I have just finished reading Speaking Mathematically, in which David Pimm explores the use of metaphor in mathematics, and the metaphor of mathematics itself as a language. It is a hugely thought provoking book, but also a book that I found problematic in some respects.
Part of the book is devoted to exploring the (largely hidden) use of metaphor in mathematics, and the subsequent problems that students may face (p.195):
There are some symbols where the meanings are closely related but not identical, and for some of these the notion of (structural) metaphor is appropriate. There is a problem of pupils not realizing that different (if related) concepts are being employed. There is no trace in the symbolism to indicate that a metaphoric usage is being employed.
Pimm cites various examples, such as the metaphor of using natural (counting) numbers to stand for positive integers. The natural number 2 is an answer to a question of the form 'how many', whereas +2 is not a sensible answer to this type of question; the metaphor, although helpful to a point, breaks down.
This problem is compounded through the use of the metaphor taking away with counting numbers for subtraction on the integers. For example, 7 - 5 is initially experienced by children as 7 (of something) take away 5 (of something), and within the context of taking away, negative numbers need not (do not?) exist. How then is a student to make sense of 5 - 7 within the model of taking away? Again, the metaphor breaks down.
There are many further examples of problematic metaphors in the book:
What of the + sign? It is a sign for the positive integers, as in +2, but it is also an operation, a 'call to action', as in the case of 2 + 3. What then are students to make of the expression 2a + 3b?
Is the square root sign a similar call to action? We encourage the conversion of root(16) to 4, but no conversion is required when dealing with surds, such as root(2).
What of the equals sign, sometimes used for 'naming' such as f(x) = 2x, and other times used as an equality between two expressions (leading to something to be solved), and sometimes as a 'construction', for example: 2 + 3 = 5? And what are students to make of 7÷4 = 7/4?
There are numerous deeply ingrained metaphors that we use unknowingly in Maths, such as the number line. Are we aware that these metaphors exist? Are we aware of the problems that might arise through our use of metaphor when teaching? Should we be aiming to avoid the use of metaphor in mathematics, a view taken by Dick Tahta?
Given that metaphor is the process of describing something abstract in terms of something more concrete (structural), is it even possible to avoid the use of metaphor when ascribing meaning to mathematical symbols?
For those wanting to find out more about metaphor in mathematics, try this article by Pimm, or the excellent article Metaphor and Metonymy by @AlfColes in MT208. Those looking to go a bit deeper might like this article by Anna Sfard.
The symbol is the object
This leads me to the part of the book that explores the metaphor of mathematics itself as a language.
Pimm bemoans the fact that many students view Mathematics as the 'rule-governed manipulation of marks on paper' (p. 174):
Sadly, I fear, this description of algebra in particular, and mathematics in general, would find a strong resonance with the experience of many pupils in mathematics classes… Because meaning is largely absent for many people in mathematics, they are forced back on to attempting to learn the features directly for generating correct grammatical expressions.
Pimm suggests that we must attach meaning to mathematical symbols. However, Pimm concedes that part of doing mathematics consists of symbolic manipulation (p.175):
It is generally accepted among mathematicians that for complex calculations, which have become routine, it is often easier to ‘disengage’ the semantic component, and operate formally, by rule, on the symbols alone. School teaching at all levels seems to have accepted the goal of symbolic algorithmic fluency, without sufficient concern for semantic re-integration.
So mathematics is symbolic manipulation according to a set of rules, but also with a semantic component. What is this semantic component? How is it 'dis-engaged', and how is it to be 're-integrated'? Is not the applicability of mathematics due to the ability to operate on symbols that are devoid of meaning?
In Pimm's view, this leads us into dangerous territory (p.159):
The very real and frequently realized danger is that the symbols themselves, rather than the ideas and processes which they represent, will be taken as the objects of mathematics, the reality to which the language and notation is pointing and referring.
Of course we must show students how the symbols of mathematics can be used to help solve mathematical problems, rather than just manipulate the symbols for the sake of it. But what, or whose, is the 'reality' to which the symbols of mathematics point?
Here are two more statements from the book regarding meaning (p.178, 195):
Mathematics is not the manipulation of symbols according to prescribed rules: mathematical activity can be both purposeful and meaningful to human beings.
The symbol is the object is an extremely powerful metaphor at the heart of mathematics, but with this power comes the potential for the destruction of meaning.
What is the 'meaning' that Pimm perceives lies underneath the language of mathematics? Perhaps this is where the metaphor of mathematics as a language breaks down?
Finally, Pimm makes this call to mathematics educators (p.203):
One focal concern of all involved with mathematics education should be to ascertain how to deny symbols pride of place as the objects of mathematical inquiry…
Is this desirable? Is it even possible? If so, how? Perhaps Pimm answers these questions in his book Symbols and Meanings in School Mathematics, written 8 years later - it is on my shelf waiting...