CComplexity in Mathematics Education
Brent DavisWerklund School of EducationUniversity of CalgaryCalgary, Alberta, CanadaEmail: [email protected] SenguptaWerklund School of EducationUniversity of CalgaryCalgary, Alberta, CanadaEmail: [email protected] Note: This chapter has been accepted for publicationin:
Lerman, S. (Ed.), Encyclopedia of MathematicsEducation, Springer. a r X i v : . [ phy s i c s . e d - ph ] M a y Introduction
Over the past half-century, “complex systems” perspectives have risen toprominence across many academic domains in the sciences, engineering andthe humanities. Mathematics was among the originating domains of com-plexity research. Education has been a relative latecomer, and so perhapsnot surprisingly, mathematics education researchers have been leading theway in the field.There is no unified definition of complexity, principally because formu-lations emerge from the study of specific phenomena. One thus finds quitefocused definitions in such fields as mathematics and software engineering,more indistinct meanings in chemistry and biology, and quite flexible inter-pretations in the social sciences (cf. Mitchell 2009). Because mathematicseducation reaches across several domains, conceptions of complexity withinthe field vary from the precise to the vague, depending on how and wherethe notion is taken up. Diverse interpretations do collect around a few keyqualities, however. In particular, complex systems adapt and are thus dis-tinguishable from complicated systems. A complicated system is one thatcomprises many interacting components, and whose global character can beadequately described and predicted by specifying the rules of operation of theindividual parts. A complex system comprises many interacting agents, and emergence of global behaviors that cannot be adequately predicted by simplyspecifying the rules of the individual agents is a central characteristic of suchsystems. Some popularly cited examples of complex, emergent phenomenainclude anthills, economies, and brains, which are more than the linear sumof behaviors of individual ants, consumers, and neurons. In brief, whereas theopposite of complicated is simple, opposites of complex include reducible anddecomposable. Hence prominent efforts toward a coherent, unified descrip-tion of complexity revolve around such terms as emergent, noncompressible,multi-level, self-organizing, context-sensitive, and adaptive.This entry is organized around four categories of usage within mathe-matics education – namely, complexity as: an epistemological discourse, anhistorical discourse, a disciplinary discourse, and a pragmatic discourse.2
Complexity as an epistemological discourse
Among educationists interested in complexity, there is frequent resonancewith the notions that a complex system is one that knows (i.e., perceives,acts, engages, develops, etc.) and/or learns (adapts, evolves, maintains self-coherence, etc.). This interpretation reaches across many systems that are ofinterest among educators, including physiological, personal, social, institu-tional, epistemological, cultural, and ecological systems. Unfolding from andenfolding in one another, it is impossible to study one of these phenomenawithout studying all the others.This is a sensibility that has been well represented in the mathemat-ics education research literature for decades in the form of varied theoriesof learning. Among others, radical constructivism, socio-cultural theoriesof learning, embodied, and critical theories share essential characteristics ofcomplexity. That is, they all invoke bodily metaphors, systemic concerns,evolutionary dynamics, emergent possibilities, and self-maintaining proper-ties. Of particular relevance is the recent emphasis on intersectionality as akey element of critical race and gender theories, which explicitly situates ourexperiences of knowing and learning in mathematics classrooms as emergentfrom our simultaneous positions of marginalization and privilege, as well asthe interplay between historical, institutional and social forces and individualdesires (Levya 2017).As illustrated in Figure 1, when learning phenomena of interest to math-ematics educators are understood as nested systems, a range of theoriesbecome necessary to grapple with the many issues the field must address.A pedagogy for knowing and doing mathematics that is epistemologicallycommitted to complexity necessitates insights in the form of multi-level anddiverse models of the complex dynamics of knowing and learning. Moresignificantly, perhaps, by introducing the systemic transformation into dis-cussions of individual knowing and collective knowledge, complexity not onlyenables but compels a consideration of the manners in which knowers andsystems of knowledge are co-implicated (Davis and Simmt 2006).3igure 1. Some of the nested complex systems of interest to mathematicseducators
School mathematics curricula is commonly presented as a-historical and a-cultural. Contra this perception, complexity research offers an instance ofemergent mathematics that has arisen and that is evolving in a readily per-ceptible time frame. As an example of what it describes – a self-organizing,emergent coherence – complexity offers a site to study and interrogate thenature of mathematics, interrupting assumptions of fixed and received knowl-edge.To elaborate, the study of complexity in mathematics reaches back thelate 19th century when Poincar´e conjectured about the three-body problemin mechanics. Working qualitatively, from intuition Poincar´e recognized theproblem of thinking about complex systems with the assumptions and math-ematics of linearity (Bell 1937). The computational power of mathematicswas limited the calculus of the time; however, enabled by digital technolo-gies of the second half of the 20th century, such problems became tractableand the investigation of dynamical systems began to flourish. With comput-ers, experimental mathematics was born and the study of dynamical systemsled to new areas in mathematics. Computational modeling made it possibleto model and simulate the behaviour of a function over time by computing4housands and hundreds of thousands of iterations of the function. Numeri-cal results were readily converted into graphical representations (the Lorenzattractor, Julia sets, bifurcation diagrams) which in turn inspired a newgeneration of mathematicians, scientists and human scientists to think dif-ferently about complex dynamical systems. Further advances in computingin the form of parallel and distributed computing and multi-agent model-ing enabled scientists and mathematicians to simulate emergent phenom-ena by modeling simultaneous interactions between thousands of interactingagents (Mitchell 2009). Through such efforts, since the mid-20th century, asmathematicians, physical and computer scientists were exploring dynamicalsystems (e.g., Smale, Prigogine, Lorenz, Holland, etc.), their work and thework of biologists, engineers and social scientists became progressively moreintertwined and interdisciplinary (Gilbert and Troitzsch 2005; McLeod andNersessian 2016).In brief, the emergence of complexity as a field of study foregrounds thatmathematics might be productively viewed as a humanity. More provoca-tively, the emergence of a mathematics of implicatedness and entanglementalongside the rise of a more sophisticated understanding of humanity’s rela-tionship to the more-than-human world might be taken as an indication ofthe ecological character of mathematics knowledge.
A common criticism of contemporary grade school mathematics curriculum isthat little of its content is reflective of mathematics developed after the 16thor 17th centuries, when publicly funded and mandatory education spreadacross Europe. A deeper criticism is that the mathematics included in mostpre-university curricula is fitted to a particular worldview of cause–effect andlinear relationships. Both these concerns might be addressed by incorporat-ing complexity-based content into programs of study.Linear mathematics held sway at the time of the emergence of the modernschool – that is during the Scientific and Industrial Revolutions – becauseit lent itself to calculations that could be done by hand. Put differently,linear mathematics was first championed and taught for pragmatic reasons,not because it was seen to offer accurate depictions of reality. Descartes,Newton and their contemporaries were well aware of nonlinear phenomena.However, because of the intractability of many nonlinear calculations, when5hey arose they were routinely replaced by linear approximations. As text-books omitted nonlinear accounts, generations of students were exposed toover-simplified, linearized versions of natural phenomena. Ultimately thatexposure contributed to a resilient worldview of a clockwork reality.However, recent advances in computational modeling have made it pos-sible for complex phenomena that are traditionally taught in post-secondarylevels, to be easily accessible to much younger learners. With the ready accessto similar technologies in most school classrooms within a culture of ubiqui-tous computation, there is now a growing call for deep, curricular integrationof computer-based modeling and simulation in K–12 mathematics and scienceclassrooms (Wilkerson and Wilensky 2015; Sengupta et al. 2015). Efforts forsuch integration fundamentally rely on learners iteratively designing, evaluat-ing and re-designing mathematical models as the pedagogical approach, usingagent-based modeling languages and platforms (e.g., Scratch, Agentsheets,NetLogo, ViMAP, etc.). In agent-based modeling, learners can simulate therelevant mathematical behaviors by programming the on-screen behavior ofcomputational agents (e.g., the Logo turtle) using body-syntonic commands(e.g., move forward, turn, etc.). Emergence, in such computational models,is simulated as the aggregate-level outcome that arises from the interactionsbetween many individual-level computational agents. The creator of the firstsuch modeling language (Logo), Papert (1980) argued that agent-based mod-eling can create space in secondary and tertiary education for new themessuch as recursive functions, fractal geometry and modeling of complex phe-nomena with mathematical tools such as difference equations, iterations, etc.Others (e.g., English 2006, Lesh and Doerr 2003) have advocated for similarlythemed content, but in a less calculation-dependent format, arguing that theshift in sensibility from linearity to complexity is more important than thedevelopment of the computational competencies necessary for sophisticatedmodeling. In either case, the imperative is to provide learners with access tothe tools of complexity, along with its affiliated domains of fractal geometry,chaos theory, and dynamic modeling.New curriculum in mathematics is emerging. More profoundly, when,how, who and where we teach are also being impacted by the presence ofcomplexity sensibilities in education because they are a means to nurtureemergent possibility. 6
Complexity as a pragmatic discourse
To recap, complexity has emerged in education as a set of mathematical toolsfor analysing phenomena; as a theoretical frame for interpreting activity ofadaptive and emergent systems; as a new sensibility for orienting oneselfto the world; and for considering the conditions for emergent possibilitiesleading to more productive, “intelligent” classrooms. In the last of theseroles, complexity might be regarded as the pragmatic discourse – and of theapplications of complexity discussed here, this one may have the most po-tential for affecting school mathematics by offering guidance for structuringlearning contexts and re-shaping disciplinary pedagogies. Three key insightshave emerged in the literature that can guide pragmatic action in the K–12classroom. First, complexity offers direct advice for organizing classroomsto support the individual-and-collective generation of insight – by, for exam-ple, nurturing the common experiences and other redundancies of learnerswhile making space for specialist roles, varied interpretations, and other di-versities. For example, participatory simulations, in which each learner canthemselves play the role of an agent in complex system using embodied,physical and computational forms of modeling, have been shown to be effec-tive pedagogical approaches for modeling emergent mathematical behaviorsby highlighting and integrating both individual and collective insight (e.g.,Colella 2000). Second, the emphasis on such participatory forms of mathe-matical modeling, in the context of modeling complex phenomena, can actas a bridge across disciplines (e.g., biology and mathematics education, seeDickes et al. 2016). A third key insight is the notion of reflexivity across dis-ciplines – that is, conceptual development within each scientific, engineeringand mathematical discipline can be deepened further when relevant phenom-ena are represented as complex systems using mathematical modeling in waysthat also highlight key practices of engineering design such as design thinking(Sengupta et al. 2013).As complexity becomes more prominent in educational discourses and en-trenched in the infrastructure of “classrooms”, mathematics education canmove from an individualistic culture to one of cooperation and collaboration,and from mono-disciplinarity towards inter- and trans-disciplinarity. These,in turn, have entailments for the outcomes of schooling as evident in move-ments from disciplinary ideas to crosscutting practices, from independentworkers to team-based workplaces, and from individual knowing to socialaction. 7
Cross References
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