THEORETICAL INCOMPLETENESS: A DRIVING MECHANISM OF EVOLUTION IN MATHEMATICS EDUCATION RESEARCH

Iskra Nunez

iskra.nunez @ googlemail.com

ABSTRACTThe current study considers four centuries of philosophical thought in influencing the expansion of theoretical perspectives brought to bear on mathematics learning and teaching. Drawing upon the AQAL model of integral theory as a heuristic, it inquires into the necessity to use multiple theories in mathematics education research (thereafter MER) through four moments in history, from the late seventeenth-century hermeneutics, to eighteenth-century transcendental idealism, to nineteenth-century pragmatic theory, to the twenty-century postmodernist theory. Each moment in the analysis of theoretical perspectives in use within MER proceeds sequentially by identifying weaknesses (or blind spots) in its philosophical precursor. This research concludes with a discussion of some of the possible contributions of this approach for twenty-first century MER.

Keywords Critical Realism, Evolution, Integral Theory, Mathematics Education Research, Mechanism, Philosophical Methods, Plurality, Theoretical Incompleteness.

INTRODUCTION

What drives the evolution of theoretical perspectives currently employed in MER? What mechanism must be at work for such a theoretical plurality to occur? Or what must be at work in a particular theoretical perspective for the next breakthrough in the history of philosophical thought to occur? We suppose that there must be a mechanism at work driving such a movement of evolution derived in philosophical thought from the late seventeenth-century hermeneutics, to eighteenth-century transcendental idealism, to nineteenth-century pragmatic theory, to the twenty-century postmodernist theory in order to transition historically from one century to the next. This assumption is thus an impetus for the present study. In fact, mathematics education researchers have been concerned for a long time with new attempts to harmonize, connect, synthetize, or integrate MER’s plurality of theoretical perspectives (see e.g. Sriraman & English, 2010). As a way to contribute to this discussion, we begin by introducing integral theory, as a heuristic framework to organize four centuries of philosophical thought constituting MER’s theoretical precursors. Then, in the methodology section, we introduce the idea of Achilles’ heel critique, drawn from the philosophy of critical realism to formalize this study’s argument for a mechanism of evolution capable of explaining MER’s theoretical plurality in temporal terms, and in terms of a real, natural necessity for the existing expansion in philosophical thought. To our knowledge, no research has yet considered integral theory to study the internal workings of the evolution and current expansion of theoretical perspectives brought to bear on mathematics learning and teaching.

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HEURISTIC FRAMEWORK

Integral theory’s characterizing feature is its all-quadrants, all-levels, all-lines, all-states, and all-types framework involving a four-dimensional (or quadrivium) perspective of the world (see Figure 1). We draw upon we idea of quadrivium as a heuristic based on integral theory. This quadrivium represents AQAL, a graphical model by which integral theorists understand reality as constituted of the all-encompassing relationship between its four dimensions: intensions (or experiences), individual (or group) behaviors, cultures, and social systems (or structures). In brief, AQAL provides a framework by which to assess, deal with, and potentially interrelate multi-theoretical insights from these four primary dimensions of knowledge into a binding resolution of problems. The all-quadrant element of the AQAL model of integral theory, for instance, has been used within the mathematics educational community in studies of inclusive pedagogy, as we shall see next.

Figure 1: AQAL adopted from Esbjörn-Hargens’ (2015) comparative introduction of integral theory

AQAL’s all-quadrant element and a pedagogy of living mathematics education

Renert’s (2011) quadrivium perspective applied to MER illustrated how integral theory may serve to integrate different types of knowledge, with what he called pedagogy of living mathematics education (PLME). Mathematics content knowledge as objective knowledge corresponds to the PLME exterior-singular dimension. Using integral theory, this dimension

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denotes the objective perspective of the single individual as outsider (or individual-exterior view); it is associated grammatically with “it” (the third person singular pronoun). Mathematical pedagogical content knowledge as subjective knowledge corresponds to the PLME interior-singular dimension. Using integral theory, this dimension denotes the subjective perspective of the single individual as insider (interior-singular view); it is associated grammatically with “I” (the first person pronoun). Mathematics curriculum knowledge as intersubjective knowledge corresponds to the PLME interior-plural dimension. Using integral theory, this dimension accounts for the multiple-perspectives between different cultures from an insider stance (or collective-interior view); it is associated grammatically with “you/we” (plural pronouns). Renert added an additional dimension of interobjective knowledge that deals with the broader social system as resources for teaching as part of the PLME exterior-plural dimension. Using integral theory, this dimension accounts for the interconnected perspectives between structures, networks, and systems at a global scale and from an outsider stance (or collective-exterior view); it is associated grammatically with “its” (the possessive form of it). Then Renert and Brent (2012) used these integral-theoretic ideas to argue for inclusion with their idea of a sustainable mathematics education, a response to current deteriorating environmental conditions.

The all-quadrant element may be advantageous for MER as an organizing heuristic, and as a simple graphical model of knowledge-inclusion. It may allow researchers of mathematics education to see, for example, that in the lower-left quadrant, “ours” (the possessive form of we) is absent. Using AQAL, it is not hard to see that what is missing from the model itself is what we could refer to as an ours-perspective to denote that the possession of the collective-exterior dimension—as it includes nature and intellectual property—also belongs to the commons, as our shared human knowledge, which deserve a space free from the processes of colonization and commercialization.1

The intentional dimension of AQAL’s all-levels element and hermeneutics

In order to illustrate the remaining elements (all-levels, all-lines, all-states, and all-types) of the AQAL model of integral theory, we consider examples from the four primary fields of knowledge in MER (see Figure 2). The all-levels element of AQAL refers to the degree of complexity within each quadrant. The research of Van Zoest and Bohl (2005), for example, may be allocated to AQAL’s dimension of intensions (or experiences). This is because Van Zoest and Bohl inquired into the experiences of being a mathematics teacher along a spectrum of increasing levels of complexity about self-beliefs, ranging from teacher as authoritarian self, facilitator self, socially empowered self, and holistic self. This type of narrative research has philosophical roots in hermeneutics as it emphasizes interpretation of individual experiences in the world. It thus highlights subjective knowledge from the individual-interior view of AQAL, to use integral-theoretic terms.

1 I am indebted to Paul Ernest for pointing out that another possible weak point of integral theory may be seen in (Figure 1) AQAL’s implicit hierarchy of features denoted by the numbered concepts in the diagonal lines representing the elements of all-levels, all-lines, all-states, and all-types. This reference to hierarchy suggests a linear trajectory of increasing complexity between its elements. For an alternate consideration of the idea of complexity in mathematics education, see Fayez M. Mina’s contribution in this Special Issue of Philosophy of Mathematics Education Journal.

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Figure 2: Examples of MER organized from an AQAL integral-theoretic view of the world

The cultural dimension of AQAL’s all-lines element and constructivism

The all-lines element refers to the different capacities that sequentially evolve through the degree of complexity of the levels. The research of Frankenstein (1983), Ernest (2001), and Skovsmose (2014) on the philosophical position of critical mathematics education, for example, may be allocated to the all-lines element of AQAL’s dimension of culture. This is because the foundational element in, for example, Ernest’s research is to problematize societal aims of, and purposes for, teaching mathematics, along the particular lines of historically evolving groups—such as conservatives, industrialist, liberal professionals, and reformers—whose particular interests include justifications for teaching a variety of mathematical capacities needed for an empowered workforce—such as the capacity for numeracy, practical problem-solving skills, and the understanding and creation of new mathematics— including thee multiple views— absolutist or fallibilist—of the subject matter. Elsewhere I have illustrated that, in the most general sense, constructivist-based research in MER, with its departure from a mere interpretative view of the world as it inquires into the possibility of understanding reality by means of model-building and authorial critique, has philosophical roots in transcendental idealism (see Nunez, 2015 for an extensive discussion of the philosophical precursors of critical mathematics education together with constructivism, social or otherwise). In integral-theoretic terms, Ernest’s research on critical mathematics education as well as, on social constructivism for MER (see Ernest, 1998), may be said to exemplify AQAL’s collective-interior view of the world.

The quantifiable dimension of AQAL’s all-types element and pragmatism

The all-types element refers to a variety of largely quantifiable, stable stylistic features and recurrent patterns that are associated with each quadrant. The pragmatic research of Yan and Kember (2004) may serve to describe AQAL’s dimension of behavior using its all-types

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element. Their best-practices study reveals two types of measures by which to understand learners’ behavioral styles: avoider and engager, during mathematics learning in small-group interaction. Further O’Boyle’s (2005) investigation may also serve to illustrate AQAL’s dimension of behavior because it considers the patterns associated with electrophysiological measures to reveal learners’ brain activity during problem-solving. Investigations concerned with behavioral-based and measure-finding explanations of mathematics learning and teaching have philosophical roots in pragmatism. This is because pragmatic-founded research tends to be associated with best-practices (or what works in practice) by means of a systematic identification of recurrent patterns of behaviors, language, interventions, and other regularities of events done in experimental or quasi-experimental conditions. Measures-oriented research emphasizes objective knowledge, and thus exemplifies AQAL’s individual-exterior view of the world.

The social-systems dimension of AQAL’s all-states element and postmodernism

The all-states element of AQAL denotes a temporary or long-lived occurrence of any aspect of reality. The research of Adler (1998), Morgan (2006), Moschkovich (2010), Sfard (2008), and Civil and Planas (2012), to name a relevant few, serves to illustrate the all-states element of AQAL’s social dimension of systems (or structures). Their work with multi-lingual, multi-cultural learners and their teachers seeks to deepen our understanding of the semiotic features in analyzing mathematical writing, the linguistic means of mathematical discourse, as well as the social-systemic resources needed to help ensure that students have an equitable access to mathematics knowledge and, in some cases, access to English-language knowledge as needed. With its consideration of mathematics as a structure and a system of social interests, their research may be said to follow in the postmodernist tradition. Using integral theory terminology, their work may be said to employ a collective-exterior view of the world.

METHODOLOGY

An inclusive understanding for explaining the necessity to use multiple theories in MER might be possible through an understanding of their respective Achilles’ heels (AH) in terms of the moments of evolution that describe MER’s development as a research field. To this aim, the notion of an AH critique, drawn from the philosophy of critical realism, serves as a productive method; this is because AH “seeks to show that it is precisely where a position seems strongest that it is actually most weak” (Bhaskar, 2008, p. 340). Elsewhere, we have argued that this idea of an AH critique is advantageous for MER (Nunez, 2015).

Although this type of methodology “might be the sole article on critical realism (also known as post-positivist realism) in mathematics education” (Atweh et al., 2016, p. 4) that includes an AQAL-informed visualization of an AH critique, we nonetheless continue to deepen the original line of research that brought together critical realism and mathematics education since its inception in 2011 (see Nunez, 2012a), and thereafter (see Nunez 2013b, 2015a), including related research on learning theories such as activity theory (Nunez 2013a, 2014, 2015b) and literary theory (Nunez, 2012b).

Building and operationalizing the four quadrants of AQAL

We begin by building the four quadrants of AQAL, along two foundational axes. The first axis separates the interior and exterior perspectives—i.e. the view from the inside and the view from the outside. The second axis separates the collective and the individual perspectives—i.e. the plural view of a group of people from the singular view of an

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individual. Then setting the first axis (interior/exterior) orthogonal to the second one (individual/collective) will produce a conceptual coordinate system of perspectives denoted as the upper-left quadrant (UL), lower-left quadrant (LL), upper-right quadrant (UR), and lower-right quadrant (LR)—to reiterate, these four are also called dimensions or quadrivium.

Now in operationalizing the AQAL quadrants, we illustrate the evolution or movement of four centuries of philosophical thought in influencing MER’s development from hermeneutics, social constructivism, pragmatism, and postmodernism, drawn from Lerman’s (2006) analysis of the plurality of theories employed in MER. Informed by Lerman’s analysis, we can find that these four theoretical perspectives constitute some of the most predominant ideas in MER as a research field. These four theoretical perspectives were firmly in place by the end of last century. In what follows we identify some of their precursors in Western philosophy.

Seventeenth-century, pre-secular hermeneutics. Hermeneutics is understood as the “study of the theory and the practice of interpreting texts. It dates back to ancient times … At issue in [pre-secular hermeneutics] was determining the correct way of reading scriptural texts … It was not until the 18th century that a secular form of hermeneutics was developed by Friedrich Schleiermacher, [who] was the first to raise the question of authorial intention … arguing that it has to be understood in context (Buchanan, 2010, p. 227). In MER, this theoretical perspective highlights subjective knowledge in terms of our situatedness in the experiential world as a process of socialization (Brown, 1991). However, it usually cannot account for subjective knowledge of the world outside experiences. Therefore, we associate the UL dimension of intensions with seventeenth-century, pre-secular hermeneutics because scientific endeavors at this time were mainly concerned with interpretations as opposed to critical self-questioning or model-building.

Eighteenth-century transcendental idealism. Kant’s transcendental idealism argued that “although the form of experience is ideal, or relative to us, this is not to deny the reality of something independent of this form” (Ameriks, 1999, p. 461). Based on his philosophy of science, scientific endeavors at this time were mainly concerned with interrogating the conditions of possibility for existence (or a priori categorization). This idea combined both intersubjective validity and also objective validity but only relative to our sense-experiences. In MER, social constructivism, with its origins in transcendental idealism, stresses intersubjective knowledge, and even its objective basis, with the metaphor of building mathematical knowledge in a social-cultural context (Ernest, 1998). Social constructivism, however, tends to lead to relativism since there are no worldly grounds independent of this perspective for testing or choosing from the plurality of rival theories. Thus, we associate the LL dimension of culture with eighteenth-century way of thinking in transcendental idealism.

Nineteenth-century pragmatism. Peirce and other proponents of pragmatism, argued for the “practical consequences” of theoretical perspectives in engendering truths on reality precisely because they saw the world as “really malleable”. In MER, pragmatism emphasizes objective knowledge in terms of the success in practice (Simon, 2009). It rightly sees the possibility of knowledge about the world independent of the level of objective knowledge as useful practicality. However, its a strategy of containment is to privilege mathematical utility over theoretical or philosophical debates. Therefore, we associate the UR dimension of behaviors with nineteenth-century pragmatism.

Twentieth-century postmodernism. Wittgenstein, Derrida, and Lyotard, included some of the proponents of postmodernism. They emphasized a variety of ways of thinking that sought to go beyond some of the deterministic views usually associated with modernist

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perspectives. In MER, this view, for instance tends to emphasize language in terms of mathematical thinking and mathematical practices as a network of power relations (e.g. Valero, 2004), and it includes a rejection of meta-narratives (e.g. Ernest, 2004). However, it tends to misconstrue the nature of the world as exhausted by concepts. Therefore, we associate the LR dimension of systems with twentieth-century postmodernism.

ANALYSISAnalytically, we proceed sequentially to the first moment of the evolution of philosophical ideas in history by identifying the Achilles’ heel of hermeneutics, then constructivism (social or otherwise) with philosophical roots in transcendental idealism, then pragmatism, and last postmodernism (see Figure 3). This Achilles’ heel is not any arbitrarily weak point but it is the point at which its proponents deem it strongest, i.e. in the UL (individual/interior) view of subjective knowledge. Another important feature of this argument is that includes a temporal element; that is, each moment in evolution proceeds sequentially in time from one century to the next by identifying weaknesses (or blind spots) in its philosophical precursor beginning with seventeenth-century hermeneutics, to eighteenth-century transcendental idealism, to nineteenth-century pragmatic theory, to the twenty-century postmodernist theory.

Figure 3: Explanation of MER’s theoretical expansion via theoretical incompleteness through analysis of AH in four centuries of philosophical thought

First movement of evolution. This AH of hermeneutics = UL is the point of vulnerability on which an AH critique of hermeneutics fastens precisely because it is the realm associated with experience and it cannot sustain knowledge outside itself, i.e. it fails to

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account for a dimension of knowledge outside narratives, interpretative knowledge, and experiences.

Second movement of evolution. In the second moment of evolution, we proceed by identifying the AH of constructivism, social or otherwise. The AH of constructivism = LL (collective/interior) perspective of intersubjective knowledge. The vulnerability of LL is precisely the realm that accounts for multiple perspectives between cultures and communities (collectives), i.e. at what appears to be its strongest point. A problem with social constructionism is that it fails to posit the independence, or at least, the prior existence and causal efficacy, of objects of scientific research—i.e. in a dimension of our knowledge about the world that is properly ontological relatively independent of our theories and discourses about the world in the epistemological dimension. Since there are no worldly grounds for testing or choosing from the plurality of rival theories, social constructivism tends to lead to relativism. It is important to reiterate that “inclusion” of UL and LL does not occur here via a simple concatenation (or preservation without replacement) of two different perspectives. Rather, an inclusive understanding is possible because we are able to provide more explanatory power with the idea of a mechanism that drives evolution, i.e. theoretical incompleteness, and the systematic identification of AH of hermeneutics and AH of constructivism. This demonstrates how the AQAL model of integral theory can effectively be used to bring multiple theories to bear on the visualization of an inclusive understanding of these two moments of philosophical evolution diachronically.

Third movement of evolution . The third moment of evolution proceeds with the identification of the AH of pragmatism. This AH of pragmatism = UR (individual/exterior) perspective of objective knowledge precisely because its proponents regard this as its strongest point. This AH of pragmatism could be further explained as a reductionist tendency in which the dimension of objective knowledge tends to be misconstrued in terms of the identification of patterns of regularities of events, behaviors, language, or functions of the brain (identified in closed or quasi-closed systems under experimental conditions). Another problem of pragmatism is its tendency to privilege utility over philosophy.

Forth movement of evolution. The forth moment of evolution proceeds with the identification of the AH of postmodernism. AH postmodernism = LR (collective/exterior) perspective of interobjective knowledge. This Achilles’ heel of postmodernism is precisely the LR dimension, which as its purportedly strong point as it encourages a world-view of networks, structures (or systems), but often misconstrues them in terms of the collectivist account of individual activity in groups or the individual activity of groups (i.e. the error of methodological collectivism (the analysis of behaviors (actions or language) of individuals in groups or the behaviors (actions or language) of the masses; that is, the analysis of groups of individuals).

OPEN DISCUSSIONThe notion of AH critique organized by AQAL as heuristic has allowed us to put forward the idea that a mechanism that drives the utilization of multiple theories is argued to be precisely the incompleteness of any particular theoretical perspective. If this mechanism exists, then it is possible to imagine MER’s development in terms of the need for an alternative view of the world, hence leading to MER’s theoretical expansion. The reader may ask: What about if the mechanism of theoretical incompleteness driving the evolution in philosophical thought that we have proposed here turns out to be either erroneous or fictitious? As a way of response,

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we consider the opposite; that is, we consider the non-realist position which posits the not-existential capacity of concepts—such as global warming, capitalism, peer-pressure, socio-economic classes, and so on—to be causally efficacious on social reality. If theoretical incompleteness is imaginary on our part, then this argument has not contributed to understanding MER’s evolution as an expanding knowledge field; otherwise, we have managed to frame the need for further research (see forthcoming Nunez, 2017), and thus we invite readers to engage with an alternative, critical realist view of the world, or as Hartwig (2015a) puts it in his invitation:

Of course, if capitalism does not exist as a real social kind but is just a convenient fiction on the part of investigators, … or if social structures are merely collectivities (groups) of people, as in integral theory, there would seem to be less to be concerned about. For an argument that social structures, understood as internally related social practices such as those between capitalists and wage-earners, are real and causally efficacious on whatever scale and geo-historical range empirically based research reveals.

To illustrate the above quotation and its correspondence to MER, we consider a non-real form of argumentation: if social relations are understood without causal efficacy then the learning and teaching of this subject matter is conceived in isolation without considering and the particular geo-historical events that shape it; that is, mathematics education conceived in value-free terms, and one that is divorced from the understanding and thus the organization of equitable societies. If mathematics is detached from the world, then the gap that separates mathematics from reality serves to open a space for mystification of its creation and practice. This non-real view has been challenged since the 1980s under the well-established program of ethonomathematics (see d’Ambrosio, 1985). We invite investigators of mathematics education to continue to challenge traditional views by considering the critical realist argument in which mathematics education research is part of a structured and differentiated world, a reality that may be seen productively as constituted of internally related, and causally efficacious social relations.

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