Chapter 4 Number Operations

CHAPTER 4

NUMBER OPERATIONS

Summary

• There are many different words for the four basic number calculations in English, +, -, x and divide. The use of BODMAS or BIDMAS ensures that calculations are completed in the correct order

•  Rounding and estimation questions are not present in some cultures, especially those that use symbols, such as Chinese

•  Word problems or topic-based work are fantastic opportunities to expose and explore cultural and language variations for ESOL learners

•  Methods for calculations vary between cultures, especially for division

• Personally, I do not interfere with learners’ methods unless they are ineffective

• If learners are going into any form of education in the UK, they will need to know UK methods

• Repetition is vital when learning new vocabulary

• Resources can be used in many ways to maximise learning opportunities and consolidate learning

Introduction

By number operations I mean what we can do with numbers in terms of adding, subtracting, multiplying and dividing. You might think that these simple processes are the same the world over, but guess what? They are not!

In English we have many alternative words for the number operations, such as ‘add’, ‘plus’ or ‘and’ for addition (Resource: Number Operations-Words). There are also many different ways of phrasing the questions, so the numbers are not in a fixed order in the sentence, and learners need to understand the sentence to ensure their answer is correct.

Challenges can arise from both interpretation and calculation points of view. For instance, if we are adding or multiplying, which are known as commutative operations, it does not matter which number we use first, as the answer to 7 x 21 is the same as 21 x 7, but we do need to understand the language, e.g., ‘seven times twenty-one’, seven multiplied by twenty-one’, or ‘seven lots of twenty-one’ are the same as ‘twenty-one times seven’, ‘twenty-one multiplied by seven’, or ‘twenty-one lots of seven’. Similarly, if we have been asked to ‘sum’ two numbers we need to know that it means add.

However, in the cases of subtraction and division, which are known as non-commutative, the language and the order of operations are both relevant to getting a correct answer. For instance, to answer ‘deduct 7 from 19’, we need to know that ‘deduct’ means subtract, and the answer is 12. ‘What is 19 take away 7?’, has the same answer of 12, but the numbers are reversed in the sentence, and this can be very confusing for ESOL learners. You might find ‘take away’ causes a few issues too!

All of our brains, ESOL or not, seem to be wired to sometimes do an easier calculation than the one they are presented with. An example of this is when we see a calculation like ‘subtract 12 from 11’. The answer learners give may often be ‘1’, but the correct answer is ‘minus 1’ or ‘-1’. Answers that involve negative numbers tend to come up in questions on temperature, and this question can be re-phrased as ‘it was 11ºCelsius during the day, and the temperature fell overnight by 12º. What was the temperature at its lowest point?’  

In other languages there may only be one way or very few ways to say a number operation, such as subtract in Italian which translates literally as ‘remove’. Sadly, in English we don’t use ‘remove’, which is odd when you think about it, as we seem to use everything else.

Also, it can be even more confusing when the same word relates to a different operation. For instance, in some countries in Asia the word 'into' tells people to multiply, whereas in English it is used for division, so the answer to '5 into 20' for some learners will be 100 and others will be four.

BODMAS or BIDMAS

The abbreviations of the words contained in the short form above informs everyone on the order that calculations must be done in for our number system to work. ESOL learners may have their own version of BODMAS or BIDMAS using their first language, and it could be worth displaying all the different versions on a poster in the classroom, firstly as a memory aid, but also to show that maths skills in the first languages are acknowledged and valued.

‘B’ stands for brackets (like this)

‘O’ stands for operation of powers, such as when we square or cube a number, such as r squared, or when we find the square root of a number. / ‘I’ stands for indices, which is the posh name for powers or roots (Examples include: 3², 5³ or √25)

‘D’ stands for division

‘M’ stands for multiply

‘A’ is for addition

‘S’ is for subtraction

Thus, using the correct order of calculations, 27 + 0.4 x 3 would be 1) 0.4 x 3 = 1.2, then 2) add 27 to the answer, so = 28.2.

If I don’t use BODMAS I would get 27.4 x 3 = 82.2; this is the wrong answer and it is very different, though in this case interestingly the numbers are transposed, i.e. the 2 and the 8 have swapped position.

Rounding and Estimating

Rounding and estimating are both concepts that can cause an ESOL learner problems. Firstly, round means a shape like a circle, doesn't it? So, what is this rounding rule? To estimate an answer, learners need to round the numbers first and then work out a rough or approximate answer. An example is £1.99 plus £6.40, is roughly £2 plus £6, so the answer is approximately £8. Working it out exactly, then rounding the answer will NOT be marked as correct in exams!

Languages that use symbols, such as Chinese, Korean or Japanese, are known as logographic languages. In these languages a symbol is used to represent an object or idea. These can present an additional challenge for learners required to do an estimate. For instance, my Chinese learners who are still translating all or most of the English text into their first language, then doing the calculation, then translating back, can be very resistant to changing the ideogram or symbol to another one before doing the calculation, because it adds enormously to their workload. Rounding and estimation do not appear in some other nations’ maths classes, and many learners are used to the idea that the exact answer is the only one that counts, so rounding and estimating answers can take some practice and are best put into practical scenarios. Shopping is ideal, as most of us will roughly add up the cost of items in a shopping basket whilst waiting in the queue.

Topic-based maths work or word problems

Topic-based work, such as maths questions focussed on a shopping trip, exposes ESOL learners to a potentially unfamiliar culture and language in a covert, or hidden, rather than an overt (‘in your face’) way, as we might find in an ESOL or a Citizenship class. In maths lessons topic-based maths problems are called word problems, because the maths calculations have to be decided upon by the learner after comprehension and consideration of the language. An example is ‘Amy has seven apples. She gives two to Billy and eats one. How many apples does she have left?’ This contrasts with ‘7 -2 -1=?’, which is clearly much easier to understand.  

Methods for calculations

In the UK we have a number of different methods ourselves, some of which learners may have seen, some not. Currently in UK primary schools there is a belief that being able to use a range of methods depending on circumstances is desirable, so those ESOL learners who have young people coming home from school with homework, and who want to be able to help, or who are hoping to help out in school, will need to see all the options. For instance, if we want to add two numbers in our head we might use the partition method, where we break the numbers down, e.g., ‘21 + 17’ is the same as ‘20 + 10 = 30 and 1 + 7 = 8’, so the final answer is 38.  (Or we might round then add or subtract to get the answer spot on, e.g., ‘21 + 17 is 20 + 20= 40’, then we have 1 more on one number, and 3 less on the other, so ‘40 + 1 – 3 = 3.)

If we are multiplying, we might use a traditional method, or a grid method, or a lattice method For examples of these see the picture on the left at the top of the page.

Working with learners, I have found the lattice method can be very successful for learners who are on the dyslexia spectrum and can help those learners to succeed where other methods have failed.

I have found that most ESOL learners use a traditional method that looks the same as ours, but the exception to this is division, which may look different. In fact, as far as I know, it is only the UK, USA, Canada, Australia, New Zealand, and Sweden that use the method generally called the bus stop method, which is illustrated on the right hand side, above.

As a general rule if the method the learner has works, by which I mean the answer is correct, I will not alter it, however bizarre it looks to me. It will not look bizarre at all to the learner, of course! If the learner wants to work with other people, such as children in school, then they will need to know the methods used in the UK. If the learner's method is not working then either you need to know what is going wrong, and another learner with the same first language could help here (if both learners are happy with that) or you need to offer alternatives that the learner feels happy with. No marks are ever lost on exam papers or in real life for using a method that works for you and gets the correct answer.

Whilst on the subject of exams, Functional Skills and GCSE learners can use a calculator in part of the exams, but they will still need a working method they can recall or remember and apply in the non-calculator sections.

Resources and ways to use them

There are worksheets covering all these options at the back in the appropriate section. The matching activity with the words and symbols for the number operations will usually generate much discussion, and I often start with this, leading into practice with work sheets of examples of the types of sentences learners could come across in real life and the exams. Learners could then practise in pairs with one asking the other the questions, and the second person giving the answer, either in a full sentence, or just the number, depending on their English level. They could then swap over. Early finishers could then generate their own questions and answers…

Repetition is vital when learning new vocabulary and using these resources in more than one way in your lessons will save you time and effort that can then be focused on the learners. Points of focus can be making sure answers are correct, that misconceptions are eradicated, and that comprehension and/or pronunciation is improving. Learners will have used reading, writing, speaking and listening skills in maths lessons, in line with ESOL Core Curriculum guidelines. The activities described are examples of how much an ESOL maths session will often differ from a maths session for English speakers, and an indication of how important a language perspective is for the teacher.

As someone who teaches both ESOL learners and English speakers in a Further Education college, I have found in over 20 years of teaching, that English speakers get just as much out of the resources that I have developed for ESOL students and I use these with everyone. Of course, I can probably leave out the pronunciation practice with fluent English speakers! In effect, I am teaching maths as if it were a foreign language, which for some of the learners it certainly is!

Use of dictionaries

Allowing learners to use dictionaries in class and in exams is highly recommended, and I will come back to this later, but not everyone will have a bilingual dictionary, and even if they do, where there is more than one definition for a word can lead to much confusion for learners. For instance, as a class with my students we were looking at a word problem about making a shelf with a piece of wood. I had to spend some time drawing pictures on the board to explain ‘shelf’, as a flat surface that we place objects on (it is OK if there is one in the room, and you can point to it). The first definition in a Chinese- English dictionary of ‘shelf’ was a flat piece of land under the sea close to the shoreline. I had to draw a ‘continental shelf’ on the board too, so that the non-Chinese learners could see what the challenge was!

The use of smart phones for translation in the classroom works well, but will not be allowed in examinations, so it is important that learners know how to use the translation devises that can be used in exams.

JMS 2026