Notations For Multiplication
Now I want to look at different notations for multiplication, and suggest one that is not currently widely used in schools but has wide currency elsewhere.
In elementary schools, by far the standard way to denote multiplication is by using the “×” sign. We write 3 × 4 = 12, and we typically pronounce that as “three times four makes twelve”. The main reason that “×” is so closely associated with elementary school is that this symbol is so easily confused with the letter “x”, and “x” is so much the domain of secondary school mathematics, as the primary symbol for an unknown amount. At least, this is the standard explanation passed on from the distant past as to how come we switch in middle school to a different notation for multiplication. This new notation is “•”, a dot. So “a•b” is the new way to write “a times b”. Unfortunately, this dot is not suitable as a universal symbol for multiplication either, since “3•4” (three times four) is very easily confused with “3.4” (three and four tenths). In other words, the dot looks a lot like a decimal point, and neither your typical kids’ handwriting nor a typical laptop keyboard will let you easily distinguish the two. So, the middle school textbooks tend to carefully use the “•” for multiplying letters (a•b), but also combinations like 2•a, but then switch to the old “×” to render “3×4”.
At roughly the same time, or later, the textbook will also introduce the idea that multiplication need not be written at all:
“ab” is read as “a times b”. This very useful convention, which has been standard for centuries now, allows us to write
to mean , eliminating any confusion between × and x.
However, this notational standard of writing nothing for multiplication (more precisely, to indicate multiplication by juxtaposition) comes at a cost. This cost is quite considerable. Yet because we adults are so used to this standard, we are barely aware of the cost. To become aware of part of this cost, you can watch kids when they are first learning this convention. Since juxtaposition doesn’t work for 12 times 34: nobody would see 1234 and be able to tell whether it meant 1234 or 123×4 or 12×34 or 1×234, a different convention for multiplying numbers is needed. Teachers will tell kids to write 12(34) to mean multiplication. The multiplication is still expressed through juxtaposition, and the parentheses are now used to make clear what the pieces are that are being juxtaposed. For kids, this is not real obvious, so here is one cost. A bigger cost, in my mind, is that it pretty much forces variables to be single letters. This is not something that kids seem to have trouble with, and not something that teachers typically see as a limitation. In the standard notation, “cost” means “c×o×s×t”, and can’t be a single variable that indicates how much something costs. And indeed, if you look at college-level math texts, you will see how many alphabets mathematicians have appropriated (Greek, Hebrew, German) in order to have enough different symbols. Even so, the convention is not used consistently. When students see sin(45) in secondary school, they know somehow that this means the sine of an angle of 45 degrees, and not s×i×n×45.
When communicating mathematical formulas to computers, whether when writing computer software or entering formulas in spreadsheets like Excel, entirely different conventions are used, born from the limitations of typical keyboards. The computer-based conventions resemble a lot more than .
Since computers require very precise communication, a lot of work has been done to come up with clear and unambiguous languages for formulas. Though there isn’t a single standard for these, there are commonalities in the conventions that have been developed. In almost all of them, a variable is not restricted to a single letter, it can be a construct like total_cost_before_taxes. In almost all of them, juxtaposition is avoided and all operations have to be explicit. Many of these computer-based conventions rely on the “*” sign for multiplication. This asterisk is found on all keyboards, and looks somewhat like a cross, a “×”. Most importantly, it is not easily confused with x.
Here, then, is my proposal: introduce the asterisk “*” fairly early on in school as a variant of “×”, right around the same time that decimal numbers come into play. Don’t make a big deal of it, don’t require its use, just get students used to seeing it as meaning the same thing as “×”. This is not a new thing to students: variant notations for division are very common: “÷”, “⁄”, “—”. So now multiplication has a variant, and this variant can gradually replace “×” and instead become a variant of the juxtaposition learned in middle school. So a×b moves through a*b to ab, whereas 3×4 moves to 3*4 and never need become 3•4 or 3(4).
Pros: no confusion, no ambiguity, easy to use on a keyboard. Kids learn this easily ( I’ve used this for years, and never had to explain it to kids beyond “this is just another way of writing ×”). Secondary pros: easy transition to Excel or C# or a number of other computer languages that use this same convention for multiplication.
Cons: teachers are generally unfamiliar with this convention, and especially elementary grade teachers.