Notations for negative numbers
This blog entry marks the start of a series about mathematical notation and its uses in school. I intend to look in detail at aspects of the standard mathematical notations, and how these help or hinder understanding of math in the various grades. I also intend to look at special notations introduced in school to aid in understanding key math concepts, and to propose additional such notations.
A simple and useful example is the use of two different “minus” signs in school (at least in the USA), the normal “-” sign for subtraction, and the raised minus sign for indicating a negative number, ⁻7 for negative seven. This notation, typically in use up to 8th grade, allows the text books to write 8 – ⁻7 rather than having to write 8 – (-7), for instance. Teachers typically think of this negative number notation as a kind of training wheels. This is akin to the use of pointing in Hebrew to indicate vowels, which is primarily used in texts for children or beginning Hebrew students. Indeed, most students are encouraged to drop this special notation as they get further along in school.
From my own observations with students up to 8th grade, this use of a distinct minus sign to mark a negative number is both successful and non-intrusive. By ‘non-intrusive’ I mean that, as far as I can tell, the notation can usefully coexist with the standard notation, and that if some 8th grader decided to keep using that notation for the rest of his or her life, the rest of the world would barely notice and would not be in the least inconvenienced. This would be true even if the student ended up in an engineering school or became a math major. The use of the raised minus sign (the official name of this symbol appears to be superscript minus) is a harmless variation out there in the world, and is demonstrably useful in the lower grades. These may be precisely the conditions under which a variant notation can spread far and wide.
In this series, I will bring up other examples of variant notations and try them out, to see if they meet these same tests: harmless in the wider world, and useful in the context of math education. In addition to looking at notation, I will also occasionally bring in examples of variant vocabulary and terminology.