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It has been a dramatic week for Guess the Misconception, upon its return after the Easter break.

Firstly, there has been an outcry, both in my maths department and across the nation as a whole, from teachers asking “why wasn’t 100 one of the answers?” or “what happened to 40?”. This highlights both an inherent problem with multiple choice questions – namely, what happens if the answer a student thinks is correct is not actually one of the given answers? – but also reinforces the importance of students giving explanations to support their answer on Diagnostic Questions. A few of our Year 11s gave explanations such as “I am going for 110, but I reckon the answer is 100 due to alternate angles being equal”. This allowed us to go beyond the data and understand this student’s misconceptions.

Secondly – and possibly even more dramatically – the least popular misconception as voted for by teachers, turned out to be the most popular misconception held by students!

This question in… well, question… on angle facts is one of the most poorly answered across the free daily GCSE Revision Streamto date, with over half of students getting the answer wrong. Let’s take a look at the question, and see the kind of misconceptions that it reveals:

Teachers felt that A would be the most common choice, and indeed this did attract around 30% of incorrect student answers, with explanations revealing misconceptions around straight lines and the type of triangle invovled, such as:

“It’s a as it sits on a straight line and angles on a straight line add up to 180 degrees so if you take it away from 180 take away from 100 it’s left with 80″

“A straight line is 180° And this triangle is an equilateral triangle meaning all angles are the same size, triangle is 180°. 180-100 = 80, meaning the other two angles are also 80°”

But the answer that attracted over 40% of incorrect student responses was D. Can you see why? Let’s read some of explanations from the thousands of students who opted for this to find out:

“Exterior angle on the bottom two corners are 100 degrees, the angle of a straight line is 180 degrees so 180-100=80 times that by two equals 160 and then take 160 from 180 as that’s the angles in a triangle added together and that equals 20″

“This is an isosceles triangle meaning that the 2 bottom angles are equal. So first you have to do 180-100 because angles in a straight line and you get 80. This means the 2 bottom angles are going to be 80 degrees. After to find the top angle I did 80+80 which is 160 and then take that away from 180 and you get 20 degrees.”

“because the angle labelled 100 degrees is on a straight time which sums to 180 and that means the angle beside it is 80 degrees and both bottom interior angles are the same so 80 plus 80 is 160 and all interior angles in a triangle make 180 so you do 180 minus 160 which is 20 degrees.”

“First work out The angles along the bottom line 180 degrees on a straight line, so 180 – 100 (which is the exterior degree) = 80 so the angle to the bottom right is 80 degrees, as is the one on the left Inside a triangle the angles add to 180 There are already two 80 degrees 80 + 80 = 160 180-160 = 20 degrees”

What do we take from this? Well, as a maths department we were a little shocked firstly at how poorly our Year 11s performed on this question, but also the range of the misconceptions they held. Some teachers decided they were going to show this question at the start of the next lesson and allow students to discuss the answers. Other teachers announced they were doing a full angle facts revision lesson. But all decided that some sort of action was required. This might be the case with your Year 11s as well :-)

With the GCSE Maths exams getting closer, now is the perfect time to sign your Year 11s students up to the free daily GCSE Revision Stream. Over 10,000 students around the country cannot be wrong :-)

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Bearings are the kind of topic that students cannot bear (sorry). The tend to sneak their way into several key topics, most notably angle facts, construction and trigonometry. Hence, if a student’s understanding of bearings is dodgy, then it could jeopardise their chances of success across other areas.

This question is one of the most poorly answered across the free daily GCSE Revision Streamto date, with over half of students getting the answer wrong. Let’s take a look at the question, and see the kind of misconceptions that it reveals:

As predicted by teachers, by far the most popular incorrect answer was C). Why would students go for an answer of 150 degrees? Their explanations are incredibly revealing, suggesting varying levels of understanding of angles, but with once again the common thread being students falling flat due to a misremembered or misunderstood rule for bearings:

“this is because angles on a straight line equal to 180 so we take (180 away from 30) which gives us 150″

“Since the angle in the triangle is 30 degrees. I cant measure the bearing of b from a physically. Because, i know that angles on a straight line add up to 180 degrees. I use this understanding to figure out that the bearing is 150 degrees(0150 degrees)”

“Starting clockwise from north, the angle has to be 360 all the way around. 180 is already shown add the 30 degrees. 180+30=210 360-210= 150 degrees from B to A”

“Bearings always go clockwise, and angles on a straight line add up to 180 degrees. So 180 – 30 will give you the bearing from A to B which is 150 degrees”

It is interesting to note that the answer of 30 degrees cropped up in both answer A) and D), which together accounted for just as many wrong answers as C alone. Where does 30 degrees come from, I hear you ask. Well, in the words of students across the world:

“because it needs to be from north so you put the north line there, this creates a z shape. This means alternate angles are equal”

“Angle on a straight line is 180 and add 30 to find the bearing”

“180 – 30 = 150 150 + 180 = 330 360 – 330 = 30″

“Bearings always have three digits and the angles are on the same line.”

And my personal favourite:“HAVEN’T GOT A CLUE ABOUT BEARINGS MISS!!”

Like the rest of the country, over half our students got this bearings question wrong. How will we deal with this? Well, each week in our Monday Maths Departmental meeting, we discuss a particularly poorly answered question from the free daily GCSE Revision Streamthat all our Year 11s complete. We then show this question to each of our Year 11 classes and discuss not only the correct answer, but some of the best ways of explaining the answer, as well as the reasons behind the wrong answers. Doing this on a regular basis (we hope!), will allow us to focus on the areas that our students need help with in an efficient way, and through discussions lead to a deeper understanding of potentially troublesome topics.

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Last week’s question on the area of a triangle brought up two key, but completely different misconceptions. And the one teachers felt was the most popular was in fact beaten into second place by another nasty one. Let’s have a reminder of the question, and then take a look at those two misconceptions:

Incorrect Answer C

Whilst it is pleasing to note that these students have clearly remembered the important fact that they need to divide the product of the base and the height by two, unfortunately they have chosen the wrong height. That becomes very apparent in their explanations:

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?

Because the formula for the area of the triangle is base x height divided by 2. 7cm is a base, 6cm is a height so the area of the triangle is 6×7 divided 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

8×7 gives the total of the area of a square then divide that by 2 to get the area of a triangle

Incorrect Answer B

Students opting for this one could not resist the temptation to include every single number in their calculation of area:

because to work out an area of a shape you times the length of each side together

Base x Height x width is how u find out the area of an isosceles

It is necessary to include the six as it tells you the height

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

What can we learn from this? Well, firstly the specific answer the students give determines the support the support they need. Perhaps students who went for C) need a visual approach as to why the perpendicular height is needed. Whereas the students going for B) are showing a lack of deep understanding of area, and perhaps need to go back to square one. The bottom line is that both answers imply a misremembered or misapplied algorithm, which is one of the most common causes of misconceptions that I see across the website.

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Last week’s question on fractions and decimals has been written by AQA as a taster to the kind of questions students can expect on the 2017 Maths GCSE. As some of you may know, AQA will have at least four multiple choice, diagnostic Questions on each of their GCSE Maths exams, both Higher and Foundation:

Teachers correctly predicted that A would be the most common incorrect answer, but the reason for it is likely to be a big surprise. Moreover, look at the popularity of the other choices! This is a question that exposes a whole series of different misconceptions.

Incorrect Answer A

Now, you may think that students here have done the classic thing and changed 0.5 to ½, and then simply added the numerators and denominators together. That is certainly what my department and I thought when discussing this during out meeting last Monday. However, having read the students’ explanations, that misconception accounted for very few of them. By far the biggest misconception was a misunderstanding of what the question was actually asking, as these explanations will illustrate.

“4/6 simplifies to 2/3 the that as a decimal is 0.66 recurring and 3/4 as a decimal is 0.75 and 0.5 so 0.66 recurring is in the middle”

“because a is 0.6 and it is smaller than 3/4 which is 0.75 and more than 0.5″

“because 3/4 equals 0.75, 4/6 is equal to 0.6 and that is below 3/4 but more than 0.5″

Students have clearly read the question as “which number lies between 3/4 and 0.5?”. How frustrating is that!!!!

The other incorrect answers revealed mistakes with place value, and adding fractions, as this selection will illustrate:

B:0.34 + 0.5 = 0.84″

C:because 3/4 is 0.75 and 1.205 is above 0.5 and 0.75″

C:0.50 add 0.75 = 1.205″

One lesson that we as a maths department are taking from this (as well as ensuring students read the flipping question!) is that in the new GCSE, more than ever before, students will be challenged to combine skills together. So, to succeed in this particular question, it is not enough to be able to convert a fraction to a decimal – students must then display competent knowledge of place value to get the right answer. Students need to ensure that having completed one skill, they do not rest on their laurels and make classic mistakes, and instead switch right back on until the end of the question.

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The hat-trick was on this week, and the teacher members of my Diagnostic Questions website certainly did not bottle it. Once again, you correctly predicted the most common misconception that students had with the following reverse mean / backwards averages question:

Backwards-mean

The most common misconception was indeed a) 18, which attracted half of the incorrect answers. What reasons did students give for this? Well, let’s have a read of some of their explanations:

58-40=18

The total is 58, so if one of the numbers is 40, the others must add up to 18.

However, I think you will be slightly surprised about the second most common misconception that students have. Whilst only 6% of teachers went for c) 162, it in fact elicited 26% of incorrect student answers. Can you see why someone would get an answer of 162? I must admit, I couldn’t. But then I read their explanations:

You need to multiply by how many numbers there are to do these sort of questions. The difference is 18 and there are 9 numbers, so you get 162

The difference is 18, and there are 9 numbers, so the mean must be 18 x 9

18×9

I hope all of this serves to illustrate what I believe is the most important point of Diagnostic Questions. It is not simply the case that students are either right or wrong. They can be wrong for very different reasons, and each reason requires a specific form of intervention. Would you agree that an answer of c) is slightly better than an answer of a)? And further still, an answer of d) is the “best” wrong answer of all? Or am I talking nonsense? We will return to this more, I am sure, in the coming weeks.

For now, check the blog on Sunday for the next round of “Guess the Misconception”!