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Words into symbols, symbols into words

Mathematicians use a lot of symbols. Everyone knows that. A symbol may represent a simple constant (C is the constant of integration) or represent a collection of difficult concepts (G is a group). Students need to be able to unpack the meaning from symbols and to do the reverse — bundle up concepts into a few symbols.

Asking students to turn words in symbols and symbols into words is a useful exercise for this. One of the best places to use it is in analysis particularly where quantifiers are grouped together in subtly different combinations. For example,

\forall \epsilon >0 \exists N\in N : n>N \implies |a_n-a| < \epsilon

is profoundly different to

\exists N\in N \forall \epsilon >0 : n>N \implies |a_n-a| < \epsilon .

Translating these into words serves at least a couple of functions. First, students get a different perspective — it is almost as if a different part of the brain is being used. Second, and this is the most important, they observe what is there. It is all too easy for an eye to glide over a symbol and not have it consciously perceived. This activity prevents that. Third, they are forced to describe what a symbol means and that requires them to know the material. Being unable to translate helps locate areas of weakness.

The activity can be used in other areas to show how mathematical concepts are tied together in a single package. For example, “Any group with order equal to the square of a prime is Abelian” can be written as “G\in \{ \text{Groups} \}: \text{ord} (G)=p^2, p\in \{ \text{Primes} \} \implies xy=yx \ \forall x,y\in G“. The former is much more compact and readable.

The aim of this example is to improve mathematical writing skills by showing contrasting statements. This can then be developed into asking for the perfect blend of words and symbols.(I don’t really believe that there is a perfect answer.) A good compromise for the above might be “For any prime p, a group of order p^2 is Abelian”. Students can learn a lot about their preferences and those of others from discussing this.

This activity should be used sparingly. It is only effective in the service of some other goal. It is particularly good for making students observe what is written, for developing mathematical writing skills, and identifying weak areas of understanding.

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