On Saturday 4th July, I was kindly invited to give the opening address at the wonderful Edexcel Maths Conference in Warwick. The title of my talk was “The 5 most interesting misconceptions in mathematics”. Quite a few people asked me for a copy, so here it is, along with some screen shots.

And as a disclaimer, these are purely my own choices for what I considered to be interesting misconceptions – ones that in 10 years of teaching I have not necessarily emphasised, but I certainly will from now on.

All the lovely sunburst diagrams, come courtesy of our Data pageon Diagnostic Questions. Please have a play around – I promise you will not be disappointed!

What are the major misconceptions that students have when it comes to probability tree diagrams? Specifically, care students able to correctly interpret the meaning of the question and select which branches they require?

Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)

Have a look at the question below.

What percentage of students do you think get it correct?

What is the most popular incorrect answer?

What explanations do students give for the incorrect answers?

Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:

At the time of writing, this question has been answered 433 times, and has been answered correctly 52% of the time. The most popular inocrrect answers given are C with 21% and B with 19%.

To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.

Probability tree diagrams are a staple of the maths GCSE. Students usually encounter one for the first time around Year 8, and keep bumping into them every school year thereafter.

When you think about it, there is a lot that could go wrong with a tree diagram question. Firstly, there is the construction of one from scratch if it is not given in the question. Secondly, there is the (potentially huge) issue of being able to multiple and add the fractions or decimals that may be involved in them.

But on top of all that, there is the concept of the tree diagram itself. Which, when you come to think about it, is rather bizarre. In the example above, students are essentially being asked to perceive four alternate realities. The first of which entails a win in the first game, followed by a win in the second game. Then, in a parallel universe, we ask them to consider what would happen if they won the first game but then went on to lose the second game. And so on. And for some students, this can be a little much to get their heads around.

This is why I like this particular question. The tree diagram has already been constructed, the branches are labelled, and the answers have been chosen in such a way as to avoid any confusion with the decimal operations themselves. Hence, we can focus all our attention on the students' understanding of the concept of a tree diagram, and specifically can they interpret what is meant by winning "at least one game".

Let's have a look at how the students themselves interpret this:

Correct Explanations for A

Many students explained the correct answer beautifully:

I could win the first and win the second. 0.6 x 0.6 = 0.36. I could win the first and lose the second 0.6 x 0.4 = 0.24. I could also lose the first and win the second 0.4 x 0.6 = 0.24 0.36+0.24+0.24 = 0.84

Looking at the tree diagram you can see that there are four outcomes (win + win, win + lose, lose + win, lose + lose). Three of these give at least one win. The probability of a win and a win is 0.36 (0.6 x 0.6 you times because its an 'and'- win And win- not an 'or'). The probability of a win and a lose is 0.24 (0.6 x 0.4). The probability of a lose and then a win is 0.24 (0.4 x 0.6). Then add up these three probabilities (add because its an 'or'- you want the probability of a win + win OR a win + lose OR a lose + win). 0.36 + 0.24 + 0.24= 0.84

At least one win= win and win or win and lose or lose and win. When its an 'and' you times (e.g. win And win) so the probabilities of each of these are: win and win=0.6 x 0.6=0.36, win and lose=0.6 x 0.4= 0.24, lose and win= 0.4 x 0.6= 0.24. Then when its an 'or' (e.g. win+win Or win+lose) you add so the probability of at least one win is 0.36+0.24+0.24= 0.84.

Go along the first branch and there are two possible scenarios of winning at least once. So 0.6 x 0.6 = 0,36 + (0.6x0.4) is 0.6 Then on the bottom branch you can only win once by 0.6x0.4 which is 0.24. Add that on to 0.6 and you get 0.84

A textbook could not have written it better. And all it takes is for one of these explanations to make sense to your students, and their misconception is well on its way to being resolved.

Incorrect Explanations for B

B attracted almost 1/5th of responses, and you may be able to guess why. This is not down to an inability to multiply decimals, it is doe to a misconception with the concept of following and understanding the tree diagram itself.

it could be win then loss which would be 0.4*0.6, or loss then win which would be 0.6*0.4, both of which would be 0.24.

Because a win then a lose has the probability of 0.4/1 and a lose then a win has the probability of 0.6/1 and those multiplied together=0.24/1=0.24

I think this is the answer because You times togethor 0.6 and 0.4 to get 0.24

follow the path of win then loss then follow loss then win then divide them

Incorrect Explanations for C

Answer C was the most popular incorrect answer. Again, I feel the misconception evident here is down to interpreting both the question and the tree diagram itself. What does it mean to win at least one game?

If he loses first and wins second 0.4 x 0.6 = 0.24 + If he wins first and wins second 0.6 x 0.4 = 0.24 ------ 0.48

I think this is the answer because it would either be 0.6*0.4 or 0.4*0.6 so if you add the answers together you would get 0.48.

If you go across the first win and second lose you get 0.6 x 0.4 = 0.24 And then do that for the second branch to again get 0.24 which equals 0.48

Multiply together 0.6 and 0.4 and 0.4 and 0.6. I then add the answers together after this which gets me 0.48

Incorrect Explanations for D

And then we have answer D. Now, I did not write this question, and to be honest I could not see why students would choose D. But a fair few did, and their reasons were rather interesting:

because 0.6/ 2 games = 0.30.

Add them all together

So, we have seen a whole bunch of misconceptions that students have when it comes to probability tree diagrams. Crucially, I feel the misconceptions are with understanding the concept of the tree digram itself, and not with operations within the tree diagram. These are two separate issues, and I like the way this questions isolates them.

This question can also be found in an Tree Diagrams quiz. Try it below and click here to assign it to your students.

Below is a video showing you how to access all the pages and information discussed above:

What are the major misconceptions that students have when it comes to calculating the area of a triangle? Specifically, can students correctly determine which lengths are needed to get the right answer?

Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)

Have a look at the question below.

What percentage of students do you think get it correct?

What is the most popular incorrect answer?

What explanations do students give for the incorrect answers?

Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:

At the time of writing, this question has been answered 471 times, and has been answered correctly just an impressive 76% of the time. The incorrect choices are split evenly amongst the three other options, with each accounting for 8% of responses.

To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.

Now, looking at the figures, it doesn't appear that there are too many nasty misconceptions lurking around this question. After all, 76% success rate is probably the highest we have had in this Question of the Week feature.

But before we go and make a cup of tea, and relax in the knowledge that the majority of the world can work out the area of a triangle, there is a,ittle more going on here than meets the eye.

I have selected this question in particular because of the explanations. The way students all around the world articulate their explanations on the website (and they are getting better week by week) allows us as teacher an unprecedented insight into their thinking. And whilst the numbers getting this question wrong might seem insignificant, the reasons for these mistakes are not.

Fortunately (and this is my favourite part about the website), students with misconceptions can go along way to resolving them themselves by reading correct explanations given by students all around the world. And what wonderful explanations we have:

Correct Explanations for A

Many students explained the correct answer beautifully:

This is because the formula for the area of the triangle is base x height divide by two. 7cm is a base, 6 cm is a height so the area of the triangle is 6 x 7 /2

to work out the area of a triangle you do 1/2 x base x height. The base is 7 and the perpendicular height is 6 so it would be 7x6 and then timesed by a 1/2 or divided by 2

To find out the area of this isosceles triangle you have to times ( for this one) 7 x 6= 42 then you have to divide 42 by 2 because you found a rectangle's area ( double the sizeof this triangle) so divide by 2 and you get 21cm2.

2 of these triangles fit into a 7cm by 6m rectangle so find the area of that by multiplying 7 and 6 together then divide by 2

The base is 7 and the perpendicular height is 6. You do not need the 8cm as this is the slanted height. You do base x height and then divide your answer by 2.

All it takes is for one of these explanations to hit home with a particular student, and their misconception will be resolved. That is why it is so important that we encourage our students to review and like their favourite explanations.

Incorrect Explanations for B

B is a classic. I think it comes from the students' belief that if a number is given in a question, then it must be used in the answer, or why would it be given to them? Giving students regular exposure to situations where some of the information they are given is not needed may be the key to helping resolve this issue.

its the method you use for finding out the are of shapes only some other shapes you use a different method but not a triangle

I think this is the answer because you times all of the numbers together

the other three answers simply don't have enough information.

on the other answers where does the 2 come from

Incorrect Explanations for C

Answer C will catch out students who do not know the importance of the height being perpendicular. Or, as it also likely, they have heard the word perpendicular, but do not actually understand what it means

I think this is the right answer because you can't just times them without using a division because that would make a rectangle. I think you times the 8 and 7 because that would make like a parallelogram which fits two triangles in it that's why you divide it by 2?

as its not a right angled triangle you cant use base times height so you have to use base times length divided by 2

I think it's C because you make this isosceles triangle into a square then you time 8 and 7 and divide by two to get the area.

Because the formula is base times height divided by 2

Incorrect Explanations for D

Answer D was designed to identify those students who do not know, or who have forgotten the importance of dividing by two, or halving. Their explanations reveal some interesting insights:

you do times the perpendicular line by the base line

You dont worry about the diagonal lines.

split the triangles into 2 so there is two right angled triangles so you have two equation of 6x7/2, but these cancel out to make 6x7

You do not halve this sum as the triangle is made of two right angled triangles

So, we have a whole bundle of important misconceptions concerning the Area of a Triangle that we need to identify in our students and then resolve. Hopefully Diagnostic Questions can help you do this.

This question can also be found in an Area of Simple Shapes quiz that I put together. Try it below and click here to assign it to your students.

Below is a video showing you how to access all the pages and information discussed above:

What are the major misconceptions that students have when it comes to Stem and Leaf Diagrams at GCSE? Surely nothing can go wrong in this lovely topic?...

Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)

Have a look at the question below.

What percentage of students do you think get it correct?

What is the most popular incorrect answer?

What explanations do students give for the incorrect answers?

Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:

At the time of writing, this question has been answered 223 times, and has been answered correctly just 22% of the time. By far the most popular incorrect answer is D, which 53% of students go for,

To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.

Now, Stem and Leaf Diagrams are one of those topics that I am often tempted to brush over during GCSE Maths revision. After all, students first meet one in Year 7, and then keep meeting them every year onwards. And they never change. You always need to put the numbers in order, come up with a decent stem, ensure you don't miss any numbers out. Oh, and remember your key. Always remember your key.

And students who will look at disdain at the sight of a quadratic question or a circle theorem, will usually smile sympathetically at the sight of a stem and leaf diagram, as they can't possibly do them any harm. Can they?

Well, I got the inspiration from the question above following a review of the June 2014 GCSE Higher Paper 1 from Edexcel. We were presented with a student's answer that was similar to the one above and asked to mark it. Almost every teacher gave full marks (including me!). But there is a mistake! And apparently it was a fairly common mistake amongst actual, real life GCSE respondents. And okay, it is only going to cost the students one mark, but that one mark could be the difference between a D and a C, or an A and an A*.

So, what did the students on Diagnostic Questions make of it?

Correct Explanations for B

Some students spotted the mistake, and explained it beautifully:

The question asks for the height, and the data represents centimetres, not children as the key suggests

The question asks for the height and the key states children which the question was not asking for

because the key should say 125cm not children as its taking about height in cm

Incorrect Explanations for A

Thinking the stem should only have one number in it is a misconception I run into a few times with students each year. Here is how they explain it:

The stem should have just 1 number in it not like this one where it had two numbers in

I think this is the answer because you are only meant to put one number down the side

Because the number on the stem should only have one digit as it represents a "ten"

Incorrect Explanations for C

Here I wanted to test if students could correctly match up the numbers in the list to those in the diagram. It is interesting what they thought the problem was here:

because the number 113 is missing

There's a 100, but 100 is not plotted on the diagram

there is not another 105

These careless mistakes cost lots of marks!

Incorrect Explanations for D

By far the most popular answer, amongst teachers as well as students. And here, I suspect it is people just glancing at the key, acknowledging its presence, and then moving on. Or maybe they are going one stage further and checking the coding 12|5 does represent 125. But are they going that extra stem and looking at the actual units? I don't think they are. The students explain this very well, and it is interesting that they all make mention of the key:

Because the key is right, the diagram is right and no numbers are missing and in the stem its a 2 digit number because it's representing 3 digit numbers

There are no mistakes - it seems incorrect due to the two numbers in the stem however this is because of the fact that the number in the stem and leaf diagram is a 3 digit number

The key works to show hundreds and the stem and leaf diagram shows all the data given

So, how do we deal with all of these? I think it is the age-old advice of telling students to check over their work. And yes, before you say it, my students never listen to me when I say that either. But when they did this question as part of the GCSE Maths quiz below, it did provide a talking point that I hope will stay with them.

This question is taken from the Edexcel GCSE Higher Paper 1 quiz that can be used as preparation or a follow up to the actual paper copy. All the questions and numbers are different, but the same skills are tested. I hope you find it useful.

Below is a video showing you how to access all the pages and information discussed above:

What are the major misconceptions that students have when it comes to combining events in probability? Specifically, when the question does not explicitly ask for a tree diagram, what strategies to students employ, and what mistakes do they make?

Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)

Have a look at the question below.

What percentage of students do you think get it correct?

What is the most popular incorrect answer?

What explanations do students give for the incorrect answers?

Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:

At the time of writing, this question has been answered 111 times, and has been answered correctly just 33% of the time. By far the most popular incorrect answer is A, which 48% of students go for,

To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.

As we all know, probability can be a minefield of misconceptions. Students are rarely sure when to add and when to multiply, and even when they can remember the rules, they rarely understand why they are doing what they are doing.

Indeed, when faced with the traditional tree diagrams probability question in the higher tier GCSE exam, many students are absolutely fine. They go into the well rehearsed routine of filling out the branches, and then multiplying the numbers going across and adding them to go down. So long as they are careful, they will help themselves to the 4 or 5 marks on offer without breaking sweat or over-exercising their brain.

But it is questions like the one above that cause problems, because there is no mention of a tree diagram. The question calls for a deeper level of understanding from the students, and each of the wrong answers teases out a key, and all too common, misconception.

Let’s see what expatiations they students gave, in their own words:

Correct Explanations for A

There are a couple of beautiful student explanations here for what is a pretty tricky concept to explain:

If you think about a tree diagram, there is a 1/4 change of getting a 2 on the first spinner, and a 1/4 chance on the second spinner. If you need both to happen, you must multiply, which gives you 1/16

There are 16 different scores you could get (4 x 4). Only one of them gives you two lots of 2. So the answers must be 1/16

Interestingly, a few students get this question correct, but via their explanations we can see that their understanding is not quite right:

i think this because there is only 1 chance of getting a 2 and there are 16 numbers altogether

I think this is the answer because there is the number 1 on each spinner and if you put the two one’s together you get 1/8

This illustrates the importance of reading students’ explanations, and not just assuming because they have got it right that they understand.

Incorrect Explanations for A

A was by far the most popular choice for students. And their explanations reveal exactly the misconception that is all too common with probability:

There are 4 bits on each spinner which make the spinners up. 4 add 4 equals to 8 and there is one 2’s on each spinner which adds up too 2. So therefore we have 2 chances in 8 of getting a two on both spinners.

i think this because there are two 2’s and there are 8 numbers altogether this means that it is 2 over 8

you have a 1/4 chance of getting a 2 on each dice and then you double that

There are 8 numbers in total 2 of them are 2

As a teacher, how would you deal with that misconception?

Incorrect Explanations for B

B tempted in a few students. And I would argue it is probably the best wrong answer, as the idea of 16 outcomes is shining through. But where does the 2 come from?

i think this is the answer because there are 2 sets of 2s and there are 16 sections

Incorrect Explanations for D

Not many go for D, but like B there is some understanding evident here. The 1 in the numerator suggests that students have correctly identified there is only one possible outcome (as opposed to two) that gives what we are looking for:

there is one 2 on each spinner, so you need to multiply these, and 1 x 1 = 1. And there are 8 different outcomes (1 to 4 and 1 to 4), so it must be 1/8

So, how do we deal with all of these? Do we resort to a tree diagram? Will that give us the understanding we need, or just fall back on the same old rules? How about a sample space diagram showing all the possible outcomes? Any other suggestions?

This question is taken from the Probability with Spinners Quiz, which provides more challenges like this one:

Below is a video showing you how to access all the pages and information discussed above: