New GCSE Grade Boundaries - my thoughts

I am going to start this blog by making the point clear, it is impossible to accurately grade pupils on the new GCSE for Maths. Completely impossible. Anyone that tells a pupil that they have achieved a particular grade is at best making an educated guess and at worst is making something up. If there is any way you can avoid giving pupils grades, making predictions of pupils eventual grades or even talking about future grades with any stakeholder then you should take the opportunity and avoid it like it is a highly contagious illness.

That said, many schools are not giving departments and heads of maths the opportunity to avoid it. There are plenty of schools out there requiring staff to predict grades for pupils (some as low as in Year 7!), or provide current working grades. Even when schools don't require this, Year 11 pupils looking at the next stage are being asked for predicted grades in English and Maths from colleges or other post-16 providers. I have been in touch with many new department heads that are struggling to answer the demands of schools, parents and pupils with regards the new GCSE grading and so this post is designed to give some support and guidance for anyone who finds themselves in this unenviable position.

You will hear people say that you cannot grade at all for the new GCSE, and I can see where they are coming from (see paragraph 1!). I do believe that it is possible to make some educated guesses about what the landscape is going to look like - we do have a reasonable amount of information to work on and one thing mathematicians are good at is building models for situations with many variables. We just have to be clear about our modelling assumptions and how that affects the accuracy of the predictions from the model. Lets start with the information Ofqual have provided:

This is probably the most viewed guide that teachers and schools have with regards the new grading. The key line in this is actually 'Students will not lost out as a result of the changes'. That means that if you have a kid in front of you that is a nailed on C for the old GCSE, they are at least a 4 on the new. Similar for A and 7, and G and 1. Of course this doesn't help with the borderline kids, but it is somewhere to start. The most updated postcard also has this information:

What this means that if you are assessing pupils (mock exams or similar), once you have set the 4 and 7 boundary, you can set the 5 and 6 boundaries arithmetically. Although it doesn't say it here, I am reliably informed (he says, waiting to be shot down!) that the same is true for grades 2 and 3; they should be set equally between 1 and 4. The upper grades can also be calculated, using the tailored approach for grade 9. The tailored approach can be summarised as:

Percentage of those achieving at least grade 7 who should be awarded grade 9 = 7% + 0.5 * (percentage of candidates awarded grade 7 or above).

By my calculations on last years figures, this will mean nationally about 15% of the pupils awarded 7+ will be in the 9+ bracket, which will end up being about 2.4% of the total cohort (based on 15.9% A* and A in 2016 translating to a broadly similar proportion for 7+). Of course if your cohort is very different to national then it shouldn't be massively far out to apply the tailored approach to your A and A* figure from last year (if you have one - I don't as this is the first year for GCSE) once you have adjusted for differences in the starting points of the cohort. This means we can have a reasonable stab at a grade 9 boundary for any mock exam we set. The grade 8 boundary should then be set halfway between 7 and 9.

Using this approach it should be reasonable to generate some grade boundaries for a mock exam by looking at kids that would definitely have secured a C, A and G on the old GCSE exams, using their scores to set grade 4, 7 and 1 boundaries respectively, and then calculating the 9 and the others using the calculations Ofqual provides.

Another approach that we (and several other groups have employed) has been to combine papers with other schools all doing the same board. This has allowed us to use proportional awards to set the 1, 4 and 7 boundaries statistically rather than through moderation - although it is still a bit unclear as to precisely what proportion will be used for the 4. This is the approach that the PiXL club among others also used, although from some points of view with varying degrees of success.

This is all well and good for individual schools and cohorts, and setting retrospective boundaries when cohorts have already done mock exams, but what can we predict about the final exams? The true answer is very little, but perhaps not absolutely nothing. Using what we know it is possible to make some predictions about the likely distribution of the grade boundaries going forward, but with a very large margin for error built in, primarily because of the very different style that the assessment has which is very hard to quantify. We do know though that the balance of difficulty will shift in both sets of papers so that 50% of the Higher tier paper will be aimed at grades 7 plus, and similarly 50% of the Foundation tier will be 4+, which is between 10 and 20% increased on the current top two grades in each paper.

We also know about the shift of material so that the Foundation tier will assess some material that is currently only Higher, and some of the material currently on Higher will no longer be assessed on Higher. Factoring all of this in we can make adjustments on current boundaries to make educated predictions at new boundaries. I will start by looking at the AQA boundaries for last year:

These are the boundaries converted to percentages for last year, and from these we can make some sensible adjustments. Given that there is now no D grade material on the Higher paper, it makes sense that the award of 3 (there is still a discretionary award of 3 similar to the current E award on the Higher) will come down towards where the E is now - around the 8 to 10% mark. The grade 4 will then have to come down as well to reflect the fact that all the D grade material is gone. With the D currently at 17.7% it is reasonable to predict that the 4 value will fall somewhere in the range of 15% to 25%. The B grade at 53.1% will also come down to nearer the current C grade - this won't translate automatically into 5 or 6, but given that B falls between 5 and 6 then 5 is likely to come in in the high 20s or low to mid 30s, with 6 likely to fall in the mid to high 30s to low 40s.

It is almost certain that the grade 7 boundary will have to come down from 71% that the current A grade sits at. When you consider the loss of the D grade material which nearly all A grade+ pupils will be scoring well on, along with the increase in the amount of material at A grade/grade 7+ then one can justify quite a dramatic drop in the 7 grade boundary - with 50% of the paper at grade 7+ it is not outside the realms of possibility that the boundary for 7 will actually be below 50%. In reality something in the early to mid 50s is probably the most likely area for the 7 boundary, and almost certainly less than 60%. The 8 and 9 are probably the hardest to predict, because of the 9 calculation, and that 8 will be based on 9 and 7 together. It would be hard to see the 9 grade boundary being less than the current A* as this would defeat the whole reason for adding the extra grade into the top of the system. Currently 5.7% gain A*, so if 9 is going to halve this figure or better, then the expectation of a boundary somewhere between 90% and 96% would seem a fair prediction. If this is the case then the 7, 8 and 9 are going to be quite widely spaced, which is expected if they are going to allow distinguishing of candidates at the top end. If we take all of this into account, and apply to a total of 240 marks, we get boundaries somewhere around the ones below for the Higher tier:

I can see these being accurate to within 10 to 15 marks at a maximum, and significantly closer in some cases (points for me if I get any of the spot on!).

Turning our attention to Foundation, we can do a similar 'analysis'. There is no reason that the grade 1 boundary should have to change much from the current G grade (except of course pupils really struggling to access the paper!) and so pupils are still likely to need in excess of 20% to be awarded a grade on Foundation (or perhaps a short way below). The most interesting here is the grade 4 boundary, with similar arguments for the 7 on Higher. There is reason to believe that this will have to come down significantly with the addition of extra, more demanding content in Foundation and the balance of the paper shifting to include more material at grades 4 and 5. A figure close to the current D grade percentage of around 55% seems rational, and it could even dip below 50% (I suspect that it won't as the balance of pupils sitting the Foundation paper instead of the Higher is likely to change so that there are more pupils that would score higher marks than currently sit the Foundation tier). Given this the grade 3 boundary and grade 2 boundary are calculable as equally spaced between the two. The grade 5 boundary at Foundation is probably the hardest to predict with any certainty as it likely to rely heavily on comparable outcomes with the Higher tier to set - if the 5 boundary for Higher has to be calculated then pupils awarded 5 on Foundation will need to be checked to make sure they are demonstrating similar understanding to those awarded 5 on Higher. I suspect it is likely to be above the current 66% for a C on Foundation, and have gone in on the low 70s. Based on this, my best guess for Foundation, with similar accuracy at all except Grade 5, looks like this:

A similar 'analysis' of the Edexcel boundaries yielded these results:

A big assumption here is that pupils continue to score better on Edexcel than on AQA, which by all accounts is not a good assumption to make. The tests from Ofqual suggested that pupils answered the AQA papers better than then Edexcel ones, so this second set of boundaries may well be less accurate than the others. Ultimately though, if you have nothing else you can use, and you absolutely must talk about grades etc with SLT, parents etc then this is the absolute best guess I can come up with; of course it remains to be seen how good a guess they are, so use these are your own peril as they come with precisely zero guarantees!

Methods of last resort 2 - Order of Operations

Teaching the correct order of operations is possibly one of the most debated topics for maths teachers. In my #mathsconf8 session I was asked 'what is my problem with BIDMAS' and proceeded to outline times when this acronym is redundant (e.g. 4 x 3 ÷ 2) or even downright wrong (4 - 5 + 6 would mistakenly be given as -7 rather than the correct answer as 5). Various diagrams have been mooted as the solution to this, and there are several examples below:

 

I have several issues with these diagrams, which can be summarised as:

(a) It isn't specific enough for all of the possible functions that can be applied to numbers (even those that include square roots don't involve higher roots, and no mention of sin, cos, tan, log etc)

(b) BRACKETS ARE NOT AN OPERATION (please forgive the shouting). This may seem like semantics but for me it is an important distinction - brackets are used to either alter or clarify the order of operations intended, but are not an operation in themselves (just a note on clarify, an example of this is 12 ÷ (3 x 4) needed clarity as without these brackets the answer would be 16 and not 1). If we are going to teach pupils to understand the maths they are doing then we need to be communicating understanding like this, and not allowing pupils to mistakenly believe that brackets are an operation themselves.

But this post is not about teaching correct order of operations (although that segue has outlined my thoughts on it quite nicely); this is about when you wouldn't want pupils teaching using the correct order of operations in the first place. The example I used in my #mathsconf8 session was:

673 x 405 — 672 x 405

Any mathematician is definitely not applying the correct order of operations in this situation; and is quickly writing down that this is just 405. With the advent of 'teaching for mastery' gaining ground in mathematics education pupils are being increasingly exposed to questions like this when looking at distributive laws, or factorisation but I am yet to see it, or anything like it, thrown into a lesson on Order of Operations as a non-example. There is good evidence out there now to back up the idea that non-examples are important in communicating a concept and so if we are trying to communicate the correct order of operations we should be highlighting cases like this as when applying the correct order of operations is not wrong, but is just wildly inefficient compared to use of the distributive laws (in this case the formal statement would be something like 673 x 405 - 672 x 405 = 405 x (673 - 672) = 405 x 1 = 405).

Some other examples of times when correct order of operations are an inefficient way to solve problems (particularly without a calculator) are:

• 12 x 345 ÷ 6
• 18² ÷ 9²
• √128 ÷ √32 (although this one does require some real mathematical understanding)
• 372 + 845 – 369

I would be exploring all of these questions prior to teaching the correct order of operations, and then including questions like it in the deliberate practice on the correct order of operations to ensure that pupils are recognising when not to apply them alongside when they are absolutely necessary.

Love teaching, love maths, love twitter.

As anyone who has known me for the last year and half will know, I love Twitter. As a medium for connecting educators and sharing practice I have not seen anything like it. I have probably had more professional conversations, attended more real CPD meetings and moved my practice on more in the last year and a half than in the previous 8 and half that I was working - and a lot of that can be attributed to Twitter. It is easy to begin to take the impact for granted once you have been used to it for a while, but then something will come along that makes you fall in love with it all over again. For me this happened very recently following the Secret Teacher article about teaching maths.

Perhaps the thing I love most of all, more then twitter (although less than my family) is teaching maths. The joy of developing real understanding in pupils, seeing pupils go from nervous incomprehension to confident understanding is a joy that I am not going to soon tire of. Which is why articles like the Secret Teacher article make me so sad, when practitioners talk about how useless maths is for all but a small minority and how teachers are wasting time trying to teach all but a narrow set of skills to the majority I really do begin to despair of the poor opinion that some teachers have of pupils and of their role.

Which brings me back to what makes me fall in love with Twitter all over again - the response from some of the colleagues, and people I now class as friends, was just brilliant. Within minutes we had responses like this from Ed Southall (@solvemymaths) which so eloquently rebuts some of the poorer arguments in the article and really brilliantly we had a movement starting on Twitter courtesy of two of our newer teachers @MissBLilley and @Arithmaticks called #loveteaching.

With the media and politicians seemingly fighting to report all of the ineptitudes and 'tribulations' (as the Guardian advertises for in its Secret Teacher blog), these two dedicated and driven young teachers have tried to take it upon themselves to be a big part of the opposite voice - the voice that highlights all of the things that we love about teaching and what is bringing and keeping those special people like these two ladies into the classroom. For me this is a perfect example of the power of platforms like Twitter to unite like-minded educators and provide a voice for the profession, and it makes me appreciate Twitter and the people I meet through it all over again.

So I love Twitter, the camaraderie and the connectedness (if that is a word!); I love maths, the wonder and beauty, the way it has of revealing deeper and deeper insights for those that are prepared to work hard at it, but above nearly all I LOVE TEACHING.

Methods of Last Resort 1 - Percentages

Following on from my session in Kettering at #mathsconf8 I will be writing a series of blogs about the areas of maths I find or figure out that might be better looked at separate to any problems that might be solved using a standard approach or a 'method of last resort'. The first area I want to look at is percentages.

Because of the multiplicative nature of percentages there are lots of questions that can be solved without having to resort to approaches such as "Find 10% first..." or "What multiplier calculates...." or other standard approaches. The point I made at mathsconf is that I would want pupils to understand why these questions can be solved quickly and straightforwardly, and that actually by exploring the special nature of some of these calculations we can deepen pupils understanding of the topics - in this case percentages.

Find 32% of 75

This is the example I used at mathsconf. There are still plenty of teachers that don't realise that 32% of 75 is the same as 75% of 32, but once they see it they understand why. What I like is that in explaining why this is true really does get at the heart of percentages and how they are calculated and so it is a perfect little 'explain why' to stretch pupils as well as then serving as reinforcement of concepts for others.

Find 32% of 100

Try it; you will be surprised how many pupils don's immediately link the % with the 100 or are unsure when they want to say 'isn't that just 32?' Again this sort of question gets at the heart of percentages as parts of 100.

Find 32% of 50

If you have built up to it these are actually now becoming quite straightforward, but encouraging pupils to talk and explain why is still powerful.

Find 32% of 200, 300, 400 etc

I probably don't need to say much more at this point.

As well as calculating percentages, equally there are similar questions for writing one value as a percentage of another. Again there are standard approaches for this (writing and converting fractions or similar) but there are questions that anyone with a real understanding of percentages would look at and solve. This set of questions comes from a well known worksheet provider; see if you can spot the ones that could be done without requiring the use of a standard approach or 'method of last resort'.

Even if you don't really know your fractions, questions 3, 5, 6, 7, 11 and possibly 12 and/or 17 can be solved using some relatively straightforward multiplication and division. Do we always teach pupils though that if they can see an obvious way to write it as 'a percentage of 100' that this will be much quicker than a standard approach, and more importantly to support them in understanding why this works which would lead to a deeper understanding of percentages as a whole.

Variation in Mathematics

I am determined not to let my blog frequency slip below once a month no matter how busy I am; I probably have enough stuff for a year's worth of blogging at this point but one topic that I did want to discuss was the idea of variation and its use in mathematics teaching.

I was lucky enough to be asked to host the twitter chat for the NCETM around this topic on 20th September and I jumped at the chance. The idea of variation is one that has interested me since I was observed teaching ratio and used these problems as my examples:

The visiting professor suggested that this series of examples showed the hallmarks of variation theory. This peaked my interest in the topic; I had known working in Oxford that Anne Watson and John Mason had done work on the idea but hadn't really had the opportunity to read any detail. I decided to look into variation a little more to see what it was all about.

The first article I read was from Anne and John written for the Open University, and to this day remains one of my favourites on the subject. Entitled 'Seeing an exercise as a single mathematical object: using variation to structure sense-making' it really does give an excellent introduction to the idea of really thinking about and structuring the variation between questions or examples to  allow pupils to make sense of how different facets of the situations effect the outcomes. One of my favourite activities from this article is:

I wont repeat Anne and John with all of the discussion, but the full article can be found here and I strongly encourage reading it.

My other favourite article about variation is this one from the Centre for Innovation in Mathematics Teaching. The focus is very much on drawing out the misconception of eastern mathematics as relying a lot on rote learning, and could even been seen as a fore-running article to much of the recent focus on mathematics teaching approaches in the highest performing eastern jurisdictions. It is this article that gives me my clearest idea of the purpose of variation theory, namely

"the central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features"

The article also outlines nicely the difference between procedural variation and conceptual variation. During the chat I shared this activity which links to my original ratio problems and quite succinctly shows the idea of procedural variation.

As a guide to implementing different types of variation in the classroom this article is about as good as it gets. It certainly influenced my design of a department activity which resulted in some of these excellent activities (which haven't been formatted for pupil use yet!)

I will definitely be using both of these articles with the work group we will be forming as part of our work as the lead secondary mastery school for the East Midlands South hub and would strongly recommend that anyone looking to ensure that every part of an activity is deepening pupils' understanding.

Iteration and the new GCSE

So my blog frequency has become significantly lower recently - believe it or not I have been even busier than normal writing and sourcing resources for our new Year 7 mixed ability course, putting together topic tests for Year 7 and Year 11 (thanks AQA for all of your work putting your own topic tests together - I have stolen most of them!) and then writing the homework booklets for all three of my Year 11 schemes for term 1. All in all today is actually the first day since we broke up (bear in mind that Leicestershire broke up on 15/07/16) that I haven't been doing school work of some description - as a reward for finishing the homework booklets a day early I gave myself the weekend off!

One of the things that I have had to sort out as part of writing the tests and homework booklets is finding sources of questions on iteration and numerical methods for solving equations, so I thought I would share some of the better ones here, and also offer some tips on designing your own.

1) Check out A-Level worksheets - I dug through some of my old Core 3 resources (unfortunately I haven't taught A-Level for the last two years since moving to my new school) and found an ample supply of iterative formulae that were used. Some of them weren't suitable (too many natural logarithms and exponential functions) but many were with just some small adaptations. In particular a lot of A-Level questions ask pupils to show there is a root in a given interval using a change of sign approach and also ask pupils to justify why a given formula will converge to a solution. As far as I have seen the GCSE will not ask pupils to use a change of sign to show there is a root in a given interval,although to be fair it wouldn't be a bad thing to do with pupils as a way of tying roots of equations, graphs and iteration together. In addition it will definitely not require  pupils to justify why a given iterative formula will converge, as this requires knowledge of calculus - although again it might be nice for the best mathematicians to look at this as a way of linking rates of change to iterative formulae. For some examples questions made from A-Level worksheets check out my Year 11 Higher or Higher+ term 1 and 2 homework booklets - there are a few pages on Iterative methods with a few exam style questions all taken from A-Level worksheets or similar.

2) Exam board website - we are using the AQA exam board and they have a multitude of resources available for use with iteration. If you don't know AQA's site http://allaboutmaths.aqa.org.uk/ it is well worth getting yourself signed up for it. Browse to the New GCSE (8300) and select the Numerical methods section under Higher GCSE Algebra resources and you will find worksheets with some decent enough questions, as well as their topic test with some more. The one I really like though is their 'bridging' material, which can again be found under the New GCSE (8300) page. They have a lovely document in there called Pocket 4, which is all about iterative formulae. Although billed as a KS3 bridging material I would definitely save some of the later activities and use them during the actual GCSE teaching.

3) Linked Pair Pilot - Although trial and improvement is not mentioned specifically in the new GCSE specifications, it is still being used under the guise of a numerical method. The Linked Pair Pilot papers, in particular the Applications 2 paper, has some nice examples of trial and improvement used to solve practical problems in geometry and other areas, which is nicely in keeping with the aims of the new GCSE. Often they have the tables printed on a separate page as well, which means you can feel free to not use them for the more confident mathematicians, just giving them the page with the question setup on instead.

4) Pixi Maths - If you haven't seen Pixi Maths TES shop yet, then I would definitely head over there (https://www.tes.com/teaching-resources/shop/pixi_17#). Pixi has created some lovely resources for a variety of topics, including iteration - https://www.tes.com/teaching-resource/iterations-11064012 although don't be fooled by the line that says trial and improvement has gone. Still there is a nice PowerPoint and activities which includes a jigsaw for the rearranging part of iteration and then a worksheet with some iterations to perform.

5) Design your own - It isn't actually that tricky to design iteration questions, although there are a couple of things to beware of to ensure the question will work. Start with a polynomial set equal to 0; cubics are good as they can't be solved using other GCSE techniques (except if it has an obvious factorisation) and are guaranteed to have at least one root. From here you can do one of two things:

(a) Use the Newton-Raphson formula:

The examples of exam questions I have seen using this formula have had the subtraction simplified to give a single fraction as the iterative formula, however I cannot see any reason why pupils couldn't be given the formula with the basic substitution already done and told to do a 'show that', i.e.

        

(b) Rearrange - the classic method for generating iterative formula is to rearrange the equation 

f(x) = 0 into the form x = g(x). This is being used a lot in the new GCSE practice and sample materials which include asking pupils to show how a given rearrangement can be arrived at:

If you use this approach to design your own question then a word of caution - not all possible rearrangements will find all of the roots. The best things do here is to check the graph of the rearranged function for the gradient in the locale of the root. The rule goes that if the gradient of the rearranged function around the root you are looking for is in the range (-1,1) then the formula will converge to the root there - if not then it wont. For example for the problem above the graphs of the original function and the rearranged function look like this:

where the red graph is the original cubic and the blue graph is the square root function. You can see that there are actually three roots to the cubic, corresponding to the three points that the root function intercepts the line y = x. However the given rearrangement wont find the root that is slightly bigger than 2, as the gradient of the root curve is greater than 1 around that point. The rearrangement will quite happily find the other roots in the intervals (0,1) and (-1,0) as the gradients are close to 0 around these points. It is definitely worth just checking this if you are going to design your own rearrangement questions as you wouldn't want to give your pupils rearrangement that doesn't work!

#Mathsconf7 - a cracking day out

I don't normally write blogs about conferences and events; there are much more eloquent people out there who normally do a great job of highlighting the key parts of the sessions they visit. Unfortunately quite a few of them were unable to attend this weekend and so the task falls to me to sum up my experiences of this wonderful weekend.

Friday night - the pre-drinks were great fun. Myself and Andy (@ColonelPrice) had a lovely dinner at the Cattle Grid (heartily recommended for anyone visiting Leeds in the future) followed a great catch-up with Mark (@EMathsUK) and the LaSalle team, Ben and some of the AQA team, Graham Cummings and @deko_j from Pearson, @dannytbrown @KristopherBoulton and @Naveenfrizvi. I also got to meet Douglas Butler (@DouglasButler1) for the first time (having missed his apparently fantastic ATM/Ma session in Leicester recently) as well as the dangerous duo of @AnandaCatterall and @MissVaseyMaths. As things wound down at Azucar we set off to find @El_Timbre, @missradders and @jennypeek to continue the drinking - nights out in Leeds are fun, but no more will be said!

Start of the conference - Following the usual messages from Mark and Andrew Taylor we were treated to a fantastic key note from professor Mike Askew (@mikeaskew26 which really highlighted some of the ways of working with pupils that actually do have impact in terms of pupils solving problems. In particular the importance of asking deep, exploratory questions like the one below. 

Another important point raised here was the relative effectiveness of front-loading the lesson with examples and then pupils practising on lots of examples with a much more interleaved approach mixing worked examples with independent practice. The research Mike quoted suggests this second approach created much better outcomes in pupils compared to what might be considered the more traditional approach. Although there is lots to mention from this session, the one other thing I really want to mention is the use of little low-stakes quizzes on prior topics done half way through a current topic in order to refresh previous knowledge and understanding.

Unfortunately due to technical difficulties I missed the speed-date but I am sure it was as useful and exciting as always.

Session 1 - Avoiding misleading assumptions

This was my first delivery of the day looking at the sometimes rather limited diet of examples that maths teachers have seen in the past, and therefore pupils see now. We played a game that tested the delegates creativity around designing examples. The full presentation can be found here and the major points of the game can be seen below:

Session 2 - Questioning and Culture.

My second delivery, along with @ColonelPrice making his #mathsconf debut. The session again seemed to be well received as myself and Andrew explored different aspects of questioning, including some brilliant responses to the request to come up with some non-standard questions to this stimulus (the idea of finding the equations of the lines given that the vertex at the bottom left has coordinate (0,0) was particularly inspired!). 

Again the presentation can be found here.

Lunch and the Tweet up - After less than inspiring fair in Peterborough we were treated to a very nice hot and cold selection for lunch. Of course no tweet up was ever going to be the same without @tessmaths there but our team of @MrBenWard, @HR_Maths @MissBLilley, @ColonelPrice, @missradders, @El_Timbre, @EJMaths and @MissBsresources really mucked in, and a great time was had by all! The puzzles proved particularly popular (with MissBLilley in particular more so then the delegates!) and congratulations to @hexagon001 for winning the triangles competition (with thanks to @EJMaths for donating the book used as the prize) and also to @KerryDunton for winning the smallest unique positive integer competition with a great choice of 6.

Session 3 - Golden Age

I always love talking about practice with @dannytbrown, I am only sad that I don't get the opportunity to do so very often. Listening to Danny talk about being present, being aware of our own awareness and noticing what leads to our actions before they happen, drawing on the work of Mason, Tahta etc, was hypnotic and Danny's clear passion but very deliberate approach is the perfect vehicle for delegates to slow down and really think about themselves in the classroom. This is possibly the only session I have not tweeted from as it was impossible to truly listen to Danny and distract yourself with a device at the same time, and would have been the antithesis of the whole session. Danny is clearly one of the deeper thinkers of our profession and everybody should take the time to listen to his thoughts and engage with the material he puts out in his excellent blog.

Session 4 - Teaching for Depth

The lovely ladies of the White Rose Maths hub (@wrmathshub) led by Beth talked about some of the work they have been doing to try and really ensure that the pupils across their area develop a really deep understanding of maths. Drawing on inspiration from Shanghai around ensuring pupils access truly intelligent practice and work with multiple representations the team have put together some excellent resources and assessments linked into their scheme of work. I will definitely be paying regular visits to their dropbox when I am doing my own KS3 re-write next year and stealing as many of their materials as I can get away with!

Of course one of the best parts of any #mathsconf is the chance to catch up with old friends and puts faces to the names of new ones and this was no exception - most of the names I have already mentioned and if I try and create a list here I will guarantee to miss someone out so I will simply say if I spoke to you today it was great to meet you/see you again and if I didn't then make sure you say "Hi" next time (particularly you @MrBartonMaths as I have seen you twice at mathsconf and haven't talked maths properly with you yet!).

I cannot lavish enough plaudits on @EmathsUK and the @LaSalleEd team for the fantastic work they do three times a year to bring these events together and having started at #mathsconf4 I hope I am still around when they are doing #mathsconf40!

Angles on 'straight lines' - tackling a key misconception.

As mentioned in my blog post a couple of days ago (I know, two in a week!) I have been teaching angle properties to a couple of different year groups. One misconception I kept bumping into is surrounding pictures like this:

The key misconception I am talking about is this, "123 + c + a = 180 because those angles lie on a straight line".

For me, it is easy to sympathise with this, as of course these angles do "lie on a straight line". I think there are a couple of issues here and have been trying to deal with both through the topic. The first is a language issue, and the second involves the diagram.

The first issue is the idea of angles that 'lie on' a straight line. To me talking about angles on straight lines actually helps reinforce this misconception, rather than preventing it. Instead I think it is better to talk about angles that "form a straight line", this allows you to demonstrate that the angle 123 and c form a straight line, but that a is not needed to form the line, it is further down the line.

The other approach I have used alongside this is to get pupils to actually mark the point where the angles come together to form a straight line. Of course with the pupils being encouraged to look for angle properties rather than chase after particular angles (see my previous post), the conversation goes something along the lines of, "where do you see angles forming a straight line", followed by "can you mark where they form the straight line", which leads to pictures like these:

These sorts of pictures really help show why the two (or three in the case of the upper line) angles are the ones that form the straight line, and that a is not involved.

My advice would be that when teaching angle properties, consider how the language you use supports pupils in identifying angle pictures correctly, and ways in supporting pupils on securing the correct angles as part of the correct pictures.

Angle properties - don't go chasing angles...

Recently I have been teaching angle properties and calculations to Year 7 and Year 9. Particularly in Year 9 we have been exploring problems that require multiple properties and steps to arrive at a solution such as the problem below:

The approach I am taking here is not to focus on finding a particular angle, but rather than trying to focus pupils on the sorts of pictures they see. This means that instead of asking questions like "can you tell me the size of this angle?", I am asking questions like "Can you see any straight lines in the picture?". I am also modelling this process in examples, for example when we went through this example:

rather than trying to find h and then trying to find i, we instead just went through the different angle properties we knew and found angles that fit, including completely useless facts like 46 +90 + 44 = 180. Altogether we wrote down:

h + 46 + 90 + 44 + 61 + i = 360 (full turn)

h + 46 + 90 = 180

46 + 90 + 44 = 180

44 + 61 + i = 180

61 + i + h = 180 (all straight lines)

h = 44

i + 61 = 46 + 90 (both vertically opposite).

Only when we had written all of this down did we talk about and look at which bits of information may be useful in helping find h and i (quickly identifying multiple ways of finding both h and i) and eventually writing down the values of both angles.

This approach is definitely having an impact in terms of pupils working through these sorts of problems as they are less hung up on the fact that they can't immediately find values of an angle and are correspondingly (nice use of terminology!) more ready to make an attempt at these problems. This, coupled with a visualisation of walking down the paths that the diagram shows (more on this in a blog to come) seems to be a real support to pupils in working with these sorts of diagrams.

Dividing Fractions - not just KFC!

Is there anything with more potential for pupils to go wrong with in the arena of fractions than division by a fraction? Whether it is turning over the wrong fraction, both fractions, or not even having a clue about it, division by a fraction does seem to be a real stumbling block for a huge number of pupils. So I thought I would share the best 3 approaches I know to dividing by fractions.

1) Multiplying by the reciprocal

This is basically where KFC comes from - although it is really important that pupils do understand the language of reciprocal and can identify reciprocals for areas of maths like functions. I like to build this by looking at unit fractions first, and definitely mixing up dividing both integer and fractional values, i.e.

                                                             6 ÷ ¼

                                                             ½ ÷ ⅓

                                                            ⅚ ÷ ⅛

Showing that these are the same as 6 x 4, ½ x 3 and ⅚ x 8 respectively is an important first step. Once this is secure we would look at dividing by a non-unit fraction as dividing by something x times bigger than the unit fraction, and so needing to divide by the unit fraction and by x i.e.

                                          ⅚ ÷ ⅘ = ⅚ ÷ ⅕ ÷ 4 = ⅚ x 5 x ¼ = ⅚ x 5/4 = 25/24

Highlighting and reinforcing the fact that 4 is the reciprocal of ¼, 3 is the reciprocal of ⅓, etc makes this approach complete.

2) Dividing term by term

Although not an approach used a lot, this can be a really nice link to multiplication provided pupils can work with the fractions within a fraction that result. The idea centres on being able to divide numerators and denominators independently i.e. 

                                                                          ⅚ ÷ ⅘ =

We can then proceed to multiply by 4/4 and by 5/5 (or alternatively simply by 20/20 if pupils will understand the reason for this in one step)

3) Using common denominators

Like addition and subtraction (and particularly if you have already worked out common denominators for addition or subtraction) if fractions are given with a common denominator then dividing them can be quite straightforward.

⅚ ÷ ⅘ = 

The idea here is if you have 25 lots of something and you divide by 24 of the same something then you have 25/24 independently of the something. So

i.e. if we have 25 thirtieths divided by 24 thirtieths you have 25/24 independent of the original thirtieths.

It may be that pupils will take to one method of dividing fractions over others, and that the pupils who grasp the concept quickly can work with all three, showing they are equivalent, choosing the optimum approach for different situations and in general working with all three to achieve true mastery of division by a fraction.

Parallel lines are the same length and other such nonsense!

Recently we have been talking about the messages and misconceptions we convey without meaning to. A colleague of mine (not in my school) put me onto one - when we draw parallel lines we nearly always draw them the same length. A quick google image search suggests that this is not just the maths teachers I know:

We can see that whilst most of the pictures do show horizontal or vertical lines, all of the pictures show parallel lines the same length. Whilst some might say that this isn't really significant, I wonder if it is not something we should be aware of anyway in terms of forcing ourselves to think about the implicit messages that we give to pupils alongside the explicit content or skills we are trying to teach.

Another example that we have come across recently is that the equation 3x = 4 has no solutions because "three doesn't go into four". There was some debate as to whether this shows a general lack of understanding of division, or is a function of the fact that most equations we being to show pupils in there initial introduction to equation solving have whole number solutions. On the subject have you ever noticed that pupils struggle a lot more with equations of the form  compared to 4x - 3 = 5? Could it be that on balance they see many more equations of the second type than the first?

Some other areas of discussion:

• Index laws using a base that is not a single term and powers that are not integers or simple fractions.
• Area of triangles where perpendiculars are horizontal and vertical.
• Fractions - only ever talking about simplification of fractions with a numerator and denominator that are positive integers.

There are lots of other patterns you can find in textbooks and other materials that teachers naturally draw on for their own examples - so my suggestion is to really think about the breadth of examples that are possible with the maths pupils are learning; and not just the typical examples you may have seen before.

Probability without numbers

"There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow. Hannah takes at random a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is ⅓.

(a) Show that n² – n – 90 = 0

(b) Solve n² – n – 90 = 0 to find the value of n."

Look familiar? This question caused massive controversy when it was released in summer 2015 as it was seen as too much like things to come - many felt that it was more like the sort of question we might expect in 2017 when the new '1-9' GCSE is first examined and had no place in the current GCSE. Whether you believe this or not, the point is clear that pupils need to understand the ideas of probability and apply them outside the realms of numerical chance. With that in mind I thought I would share some ideas about developing probability without giving (too many) values.

Probability and Proportion

I am surprised we do not see more links between probability and proportion as ultimately probability is a proportional idea, the chance of something happening is measured as a proportion of the things that are possible or as a proportion of a number of trials in an experiment. In the past proportionality has generally be pretty limited to calculating an expected number of trials that would satisfy the given condition. I think it is clear though that with anything up to 25% of the new GCSE paper content being linked to ratio and proportion I think that we will see a lot more questions linking these two topics in the future. Questions like the ones below could become much more common:

1) A packet of sweets has orange, blackcurrant, strawberry and lemon sweets in the ratio 4:3:2:1. James and Sarah both buy packets of the same number of sweets. James doesn't like strawberry and so gives all of his strawberry sweets to Sarah. Sarah gives James all of her lemon sweets in return. If James takes a sweet at random from his bag, work out the probability that James take a lemon sweet.

2) A childs' shape sorter has red, green, blue and yellow shapes. The number of red shapes is twice the number of green shapes. The number of blue shapes is twice the number of yellow shapes. In total the number of red and green shapes is twice the number total number of blue and yellow shapes. Work out the probability of a child selecting a red shape if the shape is taken at random.

To be fair it strikes me that a lot of ratio and proportion question can be adapted to give a probability question - question (1) above could just as easily be "write down the ratio" rather than "work out the probability" and there are lots of ratio and proportion questions out there that could be adapted to this vein.

Probability and Algebra

Hannah and her sweets have given us a pretty clear indication that this will be a rich source of links for examiners to mine and again it makes perfect sense: if you understand the ideas of probability and algebraic expressions/equations then there should be no reason why you cant apply the two ideas together. We have also seen in the SAMs at least one question that has purely algebraic expressions inside a Venn diagram linked to probability for pupils to work with and I am sure we will see more examples in the coming years.
1)

2) A bag of counters contains red, blue and green counters. There is one more red counter than green, and one more green counter than blue. Stefan takes a counter out of the bag and puts it on the table, followed by a second. The probability that Stefan takes a blue followed by a red is 1/9. Calculate the probability that Stefan takes two greens.

It strikes me that replacing lots of the numbers in current probability questions with letters will generate questions of this type, and so would be well worth some time in faculty meetings designing.

No doubt at this point people will be thinking "yes but probability and statistics will only be 15% or so of the content..." and of course they are right, but don't forget that 15% of 240 marks is a good 36 marks, so there is plenty of space for one of two questions of this type to creep in, particularly as they can also count towards the 20%  to 30% Algebra content or 25% to 20% Ratio content so I would suggest it is well worth building questions like this into your GCSE schemes.

Dimension and Pythagoras

My Year 10 have recently been working with Pythagoras in 3-D objects, and quite typically in my experience they were having difficulty identifying suitable triangles to calculate some of the lengths; particularly those lengths that go through a shape requiring multiple applications of Pythagoras. Being ready for this I decided to try an approach that I had been considering that links the number of applications of Pythagoras' Theorem to the number of dimensions that the line moves through. The approach met with some success and I can see how it might have real potential in linking to dimensional analysis so I thought I would outline it here.

The first and one of the key points was to ensure that pupils understood that Pythagoras' Theorem is a relationship concerning area. Although we often use Pythagoras' Theorem to solve for missing lengths, the actual essence of the relationship is between the areas of three squares where two of them meet to form a right angle. The image below is one that is typically used to illustrate this (and one I have used lots in the past).

Once my pupils understand that this is a relationship area, the discussion is then turned to dimensions. What I am hoping to show pupils is that the two shorter sides of a right triangle are lines that only move in one dimension, whereas the diagonal moves in two (as the two sides are at right angles they can be considered to be two independent dimensions). So Pythagoras' Theorem can be thought of as a relationship that starts with lines moving in a single dimension, and relates them to a line moving in two dimensions (linking to area being a two dimensional concept).

Now let us consider a cuboid like the one below:

An early job of work to do with pupils here is to make sure they can identify lines that move in one, two or three dimensions. Generally for me this leads to lots of gesturing around the room and drawing imaginary lines along walls and floors, as well as between corners across the room. Once pupils understand how these lines are moving we can start looking at which distance can be solved with only one application of Pythagoras' Theorem and which cannot. For example in the cuboid above the distances AF, BD, FC etc can all be solved directly using a single application of Pythagoras' Theorem as they are all lines that move in two dimension. 

Now let us consider a line that moves in three dimensions, for example EC. A tip from me on this, before drawing the triangle in the shape, try drawing the rectangle first (as below). For some reason pupils see this more easily that just the triangle.

The discussion we had here is that we cannot expect to solve for lengths like this with one application of Pythagoras, because this length moves in all three dimensions and Pythagoras' Theorem is a two dimensional relationship. In order to solve this problem we are going to need two applications of Pythagoras, one to relate two of the one-dimensional lines with a line that moves in two dimensions (in the above case either EG or AC will do the job) and then to use this line moving in two dimensions with the line moving in the third dimension to relate to the line moving in all three dimensions.

To some this might seem like overkill, but what it does is give pupils an objective test as to whether a line can be solved using given information - if it moves in more than two dimensions it cannot be solved by a single application of Pythagoras' Theorem using lines that only move in one dimension. Of course from here you can complicate things and look at other three dimensional shapes, begin to make judgements about whether line can be considered to move in only one dimension (right-angled to each other), two dimensions or three dimensions. I also think it reinforces area as a key concept and will provide a nice link to dimensional analysis of different formulae when it comes to looking at that concept in more depth.

So in the future I think I will definitely be talking about Pythagoras' Theorem as an area relationship, and definitely be talking about dimensions that lines move in more formally with pupils; if your pupils are having trouble applying Pythagoras' Theorem to three dimensions why not try it as well?

Proportion and straight line graphs

By now pretty much everyone involved in delivering the new Maths GCSE course in England are aware of the increased emphasis on pupils having proportional reasoning skills. Ratio, Proportion and rates of change as a separate strand of the GCSE is worth up to 25% of the assessed content of the qualification, and will appear linked with lots of other areas of maths. Already we are used to the idea of ratio and proportion appearing in trigonometry, scale diagrams, recipes, value for money, many different contexts; I want to look specifically at a couple of ways proportion appears in straight line graphs.

Most teachers of the new GCSE (and quite possibly old) will be familiar with the obvious relationship that proportion has with straight line graphs; namely the graph of two variables that vary directly with each other. Graphs of the form y = kx are a fairly straight-forward link between proportionality and straight line graphs.

In this graph, the y values and x values are proportional, with the ratio x:y being 3:5. This kind of proportion should be relatively straight forward for any pupil that really understands proportion as an idea, and for those more graphically minded may even help with being able to visualise proportion. So what about this graph:

Clearly this is not a 'proportion' graph in the sense that y and x are not in proportion to each other. However if we take a closer look...

Clearly there is a proportion going on here, but what is it? Of course in this case it is not the variables that are in proportion; rather it is the rates of changes in the variables that are in proportion. Specifically in this case that the change in y is half of the change in x (leading of course to the gradient of ½). 

This proportionality is often overlooked, or at least not made explicit, but given that rates of change is now part of the new GCSE I think it will be worth highlighting the idea of a straight line as a line where the rate of change of y is proportional to the rate of change of x and that this proportionality is where we get the concept of gradient. This may well help pupils when it comes to rates of changes of curves by applying tangents; if pupils are already familiar with the idea of gradient at rate of change because it has been made explicit when working with straight lines the the concept should come more readily when moving on to rates of change of curves.

So in order to ensure your pupils are ready for rates of change at GCSE, consider introducing them not just to graphs where the variables are in proportion, but also where the rate of change of one variable with respect to another is proportion: for if pupils can gain a deep understanding of how gradient links to proportionality then the beginnings of calculus are well within their grasp.

Introducing surds - cutting out squares...

The 'recent' changes to the KS3 curriculum suggest that we should be given our pupils a basic understanding of surds and surd calculation prior to GCSE. Some people may be wondering about how to introduce the idea of surds to KS3 pupils and so I thought I would share one of the things I do with pupils, and that is to look at drawing and possibly cutting out squares.

Typically this will start with some nice big squares. I actually quite like using inch-long squares; not because I have a real hankering to return to pre-decimalisation (I am too young to remember anything other than pounds and pence anyway!) but because I find centimetre squares too small and fiddly for this sort of work, and anyway I think there is something nice about reinforcing the concept of area by looking beyond the normal cm or metre squares. You can't buy inch square paper these days (or if you can I don't know where you can) but you can download square grids from the internet and stretch the image so that each square is 2.5 cm long. I need plenty of this paper as kids will need lots of attempts to try and fail; kids get to fail a lot here, so if you are looking to examine mindset as well this is a great activity to try.

I will start kids off by getting the to draw a square with 25 inch-squares inside, which most will do quite quickly. Next I will tell them to draw a square with 16 inch-squares inside of their 25 inch square so that they have this smaller square inside the larger square. The next part will be a discussion about what we might be able to say about squares with areas between 16 and 25 inch-squares, with the aim that pupils will realise that any square with an area between these two will have to have sides between 4 and 5 and therefore will be (a) drawable between the two squares we have already drawn and (b) have area made up part squares. Then comes the challenge (which at first to some pupils doesn't appear as much of a challenge): draw me the square that has an area equivalent to precisely 20 inch squares. Depending on the group this will proceed in one of two ways; either they will draw an attempt between the two squares they previously drew, bring it to me to measure (I keep a ruler that measures inches for this and other purposes), and then become frustrated when I show them that their area cannot be quite 20 inch-squares or alternatively (which I prefer) I will get them to draw a 4 x 5 rectangle and then cut their square up and see whether it can completely cover the rectangle (prompting to leave the 4x4 square intact and just cut the excess from around it and try and make it fit if necessary). This second approach is definitely nicer provided the pupils have the resilience to keep re-drawing the squares every time they make a mistake: of course sometimes because of small gaps between their pieces they will think they have completely covered the rectangle and you will have to show them (either by measuring the hole they cut or by talking about very small gaps etc) that they haven't got to exactly 20.

The purpose of the activity of course is to plant the idea in pupils head that the task may be impossible to perform in reality. I will often talk about accuracy of measuring instruments here as well, and get pupils to imagine rulers that could measure down to a millionth of an inch or more. What this allows me to do is introduce the idea of an irrational number in a way that speaks to pupils experience; they have seen first hand that the square root of 20 cannot be found as a decimal or fraction of the length of a real square. I can then talk about the fact that mathematically the number needs representing exactly rather than as a rounded value, and so the surd form is required, which leads to the need to be able to calculate with numbers in this form and so on. For me this approach is much more powerful than a simple calculator investigation and not just because it is more engaging for kids than just mindlessly punching numbers into a machine, but also because it really highlights the reality of irrational numbers: these are numbers that cannot be measured and cannot be represented in ways that have been used before.

So if you are looking for a nice concrete way to introduce the idea of surds that gives (for me) real insight into the fundamental nature of these type of numbers, then try getting your kids just drawing and cutting out squares.

Mode - Most unappreciated?

For me, the mode is one of the most unappreciated averages that we teach in maths education. Perhaps because of its simplicity, perhaps because sometimes we feel it can contradict the idea of 'centralcy' that we look for in a good average, or perhaps because it isn't often talked about formally in many everyday situations, but rarely does mode get significant lesson time beyond its introduction in primary school. Often it is paired with median in a unit which then devotes a whole lesson or more to mean calculations; occasionally it is lumped into a general 'averages' lesson and becomes almost a footnote when looking at lists of numbers to get pupils to look at the list ready to find median and mean. In my opinion this is a real shame as the mode can be one of the most versatile and available averages, so I thought I would share with you some things that I like to do with the mode...

1) Mode from different representations

I love introducing the idea of mode as the most frequent item of data, and then challenging pupils to identify it in lots of different representations, such as bar charts, pie charts, tally charts, bar line graphs etc - recently I gave pupils a sheet with these images on and challenged them to find the mode from each situation: 

We got a load of misconceptions out of the way here; a mode of 4 or a mode of 7 from the frequency table, a mode of 6 from the bar line chart, what happens when 241 and 242 appear the same number of times; stimulated a lot of discussion and conflict and led to some real understanding.

2) Make up a list of data.

A nice pre-cursor to more complicated problem solving is to just give pupils a mode and to ask pupils to come up with different lists of numbers that satisfy the conditions. As a simple example, the question might well be something like "4 numbers have a mode of 3, Give a possible list of the 4 numbers." This can then be complicated in the following ways:

• 4 numbers have a mode of 3. What is the maximum number of 3s in the list? What is the minimum number of 3s?
• 4 positive whole numbers have a mode of 3. All of the numbers are 3 or less. Write all of the possible lists of numbers.
• 4 positive whole numbers have a mode of 3. The numbers add up to 10. Write down the four numbers.
• 4 positive whole numbers have a mode of 3. What is the minimum total that the four numbers can have? What about the maximum total if all of the numbers are less than or equal to 3? Less than or equal to 5?

amongst other similar examples.

These sorts of questions are nice to get pupils thinking and reasoning with mode; it is lovely to see them reason that the third list cannot have two 3s or realising that the fourth list cannot sum to 8.

3) Modal mystery

Similar to above, designed to promote reasoning around the mode, these sorts of questions are lots of fun to throw at kids:

                                     2           .............         1         ..................          3

The above list of 5 numbers has 2 values missing. What could the mode be? What could it be if 3 is the highest number? If 1 is the lowest number? What about if we change the 3 to a 1? 

4) Real life modes

Although rarely referred to formally as mode, a lot of statistics encountered in real life boil down to a mode. Whether it is votes on a popular TV reality show (X Factor, Strictly come Dancing, I am a Celebrity et al...) or likes on Instagram, giving a couple of examples like this and asking for more from pupils personal experience is a lovely thing to do with mode, because once you start to think about it, you can come up with loads!

So please, when you are teaching averages, don't just skip over the mode; there is so much more to come from this most common of measures.