Some Features of the Mathematical Writing System (Speaking Mathematically)

David Pimm (1987)
Chpt6: Some features of the mathematical writing system

The author focuses on mathematical symbols and its referents in this chapter. He classifies the symbols into four main categories: logograms, pictograms, punctuation symbols, and alphabetic symbols. For instance, logograms are used only for mathematical contexts such as ＋, －, ×, or ÷ whereas pictograms are mainly invented in geometry such as △ or ∠. Additionally, he explains color, order, positioning, relative size, orientation, and repetition in which the features are exploited in mathematics. For example, in Moroccan schools, they write negative numbers in green color and positive numbers in red. According to his argument, because the symbols tend to be focused, “production of symbols” is emphasized rather than the meaning of the symbols. The symbols themselves are treated as objects in mathematics. Therefore, the idea of ‘the symbol is the object’ leads learners to calculate mathematics operations without understanding the meaning of the objects.

Although I forget whether I wrote about this in this blog, I often see many students who can write symbolic style in mathematics but cannot explain it. They might rely on rote or be trained to do machine-like operations. Actually, most workbooks of mathematics are filled with those questions which ask learners to show their memorization or accurate operation they have learned. Therefore, it should be usual for the learners to hold the idea that essential part of mathematics is memorizing formulas and how to use them.
 I often ask my 2^nd-grade students to explain and write down their idea on the paper. (The context in this “explain” is limited to use verbal style.) Some students whose mother tongue is not Japanese use mixed style (use symbolic and verbal together) to describe their thoughts. I assume that they understand symbols in their first language while they do not know how to express them in Japanese. This means that, in terms of mathematics education for language learners, aims of the teachers about the shift of style can be verbal (in mother tongue) -> mixed (in mother tongue) -> symbolic -> mixed (in additional language) -> verbal (in additional language).