Inferentialism in mathematics education: introduction to a special issue
Math Ed Res J
Inferentialism in mathematics education: introduction to a special issue
Arthur Bakker 0 1
Stephan Hußmann 0 1
0 Dortmunder Kompetenzzentrum für Lehrerbildung und Lehr-/Lernforschung, Technical University Dortmund , Emil-Figge-Str. 50, 44227 Dortmund , Germany
1 Freudenthal Institute, Utrecht University , Princetonplein 5, 3584 CC Utrecht , The Netherlands
Inferentialism, as developed by the philosopher Robert Brandom (1994, 2000), is a theory of meaning. The theory has wide-ranging implications in various fields but this special issue concentrates on the use and content of concepts. The key idea, relevant to mathematics education research, is that the meaning of concepts is understood in terms of their role in reasoning practices. In line with the anti-representationalist literature in mathematics education (e.g., Cobb et al. 1992), Brandom explains the meaning of representations in terms of reasoning practices rather than the possibility of reasoning or making inferences on the basis of representations. This view does by no means diminish the significance of representations (signs, diagrams, graphs, symbols…). Rather, understanding how representations come to be is enriched by appreciating that they gain their meaning in human activities in which, as a matter of course, people exercise reason that relies on particular inferences. Normativity is a key idea in inferentialism. Brandom suggests that Bconcepts are broadly inferential norms that implicitly govern practices of giving and asking for reasons^ (Brandom 2009, p. 120). Inferentialism offers a holist rather than an atomistic view on concepts. Formal semantics has mainly been atomistic, explaining complex expressions in terms of independent simpler ones and we see a similar tendency in teaching (noted by Bakker and Derry 2011). Brandom writes:
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By contrast, inferentialist semantics is resolutely holist. On an inferentialist
account of conceptual content, one cannot have any concept unless one has many
concepts. For the content of each concept is articulated by its inferential relations
to other concepts. Concepts, then, must come in packages (though it does not yet
follow that they must come in just one great big one).
(Brandom 2000, p. 15-16;
emphases original)
Brandom (2000)
characterises inferentialism as pragmatist, expressivist, and rationalist.
What people think or express is based on what they implicitly know how to do. We see
a good fit with the view that Bmathematical activity is human activity^
(Lakatos 1976,
p. 146; see also Freudenthal 1973)
. In our perspective, Brandom’s rationalist language
fits well with the inferential nature of mathematics.
What inferentialism is not
Inferentialism is not a learning theory or a pedagogical approach. Inferentialism is also
silent on psychological issues. Yet we think the philosophical debates which may be
connected with inferentialism
(McDowell 1994/1996; Smith 2002; Weiss and
Wanderer 2010)
form an interesting resource with which to approach thorny old issues
in mathematics education research with fresh eyes, in particular where it comes to
epistemological topics such as concepts, knowledge, or reason.
It often takes many years before philosophical ideas trickle through to educational
theory. The history of constructivism is an example of how philosophical ideas through
trends in psychology came to influence educational research and practice. Where
constructivism highlighted the need to take a student perspective on learning,
cultural-historical theories stressed the importance of culture, history, and the social.
Inferentialism—in our view—has the potential to highlight the importance of agency of
the learners who participate in a social practice by giving and asking for reasons—and
thus puts more emphasis on rationality. Reason is not restricted to the explicit and
formalised type of inference that we know of statistical inference or mathematical
deduction. Rather, Brandom’s ideas can be extended to encompass the kinds of implicit
and tacit inferences that are not immediately explicit or expressed in language: Saying
is rooted in doing
(Brandom 2008)
; knowing-that is based on knowing-how
(Brandom
2000)
.
Reason is not purely mental, cold, or disembodied
Brandom’s view on reason may at first sound contradictory to what is often concluded
from psychological research.
Tversky and Kahneman’s (1981)
experiments are renowned
for showing how people’s choices and decisions are often inconsistent and incoherent.
One possible conclusion, regularly drawn in the literature, is that people are irrational, or at
least far less rational than often thought
(Erickson et al. 2013, Chapter 6)
. However, their
experiments also show that people give reasons and feel obliged to give reasons for
whatever choice or decision they make—even if they are contradictory from a logical
perspective. Yes, one could use the term Birrational^ for logically inconsistent reasoning,
but in most (...truncated)