Philosophical Underlabouring for Mathematics Education

Abstract

The field of mathematics education has been fashioned by a diversity of theoretical and philosophical perspectives. The purpose of this study is to add to this field an analysis of the philosophical position of critical realism. To achieve this objective, the study addresses the following questions: what does critical realism have to offer mathematics education? How may critical realism underlabour for this discipline? In addressing these questions, the study provides an overview of the basic theories and the possible weak points of and arguments against critical realism and realism in general. It then draws upon the notion of a dialectical phenomenology to provide a sequence of Achilles' heel critiques of some of the perspectives that constitute broadly the theoretical landscape of mathematics education research. This critique proceeds with an analysis of didactically oriented empiricism, hermeneutics, pragmatism, postmodernism, traditionally recognized forms of constructivism, traditionally recognized theories of activity, and ethnomathematics. The last section summarizes the implications of philosophical underlabouring for a more beneficial science of mathematics education.

Keywords:

• Achilles' heel critique
• activity theory
• critical realism
• constructivism
• empiricism
• ethnomathematics
• mathematics education research

Introduction

Throughout the end of the last century and the initial years of the current millennium, mathematics education researchers have expanded their use of theoretical perspectives in an attempt to provide new ways of understanding teaching and learning. Eva Jablonka and colleagues identified that this expansion has been fashioned by the tripartite influence of ‘theories for studying social, political and cultural dimensions of mathematics education’.¹ Two major theoretical changes have been substantiated, starting with the work of Steven Lerman and his ‘social turn in mathematics education research’² in the United Kingdom in the early 2000s, followed by the work of Rochelle Gutiérrez with its ‘sociopolitical’³ counterpart in the United States more than a decade later. These changes have inspired mathematics education researchers, as observed by Bharath Sriraman and Elena Nardi, to establish new ‘developments and ways forward’.⁴ These new developments have led to the consideration, in one way or another, of what Angelika Bikner-Ahsbahs and Susanne Prediger called ‘the diversity of theories in mathematics education’.⁵ Frank Lester has argued that, with this diversity, new doors have been opened to question the ‘theoretical, conceptual, and philosophical foundations for research in mathematics education’.⁶ None of these new developments, however, have considered a critical realist perspective.

This study thus endeavours to make a contribution to mathematics education and critical realism by addressing the following questions: What does critical realism have to offer this discipline? How may critical realism underlabour for a more beneficial science of mathematics education? To address the first question, the article begins with a literature review of the conception of theory and its role in mathematics education. It then provides an overview of the grounding arguments and the possible weaknesses of critical realism. To address the second question, the article employs the notion of a dialectical phenomenology; this analysis unfolds sequentially through Achilles' heel critiques of the multi-theoretical perspectives that generally constitute the field of mathematics education.⁷ It identifies an array of multi-theoretical scotomata or blind spots that was firmly in place by the turn of the century, from didactically oriented empiricism and its legacy in mechanic-structuralism to hermeneutics and pragmatism, including their legacy in cultural psychology, whose successors follow the traditionally recognized theories of constructivism and the traditionally recognized theories of activity, including postmodernism and ethnomathematics. The study argues that critical realism may benefit mathematics education because, in contrast to these other perspectives, it provides a realist theory of ontology. The final part of the study summarizes the identified dialectical phenomenology with implications for mathematics education.

What is wrong with the concept of theory and its role in mathematics education?

The Dutch mathematician and pedagogue Hans Freudenthal remarked that ‘[w]hat is wrong is that the word “theory” — used by so many people on so many occasions with many different meanings — needs circumstantial explanations that do not contribute much factually to our subject [of mathematics education]’.⁸ In outlining ‘the historiography of mathematics education’,⁹ Gert Schubring identified that such a theory-related concern was firmly in place by the turn of the last century in the United Kingdom, the United States, and in other industrialized societies. Maria Nikolakaki went a step further to explain how this concern emerged with ‘the connection of this subject [of mathematics education] with the ideology of progress and the recognition of the rational citizen as the main political factor’.¹⁰ Since there is no unifying conception of theory in mathematics education, as we shall see, two themes serve to organize the reviewed literature on theory-related investigations: (a) studies of theory development and (b) studies of the analysis of relations between different theories. On the one hand, theory-development research usually questions the nature of theory itself; that is, the ontology and role of theory in formulating explanations. Relation-analysis research, on the other hand, usually questions the links between different paradigms.¹¹ This review emphasizes that no research that applies critical realism to theory-development or relation-analysis in mathematics education has been found to date in the literature.

In surveying over 100 studies on theories and philosophies, Bharath Sriraman and Lyn English found that theory-development research ranges from perceiving a theory as a mediator to seeing the idea of theory as an object of research with ‘its many roles in improving the teaching and learning of mathematics in varied contexts’.¹² Edward Silver and Patricio Herbst initially explained a formulation of theory as a mediator between reality and empirical data because it serves as a ‘tool that helps give closure to the corpus of data to study a problem, complementing the sources identified by the problem’.¹³ Practically, this mediating role functions as a bricolage that purportedly offers what Paul Cobb sees as ‘a better prospect of mathematics education research developing an intellectual identity distinct from the various perspectives on which it draws than does the attempt to formulate all-encompassing schemes’.¹⁴ Moreover, a formulation of theory as an object of research tends to function, as conceptualized by Alan Schoenfeld, in terms of a set of eight ‘standards for judging theories, models and results … (and [to evaluate] more generally any empirical or theoretical work) in mathematics education’.¹⁵ An immediate question (1) arises: is theory-development research in mathematics education that does not conform to Schoenfeld's set of eight standards then characterized as non-falsifiable, non-replicable, and hence non-scientific?

In relation-analysis research, for example, Luis Radford investigated a tripartite connection between (α) the implicit or explicit presuppositions of a worldview, which in turn informs (β) the methodology in order to do science, and (γ) the types of research questions that can or cannot be answered in scientific practice, and how each member of the triad or trio — that is, (α, β, γ) — provides its own ‘challenges and possibilities’.¹⁶ A subsequent question (2) arises: how do mathematics education researchers choose among the diversity of trios? Roza Leikin and Rina Zazkis focused their relation-analysis research on ‘the connections between general education theories and theories in mathematics education’.¹⁷ The approach proposed by Bikner-Ahsbahs and Prediger went a step further to test the degree of theoretical connectivity and modularity, albeit ironically with the creation of potentially infinite ‘networking strategies’.¹⁸ Other research focused on the investigation of a peaceful coexistence among competing paradigms where Anna Sfard suggested that ‘if one cannot solve the dilemma [of competing paradigms], one can and probably should learn to live with it’.¹⁹ If one does not conform to the seemingly unsolvable dilemma of rival paradigms, then the possibility of choosing among the diversity of multiple theories remains open. A binding question (3) arises: what is a theory? In the most general sense, and in accordance with the literature reviewed above, a theory may be regarded as an epistemic model to understand the world.

Although each of the reviewed themes — that is, (a) theory-development and (b) relation-analysis — provides new insights into what mathematics education researchers regard as theory, and is thus worth investigating in its own right, to our knowledge, no such themes have yet considered a critical realist perspective. To deal with such an absence in the literature, an effective exploration into whether critical realism could provide new insights for mathematics education requires a preliminary understanding of what is here meant by theory-choice.

What are the primary criteria for theory-choice?

The idea of theory-choice has advantages for mathematics education researchers because it may allow them to begin to address questions — that is, (1) and (2) posed in the previous section — related to falsification and a method by which to select a theory over other rival theories. It is worth quoting Mervyn Hartwig extensively to introduce this idea:

[i]t is vital for CR [critical realism] that theories be empirically grounded. In many of the natural sciences this occurs mainly via direct experiments under artificially closed conditions where theories are subject to a predictive test. In the non-experimental, including [mathematics education as part of] the social sciences, direct test situations are unavailable. Problems of confirmation and falsification of hypotheses and theories are therefore more acute than in experimental sciences … CR rejects actualist understandings of confirmation and falsification (and the criterion of falsifiability) prevalent in the orthodox philosophy of science, whereby laws are construed as empirical regularities confirmed or falsified by their positive instances (e.g., Popper's falsificationism). Hypotheses that do not conform to this format are said by their proponents to be non-falsifiable, hence non-scientific. However, such ‘laws’ are themselves immediately falsified in open systems along with the theories embodying them, and the history of science is littered with falsified non-existential propositions, such as those relating to phlogiston … The primary criteria for theory-choice are thus empirical adequacy and explanatory power. … [An] empirically well-grounded theory can explain more significant phenomena under its own descriptions than its rivals can explain under theirs. A social theory will in addition explain why inadequate theories continue to be adhered to.²⁰

Mathematics education researchers may take the position of theory-choice because it grounds the selection among theories on its empirically adequate conditions and holds greater power to explain more of the world than rival theories. The empirically adequate conditions of a theory make reference to the differences between the natural and the social sciences. Only the former employs the capacity of experiments to allow for the observation and identification of a single mechanism at work in closed systems. It is in this sense that experimental science allows us to empirically access the deeper levels of reality. It is important to note here that the eight standards for judging theories in mathematics education, originally proposed by Schoenfeld, do not make this crucial distinction between open and closed systems, and thus between the experimental and non-experimental sciences. No wonder that for Schoenfeld, ‘it is not unreasonable to expect that such behavior [of mathematics teachers in the classroom] can be modeled with the same degree of fidelity to “real-world” behavior as with predator-prey models’.²¹ Mathematics education, as a social science, is also concerned with understanding the world in terms of the mechanisms that produce and manifest phenomena in open systems and at various deeper levels. As rightly acknowledged by Schoenfeld, ‘the classical experimental method can be problematic in [mathematics] educational research’.²² This is because the classical approach of experimentation facilitates the study of a single mechanism and the description of its precise effects in closed laboratory conditions. Due to the open conditions of reality in which mathematics education researchers, and social scientists in general, investigate social problems, they do not know necessarily if a particular result is due to mechanisms, interacting confounders, factors, or other causes.

Informed by the above conceptions and against conformism, mathematics education researchers ought to make choices between competing theories. Critical realism offers additionally the triad of concepts: ontological realism, epistemic relativity, and the possibility of judgmental rationality, which are collectively known as the holy trinity of critical realism.²³ This basic triad of concepts, along with a consideration of critical realist ontology and its possible weaknesses, as we shall see in the next section, may provide mathematics education researchers with a further philosophical basis from which to avoid reductionisms and irrealisms in argumentation.

What does critical realism have to offer mathematics education?

A brief overview of critical realism, originally developed by Roy Bhaskar, may be delineated across three unfolding phases: (i) basic critical realism that began around 1975 by setting out to revindicate ontology,²⁴ then (ii) the dialectical turn in the 1990s,²⁵ and later (iii) a metaReality turn.²⁶ Since then, the application of critical realism in the sciences, particularly the educational sciences, has been varied.²⁷ Critical realism may be distinguished further by reference to six characterizing features: underlabouring, seriousness, immanent critique, explication of presuppositions, enhanced reflexivity, and hermeticism.²⁸ To further understand and illustrate them, I will provide examples drawn from mathematics education research where each of these characterizing features are described in the literature albeit without such a precise critical realist term.²⁹

The first feature is the idea of philosophy as an underlabourer for the sciences. This idea makes reference to the philosopher John Locke's humble desire.

The Commonwealth of Learning is not at this time without Master-Builders, whose mighty Designs, in advancing the Sciences, will leave lasting Monuments to the Admiration of Posterity … and in an Age that produces such Masters … it is Ambition enough to be employed as an Underlabourer in clearing the Ground a little, and removing some of the Rubbish, that lies in the way of Knowledge.³⁰

Apropos of a realist philosophy for mathematics teaching, Marlow Ediger wrote simply that ‘a mathematics teacher who is a realist desires objectives of instruction to be stated in measurable terms, prior to instruction’.³¹ However, it is easy to imagine that, in this age that has produced a global recession, it is ambition enough simply to be employed. A mathematics teacher who is a critical realist might desire and hope to find employment as an underlabourer of mathematics pedagogy in order to remove some of the obstacles, as Ubiratan D'Ambrosio put it, to managing to ‘survive with dignity’.³² In this manner, the reason why realism, and particularly critical realism, may be valuable to mathematics education researchers is that, as underlabourer, it provides a new theory of ontology.³³

The second feature is seriousness, a concept conceived in Hegelian fashion, referring to the harmonious unity of theory and practice, and thus the removal of dichotomies. The research of Carolyn Kieran exemplifies an application of the concept of seriousness with her argument for the removal of a ‘false dichotomy in mathematics education between conceptual understanding and procedural skills’.³⁴

The third feature is the concept of immanent critique. This idea refers to a philosophical approach that takes the argument of an opposing position and shows that it is inconsistent from within. The research of Ole Skovsmose exemplifies an application of this principle with his idea for a philosophy of critical mathematics education.³⁵ It shares elements of immanent critique because it takes the argument of value-free mathematics and sheds light on the inconsistency of the idea of neutral mathematics pedagogy. It thus urges that critical mathematics pedagogy needs to be concerned with developing pupils' competencies in all mathematical, cultural, social and political realms to lead a democratic life. Paul Ernest finds misunderstandings of immanent critique as an argument from a God's eye perspective; that is, an ominous outsider position (as opposed to one from within) based on the following assumption about critical theory:

[under the] assumption of an Archimedean fixed point, a ‘God's eye view’ from which epistemological and ethical certainties can be determined … it may be the case that we in mathematics education [ME] and critical mathematics education [CME] have not yet sufficiently learned from the insights of critical theory [CT], and are complicit in promoting instrumental reasoning through ME, despite our commitment to the ideals of CME. Perhaps we need to renew our understandings of CT and apply it more vigorously to ME. The overtly espoused goals of CME are to make the teaching and learning of mathematics empowering and liberating, rather than imprisoning and restrictive.³⁶

The fourth feature is the explication of presuppositions. This means the questioning of given assumptions. An application of this principle in mathematics education may be seen in the synergy between social and linguistic theories that turned mathematics education researchers to the analysis of discourse and the study of what David Pimm called ‘a set of meanings that is appropriate to a particular function of language’.³⁷ The research of Sfard exemplifies an application of the fourth feature with her analysis of the patterns of utterances during problem-solving. By questioning the tacit assumptions and role of language as a tool to learn mathematics, Sfard showed that ‘there is more to discourse than meets the ears’.³⁸

The fifth principle is enhanced reflexivity. The research of Freudenthal exemplifies an application of this concept with his idea of forming bonds with reality by means of its (horizontal and vertical) mathematization through guided discovery.

Horizontal mathematization leads from the world of life to the world of symbols. In the world of life one lives, acts (and suffers); in the other one, symbols are shaped, reshaped, and manipulated, mechanically, comprehendingly, reflectively; this is vertical mathematization … To be sure, the frontiers of these worlds are rather vaguely marked. The worlds can expand and shrink — also at one another's expense … The distinction between horizontal and vertical mathematizing depends on the specific situation, the person involved and his environment.³⁹

Mathematics education, as illustrated by the quotation above, may be said to be rooted in a reflexive view of mathematics as a tool with which to understand reality; that is, the mathematics that arise through mathematizing reality. However, the goal is to go beyond the idea of mathematics as a mere means to mathematize reality, and to see mathematics as a way to transform the everyday problems and understandings of the world that may allow people to take a further step forward in terms of their conscious resolution, which is what enhanced reflexivity is.

The sixth feature is hermeticism. This idea means putting our knowledge to work, or as Bhaskar put it, ‘the theories and principles of critical realist philosophy should also apply to our everyday life … [otherwise] something is seriously wrong’.⁴⁰ Schoenfeld exemplifies an application of this feature when he argues that ‘just as in the physical sciences, researchers in [mathematics] education have an intellectual obligation to push for greater clarity and specificity and to look for limiting cases or counterexamples to see where the theoretical ideas break down’.⁴¹ An exercise in hermeticism entails considerable investigation into the possible weak points of and arguments against critical realism and realism in general. Nonetheless, we begin here by delineating some of them. Jan Karlsson, for instance, suggests that an Achilles' heel of critical realism is found in its underdeveloped concept of open reality. In other words, critical realism fails to see that reality in institutions such as schools and prisons is not fully open because of a high degree of regulation and control over the daily experiences of the people within. It is thus possible, with relatively high accuracy, to predict daily events. Karlsson further proposed the idea of partially open systems; that is, reality as having various degrees of openness for better conceptualization because ‘people cannot only open closed systems, they can also close open systems’.⁴² Ernst Von Glasersfeld, the main proponent of radical constructivism, argues against realism in general. According to radical constructivist principles, the role of empirical knowledge is limited to experiences and the organization of the world and ‘not the discovery of an objective ontological reality’.⁴³ It is here that critical realism differs from radical constructivism because it does not deny the objective existence and causal efficacy of real entities in the world (e.g. causal mechanisms such as gravity, black holes, socioeconomic classes, gender relationships, and so on) but affirms the need for their theorization. With these characterizing features and arguments, as we shall see in the next section, we may prepare the ground for philosophical underlabouring for a more beneficial science of mathematics education.

How may critical realism underlabour for mathematics education?

The purpose of this study is to investigate the potential contribution of critical realism as an underlabourer for mathematics education. To achieve this objective, the critical realist notion of Achilles' heel critique is drawn upon as a method. In the words of Bhaskar, this method ‘pinpoints the blindspot in a theory, usually at what appears to be its strongest point, and a series of Achilles’ heel critiques constitutes a dialectical phenomenology'.⁴⁴ The data for the Achilles' heel critique were drawn from a selected bibliography of research studies.⁴⁵ The selection also coincides with Lerman's categorization of the theoretical and philosophical perspectives in use in this discipline.⁴⁶ It is important to note that the selected bibliography is neither purportedly comprehensive nor exhaustive. Mainly, it aims to show how the goal of an Achilles' heel critique aligns with Schoenfeld's call to find ‘where these theoretical ideas are broken’.⁴⁷ Because the selection of categories was limited in scope, the study excluded other important trends in mathematics education, such as embodied cognition.⁴⁸ However, it suffices to provide a short overview of the central line of argument in each category to identify possible Achilles' heels.

Didactically oriented empiricism

Classical empiricism, as represented by the philosopher David Hume and his subsequent disciples, emphasizes that all scientific knowledge is derived solely from the category of sense experience. It forms the backbone of didactically oriented empiricist, mechanist and structuralist visions of mathematics teaching and learning. Freudenthal himself initially identified a possible Achilles' heel and argued against these views of mathematics education and their purported strengths. It is worth quoting Freudenthal to support this point.

[To the mechanist, the mathematics learner] is a computer-like instrument that can be programmed by drill to perform, on the lowest level, arithmetic and algebraic, maybe even geometric, operations, and to solve applied problems, distinguished by recognizable patterns and processed by repeatable ones … [To the structuralist], the mathematics learner was expected to obediently repeat the master's deductions. In order to check the quality of the repetition — whether it was mere parroting or full of insight — problems were set, which in turn were deactivated by drill … To the empiricist the world is a reality where man can acquire useful experiences — a respectable point of view provided reality and usefulness are broad-mindedly interpreted … [Didactically oriented empiricism] is deeply rooted in English utilitarian education. Provided with material from their living world, learners get the opportunity to acquire useful experiences, but they are not prompted to systematize and rationalize these experiences in order to break the barriers of the environment and to expand the [theory of] reality they are familiar with.⁴⁹

Beyond the boundaries of mathematics education research, critical realism provides critiques of Hume⁵⁰ and other critiques of Humean empiricism,⁵¹ which are nonetheless important for this discipline.

Hermeneutics

Hermeneutics, with its focus on interpretation, places primacy on experience by highlighting mathematical meanings and actions in terms of processes of socialization, narratives, and personal subjectivity. Tony Brown illustrated an initial Achilles' heel of hermeneutics for mathematics education researchers when he wrote that ‘although people may believe that there are mathematical expressions that mean the same for everyone, each person places the term in the context of their experience, cultural perspective, and present intentions’.⁵² Outside the field of mathematics education, Andrew Sayer succinctly summarized yet another Achilles' heel of hermeneutics when he wrote that ‘the interpretation of meanings and actions in society does not yield to the same kind of logic as causal and structural analyses’.⁵³ Critical realism agrees with the idea that hermeneutic research is necessary as an initial step in the process to get to explanatory structures in social life, yet it remains insufficient.⁵⁴ It thus aims to avoid a prima facie account of hermeneutic research as merely enriching an ongoing human conversation without reference to the possibility of causation and structure in reality.

Pragmatism

Pragmatism emphasizes mathematical meaning in terms of the successful implementation of an algorithm, method, approach, or design, and rightly posits the possibility of knowledge outside of its own area of practice in the realm of successful practicality. Outside the boundaries of mathematics education research, David Scott summarized its Achilles' heel when he coined the idea of the fallacy of pragmatism; that is, when ‘educational researchers understand research as a practical activity which can be carried on without reference to epistemological or ontological concerns’.⁵⁵ Lester pointed out that a possible Achilles' heel of pragmatic research in mathematics education, at least in the United States, results from an infatuation with outcomes that would deliver ‘what works’ at the expense of philosophical research which ironically functions ‘to leave educational researchers with less latitude to conduct studies to advance theories and model-building’.⁵⁶ In defence of pragmatism, however, Martin Simon illustrated yet another possible Achilles' heel when he wrote that ‘mathematics education researchers cannot afford to engage as philosophers in debating established theories. Rather, we need to (when appropriate) engineer pragmatic coordination of analyses done from different theoretical perspectives’.⁵⁷ In both cases, a weakness of pragmatism is a strategy of containment; that is, tendencies that advocate the dismissal of philosophical contributions, the questioning of the status quo, and the elimination of debates over utility.

Postmodernism

Postmodernism or, as conceptualized by Frederic Jameson, ‘the cultural logic of late capitalism’,⁵⁸ rightly rejects metanarratives as ubiquitous theories and focuses on the analysis of social relations as embedded in networks of political interests and the analysis of mathematical thinking as a change in, and proliferation of, various types of mathematics-related discourses. A possible Achilles' heel of postmodernism arises when ontology is left out in favour of a uniquely discursive world. In other words, the world and indeed the objects of scientific research appear as sole constructions of society by means of concepts. For critical realism, this conflation of the dimension of ontology into language is argued to be an error in argumentation exemplifying the linguistic fallacy.⁵⁹ The issue is not that concepts and discourses are necessarily social constructions by their nature; rather, the main point is that there exist real phenomena, that is, real mechanisms with causal implications relatively or absolutely independent of our concepts and discourses about them. In mathematics education research, Brown traced correctly a possible Achilles' heel of postmodernism to its Wittgensteinian origins of ‘the meaning of a word with its use in language’.⁶⁰ That is, when the philosopher Wittgenstein wrote that

a word has no meaning if nothing corresponds to it. — It is important to note that the word ‘meaning’ is being used illicitly if it is used to signify the thing that ‘corresponds’ to the word. That is to confound the meaning of a name with the bearer of the name … For a large class of cases — though not for all — in which we employ the word ‘meaning’, it can be defined thus: the meaning of a word is its use in the language.⁶¹

Cultural psychology

Traditionally, the category of cultural psychology in mathematics education has been further subdivided into two strands: a strand that follows the work of Piaget and a strand that follows the work of Vygotsky. These strands are not mutually exclusive. Michael Cole and James Wertsch, for example, argued to go ‘beyond the individual-social antinomy in discussions of Piaget and Vygotsky’.⁶² The transition from a Piagetian strand to a Vygotskian strand may be understood as a Kuhnian unadulterated gain; that is, a new theoretical perspective changes the terms of understanding without a replacement of the old theoretical perspective. Outside the field of mathematics education, however, this transition is reconciled albeit without explanation for accumulation. Lerman illustrated a possible Achilles' heel of the transition to a Kuhnian unadulterated gain in mathematics education research when he wrote the following:

the entry of Vygotsky's work … did not lead to the replacement of Piaget's theory (as the proposal of the existence of oxygen replaced the phlogiston theory). Nor did it lead to the incorporation of Piaget's theory into an expanded theory (as in the case of non-Euclidean geometries).⁶³

The Piagetian strand

The Piagetian strand offers a theory of cognitive development through various stages. Margaret Archer summarized this strand's central line of argument in ‘the primacy of practice’.⁶⁴ In other words, Piagetian practice is a route to self-discovery and thus cognitive development. To illustrate this in practice, Piaget wrote that ‘each time one prematurely teaches a child something he could have discovered himself, the child is kept from inventing it and consequently from understanding it completely’.⁶⁵ Beyond the field of mathematics education, Vygotsky himself identified a possible Achilles' heel in the Piagetian theoretical edifice in what one may call the hiatus between learning and instruction, which Piaget omitted to account for in the influence of social interactions on learning, as illustrated with the following quotation:

Our disagreement with Piaget centres on one point only … He assumes that development and instruction are entirely separate, incommensurable processes, that the function of instruction is merely to introduce adult ways of thinking, which conflict with the child's own and eventually supplant them.⁶⁶

The identification of this Achilles' heel is important, as it is precisely in the space of the hiatus between learning and instruction that there is room to investigate, for example, collaborative interactions and other pedagogic relations in learning mathematics.

Traditionally recognized forms of constructivism

Traditionally recognized forms of constructivism include distinctions among radical, basic and social constructivism. Ernest summarized a central line of argument of ‘many of the various forms of constructivism … [with] the metaphor of construction from carpentry or architecture’.⁶⁷ Although these traditions appear differently for different proponents, they share a legacy in the work of Piaget. Von Glasersfeld, for example, grounded radical constructivism in two principles: ‘(1) knowledge is not passively received but actively built up by the cognizing subject; (2) the function of cognition is adaptive and serves the organization of the experiential world, not the discovery of ontological reality’.⁶⁸ The former principle solely forms the theoretical grounds for basic constructivism.

Robyn Zevenbergen herself identified a possible Achilles' heel of constructivism, radical or otherwise, in the omission of a socio-political dimension in the learning of mathematics:

Constructivism with its valorizing of individual, subjective constructions of meaning ignores this political aspect of meaning-making. The denial of the socio-political dimension of meaning-making is not surprising given that most of education is founded on liberal discourses that espouse the value of the individual and the dominant discourses within the field are imbued with psychology's individualism.⁶⁹

Now social constructivism emphasizes the construction of mathematical knowledge as a process rooted in a social context while taking into account the significance of subjective knowledge previously emphasized by hermeneutics. Ernest himself found the Achilles' heel of social constructivism in its purported strengths, as this view offers a circular relationship or ‘creative cycle, with subjective knowledge creating objective knowledge, which in turn leads to the creation of subjective knowledge’.⁷⁰ A problem arises when this construction of knowledge via a creative cycle of subjective–objective knowledge that seems to exhaust reality itself tends to render anything external as meaningless or non-existent. The issue at stake for realism is precisely this dimension of ontology. This is because objective–subjective knowledge revolves in a vicious cycle without considering the independence, or at least the prior existence and causal efficacy, of objects of scientific research (the dimension of ontology). The social constructivist view of subjective-objective knowledge exemplifies the epistemic fallacy, or as I like to put it, Ernest did not ‘mind the gap’⁷¹ between this creative cycle of subjective-objective knowledge (epistemology) and reality (ontology).

Cobb eloquently synthesized a related but different Achilles' heel of constructivism in ‘the central problem of traditional epistemology, that of the opposition between philosophical realism and constructivist positions that deny that ontological reality is knowable’.⁷² The issue at stake for the mathematics education community is thus the nature of being (ontology) and knowledge (epistemology), including the possibility of moving beyond relativism — the idea that all theories are equally valid — and fallibilism — the idea that theories may turn out to be incorrect. A critical realist view of mathematics education explicates why irrealism, the doctrine that denies the existence and causal efficacy of real entities in the world, is limited compared with ontological realism, which asserts the need for their theorization. Cobb went on to identify that the issue at stake for the mathematical education community is ‘the long-standing dichotomy between [relativism as] the quest for a neutral framework for comparing theoretical perspectives on the one hand and [judgmental irrationality as] the view that we cannot reasonably compare perspectives on the other hand’.⁷³ In this manner, we can see how the holy trinity of critical realism may provide mathematics education researchers with ideas by which to move beyond what Cobb sees as the long-standing dichotomy between relativism and judgmental irrationality in mathematics education research.

The Vygotskian strand

The Vygotskian strand offers a critique of Piaget via social interactions and an imaginary magnitude by which to assess the impact of socialization on cognitive development. This imaginary magnitude refers to the concept of a zone of proximal development: the ‘distance between the actual development level as determined by independent problem-solving and the level of potential development as determined by problem-solving under adult guidance, or in collaboration with more capable peers’.⁷⁴ Moreover, Archer summarized this strand's central line of argument in which ‘the private experience of a human being is shaped and ordered in learning to speak and write … This was Vygotsky's great insight’.⁷⁵ In other words, the primacy of language precedes individual development, which Vygotsky described with the infamous words: ‘any function in the child's social development appears twice or on two planes. First it appears between people as an interpsychological category and then within the child as an intrapsychological category’.⁷⁶ To illustrate this in practice, Vygotsky wrote that ‘children's learning begins long before they attend school … Any learning a child encounters in school has a previous history’.⁷⁷

The identification of two possible Achilles' heels points to a double weakness in the Vygotskian theoretical edifice. Archer identified an initial Achilles' heel in Vygotsky's disregard for the emerging sense of self when giving primacy to language over practice. In contrast, the primacy of practice over language makes reference to, as Bhaskar put it, ‘being able to walk one's talk’.⁷⁸ This is because the idea of seriousness, as a practice-language unity, may exist solely in relation to an emerging potentiality of reflexivity. It is worth quoting Archer extensively to support this point:

The primacy of practice refers both to its logical and substantive priority in human development. This is not simply a matter of it coming before anything else, though temporally it does just that; it is also a question of viewing language itself as a practical activity, which means taking seriously that our words are quite literally deeds, and ones which do not enjoy hegemony over our other doings in the emergence of a sense of self.⁷⁹

Moreover, Andy Blunden himself identified yet another potential Achilles' heel in Vygotsky's underdevelopment of a theory of society because ‘Vygotsky always focused his scientific work on interactions between individuals rather than using representations of societal phenomena and institutions abstracted from their constitution in specific forms of activity … that is to say, a social theory’.⁸⁰ Although the identification of these Achilles' heels is outside the field of mathematics education, they are nonetheless important to this discipline.

Traditionally recognized theories of activity

The work of Vygotsky went a long way in influencing a range of traditionally recognized theories of activity.⁸¹ The central line of argument of the theories of activity may be seen with the difference between the units of analysis, or as I see them, the analytic lenses by which ‘the learning process is perceived’.⁸² Outside the field of mathematics education, I have shown that ‘activity theory omits to critique the Humean legacy of Kant … [and a] problem with this omission is that if we are not critical of Humean classical empiricism, then we commit the error that Kant made’.⁸³ In the field of mathematics education, Brown focused on Radford's ‘cultural theory of learning’⁸⁴ to illustrate yet another possible Achilles' heel. This weakness refers to a tendency to homologize mathematics learning with the acquisition (or appropriation) of culturally relevant forms — of models that describe the patterns of behaviours, speech acts, attitudes, values, and so on — inherited from the remains of past cultures and histories. Brown's identification of this Achilles' heel is worth quoting:

Learning is not understood primarily as a growing alignment with more or less familiar cultural forms, or with fixed patterns of activity … The refusal to settle on any given story, and on the objects' anchoring past stories, underpins this … plea for a greater alignment between mathematical learning and cultural renewal … learning is the task of resisting and revising current cultural models … it is not enough to see the goal of education as being solely to bring students into existing practices or current conceptions of mathematics.⁸⁵

The identification of the above-mentioned weaknesses aims to critically enrich this activity-theoretical approach as it continues to expand across continents — from North America, with the research of Kris Gutiérrez and colleagues in their ‘discursive and embodied mathematical vision’,⁸⁶ to Europe, with Anne Edwards's research on mathematics teacher development,⁸⁷ and to Australia, with Merrilyn Goos's models for fostering researcher-teacher relationships for mathematics pedagogy.⁸⁸

Ethnomathematics

In the mid-1980s, D'Ambrosio introduced the notion of ethnomathematics as ‘the mathematics which is practised among identifiable cultural groups, such as national-tribal societies, labour groups, children of a certain age bracket, professional classes, and so on’.⁸⁹ Ernest summarized its central line of argument by adding: ‘a historical dimension in the teaching of mathematics can serve to counter the received Eurocentric view, and promote elements of a multicultural and anti-racist mathematics’.⁹⁰ This idea established ethnomathematics as a research programme for mathematics pedagogy, as Alexandre Pais summarized, that ‘shifts mathematics from the places where it has been erected and glorified (universities and schools) and spread it to the world’.⁹¹ Runuka Vithal and Ole Skovsmose went a step further to argue that ethnomathematics may be read as an oppositional stance and reaction not only to Eurocentrism but also to modernization theory in general. Modernization, as conceptualized by Anthony Giddens, applied the fourth characteristic feature of critical realism to inquire into the presuppositions that sustain the idea that ‘industrialism is essentially a liberalizing force’.⁹²

Dowling found an initial Achilles' heel in ethnomathematics by questioning the presuppositions of the myths of reference, participation and emancipation to illustrate that

[t]o varying degrees, all of this [ethnomathematics] work succeeds in celebrating non-European cultural practices only by describing them in European mathematical terms, that is, by depriving them of their social and cultural specificity.⁹³

Vithal and Skovsmose identified yet another Achilles' heel in ethnomathematics as a weakness inherent in the word itself because ‘the actual use of the word does not assist in resolving the uncomfortable reference to “race” but rather deepens it’.⁹⁴ Their analysis of the etymology of the word shows that the modern use of the prefix ethno- (meaning race, culture or ethnicity) could have been used to rationally justify the logic of the pedagogy underpinning apartheid education in South Africa. In other words, being pro-ethnomathematics could advocate the separation and segregation of mathematics pedagogy on the basis of an ‘ethnic’ distinction — a specific mathematics curriculum for other (non-European) ethnic majorities — and thus pro-apartheid education. What is significant about this analysis is not so much the imperfect or weak nature of a particular word, but that the emphasis on the dialogical dimension allows the researcher to see that each hegemonic discourse itself (e.g. apartheid education) can be grasped potentially as a process of the colonization, neutralization and re-appropriation of a popular non-hegemonic discourse.

Pais illustrated further a double Achilles' heel of ethonomathematics by drawing upon some of the concepts used in Slavoj Žižek's analysis of the function of ideology in capitalism. Pais's critique focused on the idea of mathematics domestication disclosed as a superficial transformation rooted in a systemic embracing of the values of multiculturalism, or what Žižek called ‘the cultural logic of multinational capitalism’.⁹⁵ In embracing multiculturalism, Pais also illustrated that ethonomathematicians tend to omit the problematization of the implicit assumptions and the role that these values have in shaping the mathematics curriculum in general.

Ideology functions by means of making effective what it officially conceals. In our case, we can say that other cultures are allowed to come into school as long as they become part of the school's culture. The system satisfies the societal demand of a meaningful education for all by importing local cultures into the curriculum while assuring that such ‘insertion’ will not actually change any of the core features of the school system. It is in this sense that Žižek says that today's capitalism needs to promote constant reforms and innovations to conceal the crude reality that core choices (such as a radical transformation of the school system as advocated by some ethnomathematicians) are not available.⁹⁶

As the ethnomathematics programme continues to develop globally — from its initial studies by Terezinha Nunes and colleagues to the daily mathematical techniques employed by young Brazilian street vendors,⁹⁷ to the more recent studies of Rex Matang's ‘pedagogical strategies for conceptual and procedural understandings of mathematics in Papua New Guinea’⁹⁸ — this investigation becomes important since it is not enough to attend to mathematics education and ethnomathematics. It is also important to shed light on its possible Achilles' heels.

Conclusion

This article has applied a series of Achilles' heels critiques, taken from critical realism, to a literature-based analysis of various theoretical perspectives currently employed in mathematics education: didactically oriented empiricism, hermeneutics, pragmatism, postmodernism, traditionally recognized forms of constructivism, traditionally recognized theories of activity, and others, as summarized in Table 1. It has outlined the potential contributions of critical realism to mathematics education research through a theorization of a realist anti-empiricist ontology and the benefits of the ideas of theory-choice and the holy trinity as a philosophical means to avoid reductionisms and irrealisms. It has further illustrated the idea of hermeticism by including the possible Achilles' heels of critical realism because, as Bhaskar put it, ‘[critical realist] theories and explanations should be tested in everyday life, as well as in specialist [mathematics education] research contexts’.⁹⁹ In this manner, it has been possible to provide critical and greater explanatory power for investigations concerned with the applications of critical realism to theory-development and relation-analysis in mathematics education.

Table 1. A dialectical phenomenology constituting the theoretical landscape of mathematics education

Author/identifier of a possible AH in a substantive theory	Description of the identified AH	Theory being considered
Karlsson 2011	• Underdevelopment of the concept of open systems	Critical realism
Freudenthal [1991] 2002	• Mechanistic, structuralist, and instrumentalist views of the learner and of the subject matter of mathematics	Didactically oriented empiricism
Bhaskar [1975] 2008; [1994] 2010	• Aporia of causality and other critiques of Hume, including Humean empiricism, rationalism, and positivism
Brown 1999; Bhaskar [1979] 1998	• Limited account of meanings and actions outside the realm of narratives• Hermeneutics as necessary in social science (i.e. it focuses on a concept-concept relationship) albeit it remains insufficient (i.e. it omits a concept-reality relationship)	Hermeneutics
Sayer 2000	• Prima facie hermeneutic research as merely enriching human chronicles and stories without an account of causality
Lester 2005; Simon 2009	• Tendency to dismiss the contributions of philosophy over utility	Pragmatism
Scott 2002	• Fallacy of pragmatism
Brown 2010	• Homology between a word meaning and its use in language without reference to its referent	Post-modernism
Wittgenstein [1953] 1981	• Homology between word and language in-use
Bhaskar [1993] 2008	• Linguistic fallacy
Ernest 2010b	• Understanding of critical theory as a God's eye perspective and unwitting compliance with instrumental reasoning	Philosophy of critical mathematics education
Lerman 2006	• Piagetian-Vygotskyan transition (or what is here called a Kuhnian unadulterated gain)	Cultural psychology
Vygotsky 1962	• Hiatus between learning and instruction	Piagetian strand
Zevenbergen 1996	• Omission of a socio-political dimension in the learning of mathematics	Traditionally recognized forms of constructivism
Cobb 2007; Ernest 1991	• Longstanding dichotomy between relativism and judgmental irrationality• Creative cycle of subjective knowledge generating objective knowledge successively creating subjective knowledge and so on
Bhaskar [1975] 2008	• Epistemic fallacy
Archer 2000	• An overlooked theory, the emerging sense of self	Vygotskian strand
Blunden 2009	• Underdeveloped theory of society
Brown 2010	• Tendency to homologize mathematics learning with the acquisition or appropriation of culturally relevant forms	Traditionally recognized forms of activity
Nunez 2013	• Omission a critique of Hume and of Humean empiricism
Dowling 1998	• Celebration of non-European mathematics by its re-description in European terms and a lack of social and cultural specificity	Ethno-mathematics
Vithal and Skovsmose 1997	• Inherent limit in the word ethno itself
Pais 2011	• Mathematics domestication as a superficial transformation, a systemic embracing of liberal multiculturalism, and lack of examination of its implicit assumptions

* Author's thesis may be classified generally as working and publishing in the field of mathematics education

Acknowledgements

This article is dedicated to the memory of Roy Bhaskar, for the countless supervisions he devoted to reading and commenting on my developing writing — with the greatest of philosophical rigor and love. It was made possible in part by funds from the National Science Foundation ADVANCE grant for Attracting and Nurturing Women Faculty at The University of Texas-Pan American. I would also like to express my thanks to two anonymous reviewers for their reassuring comments and to Aaron Wilson for his suggestions concerning the field of mathematics education.

Additional information

Notes on contributors

Iskra Nunez

Iskra Nunez is Assistant Professor of Mathematics Education at The University of Texas-Pan American.

Correspondence to: Iskra Nunez, Mathematics Department MAGC 3.210, 1201 W. University Drive, Edinburg TX 78539, USA. Email: [email protected]

Notes

 1 Jablonka et al. 2013, 41.

 2 Lerman 2000, 19. This social shift, originally conceptualized by Lerman, refers to the tripartite inclusion of theoretical insights from the fields of anthropology, cultural psychology and sociology to enrich explanations when communicating research about mathematics learning and teaching, which aimed to go beyond the study of mathematics pedagogy as the culturally recognized and valorized behaviours of pupils and teachers.

 3 Gutiérrez 2013, 37. Here the basic argument shaping the sociopolitical turn in mathematics pedagogy refers to the recognition of economics as the political power to shape education. Generally, this turn began the introduction of Freire-informed critical mathematics teaching, followed by attention to the race, language and culture of diverse student populations in the early 2000s.

 4 Sriraman and Nardi 2013, 303.

 5 Bikner-Ahsbahs and Prediger 2006, 52.

 6 Lester 2005, 160.

 7 In brief, an Achilles' heel critique (AH) identifies a point of vulnerability (a weakness) in a theory usually at what seems to be its firmest point, as conceptualized in Bhaskar [1993] 2008, 185. The entire sequence of AHs constitutes a dialectical phenomenology. The technique that this article presents was inspired by the work of Bhaskar [1994] 2010, where we find an analysis of the history of Western philosophy from the ancient Greeks to both the analytical and dialectical branches of Western philosophy, which unfolds at each stage via an Achilles' heel critique of its philosophical predecessor. Each stage in the analysis of the history of philosophical thought proceeds sequentially by identifying weaknesses (or blind spots) in its philosophical precursor — from Parmenides (i.e. in the syndrome of egocentric-monoist solipsism), to Plato (i.e. in the transposition of change as difference), to Aristotle (i.e. in the question of ousia/substance), and including but not limited to Descartes (i.e. in Cartesian rationalism), Hume (i.e. in the aporia in his theory of causality), Kant (i.e. in an implicit freedom), Hegel (i.e. in the problem of historicity), and Marx (i.e. in an underdeveloped communism). We may further illustrate AH by differentiating between an explanatory critique, or metacritique₂ (MC₂) and metacritique₁ (MC₁), and by referencing its place in a topology of critiques provided in Hartwig 2007a, 107. First, MC₂ is a totalizing concept involving dialectical (or transcendental) argumentation to explain the absences or theory-practice inconsistencies discerned in MC₁. The role of MC₁ is to identify and isolate problems. The formula MC₂ > MC₁ > immanent critique>AH illustrates the place of AH in a topology of critiques where > means that MC₂ englobes MC₁. In other words, MC₂ constellationally includes an omissive critique (a type of MC₁) that identifies absent elements in a theory; an omissive critique includes both an immanent critique (also a type of MC₁) that singles out theory/practice aporia, and AH (a kind of MC₁), which pinpoints weaknesses or scotomata in a theory.

 8 Freudenthal [1991] 2002, 225.

 9 Schubring 2014, 3.

10 Nikolakaki 2010, 8.

11 As formulated in Kuhn [1962] 1970, 4, the notion of paradigm is here understood as referring not to poor methodology, but to a process of persuasion where there are two different ‘ways of seeing the world and of practicing science in it’.

12 Sriraman and English 2010, 12.

13 Silver and Herbst 2007, 50.

14 Cobb 2007, 31.

15 Schoenfeld 2000, 646. Here Schoenfeld provided a set of eight standards to judge theories based on the following principles: i) descriptive power, ii) predictive power, iii) explanatory power, including the score and rigor of their iv) specificity, v) reliability, vi) replicability, vii) falsifiability, and viii) triangulation (or multi-evidential sources) of research findings.

16 Radford 2008, 317.

17 Leikin and Zazkis 2012, 223.

18 Bikner-Ahsbahs and Prediger 2010, 483. The idea of networking strategies refers to a scale, what is here understood and denoted with the interval (0,1), by which to assess the degree of connectivity and modularity between theories in mathematics education. At one end of this scale, the degree of connectivity is null with a strategy that simply ignores other theories; at the opposite end, the degree of connectivity is one because it unifies all theories globally. Theoretically, the middle landscape of this interval (0,1) ranges with infinite possibilities for networking strategies. To illustrate, we may suppose that each networking strategy represents a real number in this interval (0,1). Examples of networking strategies are: comparing and contrasting, combining and coordinating, synthesizing and understanding the common ground of diverse theories, methods, questions of research, and so on. Following Ferreirós [1999] 2007, 202–14, we may see the evolution of mathematical thought that shows that the interval (0,1) has unaccountably infinite numbers, but this proof is beyond the scope of this paper. For mathematics education researchers, this question remains open: how do they choose between possibly infinite networking strategies within this scale (0,1)?

19 Sfard 1998, 10.

20 Hartwig 2007c, 463, emphases altered.

21 Schoenfeld 2000, 644. A problem is that Schoenfeld is unable to conceptualize the difference between open and closed systems, which in turn leaves him to create a list of standards for judging theory-related research based on an implicit pattern-driven view of social reality; from this view, only if patterns (of behaviours) are either confirmed or falsified, then theory-related research is said to be scientific. Unlike predator-prey models of behaviour, for instance, individuals have the capacity to learn from and adapt to controlled (or quasi-controlled) conditions.

22 Schoenfeld 2000, 645.

23 Hartwig 2007b. With the holy trinity, critical realism may take both the irrealist and the postmodernist arguments forward, since its ontological realism seeks to move from the explanation of the established phenomena at one level of reality to arrive at the deeper explanation of the underlying mechanisms or structures that account for such phenomena. To move beyond fallibilism, critical realism takes the position of epistemic relativity, which is the idea that theories are historically transient social products, and which includes fallibilism. Epistemic relativity also implies that our criteria for truth and values are situated within a particular historical time. To move beyond relativism, critical realism takes the position of the possibility of judgmental rationality, the idea that agents are capable of making decisions for relative theories, beliefs, and practices. This possibility of judgmental rationality assumes both ontological realism and epistemic relativity.

24 Collier 1994. Here, we can find an in-depth examination of the constituting features of critical realism as differentiated from realism in general.

25 Bhaskar [1993] 2008.

26 Bhaskar [2002] 2012.

27 The wide variety of educational research that employs a critical realist perspective may be exemplified without being reduced to the following relevant applications: an anti-reductionist vision of educational sciences, as argued in the thesis of David Scott (2002); a passionate case for the abolition of low expectations in the primary education of Aboriginal Australians, as put forward by Chris Sarra (2011); a model of education as freedom, as conceptualized by Brad Shipway (2011); the introduction of Islamic critical realism with the aim of deepening Islamic philosophy and practice, as envisioned by Matthew Wilkinson (2013); and the introduction of a critical realist activity theory, as conceived by Iskra Nunez (2014).

28 Bhaskar 2013.

29 For introductions, see Bhaskar [1975] 2008, Bhaskar and Hartwig 2010, and Collier 1994.

30 Locke [1689] 1979, 9–10, original emphasis.

31 Ediger 1996, 8.

32 D'Ambrosio 2007, 38.

33 Following Bhaskar [1975] 2008, 11–14, this new theory of ontology begins with a double argument: (a) the initial argument that differentiates ontology from epistemology, i.e. in the difference of the inexorable and irreducible nature of causally efficacious entities or intransitive objects of knowledge (ontology), independent of our transitive objects of knowledge about them (epistemology); and (b) the inclusion of epistemology within ontology. As a way of critique, we find concepts such as the epistemic fallacy, an error in argumentation that occurs with the reduction of being (ontology) to knowledge (epistemology), developed by Bhaskar [1975] 2008, 27. A second argument follows from this dual argument, originally explained in Bhaskar [1975] 2008, 35–47. Here, we find (c) conceptual tools by which critical realism vindicates the nature of reality as both structured and differentiated, with the concept of three overlapping domains (d_r > d_a > d_e): the all-encompassing real (d_r), constitutive of causal mechanisms and structures, containing the actual (d_a), which is comprised of events, and which itself contains the empirical (d_e), comprised of experiences. As a way of critiquing one-dimensional views of the world, we find the actualist fallacy, an error in argumentation that occurs with the omission of difference and structure, which results in d_r = d_a, a flat view of reality without an account to sustain causality. A further argument is the distinction between open and closed systems. For example, reality may be understood as an open system while experimental conditions are closed systems. From these arguments, the goal in scientific practice for critical realists is to move from established phenomena (e.g. including regularities, the patterns of events, language, and behaviours, recognized or otherwise) to explain the underlying mechanisms or structures that account for such phenomena. In this manner, critical realism provides a new theory of ontology that is different from classical empiricism, in which causal mechanisms and structures are powers of all things that exist and also act independently of the flux of conditions that enable their identification, and which tend to be out of phase with the patterns of events and experiences.

34 Kieran 2013, 153.

35 Skovsmose 1994.

36 Ernest 2010b, 70–82.

37 Pimm 1987, 17.

38 Sfard 2001, 13.

39 Freudenthal [1991] 2002, 42–43, original emphasis.

40 Bhaskar 2013, 4.

41 Schoenfeld 2000, 647.

42 Karlsson 2011, 145.

43 Von Glasersfeld 1989, 162.

44 Bhaskar [1993] 2008, 372.

45 In providing a sequence of Achilles' heel critiques of a literature-based analysis of various categories describing theoretical perspectives in mathematics education, this investigation aims to contribute with the presentation of a philosophical means by which to apply the idea of theory-choice in order to make decisions that are less biased when selecting the strongest position in the global market of multi-theoretical perspectives. The selected bibliography, as organized in successive categories, is important precisely because these categories constitute the field of mathematics education research generally. The data was compiled using three databases — Education Resources Information Centre, EBSCOhost, and Google Scholar — and a double criterion of inclusion. First, the section criterion focused on combinations of some of these words: mathematics education or empiricism, hermeneutics, pragmatism, cultural psychology, theories of activity, and ethnomathematics. Second, and in order to search for possible Achilles' heels, a word search with keywords such as critique, criticism, disagreement, weakness, problem, dualism, and aporia was employed to further discriminate among the studies. With this selection criterion and a detailed reading of the literature, a selection of research publications was compiled and analysed by focusing on the identification of researchers who had claimed to find a point of vulnerability or weakness in each of the categories.

46 Lerman 2006, 10. However, this analysis is limited to publications from 1990 through 2001.

47 Schoenfeld 2000, 647.

48 See Núñez, Edwards, and Matos 1999. Other limitations with data collection were also found. For example, the study tried to find at least one Achilles' heel per category in mathematics education research; however, important contributions were found outside mathematics education. These were included in the results for their potential significance to the field.

49 Freudenthal [1991] 2002, 136.

50 Following Bhaskar [1975] 2008, 14–19, in the Achilles' heel critique of classical empiricism, we find two main occurrences: the first is the homology between world and sense-experience, and the second is the implicit use of knowledge to constitute the world itself. Using critical realist terms, this refers to the epistemic fallacy and the implicit definition of knowledge, especially in the form of the logic ‘whenever experiences of type x, then cause y would happen’, because sceptic Hume thought it correct. This cause-effect aporia leads to a flat, one-dimensional view of reality without an account of causal mechanisms, without structure, without differentiation, and without depth. Using critical realist terms, this refers to actualism. Thus, classical empiricism is incapable of providing an accurate account of science because it ignores the ontological dimension.

51 Bhaskar [1994] 2010, 154–60. Here we find that Hume's philosophy, as a turntable, is seen as a revolving platform for turning and supporting various philosophical vehicles, such as positivism, scepticism, rationalism and dogmatism. Hume is here the archetype empiricist. It is beyond the scope of this paper to provide a detailed account of Bhaskar's philosophy and his critiques of Hume and Humean empiricism. We may, however, provide an outline starting with: a) the MC₁ of Hume, which pinpoints the impossibility of sustaining causality in sense experience exclusively, a legacy inherited by Kant and others, which in turn created other problems in the history of philosophy. Then b) an omissive critique identifies Hume's non-anthropic (implicit) ontology. The reason for such aporias is that Hume's sceptical view of reality as merely sense experience is unable to conceive structure, depth, and thus change in a lack of negativity. Hume's immanent critique identifies his theory-practice inconsistencies such as in his absence of seriousness. In particular, see Bhaskar [1994] 2010, 146. Here Hume's unseriousness is exemplified within i) his announcement that it wasn't unreasonable to prefer the obliteration of the world to the scraping of his finger, ii) Hume's fideistic knowledge as a remedy to his sceptic symptoms, iii) the endorsement of existing conventions and laws, iv) the irony of searching for literary recognition and fame in his lifetime, and v) the immorality of Hume's infamous law ‘no ought from what is’, i.e. when confronted with facts, one may choose any value-position.

52 Brown 1991, 447.

53 Sayer 2000, 149.

54 In particular see Bhaskar [1979] 1998, 140–58. In brief, hermeneutics is necessary because it stresses the relations between concepts; however, hermeneutics is insufficient because this concept-concept relation cannot account for the relation between concepts and reality. Here Bhaskar provides a more complete history and a critique of the hermeneutic traditions from its Viconian roots, to Winchian insights into the idea of a non-causally generated social life, including but not limited to, the Gadamerian hermeneutic heresy when the domain of ontology is collapsed into thought.

55 Scott 2002, 2.

56 Lester 2005, 457.

57 Simon 2009, 488.

58 Following Jameson 1984, 53, postmodernism is here understood not only in terms of a break from modernism, which began in the 1950s. This break included the variations in aesthetic form in films, novels, architecture, etc. Rather, postmodernism is also historical. It tends to be described in terms of the end of utopias, politics and ideologies, including the dominance of multinational (often called global, or post-industrial) capital; it is typically characterized by the absence of a common project, an amputation of political conviction stabilized with view of the agent as consumer. Other features of postmodernism include a political neutralization, a social confusion as seen in the various forms of fundamentalism, and new decentralized networks of surveillance.

59 Bhaskar [1993] 2008, 192.

60 Brown 2010, 331, original emphasis.

61 Wittgenstein [1953] 1981, 20–21.

62 Cole and Wertsch 1996, 250.

63 Lerman 2006, 9.

64 Archer 2000, 121–53.

65 Piaget 1970, 715. From the Piagetian view of learning, children construct their understanding of the world through the process of assimilation, and fit this new understanding, as a process of accommodation, in a cognitive schemata that continues to (re)structure new knowledge recursively. The goal of the educator is then to provide organization to promote discovery, which may lead the child to adapt to their environment; the educator thus facilitates cognitive development.

66 Vygotsky 1962, 116.

67 Ernest 2010a, 39.

68 Von Glasersfeld 1989, 162.

69 Zevenbergen 1996, 103–4.

70 Ernest 1991, 84.

71 Nunez 2012, 1.

72 Cobb 2007, 4.

73 Cobb 2007, 4.

74 Vygotsky 1978, 86.

75 Archer 2000, 110.

76 Vygotsky 1981, 163.

77 Vygotsky 1978, 84.

78 Bhaskar 2013, 11.

79 Archer 2000, 121.

80 Blunden 2009, 9–11.

81 For a historical perspective of the traditionally recognized theories of activity see Axel 1997. In brief, we find three generations of activity theories, as conceptualized by Yrjö Engeström 2001, 134. First-generation activity theory (1920–1930), originally developed by Vygotsky, focused on the idea of mediation of actions and language between the subject and the object of study. Second-generation activity theory (1940–1970), advanced by Leont'ev, added the ideas of motive of activity, division of labour, and collective community. Third-generation activity theory (1980–present), originally formulated by Engeström, places primacy in the analysis of groups of collectives via the activity system, an enhanced analytic lens comprising six components — object, subject, tools, division of labour, norms, and community. It further adds the ideas of contradiction, object-relatedness, multi-voiceness, and historicity. It is also known as cultural-historical activity theory (or CHAT).

82 Nunez 2009, 7.

83 Nunez 2013, 142.

84 Radford 2008, 215.

85 Brown 2010, 340–1, original emphasis.

86 Gutiérrez, Secupta-Irving and Dieckmann 2010, 30. In developing this vision from a CHAT perspective, a key assumption is that ‘knowing the practice of mathematics involves a socially organized way of seeing, understanding, envisioning, and doing mathematics in the ways that are accountable to distinct norms of the mathematical community’, which echoes Brown's Achilles' heel critique that sheds light on a conflation between mathematics learning and a growing alignment with relatively understood cultural forms.

87 Edwards 2010.

88 Goos 2014.

89 D'Ambrosio 1985, 45.

90 Ernest 2001, 281.

91 Pais 2011, 210.

92 Giddens 1986, 137. A brief outline of the theory of modernization may begin with an initial interrogation of the presuppositions of a) the potentiality of industrialization to emancipate workers through revolutionizing technologies that would in turn create new social relations, such as self-discipline, increased leisure to practise the arts, and open possibilities to market exchanges globally, and b) the potential of industrialized societies to serve as a model for the rest of the developing world.

93 Dowling 1998, 14. Here, I provide a brief outline of the assumptions underpinning these three myths — i.e. reference, participation and emancipation. The myth of reference refers to the operation of a hierarchy of social relations based on (1) the dichotomy of intellectual and manual labour, (2) the primacy of the former over the latter, (3) the anamorphic view of mathematics as solely intellectual work, and (4) the privilege of abstraction in mathematics over its applications or didactics. The myth of participation is founded on (a) the difference between use-value versus exchange-value aspects of mathematics, and (b) a conception of mathematics as constituted by ‘everyday’ non-mathematical practices informed by its use-value while ignoring its exchange-value (as a system that creates its own knowledge); since mathematical activities are homologized with their mundane optimization of utility, this myth rationalizes its existence by virtue of (1) making relevant connections with its (less-privileged and less-able) pupils, with (2) links that accent the diversity of their background — e.g. culture, language, religion, gender, etc. — and in this sense we see that (3) participation in mathematical activity, as a toolkit and as a gatekeeper, is being allocated to manual labour, and helping to reproduce existing social structures. The myth of emancipation operates through a unification of (a) a pre-recognition of some aspect of reality — e.g. in events, discourses, behaviours, or experiences — as non-mathematical, and followed by (b) the re-description or mathematization of such a previously identified non-mathematical reality now with the use of mathematical terms, and its self-legitimization with (c) the affirmation of the unity of (a) and (b) as mathematical competencies.

94 Vithal and Skovsmose 1997, 138.

95 Following Žižek 1997, 28, multiculturalism is here understood as the archetypal ideology of global capitalism. As a form of imperial colonization by multinational capital, multiculturalism takes a type of (empty) global position. In other words, multiculturalism is a disavowed form of racism with a patronizing distance for the native cultures. It works via a double movement of appreciation and depreciation. This attitude appreciates and respects the multiplicity of cultures; simultaneously, the multiculturalist position depreciates all other specific cultures when it asserts itself, and retains a claim on its privileged stance of (empty) universality.

96 Pais 2011, 226.

97 Nunes, Schliemann and Carraher 1993.

98 Matang 2002, 27.

99 Bhaskar 2013, 4.