Why We Don't Use '×' for Multiplication

Or the 'divided by' symbol. Both × and ÷ are unhelpful symbols.

Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

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Compare these expressions:

\(3 \times 5 + 4 \times 10 + 2 \times 8 + 7 \times 3\)

\(3(5) + 4(10) + 2(8) + 7(3)\)

The second is cleaner and easier to read.

This is the first of many reasons we preference the use of brackets over the cross symbol to symbolically represent multiplication.

The same is true if we use the vinculum in place of the divided by symbol:

\(3 \div 5+4 \div 10+2 \div 8+7 \div 3\)

\(\frac{3}{5} \;+\; \frac{4}{10} \;+\; \frac{2}{8} \;+\; \frac{7}{3}\)

Further reasons for preferring these symbols:

• They reveal more structure, making it much easier to see the terms

• They automatically connect brackets to multiplication, simplifying the idea of ‘expanding brackets’ later in the curriculum

• They connect fractions to the concept of division

• They also connect substitution to prior learning, and encourage students to replace variables with numbers inside brackets, which in turn leads to fewer mistakes when dealing with exponentiation

For example, given this:

\(10x^2+7\)

Compare this substitution for x = -3

\(10(-3)^2+7\)

With these alternatives:

\(10\times-3^2+7\)

\(10-3^2+7\)

\(-30^2+7\)

With brackets:

\(10(-3)^2+7\)

It is much clearer that we have two terms, and that to evaluate the expression, one term requires exponentiation followed by multiplication.

Substitution becomes even simpler and less error prone when you start to write expressions with unknowns and variables wrapped in brackets, as below:

\(10(x)^2+7\)

\(10(-3)^2+7\)

You just swap out the x for -3. No need to write or remove any other symbols.

It does look a bit funny to us teachers at first, writing something like and it’s not conventional, but it is more overt, and so a better place to start before later covertising the brackets again.

Overtisation

More reasons to express arithmetic operations this way:

• The brackets and vinculum symbology is much closer to what students are expected to use as they advance through mathematics so there is less to ‘unlearn’ later

• It gets students very used to polynomial expressions and how to set them out formally

Most importantly:

1. It reveals that BIDMAS (BODMAS, PEMDAS, GEMS, whatever) is a lie! You do not have to work exponentials before you multiply

2. And it makes it easier to solve linear equations

We’ll be diving deeper into these last two in future posts.

For now, we’ll take a quick look at what this looks like for rudimentary ideas like times tables practice.

From there, we’ll look at a couple of ways using brackets like this can lead to problems, compared with the cross and divide by symbol, and we’ll see how to easily overcome those problems: