Monday, 24 July 2017

Developing Triangle and Quadrilateral Area

I have seen and thought a lot recently about the development for approaches to calculating area of certain polygons, particularly triangles and quadrilaterals. I haven't had much time for blogging as exams and end of year routine have dominated my waking hours however now I have some space I thought I would take the time to summarise what I have seen/thought about/heard.

Early development

I think it is fairly standard practice when introducing area calculations to start with the square and rectangle. The use of shapes on a square grid (like the examples below) to motivate the calculation of area as the square of length, or the product of length and width:






Here is where I see typically see the first deviations in teaching approaches. Some people will move from the rectangle to the triangle, using rectangles to justify why the triangle area is halved - images like the ones below are typical (in fact, taken from my own lessons, but I will be revising my approach when I teach to Year 7 next year).



Parallelograms would then follow, or be taught alongside; either as a tilted rectangle, or as a rectangle with a triangle removed and replaced as per the image below:


I was also shown this lovely image of a pile of books demonstrating the idea of rectangle and parallelogram areas being equal at the most recent Complete Mathematics Conference.

One approach I saw recently, which I believe has real merit, is to secure understanding of parallelogram area before moving onto triangle area. This is the approach I will be using from next year for one big reason - the obtuse-angled triangle. Whilst the rectangle can demonstrate right and acute-angled triangles, it is impossible to demonstrate that the area of the obtuse-angled triangle is
½ x base x height, or even adequately show what the base and height of an obtuse-angled triangle are. 
However if one is secure in parallelogram area calculation, and secure in the idea that a rectangle is a parallelogram with extra-properties, then using parallelograms to demonstrate triangle areas deals nicely with obtuse, acute and right-angled triangles. The images below make this clear:

It has always struck me as somewhat odd here that the only other shape for which knowledge of the area formula is required is the trapezium. It would seem perfectly logical to me to move on from the parallelogram to teach the area of the kite - particularly as this can also be seen as two triangles (although this is also perhaps why it isn't taught, as it can be broken nicely into two triangles)...

I would advocate strongly for this to be included when developing the idea of area as the kite, together with the rectangle is a lovely way to then discuss the duality of approach that is available when calculating the area of a rhombus.

Most pupils will have little difficulty in seeing the relationship between a parallelogram and a rhombus. Slightly less intuitive is the relationship between a rhombus and a kite. Orientation can help; having a rhombus standing on a vertex rather than an edge makes the link more visible. It is a lovely meeting point for areas covered so far, as well as enriching pupils' understanding of rhombi...


Of course then from the area of a rhombus and kite and come the area of an arrowhead...


For me, this is then the point to move onto the trapezium. There are a huge number of ways of deriving the calculation for the area of a trapezium, and although the major ones only require knowledge of rectangles, parallelograms and triangles, the manipulation of areas used in deriving the calculation for a kite, rhombus and arrowhead should be useful preparation work. We spent some time at a recent meeting looking at lots of different ways to calculate the area of a trapezium. Here are some of my favourites:

Trapezium as the sum of two triangles

Trapezium as half a parallelogram

Trapezium turned into a parallelogram


Trapezium turned into a rectangle




Trapezium turned into a triangle


Trapezium as a rectangle and a triangle


Trapezium as a parallelogram and a triangle


The benefit of exploring these different approaches is in the richness of understanding of both area calculations and algebraic manipulation that can be developed with a carefully structured teaching approach. This is definitely what I will be aiming to do with my teaching of area in the coming academic year.



Wednesday, 17 May 2017

Malcolm Swan Day

Recently mathematics education lost one of its leading thinkers, Professor Malcolm Swan. The impact that Professor Swan had on developing mathematics teaching and mathematics teachers cannot be overstated, and also cannot be adequately described in words. This post is not an obituary, I didn't ever have the pleasure of meeting Professor Swan, but despite that I have been massively influenced by his resources and the development materials he has published, primarily for me in the Standards Unit (or Improving Learning in Maths).

The purpose of this post is to highlight an opportunity to celebrate the life and work of this great Maths educator. Professor Swan's funeral is on Tuesday 23rd May, and so we are calling on Maths teachers to use Malcolm's materials in as many lessons as possible, and tweet pictures and examples using the #malcolmswanday

For those people who may not realise what we have to thank Malcolm Swan for, his materials include:

  • the aforementioned Standards Unit, which can be found on mrbartonmaths website here.
  • the Mathematics Assessment Project materials, which have their own website here
  • The 'How risky is life?' Bowland Maths project, which can be found here
  • The Language of Function and Graphs - a fantastic book, which the Shell centre have kindly provided photocopiable masters on their site here
The posts and images tweeted on the day will be collated and given to his family as a tribute from maths teachers across the country to this inspirational hero of maths education.

Tuesday, 16 May 2017

Approaches to teaching simultaneous equations

My esteemed colleague Mark Horley (@mhorley) wrote an excellent blog recently about the balance between the need for understanding when teaching simultaneous equations balanced against ensuring procedures are straightforward enough to support pupils ability to follow (read it here). Reading his reflections led me to reflect on my own approach to simultaneous equations, as well as others I have previously seen, and one that occurred to me literally as I was thinking about them. This blog is designed to act as a summary and chart my journey through the teaching of this topic.

Elimination: This is probably the first method I used, and is definitely the sort of approach I was taught at school. Very much a process driven method, I can't remember understanding much about the algebra beyond the idea that I was trying to get rid of one variable so that I could find the other. I find that the subtraction often causes problems (which is partly why Mark's idea of multiplying by -2 instead of 2 is very interesting) and of course the method doesn't generalise well to non-linear equations. I can see this being a popular approach for those people teaching simultaneous equations in Foundation tier.









                    Substitution: Another one from school,
                    this was the alternative I was taught to 
                    elimination, which was mainly because it
                    was necessary to solve non-linear 
                    simultaneous equations. I can't remember
                    it being the method of choice for myself 
                    or any of my classmates, and that is 
                    certainly borne out with my experience of
                    using it with any other than the highest 
                    attaining pupils.














       Comparison: Similar to elimination, but for me less 
       process driven and more focused on understanding the
       relationship between the two different equations. This 
       removes the difficulty around dealing with subtracting 
       negatives, and allows for the exploration of which
       comparisons are useful and which aren't, so it is a little
       less 'all or nothing' than the process drive elimination
       approach. It also copes nicely with having variables with
       coefficients that are the additive inverse of each other, for 
       example in the pair of equations above if instead of the
       approach outlined we multiply the second equation by 3 
       and get:

       4x - 3y = 9     and       6x + 3y = 21

       then the comparison would be "the left hand sides have a 
       total of 10x, and the right hand sides have a total of 30, so
       10x = 30."

       This is the approach I used when recapping simultaneous 
       equations with my pupils in Year 11 and they certainly 
       took to it a lot better than the elimination or substitution 
       that had used with them the previous year.


                                              
                                               Transformation: This approach is the
                                               one I have very recently considered, but
                                               not yet tried. The general idea is that you 
                                               isolate one of the variables, and then look
                                               at how you can transform that variable in
                                               one of the equations into the other. The
                                               same transformation applied to the other
                                               side of the equation then gives a solvable 
                                               equation. Although the equation may be 
                                               slightly harder to solve at first, I do believe
                                               this approach has merit. I would suggest 
                                               that this approach develops pupils'
                                               appreciation of the algebra and the
                                               relationships between the different 
                                               equations in a similar way to the
                                               comparison approach above. I can also see 
                                               this approach working for non-linear
                                               equations, like the one below:

           






















etc...

I will almost certainly give this approach a try when I next teach simultaneous equations - when I do I will try and blog the results!         

Thursday, 11 May 2017

Methods of Last Resort 4 - Comparing/Adding/Subtracting Fractions

Working with fractions is notoriously something that teachers complain about when it comes to pupils' understanding and ability to manipulate. As a result it often seems to me that working with fractions is a place where even the best maths teachers can often fall back into what Skemp would call 'instrumental understanding'; pupils mechanically following procedures rather than applying any understanding of the relationships between the different parts of the process or between the question and the result.

This was brought to mind for me recently when I saw the question below mixed into a group of questions about comparing fractions:

From the rest of the questions listed it was quite clear that the intention would be that pupils write the second fraction as a fraction of 30 so that the comparison between the numerators would yield clearly that the first fractions is bigger than the second. Which of course is completely apparent because the first is more than ½ and the second less than ½. Any halfway competent mathematician wouldn't even bother equating the denominators, and this is the sort of thing I would want to highlight to pupils in order to try and develop their relational understanding.

The process of finding common denominators for comparing, adding and subtracting fractions is one that can easily become automatic for pupils, and I would argue that if pupils are to really understand fractions then they need to be able to take a more discriminatory approach. The following are all examples of questions that pupils could tackle without finding common denominators:


I would argue that the first and second points are more easily done by converting to decimals than fractions (which people may or may not agree with), and that the last one certainly doesn't require a common denominator; the first is greater than ½ whilst the second is equal to ½.

So if you are truly committed to developing pupils' relational understanding of fractions then the next time you look at the sorts of comparisons or calculations that often benefit from converting into equivalent fractions with common denominators, it might be worth throwing in some examples and questions of calculations where this is a method of last resort.


Sunday, 16 April 2017

Gradient of lines - a new approach

Recently I have been teaching the idea of gradient to Year 8, and I decided to approach things quite differently. In the past I would move quite quickly through the ideas of gradient as a measure of slope, finding gradients of lines plotted on a coordinate axes, then linking gradient and intercept to the equation of a line. From my experience this is a fairly standard approach and one that a lot of teachers use. My problem is that typically not too many pupils actually get success from this approach. It occurred to me that I could do a lot more to secure the concept of gradient, and I decided to spend significantly more time than normal doing this, with some surprising results.

The first thing I did was to talk about different ways of measuring slope. Normally I would only focus on the approach I was interested in, but this time I talked about angles to the horizontal and the tangent function. I talked about road signs using gradients as ratios or percentages. Then I talked about gradient measure on a square grid. I have used different ways of defining gradient throughout my career, starting with the standard "change in y over change in x" before I realised this definition was more about how to calculate gradient on a axes rather than what gradient actually is. I played around defining gradient using ratios and writing in the form 1:n, which had some success for a while, but became cumbersome as ideas became more complex. The definition I have settled on for now is "the vertical change for a positive unit horizontal change", or as I paraphrased for my pupils "how many squares up for one square right?" The reason I like this definition is that it incorporates the ratio idea, works for square grids that may not include a coordinate axes, and I can see how it will help highlight gradient as a rate of change later on.

From here we spent quite a number of lessons learning and practising the act of drawing gradients. We started with positive whole number gradients, drawing one short line, and then one line longer, so that we got pictures looking a little like this:
What was really interesting at this point was dealing with the early misconception that the gradient of the right hand line was larger than the left, even though pupils had watched me draw both in precisely the same way. There was an idea, hard to shake, that a longer line meant a steeper gradient; I suspect because the focus was very much on 'how many squares up' the line was going. This did give me the opportunity to reinforce the importance of the single square right; this is an idea we had to keep coming back to throughout the topic.

Once drawing positive integer gradients was secure, we turned our attention to negative integer gradients. Pupils were quick to grasp the idea of negative gradients sloping down instead of up, and I was sensible enough to throw some positive gradient drawing in with the negative gradient drawing so that we didn't get too many problems creeping in at this stage.

With integer gradients well embedded, attention was then turned to unit fractions. There was a great deal of discussion about drawing 'a third of a square up'  for a single square right. The beauty of our definition of gradient here was that it allowed us to use a proportional argument to build up to the idea of drawing 3 squares right to go single square up; if one square right takes you a third of a square up, then 2 right will take you two-thirds up and 3 right will take you three-thirds (i.e. one whole). What was very quickly showed up here was a lack of security with the concept of fractions and counting in fractions (this was Year 8 low prior attainers) and so I am sure that some pupils then started adopting this as a procedure. We were then able to build up to non-unit fractions, both positive and negative, all the time drawing one line short, and then at least one line longer (in preparation for the time where we would draw lines that span a whole coordinate axis).

It was only after we had really secured the drawing of gradients of all types that we moved onto finding gradients of pre-drawn lines, which was simply then the reverse process, i.e. how many squares up/down for one square right? Again a nice proportional argument was used when the gradient was fractional. By the end of this there were pupils in the bottom set of Year 8 able to find and draw gradients like one and three-fifths.

The next part of the sequence wasn't nearly as effective. I went back to the idea of linking gradient and intercept to equations, and although pupils were identifying gradients with ease, and drawing gradients with ease, the extra bits of y-intercept and algebraic equations wasn't so thoroughly explored and the kids struggled. I almost feel like I would have liked to have left this and then come back to it as an application of the work we had done on gradient later in the year; when I design my own mastery scheme I will almost certainly separate these parts and deal with gradient as a concept on its own before looking at algebra applied to straight line geometry at a different point in the scheme.

My advice to anyone dealing with gradient would be to spend time really exploring this properly and not just rushing to using it to define/draw lines.

Saturday, 15 April 2017

The importance of evidence informed practice

I wanted to title this post the importance of evidence informed practice, but I cannot put bold words in the title unfortunately. There has been much discussion about this idea on edu-twitter recently, some of which I have involved myself in, and so I thought I would take the time to flesh my points out more fully in a blog.

One of the quotes that I have seen that created a bit of controversy around this issue was used in the Chartered College of Teaching conference in Sheffield. The session delivered by John Tomsett, Head teacher of Huntington school in York and author of the "This much I know..." blog and book series. The quote was taken from Sir Kevan Collins, CEO of the Education Endowment Foundation:

"If you're not using evidence, you must be using prejudice."

This quote caused quite a bit of disagreement, with some people very much in favour of the sentiment, and some taking great exception to the provocative language used.

I had an interesting discussion on twitter about this quote, with my interlocutor seeming to hold to the viewpoint that because all children are different that any attempt to quantify our work with them is best avoided. Their argument goes that the perfect evidence-based model for classroom practice is an unobtainable dream, and so the effort to create one is wasted. To me the point of evidence informed practice is not to try and create the perfect evidence-based model, but rather to ensure teachers can learn from the tried and tested approaches of their peers; to stop them falling into traps that people have fallen into before, and to allow teachers to judge the likelihood of success of different possible paths. To bring another famous quote into the mix, "If I have seen further it is by standing on the shoulders of Giants." (Isaac Newton). In the same vein, we don't every new teacher to have to reinvent the wheel, we want them to be able to learn from those who have faced similar challenges and found solutions (or at least eliminated possible solutions).

One of the accusations that has been levelled at educational researchers is that they are 'experimenting on kids'. This is one of my least favourite arguments against evidence informed practice as its proponents must either be ignorant of how researchers operate or be feigning ignorance in order to make a point that isn't worth making. At some level everything we try in the classroom has a risk of failure; even the best practitioners don't get 100% understanding from every child in every lesson. The big point here though is that no one goes into the classroom with anything other than an expectation that what they are going to do is going to work, and this goes for researchers as much an any other professional, and is true in fields other than education. It would seem that some of the critics of evidence-based practice see researchers as a bunch of whacked-out lunatics wanting to try their crazy, crackpot theories out on unsuspecting pupils. In fact most researchers are following up on promising research that has already been undertaken, and so in theory their ideas should have a greater chance of  success than a teacher whose view of the classroom is not informed by evidence. Even when researchers are trying totally new approaches, they are tried from a strong background and with a reasonable expectation of success. It is precisely the opposite of the view that some seem to hold, and in fact it is those who don't engage with educational research that are more likely to have some crackpot idea and then not worry so much about its success. 

One of the situations I posed on twitter was the situation of the teacher new into a school, and therefore taking on new classes. Let us further suppose that said teacher is teaching in a very different setting to that which they are used to; perhaps a change of phase, a change of school style (grammar to comprehensive may well become more prevalent), or even just a change of area (leafy suburb to inner-city say). Now this teacher has two choices in order to prepare for their first day in their new classroom. Their first choice is to read something relevant and useful about the situation they entering, They could talk to teachers in their network that have experience in their situation, including in the school they are going to be working. They could inform themselves about the likely challenges, the likely differences, and the ways that people have handled similar transitions successfully in the past and then use this to make judgements about how they are going to manage this change. Alternatively they could not, either sticking blindly to their old practice, or making up something completely random. I know which one I would call professional behaviour. 

When faced with this situation, the person with whom I was having the conversation sidestepped this choice and suggested that all would be well because they have a teaching qualification. Of course this ignores what a teaching qualification aims to do; the whole point of a teaching qualification is to lay down patterns for this sort of professional practice. This is one of the big reasons I was very much against the removal of HEI from teacher training. The idea of teacher training is to try and provide this dual access to practical experience through school placement along with skills in selecting and accessing suitable research and evidence from outside of your experience to supplement the gaps in your own practice. A teaching qualification has to be the starting point of a journey into evidence-informed practice, not the end point. One doesn't emerge from the ITT year as anything approaching the effective teachers that they have the potential to become; and the only way they will do so is by engaging with the successful practice of other teachers and using this to develop and strengthen your own practice and experience.

One other criticism levelled at those engaging with research and using it as the backbone of their practice is that the outcomes measured in order to test the success of the research are very often the results of high-stakes tests, and that these may not be the most appropriate measures of success. I have some sympathy with this point of view; I can see for example why people would baulk at the idea that the impact of using Philosophy for Children can and should be measured by their combined KS2 maths and English scores, which is what is happening in the EEF funded trial. However if we bring it back a notch we should ask ourselves what we are trying to achieve from the intervention. Ultimately I could argue that the purpose of any intervention in school is to try and make pupils more effective at being pupils, i.e. being able to study and learn from their efforts. Whether the intervention is designed to address gaps in subject knowledge, problems with learning behaviours or improve development in a 'soft-skill', the eventual intent is the same; that these pupils will be able to take what they have learned and use it to be more successful pupils in the future. Now I am not going to stand up and say that the way we currently measure outcomes from education is an effective way of doing so, but what I will say that is that however we choose to measure outcomes from education, any intervention designed to improve access to education has to be measured in terms of those outcomes. I am also not going to necessarily stand here and say that every single thing that goes on in schools should be about securing measurable outcomes for education (and I know many educators who would make that argument) but then I would argue that these things should not be attracting their funding from education sources. If an intervention is expected to benefit another aspect of a pupil's life, but it is not reasonable to expect a knock-on effect on their education (and when you think about it like that, it becomes increasingly difficult to think up sensible examples of interventions that might fit that bill) then it needs to be funded through the Health budget, or the Work and Pensions budget, or through whichever area the intervention is expected to impact positively.

Schools are messy places, subject to a near-infinite number of variables, very few of which can be controlled. It is virtually impossible to ensure that any improvement in results is due to one specific intervention; often several factors are at play. Does this mean, however, that we shouldn't experiment in the classroom, provided we have a reasonable expectation of success? Does this mean that we shouldn't attempt to quantify any success that we have that could, at least in part,be attributed to the change we made? Does this mean that we shouldn't share the details of this process, so that others can adopt and adapt as necessary, and then in turn share their experiences? To me this is precisely how a professional body of knowledge is built up, and so if teachers are going to lay claim to the status of 'professionals' then engagement with this body of knowledge has to be a given (provided they are well supported to do so). If you have the support to access this evidence, and then simply refusing to do so, then I would argue you certainly are using prejudice; either prejudice against the idea of research impacting your practice at all, or prejudice against the teachers/pupils that formed the research from which you might develop. Prejudice has no place in a professional setting, and no teacher should ever allow their prejudices to stand in the way of the success of the pupils in their care.

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...