What misconceptions do students have with the order of operations, otherwise knows as BIDMAS, BODMAS or PEDMAS?

Similar to last week, we again saw some wonderfully answered questions on GCSE Essential Skills Quiz 10, suggesting that the nation’s 15 and 16 year olds are getting better at nailing the kind of basic questions that could cost them valuable marks in the summer. However, as is always the case, a nasty little nugget has emerged that caught out the majority of students answering this quiz.

Quiz 10 is below, followed by an analysis of the Insight of the Week. Looking at the questions in the quiz, can you guess what the worst answered questions were?

Here’s how our students performed compared to the rest of the world:

 

Our students struggled a little with calculating the length of sides in similar shapes and with those tricky backwards mean questions, but like most of the students around the UK who took this quiz, our lot came a cropped with this little beauty:

Over half our students opted for answer A, and a measly 28% of them arrived at the correct answer of D. So, what is going on here? There’s only one way to find out…

 

I’ll be honest… I don’t like BIDMAS. Firstly, I do’t like the fact that everyone seems to call it something different (BODMAS, PEDMAS, PIDMAS, etc). But more importantly I do not like the confusion it instils in students. Don’t get me wrong, I completely see the need to have an order of operations to remove any ambiguity in the answer to arithmetic and algebraic questions, but as we can see from the amount of students who got this question correct compared to the other 11 questions in this quiz – there is something about the topic that students find incredibly difficult and confusing.

Let’s see if we can get to the bottom of what is going on by analysing their responses:

 

Incorrect Answer A (56% of our students chose this)

This is your classic misconception. Students choosing answer A are simply carrying out the operations in order, from left to right, as if reading a sentence in a book. But it is also worth noticing mentioning of “brackets” in students’ explanations. This suggests that problems with this question are not simply about students forgetting the rules of BIDMAS, but in misapplying and misunderstanding them. This will be a recurring and important theme across the other answers.

“13-3=10 then 10 times 4 =40 add 2 = 42”

“i think this is the answer because i did everything it was telling me to do, i subtracted 13-3 and got 10, i multiplied 10 to 4 and got 40 than added 40 to 2 and got 42 as my final answer”

“there are no brackets to tell you what to do first. i heard there was a rule for questions like this (subtraction first or whatever) but i havent been taught so i just did it in order.”

“13-3 would be in brackets and on the start of BODMAS is brackets, so you do that first. Then, Multiplication comes before addition so, next you do that. And finally, you do the addition or subtraction :-)”

 

Incorrect Answer B (12% of our students chose this)

Students selecting answer B are united in their belief that 4 + 2 is the first thing that needs sorting in this question, followed by 13 – 3. Their reasons for this are either that they believe addition comes first or – and I find this absolutely fascinating – that brackets need placing in the question:

“You must first add 4 plus 2 and you’ll be left with 6. Then you subtract 13 to 3 and are left with 10. finally you multiply with the two numbers you were left with 10 x 6 = 60.”

“you have to put the numbers in brackets and then work it out. For example (13-3)x(4+2) . You work out the sum of the brackets and then times them together. 10 x 6 = 60”

“If you put the question in two brackets it equals something like 2×30 which equals 60. It didn’t originally have brackets which makes it hard to work out.”

 

Incorrect Answer C (4% of our students chose this)

Answer C is a fascinating one. When looking at the question I could not see how students would arrive at an answer of -5. But, once again, their desire to include brackets when they are not there leads them on the path to -5:

“because you do 4+2 then multiply that by 3 and then you take-away 13 which makes -5”

“because you out 4+2 in a bracket, multiply by 3, which is 18, then do 13-18= -5”

 

An alternative incorrect answer

One of the problems with multiple choice question is that there is always the possibility that students will arrive at an answer that is not one of the available options, and hence you lose the information that that answer would have given you. Fortunately, with Diagnostic Questions, you are still able to capture that information in the students’ explanations. That is why I always encourage my students to tell me in their explanations if they believe another answer not listed is correct. In this particular question we had a number of students arriving at an answer of -1. Here is why:

“None of the answers were -1 which shows that the sum does not follow BIDMAS so if you do it from start to end the answer is 42 as 13-3 equals 10 multiplied by 4 =40+2 equals 42”

“Solve in sequence 13-3 =10×4= 40 +2 =42. Using BODMAS gives 3×4 = 12 +2 =14 .then, 13 – 14 = -1, an answer that is not an option.” .

For me, this answer reveals one of the major issues with BIDMAS – students want to do addition before subtraction. And whty wouldn’t they? After all, there is a clear, if unspoken, implication in the word itself that Brackets get sorted first, then any Indices, then any Divisions, then Multiplication, then Addition, and finally any Subtractions. Therefore, students who arrive at an answer of -1 may well be competently following the rules that they have been taught, or that they deem logical. This is frustrating for them, and potentially damaging.

 

The Correct Answer!

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. And just as importantly, please encourage your students to write good explanations – they will be improving their own understanding and helping out students from all around the world. Each time their explanation is liked, they will be notified and score valuable points on the site. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

“BIDMAS (Brackets, Indices, Division Multiplication, addition, subtraction) 3×4=12 13-12+2 13-12=1 1+2=3”

“I followed the order of operations and multiplied first, then I solved from left to right”

“According to BIDMAS, you must do the multiplication before doing the subtraction which means the question would simplify to ’13-12+2′.though, even though in BIDMAS it says that you must do addition before subtraction, both addition and subtraction are as important as each other, likewise multiplication and division. So, instead of doing subtraction first you must go left to right, just like in simple maths. This means the answer must be 3.”

“First, I did multiplication (3×4) because multiplication comes before addition and subtraction in BIDMAS. This equalled 12 which I took away from 13 – this gave me 1. I then added the 2 and my final answer was 3. The reason I did subtraction first is because it is equal to addition and came first in the sum.”

 

Tackling the Misconception in Class

Each week we discuss the Insight of the Week in our Monday Maths Departmental Meeting and try to come up with strategies for tackling it when we next see our lovely Year 11 classes. Here are a few suggestions we came up with, depending on which particular misconception our students had:

• It is interesting to see that the most popular correct explanations given by students stressed the fact that addition and subtraction have the same priority as each other, and hence after the multiplication has been carried out, the sum should be completed left to right. But, of course, that is not at all clear when students are taught BIDMAS. So, perhaps first we need to stop writing BIDMAS in a linear fashion. Or abandon the word altogether,
• One colleague suggested turning to the calculator and using the answers as an investigation. Why does the calculator give this answer? What rules is it following? Can you write a set of rules? What would the calculator say for this sum?
• I always instruct my students to put negative numbers in brackets. I find it helps alleviate misconceptions such as thinking minus four squared is minus sixteen. However, perhaps there is an argument that an overuse of brackets has led to students chucking them into sums left, right and centre! I will need to think some more about this one.

Please add any extra ideas in the Comments section below!

 

The series of GCSE Essential Skills Quizzes are available here and will always be completely free.

Quiz 11 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.