Thursday, 8 March 2018

What is x? Explaining the meaning of algebraic notation.

Recently I have been teaching linear equation solving with Year 7. We have explored various interpretations; function machines and their inverses, balancing, inverse operations, blank box etc. But one thing I did before any of this was explicitly define the role that x plays.

It is well known that pupils can struggle with algebra when it is introduced immediately as an abstract concept. Much has been made about the use of representations to support the teaching of algebra - indeed I dedicate a good proportion of my upcoming book to exploring different ways of representing algebraic expressions and equations, and how these representations can aid pupils in understanding how algebra is manipulated. However, one thing the representations struggle with is communicating the nature of the letter itself.

I once saw an excellent use of Geogebra to create a dynamic visual representation of completing the square that would allow the value of x to be varied, the squares shrinking and growing really hammered home the idea of x being a variable in the expression. However the static representations that we often use cannot convey the same meaning. So before I started working with them I decided to introduce some of the basic interpretations of letters in mathematics; in particular viewing them as variables, as parameters and unknowns.

To be fair, I didn't stress parameter too much, just a vague definition about them having a particular meaning for a value that doesn't change, like b for the base of a triangle or A for the area of a particular shape. But we did talk a lot about the difference between an unknown and a variable, and then we revisited the ideas as we went through the different lessons. At the beginning of each lesson I would ask the class what role x was playing in an equation, and that meant I went through the entire topic without once being asked what x is - which could be a first for the introductory teaching of algebra!

It seems like it really helped the pupils to understand something about the different roles that the letters can play, and in particular the role that they played in the equations we were working with. I would certainly recommend teachers introducing algebra by giving clarity over the roles that letters play in mathematics so that pupils have a sense of what they are working with before they are asked to manipulate them.

Saturday, 13 January 2018

Teaching Probability- some thoughts...

Recently I have been teaching probability and I have had some thoughts about why some pupils may struggle to remember to write probabilities as fractions (or decimals/percentages in certain contexts) and then to apply ideas about probability outside of normal contexts.

The first thought I had was that we often don't distinguish between what probability is, and how we communicate that probability to other people. If we take some typical activities from early probability lessons we might well see questions or examples like these:

Now there is nothing necessarily wrong with these questions, but for me they speak to us communicating a probability, rather than actually what probability is. The first question makes this clearer than the second, and I think the second question set should be re-phrased as "Write down the probability of picking..." in order to make clearer that this is us communicating the probability rather than implying that this is what probability actually is.

Recently in teaching probability to Year 7, I started with this:
This is by no means perfect, but what I wanted my class to understand is that probability is about our attempts to predict the future. The sentence written at the top helped with a discussion about making predictions in both simple and complex contexts - in combination with the diagram at the bottom we talked about predicting the flipping of a coin, the rolling of a biased die, winning the lottery and even predicting the weather.

As part of this section of the lesson we then actually made some predictions. First of all trying to predict the result of me flipping a coin (pupils get really invested in getting this right!), then spinning the spinner below:
I designed a spreadsheet to simulate the spinning of this spinner, and not surprisingly those that predicted 1 a lot were right more often they not! We talked a lot about the idea of equally likely outcomes at this stage, and the idea that each section was equally likely, but because so many of them are ones this makes one the most likely outcomes. We did a similar thing with a biased dice roll - in particular I wanted to stress that although we knew all of the possible outcomes, because they weren't all equally likely we couldn't just make predictions based on what should happen in theory, we needed to gather some data about what had happened before. Again I used a spreadsheet to simulate my biased die rolling 250 times, and then we made predictions based off of the data we had gathered. 

It was only at the stage that I felt the idea of predicting future outcomes was firmly linked in pupils minds with the idea of probability that I started to look at communicating that probability with them. First we just used words like "likely", "unlikely", "impossible" etc - and one of the big things I wanted to do at this stage was reinforce that prediction element again. I purposefully phrased my questions and responses to match the language I had used in the first slide; for example, I asked for each of the bags below "If I take a counter from bag ..., what is the probability that I will take a black counter?" 

Reflecting on this now I think I might change this language to "If I take a counter from bag ..., TELL ME the probability that I will take a black counter." I feel this would make the point that this is communication much more clearly.

The second thought I had is related to this, and came in the next lesson on the topic. In this lesson we were writing probabilities as fractions, and I had recently used this slide with a Year 10 Foundation tier GCSE group:
I had been considering why this was still necessary in Year 10, and came to the rather sympathetic conclusion that our teaching of probability was at least partly to blame. I reasoned that these pupils saw the act of counting the outcomes as the probability rather than just the way we communicate it. With this viewpoint either of the bottom two become perfectly acceptable - the probability is just the comparison of two counted values. I resolved when working with Year 7 to stress the difference between how we assign a value to a probability, and what that means in terms of predicting the future. I started their lesson with the use of this spinner:
and after some useful discussion around 5 still being unlikely even though it was more likely then any other number, we looked at assigning value. I reminded pupils about the two questions from last time that would allow us to discuss probabilities in theory - "Do I know all the possible outcomes?", "Are they all equally likely?" and we decided that the answer to both was yes so we could use a theoretical approach. I chose this spinner because I wanted multiple outcomes where the fractions would simplify, so I could seek to divorce the act of the counting with the probability as a prediction of the future. There were lots of sentences like "The probability of a 5 equals four-twelfths, and that means 5 should happen about one-third of the time." The big learning point here was that the counting of the number of fives and comparing that to the number of outcomes is an approach for assigning a value to the probability, but crucially it isn't what the probability is. We are going to do some more work around this next week but I feel quietly confident that this divorcing of the approaches used to measure probability with the concept that probability is how we go about predicting future events and how we communicate the surety (or otherwise) of our predictions will pay dividends as our study of this topic continues.

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.