This book is wonderful. Unfortunately (for me), he’s done a lot of the things I originally wanted to do with my own research, down to using the same examples! But fortunately (for me), he’s produced a very useful resource with some great ideas in. He cites Lakoff and Johnson, but Lakoff and Nunez do not cite him, which is unfortunate as he provides some interesting support for their ideas. Pimm inspired a whole school of thought which includes de Freitas and Sinclair’s work, Gerofsky, Rotman (I think) and more.
1. Mathematics as a language? Pimm begins by introducing the idea of mathematics as a language, noting the criticisms and emphasising that he is framing this as a metaphor. He talks about the syntax of expressions, commutativity, etc. etc.
2. Pupils’ mathematical talk. This is about pupils’ speech in the classroom; math talk organises thought. The teacher suggests that they try to explain what they’ve done for someone who isnt there.
3. Overt and covert classroom communication. Some classroom activity is about establishing terms; section vs. sector. Who is the ‘we’ that is so regularly referred to? p. 67 – 74. ‘We’ is used in baby talk, medical contexts, formal discourse eg. textbooks and schools, each for different reasons. Some are about outlining conventions. Pimm refers to the fear of involving the self in mathematics.
“…highlights differences among how I do something, how I think I do something and how I think something should be done.” p. 70
“Does the mathematics teacher as politician seem a useful perception? Certainly the same tension between the individual and the representative of a community seems present in both contexts.” p. 71
Pimm talks about the mathematical imperative with apologies to Ardrey 1971? p. 72 There’s a brilliant analysis of textbook language – potentially v. useful. p. 72
4. The Mathematics Register. Pimm refers to technical and non-technical definitions of a word. I’ve talked about this!
“Part of learning mathematics is learning to speak like a mathematician…” p. 76
Numbers can be adjectives or nouns. Pimm notes a syntactical shift in the word ‘diagonal’ in moving from ‘normal’ to mathematical use, and discusses word with different meanings in different registers and even in different areas of mathematics. He examines metaphoric images in learning, and then takes an approach to metaphor within mathematics that has something in common with Lakoff and Nunez (p. 102). There’s a word for the false ascription of one term to another: catachresis.
“Metaphors deny distinctions between things. Problems often arise from taking structural metaphors too literally. Because unexamined metaphors lead us to assume the identity of unidentical things, conflicts can arise which can be resolved only by understanding the metaphor. This in turn requires its recognition as such, and entails reconstructing the analogy on which is is based.” p. 108
5. Pupils’ written mathematical records.
“In the opening chapter, a perception of mathematics as being essentially written was alluded to. What reasons and pressures are there in schools encouraging pupils toward a written expression? What is the point of recording?” p. 113
Pimm talks about external memory and the necessity of accurate expression of ideas. He classifies styles of recording as verbal, mixed and symbolic.
“…even the London Mathematical Society’s style sheet for potential contributors to their prestige international journals for professional mathematicians includes a request that preferably no page should be completely devoid of words.” p. 121
He notes an emphasis on brevity in the prioritising of symbolic expression, and the process os abbreviation. He quotes Hewitt (1985, p. 15):
“Algebra is not what we write on paper, but is something that goes on inside us. So, as a teacher, I must realise that notation is only a way of representing algebra, not algebra itself.”
I’m not so sure about that. Try doing algebra without writing anything down.
6. Some features of the mathematical writing style. Here Pimm starts talking about symbols and referents, and the advantages of sometimes blurring the distinctions between the two. He mentions the symbolic expressions for the derivative of a function – I started off thinking about that! There’s some reference here to symbol vs. icon. This table is important:
He talks about pictograms and the origins of various symbols. Great! He distinguishes logograms, pictograms, punctuation symbols and alphabetic symbols. In “Which features are systematically exploited in mathematics?” Pimm considers colour, order, positioning, relative size, orientation and repetition.
“…not only are the positional relations of immediately to the left/right employed, but also those of immediately above/below. Lying behind the functioning of this notation is the central idea of the main line of print, against which these various deviations from a linear flow of symbols are located.” p. 153 (original emphasis)
The ancient egyptian symbol for addition looked like a pair of legs walking one way, and subtraction was a pair of legs walking the other. The bit about repetition of symbols – eg. f”'(x) – is great. Next he discusses confusions between symbol and object, eg. defining evenness by the form of the numbers rather than an ‘intrinsic definition’.
“Fluent users, who actually want to be able to confuse symbol and object when calculating (because it is so much quicker and more efficient), frequently forget that this mode of operation is a pathology, one potentially lethal to many learners.” p. 159
I used this ^ exact example in my initial proposal! He beat me to it!
“Reddy describes this confusion of form with content as a ‘semantic pathology’, one predicated upon the conduit metaphor. Quoting Ullman (1957), Redyy (1979, p. 299) asserts: ‘A semantic pathology arises “whenever two or more incompatible senses capable of figuring meaningfully in the same context develop around the same name”‘. For mathematics, this confusion is so prevalent that it could easily be called the semantic pathology.” p. 159
It’s useful for mathematics to behave as though the symbols were the objects. That’s important.
7. The syntax of written mathematical forms.
“Algebra is rich in structure but weak in meaning.” René Thom, attrib. p. 161
Here Pimm gets into some Chomsky.Some comments on how transformations in mathematics differ from natural language transformations:
“1. The meta-language of arithmetic is algebra…
3. The prevalent language of the teacher in describing such transformations is often purely in terms of the surface structure, thereby focusing attention on the form rather than the meaning…
4. Mathematical syntactic rules are prescriptive and are consciously and deliberately taught, learned and applied at the surface level…” p. 164-6
Pimm examines a number of mistakes that can be made and computer simulations thereof, and the surface nature of a computer’s engagement.
“Such a perspective is remarkably close to a formalist view of mathematics as a whole, namely that mathematics is the rule-governed manipulation of marks on paper.” p. 174
“It is… personally advantageous to have as many mathematical operations as possible at the symbolic reflex level, to ease the cognitive load.” p. 176
8. Reading, writing and meta-linguistics. Pimm begins by talking about reading from mathematical text and the necessity of interpretation.
“Finally, i would like to mention the problem of how mathematicians write for each other. Reading mathematical writing is extremely difficult due in part to the lack of redundancy in the writing system and partly to the prevailing values of professional mathematical writing. Elegance is measured in part by brevity and in part by simplicity. Accessibility plays no part. ” p. 184
The written mathematical register – notational metaphor.
Pimm talks about the extremely common use of one symbol to refer to all sorts of things – ‘2’ meaning ‘+2’, for example, and the different fucntions of ‘=’.
“These instances provide support for Wittgenstein’s perceptive observation that in mathematics processes are always identified with results.” p. 188
9. Mathematics as a language? Pimm once again justifies his use of the metaphor ‘mathematics is a language’, outlining conflicting viewpoints and citing Lakoff and Johnson as an explanation of how such a conceit can be useful. I tend to think he apologises too much.
Lakatos 1976 – mathematical literary criticism
Vygotsky 1962 and Higginson 1978 – relations between language and thought