Complexity

This afternoon, I went to a seminar given by Brent Davis (http://brentdaviscalgary.appspot.com/); currently visiting the UK from the University of Calgary. He was speaking about complexity thinking – and ways in which this approach can be applied to the study of the knowledge that mathematics teachers need for teaching.

The main point of interest for me was the idea that teachers do not necessarily know what they know. Even the very best teachers, who create the most valuable learning opportunities for chilfdren, do not draw on their mathematical knowledge in a conscious or systematic way. This is partly due to the fact that teachers of maths think in a very different way to children learning maths for the first time.

Brent gave some example of work he has done with teachers using Concept Studies – a means of revealing tacit knowledge about mathematical concepts. One example given of a Concept Study involved mulitplication. There are a number of ways to conceive of multiplication; as repeated addition, as grouping, as an area or array, as the compression or stretching of a number line, as a linear function, and so on. Different conceptions lead to very different implications for other aspects of mathematical thinking. One exmaple given was that the number 1 is a prime number under some conecptions but not under others.

So part of the difficulty of teaching and learning mathematics comes from the fact that, while teachers might be drawing on a variety of conceptions of mathematical objects and processes (even if not in a conscious way), children are often not aware either of what these conceptions are, which are appropriate for the task at hand, and what implications each might have. Brent was saying that the conduct of Concept Studies can transform a teacher’s practice by revealing these aspects of tacit knowledge – a teacher’s awareness of the metaphors being used, and an awareness of the extent to which particular conception are or are not consistent with particular tasks can tranform the experience of a learner.

This reminded me of my work on interdisciplinarity. Part of the difficulty of interdisciplinary research consists in the tacit assumptions, conventions and traditons of the discipline that a researcher carries with them. The reason that this is problematic is that it is often the case that different researchers carry tacit content that is not compatible with that of other researchers. Successful interdisciplinary work must therefore involve some revelation of this tacit content.

CERME paper

Here is a link to a paper I will be presenting at CERME7 next month… http://www.cerme7.univ.rzeszow.pl/WG/16/CERME7_WG16_Jay.pdf

It follows from the seminar I gave at the LSRI in Nottingham in May last year, and focuses on a linguistic analysis of apparently dichotomous perspectives (individual- and social-centred) in mathematics learning research.

Mathematical objects are social objects

During a conversation with Matthew Inglis, I said that I wanted to think of mathematical objects as social objects. This was in response to his saying that I had been a bit hard on mathematical Platonists during my seminar at the LSRI a few months ago. So the next step is going to have to be to explain what I mean exactly by that.

I know that I don’t mean the same as some philosophers do when they refer to Psychologism – the idea that mathematical objects are identical with particular biological occurrences in the human brain.

What I mean is most easily expressed with reference to the semiotic network that I’ve talked about before. Let’s start by assuming a post-structuralist account of meaning – that the meaning of a signifier consists in that signifier’s relationships with other signifiers.  I like this as a start, because it not commit one to any particular research perspective (cognitive, socioculturalist, etc.).

So… mathematical objects have meaning by virtue of their relationships with other mathematical objects. On the surface, this isn’t difficult to accept – the meaning of ’3′ has a lot to do with it’s relationship with ’1′, ’2′ and ’4′, for a start. But these relationships are socially defined – and that’s the different bit.

So… the meaning of ’3′ consists in the signifiers that are activated within a community in response to the perception of ’3′.  What exactly does this mean? I think this means 1 of 3 things: it’s either the union of all activated signifiers of all members of the community, or it’s the intersection of activated signifiers (activations common to all members), or it is somewhere between those two definitions. Whichever of those is the case, the social nature of the definition is clear. At the moment, I haven’t given a huge amount of thought yet to the question of where the balance is – exactly how the meaning is defined and what set of relationships constitutes the meaning of a given signifier. I have a bit of a sense that maybe it’s the wrong question to ask anyway – maybe unanswerable and maybe unaskable.

So that’s what I mean when I say mathematical objects are social objects. I’m thinking about what the implications of a definition like this are though. And one of things that I’m thinking is that in practice, a ‘social objects’ account of mathematics isn’t very different from a Platonic account.

A semiotic framework for talking about learning

This is based on my talk at the LSRI (see previous blog post). The development of a semiotic theoretical framework was my response to difficulties in working and researching in an interdisciplinary way on the subject of learning. The difficulties are primarily those of incompatible discourses (often and incorrectly, in my view, conceived of as arising due to incommensurable theoretical perspectives), and thus best addressed using a late-Wittgenstein-language-game or Derridean-deconstruction approach.

So the approach is to conceive of all thought as representations of a semiotic network. There are signifiers, and there are relations amongst signifiers. Everything that we do that we might want to call ‘thought’ or ‘learning’ involves the representation of some part of this semiotic network. The network extends within and between human minds, and includes social/cultural objects and physical artefacts.

My basic claim is that if we stick to this level of description, then there is no incompatibility of discourse, whether we want to talk about research that has been done within traditions such as cognitivism or socioculturalism, or whether we want to talk about the learning of the individual or of the group.

More on this in the near future

Platonism and Reification

I have been thinking about mathematical platonism, and wondering if there is a relationship between asserting the existence of  mathematical objects and asserting the existence of mental objects (as in traditional cognitive psychology research). It seems to me that there is an interesting connection between the two, that might help shed light on both sides.

What is a mathematical object? Mathematical platonists (or realists) claim that mathematical objects really exist, as abstract objects. There are various opinions regarding the means by which we can come to believe things about them, and how we can or can’t justify those beliefs.

What is a mental object? Cognitive psychology researchers claim that mental objects really exist. Examples include such things as the mental number line (e.g. Dehaene, Butterworth), the central executive (Baddeley) and the principles of counting (Gelman & Gallistel).

Let’s assume that no-one asserts the existence of objects without good reason. My interpretation of the motivation for mathematical platonism is that it gives us some easy answers to some questions, such as: Why is there such a close relationship between mathematics and the natural sciences? My interpretation of the motivation for the assertion of the existence of mental objects is that they help to explain some observable phenomena, such as children’s developing abilities in solving mathematical problems, and limitations in people’s reasoning abilities. The reification of both mathematical objects and mental objects serves the purpose of a kind of argumental placeholder. There’s no reason to say that they really exist is there?

LSRI seminar 18th May 2010

I enjoyed my trip to Nottingham a great deal. The main reason for going was to present a seminar on my thoughts about mathematics learning. I have been thinking about the difficulties inherent in asking and answering some questions about learning, given the dichotomy of cognitivist and socioculturalist theory. The seminar represents an early attempt to develop a theoretical framework that allows the asking and answering of some more interesting questions than can be asked from either the cognitivist or the socioculturalist perspective. A video of the seminar is available here: mms://resources.lsri.nottingham.ac.uk/Seminar_Archive/Tim_Jay180510.wmv – by the way, the archive of visiting speaker videos available at http://portal.lsri.nottingham.ac.uk/Seminars/Lists/Events/Archived%20Events.aspx is a great resource.

The name and the concept

I have been thinking about the importance of the philosophy of mathematics when trying to understand the way in which children learn mathematics. Wittgenstein has shown that names come before concepts. That is, we learn how to use the names for things as part of our natural language (through participation in language games), and then at some later time, that name becomes attached and related to other names and other concepts… and thus becomes a signifier of an increasingly rich concept. Take the example of ‘straight line’. It’s first use, for the child, is for describing certain drawn lines (on paper, or on a blackboard, with crayons or chalk) in response to an adult/other drawing and describing a straight line. At a later time, the child will learn (by example, possibly, or by correction of activity by an adult/other) that actually, a straight line can be differentiated from other lines by the fact that it was drawn using a ruler, or other straight edge. Some time after this, possibly, the child (probably as a result of formal mathematics instruction) will learn that a mathematical straight line is the shortest distance between two points, and that a line has no thickness (discovering at this time that it’s not actually possible to draw this sort of thing).

So discussions of questions like, “does conceptual or procedural knowledge come first in mathematics learning?”, seem rather misguided. The reason it’s a difficult question to answer is that it refers to a false dichotomy. What comes first is the ability to use the words in a way that allows the child to participate in language games with others (note the similarity of this with e.g. Lave’s description of situated learning – although Lave’s focus is on activity, as opposed to my focus on language). The child’s mathematical meaning-making comes later, as he/she becomes able to re-present that word/set of words incorporated into increasingly rich networks.  Using the words in a way that allows participation in language games is not either procedural or conceptual knowledge, as mathematics education researchers refer to them. In fact, ‘procedural knowledge’ and ‘conceptual knowledge’ might be best thought of as things that are only ever visible to the observer rather than the subject. It appears as if a child is behaving in accordance with (or as a result of) some combination of procedural and conceptual knowledge, but these are constructs extracted/abstracted by the observer, not constructs that exist in the mind of the subject (note that a subject can be their own observer – giving rise to reflexivity).

Interdisciplinarity in research on learning, pt.2

I am wondering again about the construct ‘interdisciplinarity’. Is there something of a paradox inherent in ‘interdisciplinary research’? What status do the results/findings of interdisciplinary research have? What body of knowledge do they contribute to? If they are to contribute to a new body of knowledge – then then constitutes the formation of a new discipline, and the work is no longer interdisciplinary. If they are to contribute to the multiple (more than one) disciplines that the research has drawn upon (probably rewritten with a veneer of disciplinary legitimacy), then the research was not interdisciplinary, but multidisciplinary. There’s the paradox.

A way to escape the paradox is to think of a level above that of disciplines and boundaries between them. Questions that demand contributions from more than one discipline (like lots of questions we want to ask of learning) are better asked from an adisciplinary standpoint. Which is to say that I think the content of research of learning and suchlike (research questions and their answers) might altogether be best thought of as adisciplinary – it is research method (with the incumbent demand for validation by an expert community) for which the community structure of disciplines is required.

Interdisciplinarity in research on learning

It seems like a tautology to say that research on learning must be interdisciplinary. Or at least that in order to work towards a true understanding of learning, the inquirer must draw on findings from more than one discipline. Disciplines including Psychology, Philosophy, Computer Science and Education all have spent sizable proportions of their time and effort adding to our knowledge of learning. Isn’t it trivial to say that any statement regarding learning ought to be valid in the terms of all these disciplines?

There are 2 main problems, though, in building an interdisciplinary picture of learning research…

1) Lack of shared language… This is the one that people always talk about. All these disciplines often use different words to talk about similar things, and similar words to talk about different things. This in itself makes interdisciplinary communicati0n difficult. But there is a deeper issue too… that is that difference in language is representative of difference in underlying (often tacit) sense of the ways in which research is (or ought to be) done and of the way in which a body of knowledge is (or ought to be) constructed.

2) Where to stop… This is possibly more tricky. I am wondering about the question, “does an interdisciplinary researcher/theorist need to be expert in all of the disciplines on which they want to draw?” I am leaning towards the answer, “yes”, because if that’s not the case that what status exactly does interdisciplinary theory have? In tandem with this question, there is the question of what disciplines to include in our interdisciplinary theorising (and the converse – which to exclude).  Can we, with any intellectual honesty, exclude any discipline that has generated knowledge of learning from our interdisciplinary theorising about learning? And if we can’t, do we have to be expert in all those disciplines? If the answer to this last question is another “yes”, then it would seem that this is not a job for individuals. Maybe that’s why the efforts of e.g. John Anderson (“How can the human mind occur in the physical universe”) and Anna Sfard (“Thinking as communicating: Human development, the growth of discourses, and mathematizing”) don’t really work for me as really satisfactory statements of learning theory.

Seminar at LSRI, May 18th

I sent a title & abstract for this seminar this week. It’s at LSRI labs, Exchange Building, Jubilee Campus, University of Nottingham at 5pm, 18th May. Hosted by Midland Mathematics Education Seminars. It will be streamed live, and available online after the event – more details @ http://portal.lsri.nottingham.ac.uk/Seminars/default.aspx

Title: Critiquing research on children’s learning of mathematics

Abstract: Research in mathematics learning is a diverse field, drawing on a large number of influences and disciplines. This interdisciplinarity seems like a good thing, until we come to try and make sense of questions that are either located at, or cross, disciplinary boundaries. A particularly difficult boundary to work at is that between ‘the individual’ and ‘the group’. A large number of theorists and commentators have between them established a rhetoric of cognitive v. sociocultural, acquisition v. participation and individual v. social. The starting point for this seminar will be a description of some of the difficulties that can be encountered when talking about children’s learning of mathematics – focusing in particular on this boundary between individual and group learning.

The rest of the seminar will deal with the question: what happens if we reject the individual/group dichotomy? Can we develop an approach to the research of mathematics learning that is genuinely interdisciplinary? And would this be a useful thing to do?

This is very much a work in progress, and hopefully will stimulate some interesting discussion. Drawing together my own experience in psychology, philosophy, computer science and education, I’m looking forward to the opportunity to discuss the issues in this seminar with an LSRI audience.