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).
Tags: language, learning, mathematics, philosophy, semiotics, Wittgenstein
