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.

When I read this chapter, I thought up the meaning of “=”. Teachers might find many students who misunderstand the meaning of this symbol and incorrectly operate it when teaching equation. This common symbol even can be seen in grade 1 textbook; thus, it is familiar to every student. Interestingly, we call “=” equal but it changes when it is used in mathematics sentences as this chapter shows. In the Japanese language, we can read 2 + 3 = 5 “2 tasu 3 wa 5”: “tasu” means “plus (add)” and “wa” is the particle having a similar function to “be-verb” in English. Although this “wa” is not a verb and there are no verbs when reading this mathematical sentence in Japanese, the sentence fragment in mathematical sentences are largely accepted in Japanese mathematics class. However, when the mathematics teachers explain this sentence, I suppose they often say “2 tasu 3 wa 5 ni-naru” that means two plus three “become” five, so the meaning of this sentence has slightly changed without the teacher’s consciousness. It could be one of the reasons why many students misunderstand the meaning of “=”. Although many of Japanese start to read “=” equal in Japanese (using the English word as the Japanese language) in secondary, many students still get confused to operate and write this symbol. This phenomenon might support the idea of Pimm in this book, so people recognize mathematics sentence as symbols without understanding the meaning.