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.

If asked to calculate

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

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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If we take an inventory of how much mathematics a person needs to know for their everyday needs, our list will hardly go beyond the content of the modern primary curriculum and could be labelled as “Numeracy skills”. The topics that are currently studied in the primary schools in England cover practically all useful numeracy and more.

When it comes to the secondary school curriculum, a serious problem regarding the content of the curriculum arises. Once the easy-to-agree-on goal of equipping all children with necessary numeracy skills is achieved, which presumably should happen by the end of the primary school, where to go next? What is the purpose of maths lessons at the secondary level?

The goal as I see it should be two-tiered. Firstly, the lessons should give a broad overview of mathematical methods and applications, including their history and (relatively) recent developments. Additionally, for students interested in further study of mathematics and sciences, the lessons should provide an introduction into mathematical reasoning and an opportunity to go deeper into technicalities.

The obvious need for two different strata of the curriculum is recognised in the existence of two tiers of GCSE Maths exams that are set for two different categories of students who in preparation for exams are taught separately, Foundation tier topics being assumed to be easier than Higher tier ones.

However, this distinction does not reflect the principal educational goals stated above. The Foundation tier topics are too narrow and detailed for students who will not need mathematics beyond primary level in their life. These students will unlikely to be able to see the wood of mathematics for the trees of exercises on disconnected topics, and the few things they would probably remember are some ludicrous mnemonics . As for the Higher tier, for students who are willing to become engineers, scientists and mathematicians, the content is too shallow and patchy; it does not lay a solid foundation for further study.

This mismatch between the needs of the students and the curriculum content manifests itself in two well-known facts. One is the enormous amount of stress and struggle the students without much interest in maths have to go through, with about 40 percent of them failing to get an official GCSE benchmark grade every year (currently grade 4, previously known as grade C). This means hundreds of thousands of 16-year olds being let down by the system year after year.

At the other end of the spectrum, there is a substantial gap between GCSE and A Level, and students continuing to do mathematics and maths-heavy subjects at A level often get overwhelmed by the pace of the course. Many of them are unable to make sense of new concepts due to the lack of adequate preparation that should have been provided by GCSE Maths.

Therefore, the current maths curriculum, aimed to serve everyone, is hardly suitable for anyone. This has been the case for years, and it will continue until the problem is recognised, articulated, and a fundamental change of the content is called for.

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The abbreviation is intended to help students to memorise the meaning of sine, cosine and tangent of an acute angle in a right-angled triangle as ratios of the sides of the triangle. Its purpose is similar to that of a baby walker: help to make first steps in trigonometry. However, unlike with children’s walkers, the SOHCAHTOA walking frame becomes indispensable and effectively paralyses their progress.

The problem is that for most students SOHCAHTOA appears out of the blue. There is no discussion about the similarity of right-angled triangles with equal acute angles, and the convenience of having the ratios of the sides packed into values of trigonometric functions is not recognised. It is not surprising then that, having been introduced to the magic mnemonic as a replacement for understanding, students are stuck with it forever.

Knowing only as much trigonometry as SOHCAHTOA can hold might be enough for practical purposes of answering GCSE questions, but the lack of understanding is not only regrettable as such, but becomes a real hindrance if students continue with Maths and Physics at A level.

Some teachers lead their students even further away from proper understanding by offering them ridiculously meaningless rules for memorising the values of trigonometric functions of some common angles. Using these rules, a student trying to recall the value of \( \mathrm{\tan {45} ^{\circ}}\) will associate it with finger-counting rather than thinking about equal sides in an isosceles right-angled triangle.

I assume that SOHCAHTOA was a result of a well-meaning attempt to simplify school mathematics and help struggling students. But its overuse made it a surrogate for real mathematics. SOHCAHTOA, as well as FOIL, “Speed triangle” and many other “tricks” became shallow substitutes for the rich mathematical content, and the quasi-maths they produce is not really worth learning.

]]>Ten years ago, the problem with reading the clock did not seem to exist: children could see analogue clocks everywhere, not just in school lessons, and learned the skill of telling the time by practising on the go. Some did struggle, but sooner or later everyone got to master that essential skill. Now, as analogue clocks are dying out, the important part of the learning process, the on-the-go practice, is missing. It’s not surprising then that children are not able to read the analogue clock despite all formal teaching in the classroom. However, a question arises about school’s contribution to children’s learning. The amount of time the primary schools spend on teaching analogue clock is no less than it was ten years earlier, but since the outcome depends mostly on whether or not children have a chance to learn about the analogue clock outside the school, teaching it in school seems to have little effect.

Analogue clock is a very unusual example of a measuring device. Although the hour hand would be sufficient to show the time, it is complemented by the minute hand that clarifies the hour hand’s position, so that in order to tell the time one needs to take account of them both. This is further complicated by having a single scale for both hours and minutes, with numbering usually only provided for the hours reading. So reading the clock is certainly much harder than reading the weighing scales or measuring cups, and teaching the topic has never been simple.

Perhaps, teachers would feel relieved if analogue clock disappeared from the primary curriculum, but I think it would be sad to leave it out. The lack of practical use cannot justify expelling the topic from school lessons. Be it in maths, science, history or design lessons, the analogue clock could serve as a fascinating example of ingenuity of its inventors who came up with this quirky piece of machinery with a hidden intricate design. Since children do not see many analogue clocks in their everyday life now, it is for the school to introduce pupils to something that they would miss otherwise. And the things that are part of students’ routine experience do not need to be taught, as the accidental experiment with analogue clock teaching showed.

]]>A few months ago his school started preparing the children for the new primary mathematics test. As in every other school in the country, the students did countless sample tests, practising written methods of calculation again and again, at speed and with full workings shown as a proof.

With tests completed last week, children can now breathe a sigh of relief. But the damage has been done: the boy does not do mental calculations any more. Not at all. When asked to divide 568 by 2, he does the “bus-stop” division, which is way below his skill level. He does not even try to think whether it would be faster to work out the answer mentally. The flexibility of manipulating the numbers in a way that is most suitable for a calculation at hand, the valuable skill that he used to possess, is gone. And I can only hope that, with my encouragement and his effort, it will return.

]]>He gives an example:

… a text book first published in 1960 still introduces the solution ofsimple equations with the words: ‘We use the rule that when we change the side we change the sign’

One can hear a hint of despair in the word “still” written in the 1960s. Fifty years on, open a modern textbook, and you are very likely to come across the same dry and authoritative style that does not allow for any discussion or reflection, nor offers any reason for the rules used. In fact, the vast majority of the textbooks do not deserve that name at all, as they are no more than sets of worked examples with a few comments similar to those mentioned by Skemp. The exercises themselves are meant to provide a close match to the curriculum, any steps off the beaten track are discouraged, and no attempt is made to build up a bigger picture. The presumption that the students do not want and do not need to know more is indeed insulting.

]]>I asked him a few questions, starting from what would be the most recent things learned at school. When he failed to give any meaningful answers, I had to turn to much simpler tasks at primary school level. Unfortunately, he wasn’t able to say anything sound.

Trying to find a starting point for a talk, I asked what would appear to be a very simple question: “What is the difference between 100 and 99?” To my amazement, the boy wrote down 100 and then 99 underneath, properly lining up the digits for column subtraction, carefully went through the borrowing process,and presented me with the answer. “3”, he said.

Puzzled, I started to explain that when counting up, we say “97, 98, 99, 100”, so numbers 99 and 100 are neighbours, and the difference between them is 1, not 3. I also suggested to look for an error in his calculation. However, he didnot think that he had made a mistake and my explanations did not make much sense to him.

Apparently, he had been well trained “to do maths”, where “doing” actually meant to translate the words of a task into steps to take and then to carry out calculations following memorised rules and procedures. The basic sums had been thoughtlessly put into memory as well, and had he been able to recollect them perfectly, he would have answered my question correctly. So, from his point of view, he did nearly everything right, with a minor slip of forgetting what is 10 -7. For him, using common sense and reasoning had never been a part of doing maths, so he did not apply them this time either.

I decided to switch to a different topic and try a question on geometry. Talking about perimeter, area, symmetry and other school stuff would probably only trigger his recollection skills, while I wanted to evoke his common sense, so I gave the boy two identical cardboard cutouts

and asked him to make this square with a hole in the middle:

tried hard, turning the pieces around and placing them next to each other in different ways, but without success.

This was the moment when it first came to me that it might not be just a coincidence that both arithmetic and spatial reasoning skills were lacking. It appears that if one has a clear mental image of numbers in a fixed order, then doing sums is associated with moving between different positions on a number line, making small steps or big leaps, zooming in and out as necessary. Practising spatial tasks helps to develop this kind of mental activity, so it can be argued that puzzles with shapes to move and arrange lead to better arithmetic skills.

Over the years I had many opportunities to test this connection between arithmetic and spatial reasoning, and successfully used geometry-based tasks to improve my students’ mental maths skills.

As for this particular boy, the only way for him to grasp the school maths concepts was to start from scratch and go back to counters, and at his age he was too embarrassed to do that. Unfortunately, he became one of a few students I was unable to help.

]]>The obsession with memorising times tables has always puzzled me. Weekly times tables speed tests, the practice of chanting the tables during the lunch breaks, and times tables competitions give the impression that learning them by heart is considered the most important part of school maths education.

In “Seven Myths about Education”, Daisy Christodoulou emphasisesthe need for memorising times tables:

“Just learning that 4 × 4 is 16 will be of limited use. But learning all of the 12 times tables, and learning them all so securely that we can hardly not think of the answer when the problem is presented, is the basis of mathematical understanding. If we want pupils to have good conceptual understanding, they need more facts, not fewer”

The phrase “all of the 12 times tables” makes me smile. Numbers up to 12 is not all there is, so why stop at 12? Would memorising the facts up to 27 x 27 be even better? Back to serious mode, I should note that there is hardly any practical use for remembering times tables beyond 10 x 10, unless you are going to compete in “Countdown”.

It might be a good idea to base conceptual understanding on facts, but times tables facts on their own do little to contribute to that understanding. What matters are patterns in the sequences of answers that reveal the commutative, associative, and distributive laws of multiplication. Without recognising these laws, the pure knowledge of times tables facts is useless.

If a child is lucky to have spotted the regularities in the times tables and is encouraged to use these patterns, then they would grasp the concept of multiplication and apply it successfully. However, focusing on memorising the answers is often too stressful and distracts from exploring the patterns, thus doing more harm than good.

Knowing times tables by heart (be it “all the 12 times tables” or just those up to 10 x 10) is neither necessary nor sufficient for good conceptual understanding of multiplication. I have met many children who had learnt their times tables brilliantly, but still did not realise that, for example, 5 x 7 is half of 10 x 7, or failed to use their knowledge of 12 x 12 to work out 13 x 12. I have also met children who were deemed to be the bottom of the class in terms of recalling their times tables, but who demonstrated an excellent grasp of the properties of multiplication and division, and could apply them successfully for solving problems.

Given that this unhealthy obsession with times tables is ubiquitous, I cannot see what the government’s intention to reinforce it by introducing the tests would achieve, apart from extra pressure on children and an opportunity for the ministers to issue pathetic statements about improvements in education. The idea is a lasting legacy of Gove’s damaging reforms and a response to a consultation on primary assessment.

The DaE paper says: “There is strong evidence to show that being able to recall multiplication tables with fluency plays a crucial role in being able to solve more complex mathematical problems involving division, algebra, fractions and proportional reasoning”.

The evidence they mention is not provided, so it is impossible to discuss its credibility. However, whatever research – if any – was the basis of the decision, it probably shows no more than correlation between remembering the times tables and ability to solve mathematical problems. The proponents of times tables testing might know their own times tables, but they seem to have forgotten another principle taught in school that is much more important for policy-making: correlation is not causation. Both good knowledge of times tables and ability to solve mathematical problems are affected by various factors such as interest in dealing with numbers or memorisation skills not to mention the quality of teaching, so the correlation is not surprising. However, it is wrong to assume that demanding rote learning of times tables is the best way to improve children’s ability to solve problems. After all, the causation, if any, might be the other way round: solving mathematical problems could require a frequent use of times tables and results in learning times tables by heart.

As for the times tables being “critical for everyday life”, I can honestly say that my excellent knowledge of times tables is of no use in my everyday life. Seriously, how often would one need to recall that 7 x 7 = 49, that is, 7 + 7 + 7 + 7 + 7 + 7 + 7 = 10 + 10 + 10 + 10 + 9? And if the special occasion requiring the knowledge of this fact does arise, it will not be difficult to work it out or reach for a calculator.

Knowing that 7 x 7 =49 **and** understanding the distributive law would help me to work out something more complicated like 17 x 7 = 119, but then again I would hardly need it in the first place. At the same time, the laws of multiplication go well beyond practicalities of everyday life of an average person. They open a door to a world of abstract mathematics, and introducing children into that world, showing its beauties and peculiarities, should be the true purpose of mathematical education. Sadly, with initiatives that prioritise rote learning of pointless facts, the door to this world remains shut.

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One would say that there was no way to learn times tables in 100 days, given that she had failed to do that in the previous seven years, but I had a trick up my sleeve. Instead of focusing on memorising the multiplication answers that apparently were her weak spot, I printed out a compact version of times tables and allowed her to use them whenever she needed. Every time the question required knowledge of 7 x 6, she would just look it up rather than try to retrieve the answer from memory. Free from the constant fear of forgetting the multiplication facts, the girl did very well: she worked quickly, enjoyed the challenge and solved more problems than ever before.

After a few weeks of successful practice, she started to rely on that piece of paper less and less as the answers seemed to get stored in her memory effortlessly. By the start of her exams, she knew by heart most of the times tables and was able to work out the rest quickly. She got a C.

Despite being repeatedly told how bad her maths was based on the results of times tables tests, the girl was not put off and did not stop learning maths. All she needed is a relief from the stress of times tables memorisation, so the story has a happy end for her. But I can imagine how many children give up on maths altogether once they fail to satisfy the requirement of learning the times tables. As long as the memorisation hurdle is seen as absolutely necessary to overcome, thousands of children who find it hard will stop there and never move on.

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