Why is multiplication taught using cross notation at first?

Alert: I am not a math educator.

It seems to me that multiplication is first taught using the cross notation, for example $3\times 5=15$.

First question - is that even correct? Maybe not all schools in every country even teach this?

Later on in education and work this notation is almost never used - by grade six it's $3 \cdot 5=15$ and soon after you only use any sign if you multiply naked numbers (it's $3.1\cdot 10^{-7}$) but mostly you only write that you want three x'es, like $3x=15kg$.

So why confuse people and introduce them to notation that is only used briefly and soon overtaken by other notation while the x becomes unknown instead of multiplication for further confusion?

I do understand that the $\times$ is used on advertisements, cashier calculators and some other material aimed at general public, but that shouldn't be the cause for such choice. It's a consequence.

• I tend to avoid $\times$ as it looks too much like $x$; so I often just use $*$ if I need a multiplication symbol. A worthwhile reference for the history of math symbols: jeff560.tripod.com/mathsym.html – Adam Dec 30 '17 at 19:58
• It is worth noting that in $\LaTeX$ the name for the $\times$ symbol is \times. – mweiss Mar 8 '18 at 16:32
• X is used in the real world: 3"x4" dimensions, scientific notation, mxn matrix, row x column, etc. – Lenny Nov 16 '18 at 2:02

In the U.S. at least, we do still use the $\times$ in scientific notation, so your example would be $3.1 \times 10^{-7}$.

I think, for young children, the dot looks too much like the period we use (in the U.S.) for a decimal point. Even adults will sometimes misinterpret. (They see $3\cdot7$ and think it's $3.7$.)

• The MathJax accepts TeX notation. Just write $3.1 \times 10^{-7}$ or $3 \cdot 7$. – Džuris Dec 30 '17 at 16:39
• There is a step in the logic that is not spelled out explicitly in the answer. Math education in childhood concentrates exclusively on computations using concrete numbers, whereas secondary and college math education brings in a lot more symbolic and abstract ideas. In the concrete numerical context, $\cdot$ looks too much like a decimal place, while in the abstract context, $\times$ looks like an $x$, and in any case we can usually use implied multiplication. – Ben Crowell Dec 30 '17 at 19:37
• German here. We're unfortunately are still using comma as decimal separator in everyday life (while in science it's uncommon). I remember school started with "×" here, too, although there is not much ambiguity of decimal comma and "∙" multiplication operator. Baffling. – jvb Dec 30 '17 at 21:15
• @jvb I don't understand why some people consider the . better than , as a decimal separator. – user5402 Dec 31 '17 at 12:34
• Different issue, of course. But since a period in written text is a bigger stop than a comma, it seems the commas for thousands etc and the period for decimal separator makes more sense. But then, it's what I grew up with. We are all likely to feel that the way we gew up with "makes more sense". – Sue VanHattum Dec 31 '17 at 18:53

So why confuse people and introduce them to notation that is only used briefly and soon overtaken by other notation [...] ?

I think that the $\times$ symbol is used to teach multiplication the first time it's seen because that symbol relates multiplication to area. I recall learning both of these concepts around the same time. So when you see the notation $3 \times 5$, instead of "three times five" read it as "three by five". Some children may already be familiar with this language since it is common in stuff "aimed at the general public" as you mention. It's also used in work environments (dimensions of lumber), and is very important while playing with Legos®. But in general, instead of talking about repeated addition, you could teach a child to think of $3 \times 5$ as the number of small ($1 \times 1$) squares in a rectangle that measures three by five. This $\times$ notation can help students make this connection between multiplication and the concept of area.

I do understand that the $\times$ is used on advertisements, cashier calculators and some other material aimed at general public, but that shouldn't be the cause for such choice. It's a consequence.

I disagree. That should be cause for such a choice, and whether it's a good precedent is beside the issue. When young'uns are first learning multiplication, I think the goal of getting them to learn what multiplication is is more important than setting a good precedent for their future math and science education. Simply getting them to think about multiplication correctly is the most important thing. And if it helps them learn it by being able to relate what they're seeing in class to examples they've seen on calculators and in advertisements, then that's for the better.

• In my language there is no "times" and "by", there is only "reiz" that is used both for multiplication and describing area. Anyway, I don't really understand how $\times$ is connecting the concept to area, it's not like a rectangle or anything, it's just a cross. – Džuris Jan 2 '18 at 11:04
• @Džuris The point is that students may connect the $\times$ symbol to areas because they've seen it before in reference to areas. E.g. A $2\times 4$ piece of lumber, a $4 \times 4$ Lego® brick, etc. – Mike Pierce Jan 2 '18 at 17:10
• So you are saying that the only real reason to teach using $\times$ arises from the fact that the previous generation was taught using $\times$? – Džuris Jan 3 '18 at 1:05
• @Džuris Yeah, pretty much. That, and each of the possible alternatives has its own drawbacks. But the cross notation isn't bad. In fact, if we weren't entrenched by the precedent of denoting multiplication with a cross, and I were given a choice of how to first show multiplication to young students, I would certainly choose a cross over either a dot or parenthesis anyways. – Mike Pierce Jan 3 '18 at 9:37

In the UK, we never use the dot notation in education up to 18. Instead, you use the x sign until you start to do algebra and want to save space by using bracket instead like $3(5) = 15$

I think this is done to keep things more consistent so that students don't get confused by a basic symbol changing part way through their education. Even in my advanced maths class, we use $\times$ to denote multiplication.

As a British student, the first time I encountered the dot was on an exchange in Belgium, where I was rather confused until I realised it was another sign for multiplication.

First question - is that even correct?

“$\times$” denotes another similar operation, the Cartesian product of sets. For finite sets, cardinality maps the Cartesian product of sets to the multiplication of natural numbers. So it is definitely correct, I mean, it is a sensible choice of a sign.

Denoting multiplication by juxtaposition is the worst choice. Every operation is denoted by a sign, so why the multiplication is “better” than the others? Well, the power is denoted by the superscript, but this is also a bad notation because a small font is less legible.

Many mathematical notations are hardly distinguished in handwriting. I suppose that this is a problem of handwriting.

• I believe juxtaposition follows from language. You don't say "three times an apple" or "multiply three and an apple". You just advertise "three apples are two bucks" => $3a = 2$. – Džuris Feb 13 '18 at 19:01
• @Džuris: I can't help but notice that apples are not numbers. – beroal Feb 14 '18 at 20:01
• My theory is that multiplication is the most common operation, hence removing the sign from factors is maximally concise (generally the point of algebraic notation in the first place). – Daniel R. Collins Feb 15 '18 at 22:11

In my experience (Catalonia), we use × as the multiplication sign until we introduce equations, where the cross an cause confusion with the letter x. In fact, "×" is the multiplication sign, and it seems that the natural question is why we stop using them in advanced courses.