Reading, Writing and Meta-linguistics (Speaking Mathematically)

David Pimm (1987)
Chpt8: Reading, writing and meta-linguistics

As I commented on the previous chapter, this chapter mentions reading of mathematics. The author introduces several examples of reading in mathematics texts. There are a symbol of “+”, dy/dx, number of multi-base, “:” in ratio, and matrices. The author acknowledges both spelling pronunciation and interpretative pronunciation are accepted in mathematics, and he insists spelling pronunciation can be shorter than interpretative one such as “sigma” or “integral”. However, this spelling pronunciation can benefit for mathematician while it seems like “the symbols are the objects of mathematics” for the mathematics learners. Therefore, reading mathematical wiring is difficult to both decode and understand it for the novice because simplicity is considered more important than accessibility in mathematics.

 Additionally, the author shows the case of Benny who 11-year-old students and views fraction and decimals as symbols. He gives priority to the rule of combination of the mathematical symbols, and he believes it can lead to the correct answer. Therefore, the author suggests that mathematics teachers should understand what kind of activity is believed and engaged by the students.

The Syntax of Written Mathematical Forms (Speaking Mathematically)

David Pimm (1987)

Chpt7: The syntax of written mathematical forms

In this chapter, the author explores grammatical aspects in mathematics standing with the idea of Chomsky’s theory (1957, 1965) which comprises a phrase structure grammar and transformation in natural language. He introduces several examples of grammatical structure and transformation in arithmetic and algebra. For instances, additions can change the order with keeping the correct answer (a + b + c = c + a + b) whereas subtraction cannot result in some cases (e.g., a – b – c ≠ c – a – b). In equation, 7 – x = 4 can transform into 4 + x = 7 such as synonym in natural language. We can find several similarities between natural language and mathematics. As he mentioned in previous chapters, the author claims that many people make light of meaning in mathematics and focus producing mathematical expressions with correct grammar in this chapter. Finally, he shows several computer software that finds solutions from given mathematics sentences.

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).