So by now you’ve surely heard in some capacity about Thursday’s “tough” Edexcel GCSE examination paper. I’m not interested in focusing on the paper per se here, but more on the implications for teachers regarding the new GCSE.

I have no doubt that this exam was written with one eye on the new, tougher papers to be sat in 2017. They’re already being rewritten by all boards (see: fiasco), as many people noticed they’re perhaps rather too difficult in their current form. That mess aside, what was incredibly obvious from the responses from students on Thursday, was that many could not cope with the new style of questions.

This has merely confirmed what many of us in the teaching profession are already not-so-secretly worried about: some students won’t even attempt the questions.

Now of course, put in context, some questions on exam papers are specifically intended to sort the wheat from the chaff. No issues there. However, there is almost certainly going to be a large inrease in examination questions that do not give explicit guidance on how to solve them. Does this sound familiar? It’s probably because that’s almost my entire ethos with the questions I write here. Although I’d be horrified if my questions ever appeared as the majority on a GCSE paper!

Let me be clear here: I am completely in favour of this new movement. I am completely for having tougher questions, and questions that require students to think and analyse before going head first into a solution. But Thursday’s paper has highlighted how far away the students are from being able to handle these kinds of questions. For a long, long time now, students have been given part a), part b), part c) – all of which hold their hands carefully until they get to the toughest bit, which they then use part a) and b) to help them with. That looks like it’s going now. Great. But the pressing question is this: how do we get students to succeed with these tougher, less clear questions?

A visual example is below, and of course, I’m going to use geometry 🙂 :

I included this question in a Year 10 (top set) end of unit test just a couple of weeks ago. I (correctly) anticipated that some students would either not attempt it, or get it completely wrong. I mentioned this online, and a few people (not teachers incidentally) suggested this question was far too easy for top set students to be branded as ‘difficult’. My argument wasn’t that students couldn’t do the maths behiind it, it was that they wouldn’t see how or where to begin.

Consider the slightly altered version of the diagram below:

If this was my diagram, I am confident that every student would have answered the question, and answered it correctly. Fortunately a lot of my students *did *see the line, but not everyone. So the point here, as with the Edexcel exam is this:

*How do we teach students to see the line?*

Or more generally,

*How do we teach students to start a question when there’s no immediately clear path?*

This kind of question has often been labelled ‘teaching students problem solving skills’. I feel that is far too general (2 + 3 = ? is a ‘problem’ of sorts) and open to all sorts of fuzzy interpretations, so I’m simply not going to label this ‘skill’. It’s important to note that it is not based in any particular area of maths either. The above could just as easily have been a written word problem (perhaps the most commonly skipped question by students) or data etc. The topic is largely irrelevant.

Anyway, here are my suggestions to help encourage students to confidently attempt questions with no step-by-step solution path:

**1. Subject Knowledge**

An eye-rollingly obvious first choice. But I’m writing it to ensure there is no misinterpretation here. Without a good foundation in mathematical ability, the rest of this list is arguably rather futile. If a student is poor at multiplication, or has no memory recall of pythagoras or the parts of a circle, what use is ‘seeing the line’ in the above question? Again, to avoid misinterpretation, I’m also not suggesting that this comes first in a linear teaching process either. That is, the rest of this list should be taught *alongside *essential subject knowledge, not entirely *after* it is considered completely secure.

**2. Increasing discussion time**

If you’re not talking to students, prompting and guiding their thought process with your expert knowledge, then how are you expecting them to start to think like a mathematician? By accident? I worry that the age-old advice of reducing teacher-led time and increasing student-led time to an often impossibly unrealistic ratio may well have inadvertedly contributed towards a lack of analytical skills in our students. I would love to know your thoughts in the comments if you agree / disagree.

I really enjoyed this post about subitizing, aimed at primary school students. In it, Steve Wyborney asks the questions “what do you notice?” followed by “what else do you notice?”. Simple but effective questioning to prompt in-depth discussions, just using the right visual prompt.

**3. Question Variety**

I’ve spoken before about the need for students to be exposed to different types of questions within a given topic, and I’ll stress it again here.

Students aren’t “finished” with area when they can answer these:

**4. Questions out of context**

It’s all too easy to throw in a tougher, slightly more obscure question at the end of a lesson, or (very often) at the end of a textbook chapter, but the issue here is that students already know what the topic is, and are aware that the approach will involve the things they’ve been doing that lesson, or that series of lessons.

If you really want to test their adaptability, throw in an unrelated question as a starter, or a homework. Or spare yourself five or ten minutes at the end of a lesson to do the same. “Oh but what about my plenary??” I think it’s about time we all moved away from such prescriptions. Review learning all the way through. Job done.

**5. Tell me everything you know**

This can be a really useful exercise. Give students a visual promopt, a diagram, or a written question. Take the emphasis away from ‘finding the answer’ and just get students to write or say all the things they know from the information they are given.

Better yet, don’t have a question at all!

**6. What can you find out (regardless of the question)**

This is a continuation of point **5.** The main difference is that you’re extending the idea of what you are given in the question, to what you can do with it. Again, removing the question is a good idea, as it takes the pressure off a given direction. Also, once again, discussion is key.

**7. Cut the bullshit**

I suspect you may need to rename this idea for use in your classroom (“Cut the Chaff”). Contextual questions are almost entirely awful. But they are in examinations, and that isn’t changing any time soon. Make light of them and use them as a teaching exercise. Again, ignore the answer initially, instead, make the aim to rewrite the question as pure maths, with no mention of Billy, Graham, Sandeep or any of their bizarre mathematical hobbies and processes to find the volume of a block of cheese. Below is a good example of a question that can have the context removed completely and very easily:

**8. What information is missing that you NEED?**

This lends itself particularly well to Geometry questions. And again, subject knowledge is also key here.

Going back to this question:

What information is missing that you need? Well, I **need the radius**. I simply cannot do this question without a radius. So then the question is less about finding the shaded region, and **ALL** about finding a way to get that pesky radius. Incidentally I don’t think we help by providing circle questions where the radius is *always* given from the centre to the right, like so:

Are we inadvertently making students always look for this **exactly positioned** line to find the radius?

Anyway…

**9. Here’s the answer, what was the question?**

A great idea highlighted on Don Steward’s blog. Another good example of getting students to think around the information they are provided with.

Another point of note here, is that we often show only one way to find a solution. Encourage different ways! That will at least help students understand that they’re not looking for a single approach, but just a way in.

**10. Stop using past papers**

Not entirely, but as many students highlighted last week, the past papers did not prepare them well for the examination. In fact, some suggested they made it worse as they were expecting specific question types and questioning patterns to appear, and they did not.

This is a new style of GCSE examination. You’d do better to create your own revision papers and assessments that are more in line with these new question styles.

The (branded too difficult) old spec papers for 2015 are a good source to begin with (until the new…new ones come out).

**Further Reading:**

Polya’s book ‘How To Solve It’

Thoughts on Teaching Mathematical Problem Solving (greatmathsteachingideas.com)

Your semicircle inside a rectangle reminded me of a problem I used with my Year 10s; I didn’t give them a question though – just a diagram, we then decided what the question might be! See circle problem here

https://colleenyoung.wordpress.com/2013/01/13/heres-the-diagram/

I frequently encourage students to draw extra lines on diagrams, it is often an excellent technique.

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Ok, I got this in about 23 seconds, but then I analysed my thinking.

My brain asked questions.

1. It’s about a circle.

2. What do I know about circles? area, radius, diameter, centre

3. Any of this any use? and it went diameter, then centre, and then the centre appeared, at the middle of the bottom of the rectangle.

4. How big is the circle? need the radius

5. Can I see anything about radius? YES !! from the centre to the circumference and I have a point on the rectangle and the circumference

6. Whoopee! I have a right angled triangle with two sides of length 3 and 4, and the radius is the hypotenuse

The rest is memory and manipulation.

I have no idea how this question is marked, but in my opinion the finding of the radius on the diagram is the maths, the rest is applying some formulae which may or may not be remembered correctly.

I appreciate your point about the radius being the length of the horizontal line from the centre to the edge, in the right-hand direction !

The kids have to move beyond the “how can I make use of the information given” to “what are they NOT telling me”

More thoughts:

1. How did they draw the circle? This one from those with a geometrical construction background.

2. What if I chopped the picture in half, it is symmetrical ?

3. What if I drew the reflection in the horizontal line? (after seeing that it is a diameter)

2 and 3 from those who “got” rigid motions, and they both turn the problem into a simpler one.

Also, if I, as a student, saw that the two given sides made up half the circumference would I get any marks for writing 3z = 12 for my equation, or is 2(z + 2z) = 24 the only acceptable equation?

And today’s stupid thought: The Americans should call the perimeter the periyard.

Just wondering if you intended two different answers in the semi-circle question. Although one side of the rectangle is clearly longer than the other, diagrams are not always drawn to scale, so the 6cm could be the vertical length and not the horizontal.

I think this is a lovely problem. Thanks.