The New GCSE #7 Non-Calc Side by Side

Ok, these are the new specimen papers (the last ones have been thrown out essentially!)

Each page is side by side for the 3 exam boards (there’s a fourth exam board, but they have 2 longer papers so the comparison is a bit more difficult, so i’ve omitted them). On hand is Michael Gove to add his own facial commentary. I hold no judgement on any of these papers officially. Make your own minds up.

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At this point two papers have finished. AQA goes on for 6 more pages:

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The Abacus Reigns #2

So my seven year old is being introduced to “the column method” for addition at the moment. He came home very upset yesterday because he couldn’t do it. After a few questions we realised part of the problem was that he didn’t know what the word “column” meant,  (take note!).

Being the annoying maths dad that I am, I asked him to get the trusty Base-10 Abacus out. He’s not quite old enough to groan in response but it won’t be long now!

We went through the idea behind column addition and lo’ and behold it started to click. Here’s how we did it:

Part 1

Here’s his sum (above). I highlighted the units / tens / hundreds with dotted lines, and we talked about the value of each digit. We also reproduced it on the abacus.

part 2

Next we added together all of our units. With the abacus there’s a clear “oops, no more units left” kind of moment, so we talked about exchanging our ten units for a single ‘ten’ in the next column over, because they’re worth the same. This in turn brings our units to zero, as shown on the abacus above.

part 3

Now we’re adding 4 tens. Well we have 3 tens at the moment, as we just exchanged ten units, so this will take us up to 7 tens. My son slid 4 beads up and counted how many there were altogether in the tens column. And how much is that worth? “70”. Goodo.

part 4

Finally we added two hundreds to our existing one hundred, to make three hundred. (not “three”, three hundreds, which is three hundred.). Finally we read out the complete answer as three hundred and seventy.

He insisted he didn’t need the abacus for the next question, and put it to one side.

Immediately he made this mistake:

final part

He’d got into the habit of exchanging ten units for a single ten, even if there weren’t ten units to exchange in the first place! So we went back to the abacus, and guess what? No problems, because we didn’t get past six beads. There’s no mistake in the answer in the picture above because I pointed out his error mid-sum.

I’m a big fan of spending time over the basics with physical representations of stuff. Maths is tricky, and if your confidence gets shot early on, it’s not exactly helpful. He came home today and demanded we do “ten column additions!!”. Good stuff.

 

The Edexcel Paper Indicator

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 🙂 :

tough q

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:

tough q v2

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:

area-questions

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:

martha

zq

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:

tough q

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:

radius-of-a-circle

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.

Picture1

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’

(summarised in free pdf here)

NRICH

Inquiry Maths

APlusClick.com

Thoughts on Teaching Mathematical Problem Solving (greatmathsteachingideas.com)

Division of Fractions using Bar Modelling

I was shown this innovative way to perform fraction division using bar modelling today. I previously thought bar modelling for this topic didn’t really work. As did many others.

bar

I think it’s pretty clever. Convert the fractions into equivalents with a common denominator, draw out the first fraction by shading in a bar model, then circle equal sized groups of the size of the numerator of the second fraction (in this case, 5) up to and including the last shaded cells. In this case, we have 2 wholes (2 entirely shaded groups) and 2/5 – because 2/5 of the final group was shaded.

Fractions, Fractions, Fractions

I’ve been creating a lot of resources for a fractions unit in Year 7 over the last month or so. I think they’re just about finished now, so I thought I’d share them.

Part 1: Shading Fractions

3 tiers of difficulty (Sup = supported, Med = Medium, Higher = Higher…duh!).

Fraction Shading_Sup 2

Fraction Shading_Med 2

Shading Fractions Higher 2

I tried to write these in such a way that the visualisation of fractions as a proportion rather than a literal size comes across.

shade 2

shade 1

Part 2: Equivalent Fractions

Again, emphasis on pictoral representation initally, with visually challenging versions of fractions for higher groups. Abstract examples also include algebra – which incidentally I feel should be generally integrated into all topics rather than completely stand alone.

fraction ordering H 2

fraction ordering Med 2

fraction ordering Sup 2

ordering 2

ordering

Part 3: Equivalent Fractions

I only made one worksheet for this, as it’s diferentiated within the questions (columned tiers of difficulty). Similar to the first 2 sections, starting with an emphasis on pictoral representation, switching to abstract, switching to algebra.

Equivalent Fractions_H 2

equiv

equiv2

Part 4: Adding / Subtracting Fractions

Adding Fractions_H 2

Adding Fractions_Int 2

Adding Fractions_Sup 2

Adding and Subtracting Fractions

3 tiers of worksheet, followed by a different sheet altogether that differentiates through columns of increasing difficulty. Bar modelling of sorts is used for pictoral introductions.

add1

add2

add3

Part 5: Multiplying Fractions

2 sets of near identical tiered worksheet. One version uses bar models in one style, the other uses a perhaps more favourable ‘area’ style version. (see below for examples)

v1

Multiplying Fractions_Higher

Multiplying Fractions_Int

Multiplying Fractions_Lower

(Above is one way of multiplying using bar-style representation)

v2

Multiplying Fractions_Lower_ALT

Multiplying Fractions_Int_ALT

Multiplying Fractions_Higher_ALT

(Above is the second style that I think is maybe more intuitive?)

area

Part 6: Dividing Fractions

Dividing With Fractions_Int

Dividing With Fractions_Sup

Just two versions this time. I suspect the intermediate version will be sufficiently difficult for higher groups if you lead which questions they attempt, rather than go linear start to end.

divide1

divide2

Part 7: Fractions of Amounts

Fractions of Amounts_Sup

Fractions of Amounts_Med

Again just two versions for the same reason as before.

amounts1

mounts2

Part 8: Extras

I didn’t make these, but they’re useful:

Number Line by Paula Krieg:

number line

Fraction Strips Templates

Interactive Fraction Strip Tool:

fr

My ‘Complements’ post about fractions

recip

Enjoy 😀