Wednesday, 1 March 2017

Methods of last resort 3 - Straight line graphs

The linear relationship is probably one of the most fundamental relationships in all of mathematics. Functions that have a constant rate of change are the basis of our most rudimentary geometrical transformations, conversions and correlations. It should be fair to say that ensuring pupils have a proper grasp of linear relationships should be an important part of any mathematics curriculum; and yet many pupils are only given a very narrow view of these key mathematical constructs.

Most pupils first view of the graphs of linear relationships between two variables are through algebra in the form y = mx + c. Pupils will be given equations of this form, and asked to substitute to find coordinates and then plot coordinates to draw lines. Some pupils may be given the opportunity to draw parallels between the equation and the relationship between the variables x and y but not all. Eventually concepts like gradients and intercepts will be taught, and here is where the narrowing will begin. Most pupils will be given an algebraic definition of gradient, such as "change in y over change in x" or similar. Can we first be very clear from the start please that this is not what gradient is, this is just one way to find the gradient if you happen to know the horizontal and vertical distance travelled (for those people who think I am being picky, another way to find the gradient is to take the tangent of the angle the line makes with the horizontal, which is seldom taught in this way).

What gradient actually is is the vertical distance travelled for a unit increase in horizontal distance. Dividing a given vertical by a given horizontal will calculate the the value, as will applying the tangent function to the angle made with the horizontal, but neither tell you what it actually is. Pupils should have a proper understanding of what gradient is, before they begin calculating it (in my opinion). But this is not actually the point of this blog post so I will get back on track...

Once gradient is 'taught' the link between its value and the value of m in the formula given above is very quickly highlighted, often either explicitly or through some form of 'discovery'. Here comes the second narrowing - from this point onward virtually every attempt to ascertain the value of the gradient of a particular line when given any form of linear algebraic relationship invariably leads back to writing the equation in the form y = mx c. Remember lines are very often defined in a different form; x + y = 5, 3x + 2y + 4 = 0 etc. Ask any competent school age pupil to find the value of the gradient of these lines, and I will guarantee that the vast majority of the time a rearrangement into the form y = mx c is attempted if the pupil is even able to attempt the problem at all. And while this approach is perfectly correct and if done well will reveal the value of the gradient, it isn't the only approach; many pupils labour in ignorance when better methods may be applied.

Take the line x + y = 5 for example. Now for most mathematicians it would be straightforward to rearrange this to give y = -x + 5, and hence find the value of the gradient of -1, and the y-intercept of (0,5). However I would argue at least equally straightforward would be to say "the points (0,5) and (5,0) are on the line, and so the value of the gradient = -5/5 = -1 and the y-intercept is (0,5) [and, by the way, the x intercept is (5,0) - which is not nearly so often asked about]. To be fair, there is probably not a huge difference in the mechanics, but as Anne Watson highlights in her blog (see postscript below) there is perhaps a difference in pupils understanding of what this line actually looks like, as well as providing more of an opportunity to reinforce the idea of vertical distance travelled for unit horizontal distance.

If we then take the line 3x + 2y + 4 = 0, the rearrangement is a bit messier - I know plenty of pupils that wouldn't be able to rearrange successfully. However it is still a rearrangement that you would want pupils to be able to do and expect that they could if they had the proper grounding in inverse operations etc. The other side of this though is that I can quite quickly see that the point (0,-2) is on this line, and that the point (-1⅓, 0) is on this line. So I can also calculate the gradient as -2/1⅓ = 1½, as well as tell you about the x-intercept and y-intercept. Perhaps even more straightforwardly I could have told you that the point (1, -3½) is on the line, and so arrived at the value of the gradient immediately, I have gone 1½  units down when x increased by 1 (from 0 to 1).

Whether you want to consider rearrangement to the form y = mx + c as a 'method of last resort' or not is up to you; clearly it is an important mathematical idea that relationships can be expressed in different forms. However I would suggest that it is not the only idea that pupils should be able to draw upon when talking and thinking about finding gradient values, and that we should be aiming to give pupils a range of strategies linked to a deeper understanding of what gradients, and lines of constant gradient, are.

Postscript: Emeritus Professor of Education at Oxford University Anne Watson recently released a blog about a similar topic (and actually using one of the same equations!) here. I have actually been writing this blog post since late January and was just trying to find time to finish it off, so wanted to go ahead and publish it anyway!

Saturday, 28 January 2017

Multiplicative Comparison and the Standards Unit diagram

Recently I have been doing quite a lot of work with proportion (one way or another) across a lot of my classes. My Year 11 classes are looking at rates of change (gradient is a proportional relationship between change in x and change in y) and probability (the proportion of outcomes that fit a criteria) respectively. My Year 8 classes are working on probability and unit conversion. My Year 10 are working on compound units. I have been realising how versatile this diagram is:

For those that don't recognise this picture, it is from N6 of the Standards Unit, which is about developing proportional reasoning. I call it 'The Standards Unit Diagram' whilst a Twitter colleague (@ProfessorSmudge) calls it a ratio table. It is probably also the best diagram I have ever seen for multiplicative comparison, which is pretty much the basis of all division and proportionality.

Lets say I want to convert between cm and metres, in particular 350 cm into metres. The diagram might look something like this:
This diagram really nicely shows off the twin relationships that are present in all proportional relationships, i.e. that one variable is always a certain number of times bigger than another (the conversion factor or rate of change, in this case 100 cm/metre) and the fact that any multiple of one of the variables is matched by a corresponding scaling in the second variable (in this case, the fact that the number of cm has been multiplied by 3.5 implies that the same also happens to the number of metres). Notice that it is only strictly necessary to find one of the relationships to solve the problem, but nonetheless it is clear that two exist (in this case, depending on the level of the pupils, the focus may be on the use of 100 rather than the scaling here). 

There is some anecdotal and written evidence (I remember reading an article but honestly can't remember what it was called) that most people will naturally focus on the scaling in a proportional problem, particularly if the scaling is obvious (the variable gets doubled or trebled for example), but what is nice about the diagram above is that it gives equal focus to both relationships that exist.

Below is one of the diagrams I used to highlight the commonality behind representation that was possible using this approach in my recent talk to Heads of Maths at the LaSalle Education HOM conference, sponsored by Oxford University Press. This diagram was used to solve the percentage problem "A jacket costs £84 inclusive of VAT at 20%. Work out the price before VAT." which is a fairly classic GCSE reverse percentage question.
Now, as was pointed out in the session, what is clear from this diagram is that the most 'efficient' way to solve this problem is simply to divide 84 by 1.2. However the diagram does highlight a possible alternative, and more importantly highlights the commonality in the relationship here which is the essence of all proportion and division, namely "100 is to 120 as 5 is to 6 as what is to 84?"

I would argue quite strongly that very few pupils actually understand division and proportion as they don't understand that this comparison is at the heart of all of these types of relationship. Every division, every proportion are basically saying "If a is to b as c is to d then a proportion exists". The one that definitely caught the eye at the aforementioned head of maths conference was this demonstration of using the diagram to highlight the commonality of relationship when dividing with fractions, in this case solving the fractional division ¾ ÷ ⅚
This can be summarised as "I don't know how three-quarters relates to five-sixths, but I know that it is the same as how 3 relates to twenty-sixths, which is the same as how 18 relates to 20, which is the same as how 9 relates to 10." The conclusion is that ¾ ÷ ⅚ = 9/10.

This way of viewing division as a proportional relationship, that can be manipulated in the same way as other relationships (i.e. as a multiplicative comparison) is a powerful interpretation, and one that I would argue that no pupil should be without. Even regular division of two integers can be seen in this way, particularly given the understanding that regular division is a multiplicative comparison to 1:


So this is literally "75 is to 15 as what is to 1?" with the 'what' being 5, and similarly with the second "23 is to 5 as what is to 1?", with the 'what' this time being twenty three-fifths or alternatively four and three-fifths. Indeed, the earlier fractional division could well benefit from a final line showing the equivalent relationship to 1 as '9/10 to 1'.

Even if you don't ultimately like the diagram or the approach, I would argue that no pupil's (or teacher's) view of proportionality or division is complete without understanding this idea of multiplicative comparison. However you choose to represent or structure it, giving your pupils an insight into this aspect of division is pretty much guaranteed to give them a deeper insight into what it it means to think multiplicatively.


Friday, 20 January 2017

Christmas Mock Grade Boundaries - our story

Ok, I was wrong last time, this is definitely the most dangerous blog I have posted; if by dangerous I mean fraught with the capacity to be wrong and inconvenience a lot of people. So I will preface by saying I am very sorry if you base anything off of this post and it turns out to be wrong; we are all just guessing here really and guesses can go wrong. Still if it helps people clarify their own thinking, or supports people that wouldn't otherwise have a way of meeting the demands of their senior teams or other stakeholders then I suppose it is worth a little egg on the face if it turns out wrong. So here is the story of our Christmas Year 11 mocks and grade boundaries:

We sat mock exams just before the Christmas holidays,which meant that by about a week after we returned from Christmas I had pretty much all of the results from our 285 pupils (of which about 250 or so actually sat all 3 papers). By this time I also had the results of one other school with about 180 pupils in Year 11, with about 160 that had sat 3 papers, and another much smaller school that were only going to sit two papers. I used a similar process that I had at the end of Year 10; I apply a scaling formula to the Foundation paper to make it directly comparable with Higher which has worked well in the past, and then applied the proportions and other boundary setting details which have been well publicised by the exam boards and great people like Mel at @Just_Maths. This led to this set of boundaries, which we applied to our pupils:


I wasn't completely enamoured with these - I knew for example that the 9 and 8 were lower compared to where I expect them to be in the summer, and in general I thought that maybe all of the higher scores were a little low (although as you go down the grades I expect them to be closer to the real end values in the summer of 2017). I did like the Foundation ones, they seemed to sit well with what I was expecting. Given that pupils still have 5 months before they sit the real thing though, I thought these were acceptable for now. At the time I couldn't make them public, as our pupils were not given their grades back until their mock results day today.

Literally two days after we had inputted mock grades, AQA released the population statistics for the cohort. I was pleased to see that our Higher pupils had scored above average compared to the population, and our Foundation had scored lower. I took this to mean that our tiering choices were about right, although as any good statistician knows making judgements based on averages alone is a dangerous thing to do and we did have to look at the pupils at the lowest end of the higher paper scores as we had a large range of values.

Although we had already set boundaries I work with a group of 5 other schools, many of which were doing their mock exams after Christmas and so would be needing boundaries - originally the plan would be to collect all of their results and set the boundaries (which would have given us a cohort over 1000, and so had at least some hope of being reasonable). With the support of some excellent colleagues who will remain nameless I managed to get hold of some data about the population rankings that were attributed to certain scores for Higher and Foundation. This allowed for the setting of the grade 7 and 9 (and therefore also 8) at Higher, based from last year's proportions and the tailored approach as outlined in the Ofqual documentation as well as the grade 1 at Foundation. The grade 4 proved more problematic, as there was no detail about how the Higher and Foundation rankings compared to each other (I am reliably informed that it is impossible to accurately do this without the prior attainment from KS2, although my scaling formula does seem to produce quite similar results).

I was able to get hold (from a source who will definitely remain nameless) of the proportions of C grades that were awarded to 16 year olds last year for the separate tiers and based on this I was able to map out the separate values for grade 4 on the Higher and Foundation tier. This also allowed the setting of the 5 and 6 on the Higher tier, and 3 and 2 on the Foundation tier. Although it is still up for consultation (I believe), I also awarded the 3 using the approach that has been used in previous years for setting the E grade boundary on Higher, namely halving the difference in the grade 4 and 5 boundary, and then subtracting this from the grade 4 boundary.The trickiest one was actually the 5 boundary on Foundation, as there is no real guidance over this one; in the live exam I believe this will be set based on comparison of pupils scripts and prior attainment (although if anyone knows more about this I would be happy to be corrected). In the end I did have to make a bit of educated guess work with comparison back between my own papers, and ended up with boundaries for the whole AQA cohort that look like this:


I was quite pleased with the similarity of these to our boundaries, although it would appear my scaling formula is a little harsh to the Foundation pupils for mock exams (it does work quite well for real exams though). At this point though I should pass on some major health warnings and notices:

  • These boundaries are NOT endorsed by AQA, and they will rightly maintain that it is impossible to set grades or boundaries for exams without prior KS2 pupil data. Although this does use data available on the portal from the AQA portal, it is only my interpretation of it.
  • There are two big assumptions used to make these boundaries, which are unlikely to completely bear out in reality. In particular, there is an assumption that the proportions highlighted in the Ofqual document are going to pretty much repeat from last year to this year; i.e. that the cohorts from Year 11 in 2016 and 2017 are pretty similar. In reality we are told that Year 11 2017 have slightly higher prior attainment than those in 2016 (although the published data does say that the two are not directly comparable). The other major assumption is that the proportions of grade 4 at Higher and Foundation will roughly match the proportions of grade Cs awarded at Higher and Foundation last year. This assumption is certainly unlikely to be true, we are already hearing that schools are entering significantly more pupils at Foundation tier (myself included compared to the proportion I used to enter in my previous schools), which is likely to raise the quality of candidate at both Foundation and Higher tier. If this is the case for the current mock data it would have the effect of lowering the Foundation boundaries (although they seem to fit too nicely for me to believe they will go lower - just a gut feeling though) and raising the Higher boundaries (which seems likely in reality).
  • We mustn't forget that a lot can happen in the next 5 months, and I would expect most of the cohort to improve their scores; I would still expect the 9, 8 and 7 to be noticeably higher than these values in the summer, although I don't think the 4 boundary will shift up by as much as some people might think. In reality these boundaries are useful in the very specific circumstance that a pupil has completed all 3 papers from AQA practice set 3, and that they have done so after about a year and a bit to a year and half of GCSE course study.
So that is our story, up until about 2 or 3 hours ago. If it helps people then great; if you disagree then fine; if you use it and it turns out wrong, well you were warned...

Thursday, 24 November 2016

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!

Sunday, 6 November 2016

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.

Friday, 21 October 2016

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.

Thursday, 6 October 2016

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.