Expressions and Formulas: Slight Variations on the Standard
In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. The focus here is on what teachers can do, even if their textbook sticks to the standard notation, to help disambiguate the standard notation for students. In the previous post, I reviewed the standard notation for expressions; in this post I’ll review common variations. In future posts, I’ll look at less familiar – but powerful – notations for expressions.
Exponentiation: one commonly seen alternative for the raised exponent shown in a smaller font is a version of the up-arrow.
In the top left, we see the standard notation of x raised to the nth power. In the top middle, we see the up-arrow (↑) used as a way to suggest the n should be raised. The up-arrow was available on specialized keyboards in the past, but keyboards for laptops etc don’t tend to have this character. People have used the caret character (^) as a keyboard substitute for the up-arrow, and by extension, for the exponentiation. Several computer languages do this very thing.
There is an obvious advantage for being able to put an expression as a linear sequence using a standard keyboard, but that advantage comes with its own disadvantage. The bottom row of examples indicates this. On the bottom left, we see x raised to the power n-1. No parentheses are needed to express this grouping: the entire n-1 is rendered in small font and raised. To get the same effect, the bottom middle expression puts the n-1 in parentheses, and so does the bottom right.
Square Root: I’m showing common alternatives below:
On top are the square root symbols with the long horizontal bar, on the bottom you see a square root symbol without a horizontal bar (√) available on some keyboards. On the left, where it shows square root of n, both the symbol with the horizontal bar and the symbol without will do fine. In the middle, where we show square root of n-1, the symbol with the horizontal bar indicates the grouping clearly and effortlessly, whereas the √ symbol requires parentheses to make the scope of the square root clear. On the right, we’ve shown a variant that’s easy to produce on the keyboard but requires both parentheses and a dedicated name, “sqrt” for square root. The “sqrt” designation is common in computer languages and is seen elsewhere as well; obviously this notation doesn’t generalize easily for third roots etc., but then neither do the variants based on the √ symbol. I get the impression that there is little demand for special symbols for third roots, n-th roots, etc., since these roots can all be rewritten as exponentiations with fractional exponents.
Division: Some common alternatives are shown below:
The ÷ symbol for division, as shown on the left, is very common in the early grades, yet rarely used past elementary grades. This, in itself, is a pretty good indicator that it is quite possible to introduce notations for lower grades that are gradually phased out without in any way damaging a student’s eventual understanding or skill or fluency. However, though rarely used by adults, adults do almost universally recognize the ÷ symbol, and this makes it suitable for use on calculators. Most calculators, by far, use ÷ to mark their division key.
The symbol seen in the middle, the slash (/), is also universally recognized as a division; on the bottom we see the case where parentheses are necessary. On the right we see a horizontal line used as the dividing line between top and bottom (here known as numerator and denominator). In neither the top nor the bottom situation are parentheses needed. The horizontal line is often pronounced as “over”. So it is n over 2, or n over n-1. This horizontal line seems like it would be very confusing with the line used for fractions. In practice, it is not confusing at all. Many students insist on fractions and divisions being not at all alike, and they pride themselves on being able to tell them apart. For them, “two fifth” is one thing, and “2 0ver 5” is something else altogether. From my perspective, the more kids see a fraction as a division, the better off they’ll be. Here is an example, I think, of a notation that is trying very hard to suggest that fractions and division are very closely related – and failing!
Note: In this account of notations for division, I’m deliberately leaving out notations for divisibility, e.g. 5|30 (read: “5 divides into 30”, or “5 is a factor of 30”) and I’m also deliberately leaving out the symbol seen in the long division algorithm. I’m also leaving out complications having to do with division (in the lower grades) resulting in a quotient and a remainder.