An essay I wrote for my Preliminary examination at Texas A&M University, Fall 2009.
The academic enterprise par excellence is the production of knowledge, an activity that we at the university usually call research. Sharan Merriam (1991), p. 43 defines it as “systematic or disciplined inquiry.” One of the structural characteristics of the university is the tacit presence in research of a hierarchy of value. The highest worth is assigned to experimental science and the lowest to the humanities. This corresponds roughly to how much of the research activity can be quantified, handled in a numerical form that can be graphed, tabulated and statistically analysed, and where the final output can be expressed as a yes/no or true/false dichotomy. Why does this hierarchy of perception of value exist? Or more precisely, why are certain kinds of research ranked higher than others? I see two reasons. The first one is that we value more research that society perceives, correctly or not, to have the greatest financial benefit. Secondly, I notice a correspondence between the ‘remoteness’ of the research from studying what being human is and the value given to the research, which in a certain sense is completely paradoxical. Maybe we are afraid of looking at ourselves and of realizing what kind of society we have created. Do we avoid becoming aware of what choices we make, what really drives us day by day, and what it all may, or may not, mean at the end?
Paradigm and its Axioms
Our minds can not but operate according to a more or less coherent set of beliefs about who we are, where we are, and what we interact with (Denzin & Lincoln, 2005c, p. 183). We can also call it an axiomatic system (Guba & Lincoln, 2005, p. 192) or a net of premises (Denzin & Lincoln, 2005a, p. 22). Researchers during their day to day activities are mostly concerned with their research methods. The methods are the source of the “stuff” their work consists of. However, methods are conceptually at the lowest level of a series of linked concepts that increase in abstractness as we move up from methods to methodology, epistemology, ontology, and lastly axiology. For a recent and thorough discussion of paradigms in education see Howe (2009a), 2009b), Johnson (2009), Bredo (2009), and Tillman (2009).
Briefly, the objective of my proposed study is to document and analyze the justifications given by the national institutions of the United States for the funding of mathematics education. This analysis will be performed in the light of two social conditions. The first one is the effects of the mathematics (and science) scores of the Programme for International Student Assessment (PISA)1 and of the Trends in International Mathematics and Science Study (TIMSS)2, and the second one is existence, as well as stubborn persistence, of the achievement gap.
The work will consist of three parts. In the first one I will build a historical and theoretical framework of the relationship3 between the teaching and learning of mathematics and the perception of the benefit for the state of mathematical knowledge which manifests itself politically in a concrete support of the teaching of mathematics. The second part will analyze public statements about the significance of the PISA and TIMSS mathematics scores for the state and future of the nation. The last part will be similar to the second, but with reference instead to the achievement gap.
Education researchers operate according to a variety of paradigms. However, the main two are positivist/post-positivist and constructivist. According to Merriam (1991), p. 44 the positivist worldview assumes “a single, objective reality – the world out there – that we can observe, know, and measure.” While Denzin and Lincoln (2005b), p. 184 state that axioms of constructivism are “relativist ontology (relativism),” “transactional epistemology,” and “hermeneutic, dialectical methodology.”
In order to identify the paradigm that frames my work, I would like to proceed inductively, by moving ‘upward’ from methods all the way to axiology, and by analyzing each axiomatic component according to the classification provided by Guba and Lincoln (2005), p. 192 and Gough (2000). However, we should note that Guba and Lincoln wrote that the “boundaries between paradigms are shifting” (p. 197).
A research method is a technique for (or a way of proceeding in) gathering evidence. One could reasonable argue that all-evidence gathering techniques fall into one of the following three categories: listening to (or interrogating) informants, observing behaviour, or examining historical traces and records. In this sense, there are only three methods of social inquiry.4
During the first part of my I research will analyze historical data and study the works of the 20th century continental scholars that are relevant to my research objective. These authors are scholars Roland Barthes, Claude Lévi-Strauss, Michel Foucault, Antonio Gramsci, Chantal Mouffe, Pierre Bourdieu, and Slavoj Žižek. Among the historical data that I would like to analyze are treatises, official documents, and correspondence. I intend to find these sources by performing library database searches, and following references from my readings. If possible, the selected text data will be entered into a qualitative data analysis software system.5
Gough (2000) describes methodology as the “reasoning that informs particular ways of doing research, or the principles that inform the organization of research activity” and also as “conceptual framework or the assumptions that guide their research” and “reasons for using such techniques in relation to the kind of knowledge or understanding the researcher is seeking.”
The study of social perceptions as evinced from public statements is obviously a study not of “objects” and not even of “behaviour,” but of ideas and of discourses. Whether the statement itself is true or not is not the issue, and even its meaning is not relevant. I am interested in understanding what counts and what does not, what is perceived as correct and what is not. But I intend to go beyond this. I also want to study what is not said, what is missing, or more precisely, what does not have to be said. This is the domain of “discourse analysis,” especially the types of discourse analysis developed by Michel Foucault.6 Peräkylä (2005), pp. 871-872 explains that according to Foucault a discourse creates subjects and objects that are explored in a historical context through methodologies that Foucault named archaeology and genealogy (Scheurich & McKenzie, 2005). Hook (2001), 2005) explicitly considered the discourse analysis performed by using Foucault’s genealogy and archaeology a ‘methodology.’ 7 Knight, Smith, and Sachs (1990) and Kenway (1990) analyze the truth claims in official state policies that are in a discourse competition with statements from pressure groups and the media. They weave the practice of deconstruction8 with the concepts of hegemony and competing truth claims, dominant ideology and social myths.
Is the study of “How do we know what we know? ” and “How can we increase our knowledge? ” A related question is “How can I be sure that what I believe to be true is actually true? ” In the field of education, the two main epistemologies are post-positivist and constructivist.
Positivist epistemology supposes that research is performed by a “objective, detached observer” (Gough, 2000, Table 1). Moreover, the researchers can be highly confident (positivism) or probabilistically confident (post-positivism) that the results they obtain have a direct correspondence with the components of reality.9
Constructivist thought in the West has been present sporadically from the time of the pre-Socratics. Statements that align themselves with contemporary constructivist philosophy have been expressed by Protagoras10 and Giambattista Vico.11 Contemporary scientists who have contributed mostly to constructivism are Jean Piaget (Dimitriadis & Kamberelis, 2006, pp. 167-177), Lev Vygotsky (pp. 191-199), and Ernst von Glasersfeld (2000). von Glasersfeld describes this epistemology as one that “holds that knowledge is under all circumstances constructed by individual thinkers as an adaptation to their subjective experience.” (p. 4).
As we have seen by moving up from methods (analysis of texts) to methodology (archaeology, genealogy, and deconstruction), we are not working with data that can be measured, manipulated, and tabulated. We are not generating an input for a conceptual device that is able to verify or falsify causal claims.12 With this in mind, we can use Table 8.3 of Guba and Lincoln (2005) to find the type of epistemology of my research. If we search under methodology for what best corresponds to my research, we find “Hermeneutical/dialectical” under the Constructivism paradigm. In addition, the authors connect constructivism, as well as post-structuralism and postmodernism, explicitly with genealogy and archaeology (p. 204).
Does this all make sense with respect to my research questions? How do I obtain my answers following a Foucaultian methodology? Not by ‘finding’ them somewhere ‘out there.’ I do build them in my mind while I analyze text that I have gathered (method) using genealogy, archaeology, and deconstruction (methodology). This process is only intelligible within a constructivist worldview. The answers are indeed ‘my’ answers.
Is the study of “what is.” It asks what the relationship between our mind and our environment is. Historically, philosophy based on the Greek tradition maintains that there is a reality that is independent of our minds. Platonists go even further and claim that even our abstract ideas have a sort of autonomous existence. The modern conception of reality mostly follows the dualism of René Descartes13, who encapsulated this idea in his famous statement “Je pense, donc je suis” (I think, therefore I am) in his Discours de la méthode (1637, Part IV).
There is a different philosophical tradition that arose in India approximately at the same time, Buddhism. One of its main concepts is the theory of emptiness that states that “all things lack intrinsic reality, intrinsic objectivity, intrinsic identity or intrinsic referentiality” (Hixon, 1993, p. xvii). While I have not read it anywhere, I do see a conceptual connection between the theory of emptiness and constructivism.
While Denzin and Lincoln (2005b), p. 184 claimed that the constructivist ontology is “relativist,” but von Glasersfeld (2000), p. 4 stated that constructivism is “without reference to ontology.” In my case I do not think that having a precise ontology is critical for my research. The ‘objects’ of my study are ‘ideas through history.” Except for a Platonist, no one, not even the most positivist realist is going to think that those ideas exist ‘somewhere’ in the universe or elsewhere.
Is the study of values (Gough, 2000), ethics (Denzin & Lincoln, 2005c, p. 183). In the exact sciences it does not, at least explicitly, play any rôle. In social sciences its status is disputed. Positivist social sciences share with the exact sciences the claim that their research process is value-neutral (e.g. Guba & Lincoln, 2005, Tables 8.2-4). This is strongly disputed by social researchers that do not accept the positivist view of their research (Howe, 2009a). It is impossible to do social research and not be concerned about values. Kenneth Howe stated “Just as social research is theory-laden, it is also value-laden.” It is inescapable. After all, the statement “my research is value-neutral” is a value statement.
Looking at ethics from a professional point of view, my work will follow the Standards for reporting on humanities-oriented research in AERA publications (Barone et al.., 2009). Having said that, I want to state that I completely agree with Guba and Lincoln (2005), p. 200 that axiology is “part of the foundational philosophical dimensions of paradigm proposal.” My work intends to assist, even in small measure, the reader to recognize in mathematics education what Pierre Bourdieu calls symbolic violence (Sabour, 1999), that the knowledge it produces is a product to be sold (Lyotard, 1984, p. 4), and that what appears to be natural, normal, a fact of life, is actually a socially inculcated way of thinking. Thus, we can reclaim some of our freedom of action.
The majority of the attention in this paper has been on methodology and axiology, rather than on epistemology and ontology. The reason is that, once we have identified their correct type, which, using the tables in Guba and Lincoln (2005), is not difficult a task, not much can be said specifically about the ontology and epistemology particular to my research project. On the contrary, its methodology concerns the reasons for how to proceed during research, and its axiology gives me guiding principles and an appreciation of the ultimate value of the research itself. The effort I devote to this research project is inspired by the statement of Michel Foucault “To change something in the minds of people-that’s the role of an intellectual.” (Martin, 1988, p. 10).
Ball, S. J. (1990). Introducing monsieur foucault. In S. J. Ball (Ed.), Foucault and education: Disciplines and knowledge (pp. 1-8). New York: Routledge.
Barone, T., Cochran-Smith, M., Howe, K., Perkins, L., Popkewitz, T., Sroufe, G., . (2009). Standards for reporting on humanities-oriented research in AERA publications. Educational Researcher, 38(6), 481-586.
Bredo, E. (2009). Getting over the methodology wars. Educational Researcher, 38(6), 441-448.
Denzin, N. K., & Lincoln, Y. S. (2005a). The discipline and practice of qualitative research. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (3rd ed., pp. 1-32). Thousand Oaks, CA: SAGE Publications.
Denzin, N. K., & Lincoln, Y. S. (2005b). Paradigms and perspectives in contention. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (3rd ed., pp. 183-190). Thousand Oaks, CA: SAGE Publications.
Denzin, N. K., & Lincoln, Y. S. (Eds.). (2005c). The SAGE handbook of qualitative research (3rd ed.). Thousand Oaks, CA: SAGE Publications.
Dimitriadis, G., & Kamberelis, G. (2006). Theory for education. New York: Routledge.
Glasersfeld, E. von. (2000). Problems of constructivism. In L. P. Steffe & P. W. Thompson (Eds.), Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld (pp. 3-9). New York: Routledge.
Gough, N. (2000). Methodologies under the microscope. Paper presented at DUPA research conference.
Guba, E. G., & Lincoln, Y. S. (2005). Paradigmatic controversies, contradictions, and emerging confluences. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (pp. 191-215). Thousand Oaks, CA: SAGE Publications.
Hixon, L. (1993). Mother of the Buddhas: Meditation on the Prajnaparamita Sutra. Wheaton, IL: Theosophical Publishing House. (Foreword by Robert A.F. Thurman, PhD)
Hook, D. (2001). Discourse, knowledge, materiality, history: Foucault and discourse analysis. Theory & Psychology, 11(4), 521-547.
Hook, D. (2005). Genealogy, discourse, ‘effective history’: Foucault and the work of critique. Qualitative Research in Psychology, 2(1), 3-31.
Howe, K. R. (2009a). Positivist dogmas, rhetoric, and the education science question. Educational Researcher, 38(6), 428-440.
Howe, K. R. (2009b). Straw makeovers, dogmatic holism, and interesting conversation. Educational Researcher, 38(6), 463-466.
Johnson, R. B. (2009). Toward a more inclusive “Scientific research in education”. Educational Researcher, 38(6), 449-457.
Kenway, J. (1990). Education and the Right’s discursive politics: Private versus state schooling. In S. J. Ball (Ed.), Foucault and education: Disciplines and knowledge (pp. 167-206). New York: Routledge.
Knight, J., Smith, R., & Sachs, J. (1990). Deconstructing hegemony: Multicultural policy and a populist response. In S. J. Ball (Ed.), Foucault and education: Disciplines and knowledge (pp. 133-152). New York: Routledge.
Lyotard, JF. (1984). The postmodern condition: A report on knowledge. Minneapolis, MN: University of Minnesota Press.
Martin, R. (1988). Truth, power, self: An interview with Michel Foucault – October 25, 1982. In Technologies of the self: A seminar with Michel Foucault (pp. 9-15). Amherst: University of Massachusetts Press.
Merriam, S. B. (1991). How research produces knowledge. In J. M. Peters & P. Jarvis (Eds.), Adult Education Quarterly (pp. 42-65). Lanham, MD: Jossey-Bass.
Peräkylä, A. (2005). Analyzing talk and text. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (pp. 869-886). Thousand Oaks, CA: SAGE Publications.
Plato. (1921). Plato in twelve volumes, Vol. 12. Cambridge, MA: Harvard University Press. (Translated by Harold N. Fowler)
Sabour, M. (1999). [review of the book The state nobility: Elite schools in the field of power]. Acta Sociologica, 42, 188-190.
Scheurich, J. J., & McKenzie, K. B. (2005). Foucault’s methodologies. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (3rd ed., 33). Thousand Oaks, CA: SAGE Publications.
Stanford Encyclopedia of Philosophy. (2009). Giambattista Vico. Retrieved October 10, 2009, from http://plato.stanford.edu/entries/vico.
Tillman, L. C. (2009). The never-ending education science debate: I’m ready to move on. Educational Researcher, 38(6), 458-462.