Learning, commognition and mathematics [Discursive learning as the primary source of all things human] by Anna Sfard

https://www.academia.edu/9739485/Learning_commognition_and_mathematics

This looks to have been published in the SAGE Handbook of Learning.

Sfard outlines the history of behaviourism, acquisitionism and participationism in theories of learning.

It was with this latter issue in mind, among others, that some researchers, especially those coming from the sociocultural tradition initiated by Vygotsky, renounced speculating about what was happening inside the human head and shifted their attention to its “outside”  – to what people say and do while going about their daily affairs. The resulting participationist line of inquiry brought the study of learning much closer to the pole of the uniquely human, but its nonstandard methods left many traditional thinkers unconvinced about its “scientificity”.

Its methodological struggles notwithstanding, participationism may be considered as the best option, so far, for those who want to know what makes human learning special. Above all, it  seems to have succeed in accounting for a centrally important trait of human learning that the former approaches left unexplained: it provided an answer to the question of how it happens that human activities evolve and grow in complexity from one generation to another.p. 2-3

The term discursive learning can be interpreted as referring to the activity of becoming able to tell and produce ever new stories about the world, and can thus be seen as the participationist counterpart of the acquisitionist term knowledge building. As such, it can be contrasted with practicallearning,[1] the aim of which is to reorganize and transform concrete objects.

[1] In choosing this name, I was inspired by the Vygotsky’s term “practical intelligence”, which he contrasted to “sign use” (Vygotsky, 1978, p. 24). p. 4

She writes in terms of mathematical narratives and stories, which is interesting.

The idea that both scientific research and school-type learning can be thought of as dealing with discursive activities has its roots in two seemingly independent developments. First, it was made quite explicitly by postmodern philosophers.  Consider, for instance,  Lyotard’s declaration that “scientific knowledge is a kind of discourse” (Lyotard, 1979, p. 3),  […]

In the second half of the 20th century, such were the voices of thinkers interested in societal phenomena and historical changes, that is, in societal learning. The idea that also school-type learning is an activity of shaping and extending communication was born even earlier, albeit without being presented in so many words. […] both [Wittgenstein and Vygotsky] repeatedly stressed the inseparability of thought and its “expression”, either verbal or not. Wittgenstein vehemently rejected the view of  thinking as “incorporeal process which lands life and sense to speaking, and which it would be possible to detach from speaking” (§339, Wittgenstein, 1953, p. 109)… p. 6

Now I wish to stress the all-important contribution of the verbal mode: It is mainly thanks to the linguistic ingredient of our communication that our discourses function as means for preserving the results of change and accumulating complexity.  Below I argue that our capacity for piling one complex innovation upon another stems mainly from the reflexivity of the verbal communication, that is, from our capacity for talking about anything, including the talk itself.

The first fact to consider while trying to explain and substantiate this last claim is that the development of mathematics has been guided all along the way by the mathematicians’ attempts to say as much as possible in as little words as possible. The best discursive means for saying more with less is the discursive construct known as mathematical object, one that appears when we replace talk about processes with talk about things. This way of talking, however, should not be taken for granted. At a closer look, number words do not begin their existence as signifying objects. These words are first learned by the child as mere labels to be used in the processes of counting. At this point, they are hardly a part of speech, as they do not appear in full sentences. For the child, to reify number means to start using these words within the same language structures as those we apply while speaking about independently existing concrete objects. p. 9

One  other important conclusion that can be taken from what has been said so far is that numbers, and in fact all other abstract objects, are discursive construct, that is, emerge from out talk rather than preceding it. p. 10

Two levels of discursive learning. The changes that happen as a result of discursive learning can be divided into two categories, object-level and meta-level. In mathematics, object-level development is one that expresses itself in the expansion of what is known about the already existing universe of mathematical objects.  Object-level growth, therefore, is mainly accumulative.  In contrast, meta-level developments are those that change the rules of the discursive game rather than simply alter the amount of endorsed narratives. This kind of learning happens when some newly introduced mathematical objects engender apparent contradictions with previously endorsed narratives. p. 10-11

In contrast, meta-level learning is supposed to lead to a change that cannot be attained by pure logic.  There is an element of contingency and of human choice in every meta-level transformation. For instance, while proceeding from unsigned to signed numbers, the mathematicians had to decide which properties of numbers that had been in force so far should be preserved and which of them could be compromised.  Historically, these decisions were hard going and time consuming, and when eventually made, they were grounded in the mathematician’s strong intuitions with regard to their prospective advantages. These intuitions were byproducts of the decision-maker’s discursive experience. p. 12

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s