Commentary on Theories of Mathematics Education

8 B. Sriraman and L. English
At the outset, it is also important to remind the community that Jean Piaget (cf.
Piaget 1955) started from Emmanuel Kant’s paradigm of reasoning or “thinking”
and arrived at his view of cognition as a biologist viewing intelligence and knowledge
as biological functions of organisms (Bell-Gredler 1986). Piaget’s theories
of knowledge development have been interpreted differently by different theorists,
such as von Glasersfeld’s notion of radical constructivism (von Glasersfeld 1984,
1987, 1989) or viewed through its interaction with the theories of Vygotsky by theorists
like Paul Cobb and Heinrich Bauersfeld as social constructivism. However
another major influence on these theories of learning and developing a philosophy
of mathematics of relevance to mathematics education is Imre Lakatos’ (1976) book
Proofs and Refutations (Lerman 2000; Sriraman 2009a). The work of Lakatos has
influenced mathematics education as seen in the social constructivists’ preference
for the “Lakatosian” conception of mathematical certainty as being subject to revision
over time, in addition to the language games à la Wittgenstein “in establishing
and justifying the truths of mathematics” (Ernest 1991, p. 42) to put forth a fallible
and non-Platonist viewpoint about mathematics. This position is in contrast
to the Platonist viewpoint, which views mathematics as a unified body of knowledge
with an ontological certainty and an infallible underlying structure. In the last
two decades, major developments include the emergence of social constructivism
as a philosophy of mathematics education (Ernest 1991), the well documented debates
between radical constructivists and social constructivists (Davis et al. 1990;
Steffe et al. 1996; von Glasersfeld 1987) and recent interest in mathematics semiotics,
in addition to an increased focus on the cultural nature of mathematics. The
field of mathematics education has exemplified voices from a wide spectrum of disciplines
in its gradual evolution into a distinct discipline. Curiously enough Hersh
(2006) posited an analogous bold argument for the field of mathematics that its associated
philosophy should include voices, amongst others, of cognitive scientists,
linguists, sociologists, anthropologists, and last but not least interested mathematicians
and philosophers!
Imre Lakatos and Various Forms of Constructivism
Proofs and Refutations is a work situated within the philosophy of science and
clearly not intended for, nor advocates a didactic position on the teaching and
learning of mathematics (Pimm et al. 2008; Sriraman 2008). Pimm et al. (2008)
point out that the mathematics education community has not only embraced the
work but has also used it to put forth positions on the nature of mathematics
(Ernest 1991) and its teaching and learning (Ernest 1994; Lampert 1990;
Sriraman 2006). They further state:
We are concerned about the proliferating Lakatos personas that seem to exist, including a
growing range of self-styled ‘reform’ or ‘progressive’ educational practices get attributed
to him. (Pimm et al. 2008, p. 469)
This is a serious concern, one that the community of mathematics educators has
not addressed. Generally speaking Proofs and Refutations addresses the importance