Monday, 4 December 2017

Methods of Last Resort 6: Right-angled Trigonometry

I must admit to having reservations as I write this blog post. Not because I am unsure as to the approaches I will outline, but rather to do with the categorisation as a 'Method of Last Resort'. Before now I have typically suggested that methods of last resort should be the things we do when our understanding of a situation doesn't allow us to take a more efficient approach - for example considering order of operations a 'Method of Last Resort' as it is the sort of thing we consider when we can't simply work left to right (as in 5 x 6 ÷ 10 for example) or when we can't simplify a calculation (as in 23 x 6 + 7 x 6 = 30 x 6, or 172 – 32). I am not completely sure that what I am going to outline falls into that category, but nonetheless here goes...

I am going to propose that SOHCAHTOA is a method of last resort. By SOHCAHTOA I don't mean the mnemonic, I mean the idea of treating the trigonometric ratios as formulae:

So what is the alternative? Well the obvious one is the unit circle, but that might be a bit much for the first introduction of trigonometry. Instead I wanted to outline an approach around similar triangles.

Let us first take sine. Sine of an angle between 0 and 90 relates the opposite to the hypotenuse in the following way:

This is all the basis we need to find missing sides in any right triangle with the angle θ. Consider now the triangle below:
This triangle is an enlargement of the first triangle, using a scale factor of 13. This implies that the opposite side is simple 13 × sin θ. Even looking at the triangle below:
This triangle is still an enlargement of the first triangle, but with a scale factor of 13/sin θ. So the hypotenuse must by 13/sin θ.

This approach also be used to find angles. Consider the triangle below:

This triangle is still an enlargement of the original triangle, again by scale factor 13. This would mean that the opposite side of the smaller triangle is 5/13. But remember, in the smaller triangle the opposite side is sin θ. So we have that sin θ = 5/13. This leads of course to θ = sin-1 (5/13).

Virtually identical approaches can be used with reference to the adjacent and hypotenuse sides, with the cosine function and the tangent function with the opposite and adjacent sides. Importantly, this approach arguably requires a deeper understanding of how trigonometric functions relate sides of a triangle than the formulae provided at the beginning of this blog post, and it is for this reason why I wonder if the formulae couldn't be considered a 'Method of Last Resort.'

Sunday, 1 October 2017

Teaching Apprenticeships - a last desperate attempt to solve our recruitment and retention crisis on the cheap.

I have read with dismay the recent news coverage of the teaching apprenticeships in England. Whilst it is welcome to hear that full Qualified Teacher Status will continue to be the domain of degree holding applicants, this is by no means enough to satisfy I or many of my colleagues with this erosion of our professional status.

Apprenticeships are a great route for many things. An old friend of mine trained to be an electrician this way - he spent 4 days a week apprenticed to a qualified electrician, and then one day a week in college learning the technical aspects of his trade. When he struggled with some of the maths he would come to me and I would give him a little extra help and support. He grew up to be an excellent electrician, owns his own company and now employs other electricians and apprenticeships. This worked for him because he could learn as much from watching and helping an electrician do their job as he could from the classroom; because the time scale was relatively short; and because he hadn't been particularly enamoured with school and was therefore very reluctant to commit to continue with full-time education.

None of these states can be applied to teaching, or to the proposed teaching apprenticeship. There are so many things about being a teaching that you cannot learn from watching teachers. Peer observation is important for teachers, but to even know what you are looking for takes knowledge and understanding that needs input first. The amount of technical input needed to be successful in the classroom at huge. Remember this is at a time when people question how much a full time PGCE or a BAEd imparts the necessary knowledge for the classroom. This is when the Institute for Teaching is planning for an examined two year training programme for teachers because it believes in the need for further training and rigour. This is a time when everyone with an interest in developing anything in education from subject specific pedagogy to overall classroom management bemoans the lack of time given to focus specifically on their 'thing' during training. You can't learn these things just by watching the handful of teachers you might have contact with whilst being apprenticed at a school. You need access to research, regular reading and development tasks, and access to people learned in not just what works for them, but with the experience of having supported hundreds upon hundreds of people entering the profession.

Now you could do all of these things on a teaching apprenticeship, but they would take time. A lot of time. The length of a degree level apprenticeship is up to 6 years. 6 YEARS!! In a profession where 20% of new teachers leave after 2 years, and only about ⅔ last 7. This timescale for training an apprentice in teaching is staggeringly long. I cant see too many schools being able to commit to a 6 year training programme for apprentices. I mean, 7 years is only about the average length of time that a fully qualified teacher will spend in a school, never mind someone training to be fully qualified. Indeed, it would be difficult to see how a school with many apprentices would be able to mentor them through the process without significant changing to the supporting staff between the start and end of the apprenticeship.

Probably the second biggest issue I have with the whole idea though is that teachers are supposed to be the front-line in inspiring a continuation of education. Whilst I can see some value in having people in schools that can reassure pupils that vocational routes can lead to success, I have this quite strong feeling that the people working with youngsters in the classroom should be clear role models of the success of academia. While I sympathise with those people that are desperate to work with young people but for reasons in their own academic history weren't able to go to university, I don't see that as adequate reason to give the message to young people that there are 'workarounds' for everything if you end up not doing well in the classroom. There are other routes to securing degrees whilst working, from Open University, part-time degrees or night classes. And yes I know some will make the argument that not everyone can afford these, or indeed will ever be able to afford to pay for a degree, but I see this as an argument for not charging fees for education related degrees, or for providing loans for an undergrad degree that can truly cover for the expenses of single parents or others with more responsibility than the typical undergrad. Ultimately it might be that not everyone that wants to can teach. I suspect that not everyone who wants to be an astronaut achieves that dream either, or a lawyer, or a doctor. The fact that we don't perhaps have as many problems recruiting astronauts or lawyers as we do teachers doesn't mean we need to open up routes into teaching that aren't suitable, it means we need to make those routes that are suitable more attractive. As my friend and much admired professional colleague Mark McCourt often proclaims, teachers should be towering intellects capable of inspiring pupils with the joy and fulfilment that comes from lifelong dedication to learning and academia. I can't see how someone who couldn't get themselves too and through university can lay claim to this, however harsh this sounds.

I said that the idea of having teachers in classrooms that ultimately weren't successful in classrooms was my second biggest issue with the idea. My biggest is one that is conspicuous by its absence - the notion of this adding value to the profession. I read articles where politicians claim that this won't make the profession worse or devalue it in any way; I don't read the same people claiming that this is a step-forward for teaching. There is a simple reason for this of course; because it isn't. If it was, we would have people making the argument for degree apprentice lawyers, or doctors, or astronauts. And we don't. Some may argue it is because of how new this level of apprenticeship is, but I can't see it ever being something that those professions clamour for. Indeed a quick search of degree apprenticeships available would seem to confirm that the majority available are in those technical and scientific areas that require the much more specific technical knowledge that this model can provide, certain careers in engineering, surveying etc. And whilst I am certainly not saying that teaching is better than these areas, I am saying that we want different things from our teachers than we do our engineers; a different type of knowledge, different skills. Our engineers need a very technical set of knowledge and skills directly related to their field, teachers need a myriad and multitude of overlapping skills and understanding to fulfil the roles of knowledge developer, pastoral carer, life enricher and everyday role model that are required in schools.

I don't think there are many teachers our there that don't see this move from government for what it is, and what I said in the title of this post, a last desperate attempt to ensure our schools have enough teachers without spending the money it would take to actually do this properly. With the minimum wage of a first year apprentice being £3.50 an hour this means schools could feasibly get a teacher in the classroom 4 days a week for as little as £4427.50 in wages, assuming apprentices would get paid for the same 1265 hours of directed time that is still commonplace in many schools. Of course some schools may offer more than the minimum wage, but in reality this is likely to be just so they don't lose any money in the apprenticeship levy - I can completely understand schools taking the attitude, "we have to spend this money on apprentices so we will pay ours a little more". I suspect though that even this will be unlikely - schools will probably just try and secure more apprentices and only resort to paying more if they are facing losing the money anyway. What then happens to these apprentices once they qualify and become more expensive is of course a different matter - as a cash-strapped school will I employ one of the apprentices I just trained but will now cost me a whole load more money, or will I just let them go at the end of their apprenticeship and take on a new apprentice? I have already seen this happen time and time again with apprentices in the back office or site team, and I have no reason to believe that some school leaders wouldn't behave in the same way with apprentice teachers.

If this government really wants to get more teachers in the classroom, and make sure those teachers are of sufficient quality and qualification background to do the job, then I suggest it remove schools from the apprenticeship levy so that they can invest their money in the training and intellectual stimulation that is crucial in retaining high quality teachers, whilst simultaneously investing real time, effort and funds into making teaching the really attractive graduate profession that it could and should be by investing in ITT, raising wages, and securing a guarantee for meaningful CPD throughout a teacher's career. Provide the sector with the money and time it needs to reduce teacher workload, address the issues with our assessment and accountability systems, and ensure that a visit from Ofsted doesn't mean the end of a career. Maybe then we will have a truly attractive graduate level profession that people strive to enter and that will make it worth getting that degree for.

Monday, 21 August 2017

Methods of Last Resort 5: Median and Mean.

Lets see if these seem familiar:

Median = middle number in a data set when the set is ordered.
Mean = total of the data set shared equally between the number of data points (or possibly "add them all up and divide by how many there are", but if you still use this, then see my blog here).

In the main, perfectly acceptable approaches to finding median and mean. Note I don't use the term average here: I think a lot more work needs to be done to separate the finding of mode, median and mean with the concept of average, and will blog about that at some future point. For now I want to concentrate on the process of finding median and mean rather than any link they have to the concept of average. Now consider the following:

1) Find the median of the list 3, 5, 6, 7, 8, 13, 10.
2) Find the median of the list 3, 2, 1, 6, 10, 9, 8.
3) Find the mean of the list 7, 9, 10, 11, 13.
4) Find the mean of the list 106, 104, 108, 107, 108.

To anyone that understands the ideas of median and mean,  these questions are a bit different, in that they don't require the definitions provided above. Let us tackle them in pairs.

Firstly the median. In both of the cases above the middle value is the median, and the fact that the lists are not in order makes precisely 0 difference. Now I can hear the arguments already, "yes but these are very contrived data-sets", "yes but that won't work a lot of the time" and I understand where they are coming from. But the point is, as a competent mathematician I get that in these cases there is no need to order. If our goal is to produce competent mathematicians in our pupils, to have pupils that understand these concepts properly, then surely they should understand this as well? And it can't be blamed on my education beyond GCSE - I did precisely no study of statistics beyond GCSE. I had choices for my modules at A-Level and so did all Core and Mechanics, and then my Degree was all in either pure maths or maths modules that linked to classical mechanics and physics. There was no statistics content at all.

A possible solution to this is to re-define the median as something like "the value in the middle position of a data set if all positions below are numerically smaller and all positions above are numerically bigger". Honestly though this definition seems overly convoluted for such a simple concept. There are plenty of times when re-ordering the list is the best strategy, even if it wouldn't be completely necessary (for example 3, 2, 1, 8, 6 only requires the switching of the first and third digit). The point I think is that pupils need to understand what the ordering is trying to achieve, and are shown explicit examples of when this isn't necessary. The ordering of the list can then be treated as a 'Method of Last Resort', something you do when the median is not already in the correct position or very close to the correct position.

Now questions 3 and 4 on the mean. Again as a competent mathematician I understand that I don't need to find the totals in these questions. In the first I can see that 7 and 13 are equally spaced from 10, as are 9 and 11, so these differences are going to even out and make the mean 10. Interestingly, I am not sure I would make the same argument if the list was 13, 9, 11, 10, 7 - I think if presented with this list I would begin to total it and then probably see that the 13 and 7 will combine nicely along with the 9 and 11. In question 4 I can see that I only need to total the 6, 4, 8, 7, and 8 and then find the mean of these 5 numbers before just adding the mean to 100 (to be fair this is something I came across when teaching myself the MEI S1 and S2 units so I could teach my Further Maths A-Level groups - it is called linear coding). Whilst this might mean we could choose to avoid highlighting this particular property of mean at GCSE (although I can't see a good argument for doing so really) it still illustrates that there are other ways of calculating the mean. Again we could solve this by re-defining what we mean by "mean" to better capture the 'evening out' idea, but this would see to again be a bit of overkill. I think the point here is that we should aim to secure understanding of mean to the point where pupils are able to identify whether the total needs to be found or not - totalling becomes a method of last resort to be used if other more efficient methods are not easily identifiable.

As I have been writing this blog, this has highlighted to me what appears to be a subtle difference between the ideas of median and mean and the accepted process for finding them. The idea of median is this idea of centrality, and an accepted process for finding it is ordering. The idea of mean is the idea of evening out the distribution, and totalling then dividing is one way of accomplishing this. I need to consider more what this means for my teaching practice. In the meantime what I will say is that I definitely think we need to be trying to secure the understanding necessary in pupils so that they can discriminate between times when the accepted process is the best, and when it isn't

For those that may not have followed this blog sequence from when I started it following my session at mathsconf, I will reiterate what I have said before - I am not saying whether you should lead with this, or lead with the standard approach before pointing out these special cases. That judgement needs to be made for classes by the teachers that work with them week in and week out. What I am saying is that I passionately believe that our pupils deserve to see these sorts of examples at some point rather than not at all. If we are truly going to teach to develop understanding in our pupils then we need to include this as part of the understanding of median and mean.

Sunday, 20 August 2017

Solving Linear Equations: some thoughts

This is quite a difficult one to examine for a methods of last resort blog post, as many methods exist for solving linear equations and different teachers will use different approaches. Probably the two most common are some form of balancing approach, similar to this:
and then the use of function machines, similar to this:

Now inherently there is nothing wrong with either of these approaches (except the function machine solves 3a - 5 = 19, not the given equation 3a - 5 = 10) provided pupils understand why they are carrying out the operations they are, or how the function machine relates to the equation they are solving, and then how the inverse machine relates to the original equation. 

The slight problem I have with both of these is how open they are to a more procedural approach. I can imagine a lot of teachers falling into the trap of teaching how to find the inverse function as a procedure rather than with any real understanding. I can imagine lots of teachers showing pupils how to manipulate both sides of an equation whilst keeping them in balance, but without imparting any real sense of why what they're doing works or what the purpose of the whole affair is. Equally I can imagine this not being the case and these methods both being taught well. 

Recently I have begun to consider a different approach, although I haven't really used it extensively yet. In the main I have drawn attention to it when using balancing as a way of developing understanding, or when pupils have suggested incorrect statements when solving an equation. The approach I have used looks at a sort of 'If...then' or 'what follows?' kind of approach. I will illustrate below with and example:
Solve the equation 3a - 5 = 10:
If 3a - 5 = 10, what follows?
Well if 3a - 5 = 10 then it follows that 3a = 15.
If 3a = 15, what follows?
Well if 3a = 15 then it follows that a = 5.

I am very aware of a couple of big points when it comes to this:
              1) What if a pupil gives something that does follow but isn't useful, for example, if 3a - 5 = 
                   10 then 3a - 15 = 0?
              2) What if pupils do not have the understanding of relationships and operations necessary to
                    see what follows, for example if 3a = 14 then a = 4⅔.

In response to the first I would (and frequently do, even when using balancing) allow this to go. I would then explore the consequences of this and try and eventually show how that wasn't a useful step. Over time I would want to develop an understanding in pupils of what the next useful thing to write would be, but in the beginning I wouldn't necessarily be too worried about this. I wonder if allowing pupils to explore (under very controlled conditions obviously) the consequences of making true but not useful statements would actually help them develop their understanding of the concept of equality and equation solving. It would concern me if we always limited pupils to the correct next step in the reasoning, as this would seem to then smack of becoming a procedure we expect pupils to follow. In fact I would strongly consider having an entire lesson early in secondary school where rather than solve equations, pupils simply have to write other true equations based on the original. I have seen activities and sessions like that being used already and I can definitely see the merit in them.

In response to the second, I would simply say that this is worthwhile diagnostic information, as it points to a gap in pupil understanding of a more basic concept. If this was the case it is a clear indication to me that I need to go back and do more work on fractions and inverse operations as the pupil in question clearly doesn't have the requisite procedural fluency in these areas. Hopefully with the advent of mastery teaching situations like this would become rarer as times goes on.

Ultimately no matter what approach you are going to use for solving linear equations, I would urge you to be wary of falling into the trap of explaining the 'how' without ever getting near the 'why'. There are ample opportunities when using balancing to explore why one statement leads to the next, and in function machines why the different machines link to each other, as well as to the original equation. However for me equations are about the relationship between two equal quantities, and I wonder if the focus on operations that is part of both balancing and function machines obscures this somewhat, so I will be exploring the use of the approach I have outlined - and I would welcome feedback from others who may be using or thinking of using similar ideas.

The Teacher Transfer Window?

Outside of education (particularly maths education) and my family, one of main interests is sport. Like many sports fans, I have been following the dealings in the football transfer window with fascination - I must admit I wouldn't have ever thought that a player would be worth one-fifth of a billion pounds.

My second in department and I, along with a couple of members of my team, have in the past joked about the idea of a teacher transfers. With the advent of performance related pay and some teachers in a school therefore being paid more than peers who may have joined the school/profession at the same time, we have occasionally laughed at the idea of a school making contact with our head teacher to try and 'buy' someone, with compensation being agreed between the schools and possibly even swap deals being done. Seeing the behaviour and dealing of some clubs during the transfer window, I started to wonder how long before some of the schools and trusts out there began to behave in a similar way.

We all know of those schools and trusts that offer incentives for 'the right candidate' when advertising for certain positions. Those TLRs, R&R allowances and 'Market forces payments' (as I saw advertised by one school) that are designed to attract people to a school that feels like it might otherwise struggle to appoint a candidate of suitable quality. Personally I have never approached someone working at a school to try and convince them to join my school, but I know of instances where others have been 'tapped-up' (to use a football parlance) to see if they are interested in changing school, or what it would take to get them interested.  A lot of this mirrors the extra wages, bonus structure, guarantees of first-team football or other approaches that teams will use for players they want to recruit. 

In education, this behaviour is still quite limited. The standard practice is still to advertise a job, see who applies, and then choose the best of those who do. Whilst an increasing number of these adverts will offer incentives, it is not yet standard practice to go out and actively recruit certain people. I do wonder though whether this will change. I wonder when it will become more standard for schools, like football teams, to scout particular talent from other schools or ITT institutes and approach them with offers rather than just encourage them to apply. I wonder if or when schools will actively building teams of particular people, rather than choosing from those that show interest. I wonder if it will ever come to the point where schools will 'compensate' other schools if they allow their staff to move before the end of a notice period (in fact I know this has happened at least once already) and a big thing I wonder is whether it would be a bad thing?

Typically I am not in favour of market forces being at work in education. I am generally of the feeling that all of us in education should collaborate with each other rather than compete, share our time and resources freely rather than compete with each other or try and make money from each other. This is why my TES shop is and always will be free for any of my resources. However we all know about the difficulties that schools in certain urban areas, coastal areas or more removed areas have in recruiting. Some (but admittedly not all) of these schools and trusts will have more money than average - they will have larger numbers of disadvantaged students or will be federated and making savings from economies of scale. I wonder if it would be a bad thing for them to be able to scout teachers (the TES talent bank might allow for this in part at least but it would probably need more performance data included, as well as more teachers signed up). I wonder if it would be a bad thing for them to be allowed to buy teachers out of their contract with an appropriate compensation package for their schools. 

Mainly I wonder if this is at some point inevitable. 

If increased autonomy for schools and trusts is to become the norm, including the ability to set pay and conditions as they see fit, I think we must at some point get to very highly effective teams being paid beyond the main and upper pay scales that most state schools still adhere to in some form. It wouldn't surprise me if, at some point where I am still teaching, we see schools or trusts begin to approach high performing schools to enquire after their staff, or having whole departments being offered improved terms in order to stay. I can see no good reason why the TES talent bank, or even a government website couldn't hold performance data for teachers exam classes alongside details of CPD and other contextual information, allowing schools to try and tempt the highest performing staff. Some people reading this will no doubt be saying to themselves things like 'yes but it is easier to get good performance data with higher attaining classes' or 'yes but just because you do well with classes in one school doesn't mean you will be to do it in a different environment'. I accept this completely, but then how many stories do we hear of footballers moving clubs and failing to perform as expected, or reach the heights that they seemed capable of (for those of you that don't follow football, it is a lot). In this sort of system there are always risks that the change will impact performance, and one would assume that schools and trusts would be aware of this. 

Maybe I am completely crazy, and have this completely wrong. Maybe there are good reasons I am not seeing why this model wouldn't work in schools, or would be wildly unpopular. But given the amount that schools spend each year on advertising for positions, particularly when they have to go 'into the market' 2 or 3 times for the same position, I wonder if this money wouldn't be better used as incentives or compensation to secure the workforce they need. I wonder if schools arranging 'transfers' or even 'loans' might not be better than a school being left in difficulty because one of their staff decides to hand their notice in on the last allowed day and they have no time to secure a replacement so have to rely on expensive agency staff of (possibly) dubious quality. More and more I wonder how long it will be before some schools and trusts decide to try this approach in earnest, and if it ends up being successful, how long before it becomes the standard practice for schools.

How important is it to teach maths for understanding?

Over the summer I have been reflecting on the 9-1 GCSE papers that were sat back in June. In particular I was remembering hearing about and talking to people back in 2013 and 2014 when we were getting the first details of the 'new' GCSE and one of the key aims being to try and make sure pupils are understanding maths rather than just being taught certain procedures in order to solve certain questions. One of the questions that struck me as evidence of this appeared in the AQA Non-calculator papers:
Those people who have taught GCSE Maths for a while will be familiar with the more typical question about averages from grouped tables from the previous specification, which looks a little more like this:
Both of these questions are worth 4 marks but the way those 4 marks are earned is very different. In the second question from the older spec, the marks are given for:
(1) identifying the midpoints of each class as the average time taken for each person in the group, 
(2) multiplying the midpoint of each class by the frequency of each class to work out an estimate for the total time taken for each class of people, 
(3) adding these estimates to give an estimate of the overall time taken for all 40 people, then
(4) dividing the estimate of the total time taken by 40 to give an estimate of the mean time taken.

The point here is that many teachers, and I include myself in this during my early career days, would approach the teaching of this concept without any of the explanation I have given above, simplifying the whole thing to a straightforward procedure:
(1) Write down the midpoints of each class
(2) Multiply the midpoint by the frequency
(3) Add your answers together
(4) Divide by the total of the frequencies.

The point is that in the past, maths teachers could get away with this because every question that asked about mean and grouped data was structured in precisely this way, even if the values were different. There was therefore no incentive for teachers (beyond their own intrinsic wish to teach pupils good maths rather than teach them to pass exams) to teach any semblance of understanding for this concept - provided pupils can remember the four steps they can answer the question on this topic.

Contrast that with the first question for the'new' 9-1 GCSE. Provided a pupil is not going to simply guess at the correct answers (which I admit is a possibility), then what should be clear is that the level of understanding of mean and range required to answer the question is significantly greater than the second question. To confidently answer the new question a pupil needs to have quite an understanding of how mean and range link to distribution, what can be inferred about the distribution from the grouped table, and also what mean and range measure about a distribution. If a pupil were to carry out the steps above (getting a correct answer of 34 minutes) they might even come to the mistaken conclusion that the only place the mean could be would be in the 20-40 class. This might be enough to perhaps score 1 mark, but certainly not the 4 marks it would have secured in the past.

This, for me, illustrates the importance of teaching maths for understanding rather than just as a set of procedures. Of course it would be quite right (in my opinion) to say that it was always important to try and teach maths for understanding, and that as teachers we should always be trying to develop understanding in our pupils. What is nice now though is that what many people see as the ultimate 'end-goal' of our teaching, the pupil securing a good GCSE grade, doesn't allow for recourse to procedural only approaches. There have been many critics of the new 9-1 GCSE, and for certain things I have been amongst the most vocal of them, but I will consider it all worth it if it means that teachers have to move away from teaching 'maths' as answering questions by following a sequence of steps and begin to try and teach maths 'these are the concepts, skills and knowledge you need and these are how they relate to each other'.

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:



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!