Mathematics education uncovered and recovered



Some time at the end of primary school or at the beginning of secondary school, children start dealing with arithmetic expressions that involve two or more operations. At that point they are told to memorise BIDMAS rules for the order of operations, which is supposed to help them to make sense of increasingly complex calculations they face in their maths lessons. Unfortunately, as with other acronyms appearing out of the blue and thrown at students, BIDMAS fails to become a useful tool to navigate arithmetic and algebra. A short abbreviation simply cannot capture the whole “grammar” of mathematical expressions, and BIDMAS is packed with potential confusion and misunderstandings.

Contrary to the most common interpretation of BIDMAS, it is not true that division should be done before multiplication, addition before subtraction and indices before any other operation. Expressions in brackets also can wait for their turn, there is no need to evaluate them before everything else.

$$\mathrm{ 7 \times 4 – 15 \times 2^3 \div 5 + 9 \times (10 – 6 ) }$$,

I could first do $$\mathrm{ 7 \times 4 = 28}$$,

then $$\mathrm{15 \div 5 = 3 }$$,

and $$\mathrm{3 \times 8 = 24}$$.

Then it would be $$\mathrm{10 – 6 = 4}$$

and $$\mathrm{9 \times 4 = 36}$$,

so it would end as $$\mathrm{28 – 24 = 4}$$

and  $$\mathrm{4 + 36 = 40}$$.

I do not need BIDMAS to find the answer, in fact, I would feel rather constrained if required to follow the order that BIDMAS prescribes. Also, at the last step, it would be completely wrong to do $$\mathrm{24 + 36 = 60}$$ and then $$\mathrm{28 – 60 }$$, although in BIDMAS, A(ddition) stands before S(ubtraction).

I believe that, even interpreted correctly, BIDMAS does more harm than good. Instead, a gradual introduction of the order of operations would be more productive and less prone to error.Here is, in brief, my suggestion how it should be done.

• Establish that additions in sums like $$\mathrm{14 + 15 + 26 + 9}$$ can be done in any order.
• Try possible rearrangements of the terms in sums like $$\mathrm{25 – 17 + 13 + 11 – 8 }$$, moving the numbers with the preceding signs around. Later, when children have become familiar with negative numbers, it makes sense to revisit this to underline that all subtractions can be replaced by additions.
• Explore the effect of inserting brackets into sums, for example,   $$\mathrm{34 – (17 + 15) = 34 – 17 – 15 }$$ and $$\mathrm{34 – (17 – 15) = 34 – 17 + 15 }$$ .
• Introduce multiplication as a shortcut for sums: $$\mathrm{3 + 3 + 3 + 3 = 4 \times 3 }$$ and $$\mathrm{5 + 5 + 5 + 7 + 7 = 3 \times 5 + 2 \times 7}$$ .
• Work with multiplication and division in the same way. Start with products $$\mathrm{ 15 \times 3 \times 4 }$$ and expressions like $$\mathrm{ 24 \div 2 \times 8 \div 3 }$$ to see whether the result is affected by rearranging the numbers with preceding  $$\mathrm{ \times }$$ and $$\mathrm{ \div }$$  signs attached.
• Observe $$\mathrm{ 36 \div 6 \div 2 = 36 \div (6 \times 2) }$$ and similar examples. Once students have learned how to work with fractions, they would see that division is a variation of  multiplication.
• Exponentiation as a shortcut for multiplication: $$\mathrm{5 \times 5 \times 5= 5 ^ 3 }$$  and $$\mathrm{4 \times 7 \times 7 \times 8 \times 8 \times 8 = 4 \times 7^2 \times 8 ^ 3 }$$ .

By this point the students are ready to work with mixtures of operations, with only one new idea to digest, namely, that each expression is simply a sum of terms. In practice it means that, looking at an expression, they need to see pluses and minuses as signs separating different terms, and these terms, naturally, should be evaluated before working out the whole sum.

Most students manage to get this idea of the sum of terms later, when working on algebraic expressions. Algebraic notation on its own helps interpreting expressions correctly. Nobody thinks that  $$\mathrm{5a + 3b }$$ requires adding a and 3, and it is the  conventions of notation rather than BIDMAS that students rely on.This is why it is quite common that a student would easily evaluate an expression above for specific values of a and b,  but presented with the same expression with numbers replacing the variables, $$\mathrm{ 5 \times 7 + 3 \times 4}$$, they would be likely to make a mistake.

A slow, years-long introduction to the convention of the order of operations, with careful consideration of chains of addition and of multiplication would lead to proper understanding of the structure of expressions and BIDMAS would become redundant.

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