Paradigms are overarching frameworks that shape our whole approach to being in the world (Kuhn 1962): they shape our perceptions, conceptions, and actions. Kuhn (1962), for instance, showed that normal scientific research takes place within such an overarching framework and that various forces work to cohere (consciously and unconsciously) the fundamental philosophical assumptions of the framework. However, from time to time the overarching framework shifts in revolutionary fashion as new philosophical assumptions are used to make better sense of particular phenomena.
Nowadays various such overarching frameworks underpin mathematics education research. The purpose here is not to judge, but to elucidate the philosophical assumptions of these paradigms. The overarching frameworks under consideration in this section are the positivist paradigm and its successor the post-positivist paradigm (that are often referred to as modernist worldviews) as well as the interpretive paradigm and the transformative paradigm (that are often referred to as post-modernist worldviews).Footnote 1
Higginson (1980) argued that
[a]ll [human] intellectual activity is based on some set of assumptions of a philosophical type. […] Reduced to their essence these assumptions deal with concerns such as the nature of ‘knowledge’, ‘being’, ‘good’, ‘beauty’, ‘purpose’ and ‘value’. More formally we have, respectively, the fields of epistemology, ontology, ethics, aesthetics, teleology and axiology. More generally we have issues of truth, certainty and logical consistency. (p. 4)
For the purposes of this chapter, the focus is on the ontological, epistemological, axiological, and methodological bases of a paradigm. In summary:
the ontological base concerns issues about the nature of reality or being in itself (e.g., ‘What is the nature of reality?’);
the epistemological base concerns issues about the nature of knowledge (including what forms of knowledge are considered as ‘scientific’) (e.g., ‘What is the nature of knowledge and the relation between the knower and what can be known?’);
the axiological base concerns issues about values and ethics (e.g., ‘What knowledge is intrinsically worthwhile and what is it about it that is valuable as an end in itself?’); and
the methodological base concerns issues about ways of studying phenomena in the world (e.g., ‘How can the knower obtain knowledge?’).
These four methodological bases are intricately related and mutually informing.
In the following subsections, the (post-)positivist, interpretive, and transformative paradigms are contrasted on the basis of their ontological, epistemological, axiological, and methodological assumptions. To do so, the more radical philosophical assumptions within each paradigm are foregrounded, not only because the controversies concerning their intellectual legitimacy often take place at the edges of those paradigms, but also because those edges are the intellectual, theoretical, and practical space for dialogue.
Table 27.1 lists assumptions that the (post-)positivist, interpretive, and transformative paradigms make about the nature of reality (ontology), the nature of knowledge (epistemology), the value of knowledge (axiology), and the nature of inquiry (methodology).
27.2.1 (Post-)Positivist Paradigm
Positivism and its successor, post-positivism, view the world as external to humans and independent of human experience. This view relies on the existence of reliable knowledge about the world that research strives to gain. In this perspective, there is a world independent of human experience, containing objects which behave in accordance with a set of natural laws. Positivist researchers can discover knowledge about these real things (knowledge that is certain, valid, and accurate), and determine the mechanisms and relations governing their behavior, through the use of the scientific method of reason, logic, and empirical inquiry (that is, experimentation and measurement of what can be observed). As there are many critical human phenomena that are not observable (e.g., mathematical thinking), post-positivists reject the positivist position that what can be studied is limited to the observable. Post-positivists are similar to positivists in that they believe the social world, like the natural world, can be studied in a value-free way that provides causal explanations for phenomena (Phillips and Burbules 2000). However, they differ from positivists in their belief that researchers should base claims regarding truth on probability rather than certainty.
In both a positivist and post-positivist view, researchers strive for uncovering a single view of reality, a reality that is separated from the mind: the rational researcher can come to know the objective world by employing analytical thought and experimental methods. This is the cornerstone of a modern worldview concerned with objectivity, prediction, generalizability, linearity, and absolute truth (Harvey 1990).
Ontology Post-positivists hold that the world is real, is structured, and that this structure can be modeled. The positivists hold that a ‘real’ reality exists and that it is the researchers’ job to discover that reality (naïve realism) (Guba and Lincoln 1994). Post-positivists concur, but add the qualification that reality can merely be known imperfectly because of the human limitations of researchers (critical realism) (Maxwell 2012). Therefore, researchers can discover reality only within a certain degree of probability. By eliminating alternative explanations for phenomena, they can strengthen existing theories that account for these phenomena, but never prove them beyond doubt.
Epistemology (Post-)positivists hold that there is an objective reality that researchers are expected to ‘mirror’ or ‘replicate’ in their models and theories. The role of research is to discover or uncover the real world and its structure. This position assumes that knowledge about the world is (or should be) objective and scientific findings can be determined reliably and validly, given that biases and values of the researcher and others involved in the research process are eliminated.
Axiology The (post-)positivist paradigm regards knowing the ‘truth’ in propositional form as an end in itself—and as the only end in itself (Lincoln et al. 2011). The role of the researcher is to be as objective as possible in order to ensure that scientific findings are obtained through a neutral process; that is, the research process is seen as largely apolitical and separate from a world of individual and group interests. The researcher has an ethical obligation to conduct research that is intellectually honest, suppresses personal bias, and avoids harm. Such research should entail careful collection and accurate reporting of data, as well as candid evaluation of the limitations of the study.
Methodology In general, researchers in this paradigm assume that they can obtain an accurate portrait of the ‘true’ nature of reality through the scientific methods of reason, logic, and empirical inquiry. The research designs used to accomplish this goal are largely deductive, with an emphasis on determining which variables explain or predict outcomes for a phenomenon of interest. Positivists borrow their experimental methods from the natural sciences. Post-positivists, in contrast, modify these methods in order to apply them to people, developing quasi-experimental methods. These predominantly quantitative methods privilege experimental, randomized-sample, hypothesis-testing studies.
27.2.2 Interpretive Paradigm
The interpretive paradigm (or constructivist paradigm) views all meaning, including the meaning of research findings, as fundamentally interpretative. This paradigm assumes that all knowledge and meaning is constructed by those active in the research process (including participants and observers) and that researchers should attempt to understand the complex world of lived experiences from the viewpoint of those who live it (Schwandt 2000). The interpretive paradigm emphasizes that research is a product of the theories and values of researchers and cannot be independent of them. As Schoenfeld (2007) underlined: “One’s explicit or implicit theoretical biases frame what one looks at, how one characterizes it, how one analyzes it, and how one interprets what one has analyzed” (p. 93).
Ontology Interpretivists reject the notion that there is one ‘real’ reality that can be known, but take a relativistic stance: that is, reality is a social construction that is experienced subjectively by different individuals. As there might be multiple socially constructed realities, “truth is the best-informed construction about which there is presently consensus” (Galbraith 1993, p. 74). Researchers in this paradigm try to understand the knowledge of others, as the others perceive it. In its most trivial version, interpretivism (or constructivism) takes the stance that the mind is active, not passive, in the construction of meaning and knowledge (von Glasersfeld 1995). Interpretivists do not so much discover knowledge but construct it.
Epistemology As there are multiple possible versions of reality (depending on the perspectives and values of individuals) in the interpretive paradigm, this paradigm replaces the concept of objectivity that is prominent in the (post-)positivist paradigm with the acknowledgement of subjectivity, inter-subjectivity, and live truth (i.e., truth in human terms) (Ernest 1998). The interpretive paradigm “challenges the traditional projection of epistemology, that of identifying a universal method for determining whether a particular theory or conceptual scheme matches or corresponds with external reality” (Cobb 2007, p. 10). The goal instead is, “to identify the variety of constructions that exist and bring them into as much consensus as possible” (Guba 1990, p. 26).
Axiology Interpretivists concern themselves with the meaning people derive from social interaction (Bryman 2016). The interpretive paradigm recognizes that experience is shaped by culture and context and filtered through individuals, and that there is not a singular reality that can be captured through research. Thus, multi-perspectival knowing is valuable as a means for balancing representations of diverse views. Interpretivists differ from (post-)positivists in that their model of ‘reality’ is contextual and situational. As such, alternative criteria are used to assess the validity of results, such as trustworthiness and authenticity.
Methodology In order to obtain contextual knowledge and to create a shared sense of reality, researchers in this paradigm typically use inductive research designs, which provide opportunities for findings to reflect context-specific, constructed meanings. Ethnography, case studies, and mostly qualitative forms of inquiry are used in this paradigm, with attempts made to better interpret meaning by obtaining, comparing, and contrasting multiple perspectives. The resultant exchange of conflicting ideas forces the reconsideration of previous positions and assists in the triangulation of multiple viewpoints (Ernest 1998).
27.2.3 Transformative Paradigm
The transformative paradigm (or critical paradigm) arose in response to the historical disadvantage, oppression, or discrimination faced by individuals belonging to minority groups, and seeks the intellectual, ideological, and spiritual liberation of such individuals (Tyson 2015). As such, it centers on the lives and experiences of those who experience oppression, discrimination, or inequality, including the following: women; people of color; immigrants; indigenous and postcolonial people; lesbian, gay, bisexual, transgender, and queer individuals; members of minority religious groups; and people with disabilities. Transformative researchers place priority on empowering those without power to bring about social transformation (see Horkheimer 1972; Mertens 2009).
The philosophical basis of the transformative paradigm is diverse, reflecting multiple approaches, theories, and positions represented in that paradigm, such as critical theory, critical race theory, feminist theory, queer theory, disability theory, and Indigenous theory (see Tyson 2015). For instance, feminist theory studies the systems and means employed by one group of people to structure and legitimize their domination of another group of people, and the strategies employed by the latter to resist this domination (see Hesse-Biber 2014; Lather 1991). Kaiser and Rogers (1995), drawing on McIntosh’s (1983) phase theory of curriculum reform, argued for “loosen[ing] curriculum from a male-dominated, Eurocentric world view and to evolve a more inclusive curriculum to which all may have access” (pp. 1–2). On the other hand, queer theory challenges the binary notions of male and female that facilitate dichotomous conceptions of gender and sexual identity (see Dodd 2009; Mertens et al. 2008).
Ontology Similar to the interpretive paradigm, the transformative paradigm recognizes multiple versions of what is perceived to be real. However, those working within a transformative paradigm argue that “those working within an interpretative framework are too passive in that the framework itself is not critically examined for distortion and bias, i.e., crucial problems of conflict and change, are passed by through the acceptance of existing reference points” (Galbraith 1993, p. 76). That is, the transformative paradigm stresses that accepting different perceptions of reality as equally legitimate is dangerous, because it ignores the damage done by the social, political, cultural and economic factors that help privilege one version of reality over another. Besides, the transformative ontological position emphasizes that what seems ‘real’ may instead be an abstraction that was reified due to the influence of social and historical factors. Thus, before accepting something as ‘real’, those using the transformative paradigm critically examine that thing’s role in perpetuating oppressive social structures and policies.
Epistemology The transformative epistemological assumption centers on the meaning of knowledge, as viewed through a variety of cultural lenses, and the power issues involved in determining what knowledge is legitimate. Objectivity in the transformative paradigm is achieved by reflectively examining the values and social positions implicit in research questions, hypotheses, and definitions.
Axiology The transformative paradigm emerged as a consequence of dissatisfaction with research conducted within other paradigms, which was perceived to be irrelevant to, or a misrepresentation of, the lives of people who experience oppression, discrimination, or inequality. Valuable research in this paradigm is defined by its fostering of social justice and human rights, and the role of a researcher is to be an agent of prosocial change. Transparency and reciprocity are essential principles of the axiological position in the transformative paradigm. An explicit connection is made between the process and outcome of research and the fostering of a social agenda.
Methodology Transformative researchers use a diversity of methodologies to understand and analyze the experiences of study participants. Many use qualitative methods such as critical hermeneutics, as well as reflexive and deconstructive ethnography (see Kincheloe and McLaren 2002): critical hermeneutics seeks to understand how research works to maintain existing power relations and reflexive and deconstructive ethnography seeks “to free the object of analysis from the tyranny of fixed, unassailable categories and to rethink subjectivity itself as a permanently unclosed, always partial, narrative engagement with text and context” (Kincheloe and McLaren 2002, p. 121).
While some researchers working within this paradigm use quantitative and mixed methods, they stress the importance of being cautious in following existing methods to avoid racist, sexist, or otherwise biased results. Despite some variety, a common theme in transformative methodology is the inclusion of diverse voices from the marginalized. This inclusion takes the form of the involvement of participants in all stages of the research process, including planning, conducting, analyzing, interpreting, using, and benefiting from research.