symbols â3,â â2,â and â5,â as well as the oper...