**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.

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Personal notation or field specific notation (as exemplified by superscript minus) should not be promoted or forced on our society’s young minds as a crutch (or “training wheels”) to learn proper mathematical notation. I have observed errors in student work (and confusion) as a consequence to the introduction of superscript minus. Common negative number notation is simply to place a negative sign in front of the number (as in -7) or to use parentheses (as in (-7)) for clarification or to eliminate ambiguity. To introduce poor notation as a crutch when not needed and then defend it as appropriate for use in scientific fields where traditional notation is simple and widely used leads the reader to wonder if other cumbersome new notations for simple mathematical concepts are soon to be introduced by “learning consultants.”

When people talk about “proper mathematical notation” or “common notation” or “traditional notation” they are talking about something that is time-bound. As a trivial example, 700 years ago, notation for negative numbers was very different. We tend to think that what we learned in textbooks when we grew up – or what we see people use in particular fields – that must be proper notation.

When it comes to the raised minus sign, this has been the norm in US math textbooks for decades now, particularly at the middle school level. You are kind of late to this particular party.

By the way, take a look at many calculators: the key for subtraction “-” is different from the key for changing the sign of a number. The latter key may be labeled “+/-” or “(-)”. Separating these keys fulfills a practical need.

People studying the history of mathematics have noticed that when they study Leonhard Euler, there is a lot of material that doesn’t look particularly earth-shaking – stuff an undergraduate student in math might recognize and be familiar with. But when they look at texts of mathematicians before Euler, it all looks very different. Much of our familiar notation was introduced by, and popularized by, Euler. There is nothing sacred about any of these notations – they are a way to communicate intentions.

Much of the mathematical notation currently in use is very clear and precise and concise in communicating intentions, yet some of it is notoriously confusing to students. Generation after generation bumps into standard f(x) notation and thinks it has something to do with f times x. Grouping symbols like parentheses are widely re-used for all kinds of different meanings and it takes a familiarity with the context to disambiguate their use.

Designers of programming languages are very familiar with these issues, their technical term for this phenomenon is ‘overloading’.

I ran into the “minus” ambiguity a day or two ago, when keeping score in a card game.

When keeping score using pen and paper, how am I to indicate that a minus sign refers to a negative score, rather than to a subtraction? Should I sidestep this issue by making a “house rule” that scores start at 100 points rather than 0 points?

By the way, there is a “scoring app” which I made for card games. In it, black figures are for positive scores, and red figures are for negative scores. (A score of exactly zero is shown in black. So sue me.) The only indication of “negativity” is color: minus signs are never shown. This is because it is much easier just to paint everything red than it is to futz around with proper typographical placement of a minus sign. I consider myself in good company: some old calculating machines have this same “philosophy” regarding negative numbers.