Wednesday, 18 April 2018

In defence of the Chartered College

Before I start properly I would like to make a couple of disclaimers:

1) I am writing a personal piece here. I am not writing on behalf of the Chartered College and nothing I write can be considered to be representative of the Chartered College or its members.

2) I am a Council member of the Chartered College.

My association with the Chartered College goes back to some way. I was an respondent to the original Princes Trust consultation back in about 2012 and attended the launch in London (I still have the document from that day somewhere with my response in it). After that I lost touch with it - I missed the crowd-funding situation (or I would have donated) and I didn't really see a lot about it until I saw the advert for new trustees later in 2016. I honestly thought the idea had petered out; it was ambitious at the time with the GTC so freshly in people's heads to say the least. I was so pleased when I saw it was still going that I immediately volunteered and was lucky enough to be one of the 7 selected to join the council. I have been a council member for a year and a half and I have felt privileged every moment.

Recently the Chartered College has been the subject of criticism. Some of this is not new, but two of the more recent ones I feel are unfounded and as a supporter of the Chartered College I wanted to redress this.

The first criticism has surrounded the review process of articles for the Chartered College journal, Impact - specifically an article written by Greg Ashman. Now I want to take this opportunity to publicly state I have absolutely nothing but the highest respect for Greg. I read his blog whenever I can (the man is so prolific it can be impossible to catch everything!) and I think he speaks a huge amount of sense on a lot of issues. I read the article in question and I thought there were some interesting points raised, and I think it could be a useful read to spark debate. The article can be found on Greg's blog here for the interested reader. I do respect the opinion of the reviewers and the people at the College that put the journal together, and they decided that the tone of the article wasn't in keeping with the style of the journal. There have however been some implications of a bias, and that the article was blocked because people found the content of the article unpalatable. I was not involved in the review process in anyway, and nor am I part of the committee that oversees the journal on behalf of the council, but I know those people. I have worked with them, talked to them, shared hopes and dreams for the Chartered College with them, and I can categorically state that there is no bias in them. These people are teachers, as I am. They spend their weeks in the classroom or in schools working to educate young people. They are people like Natalie Scott (@nataliehscott), Jemma Watson (@thefinelytuned ) and Aimee Tinkler (@aimeetinkler). There is no agenda behind us and no wish to exclude from debate, and it makes me unhappy to think that people might believe that of us.

Some will believe I am being naive at this point, and if so fine, but I would rather believe that these people are doing what they think are the right things and for the right reasons than look for hidden motives behind these decisions.

The second recent criticism has been around the launch of the Fellowship. There have been a couple of comments about this. The first is the idea that affiliates and not just members can nominate and be nominated for Fellowship. I personally don't take issue with this. I think we have to recognise that, while the contribution of teachers to our schools is immeasurable, it is not the only contribution. There are many people I can think of that have added immense value to my career as an educator but that no longer work in the classroom. I would personally nominate Professors Anne Watson and John Mason immediately; their contribution to maths education has been incalculable and they are two of the most passionate and dedicated people I have ever met when it comes to trying to ensure that our young people develop a deep understanding and appreciation of mathematics. They are just two of about 10 people I could immediately think of that would be worthy of the honour.

Another criticism around Fellowship is the idea that "As a Founding Fellow, you will be encouraged to support members and other teachers to engage with and promote the use of evidence." People out there are seeing this as asking people to pay to do extra work. This saddens me greatly. As a teacher with 12 years experience I see it as my moral and professional duty to support other teachers and support them in finding approaches to teaching that can help them overcome problems they may be facing. Granted I don't need to pay the Fellowship fee to do this, but I believe that if we are going to approach the standards of other professions then the Fellowship as a mark of someone who has the experience, skills and knowledge of our profession is an important milestone. For me, those people who deserve to be fellows would see the opportunities to support other members as a positive. As teachers I believe we must be an outward facing group if we are to solve the problems that currently plague our daily work.

A further criticism of Fellowship are some of the additional benefits, such as the Fellows roundtable and the reservation of certain Council positions as Fellows only. I am not going to go into the details but I can tell you as a member of the Constitutional Committee that we considered this very carefully. In the end it was concluded that to be effective in these roles one would need to meet the criteria for Fellowship, and that it would need someone committed enough to the ideals and ethos of the Chartered College that they would seek out that sort of role. It is hard to imagine how a teacher of less experience or less passion could effectively lead the body that holds the Royal Charter for our profession. That isn't to say that the views of newer teachers aren't important, in fact they are crucial to ensuring that the College is representative of the views of all its members. This is why there are council member positions open to all. But those positions that are required to drive the College forward, to ensure that the governance of this body is robust, are those that need to be filled by people that have the experience, knowledge and skills developed over time to fulfill that need.

The final criticism I will address is the funding. It is no secret that the College are currently funded by the government, to the tune of £5 million. This naturally raises questions about independence - how can a body be funded by government be able to criticise policy and practice? People may not believe this but I can honestly say that it doesn't really feature in our discussions. Hard as it may be to believe, but for all their flaws the DfE recognise that a strong, well-connected and informed teaching profession is a positive thing. There are people there that care as passionately about young people as we teachers do. As for criticism of policy and practice, we have always been clear that we want to work with and not against. For the profession to lead the way we cannot be a group that shouts and screams when things happen that we don't like. That is not to say there is not a time for anger, nor a time for action, but always first should be an effort to reach out, to work with, to influence by being a calm and well-informed voice.

I believe that teachers deserve to be a well-connected and authoritative body when it comes to the practice and standards of those who choose the profession. I think this is essential to us being universally considered a profession. I don't think we are there yet, but I believe that signs are hopeful. Above all, I believe that the Chartered College has the potential to be a force for good in this regard. Mistakes have and will happen along the way; we are human and not immune to them. And that is really the key - we are human. We are teachers, like many others, and we are trying. I hope that others will see that and lend us their support. And I hope that those who doubt us will either join and become part of the influence, or at least give us the benefit of that doubt while we keep trying.

Friday, 6 April 2018

The Power of Interpretations

Probably the most celebrated mathematics of recent years (1994 seems a long time ago now, but it really isn't) is probably Andrew Wiles proof of Fermat's Last Theorem. In fact, it seems likely that anyone will be able to say something similar to this until such time as the Riemann Hypothesis is proved. What people forget is that Wiles' proof was not actually directly of Fermat's Last Theorem. Wiles' proof was concerning two (at the time) separate branches of mathematics, elliptic curves and modular forms. The real wonder in Wiles' proof is that it suddenly showed that elliptic curves, which had been worked on in one way or another since the time of the Roman Empire were linked to modular forms, an invention only a hundred or so years old at the time. The fact that this also proved Fermat's Last Theorem was almost incidental (although it clearly was the focus of the media coverage) - for mathematicians the power was that Wiles' proof had allowed them to reinterpret problems in one area of maths as an equivalent problem in the connected area. Problems that had gone unsolved in modular forms could be reinterpreted as a problem in elliptic curves, and all the understanding from that area could be brought to bear on solving it.

I first read this story about 7 or 8 years ago in Simon Singh's book, "Fermat's Last Theorem", and it resonated with one I had heard at university as a undergrad*. Both spoke of a key message in mathematics; that often a different interpretation of a problem can drastically change our ability to solve it. Indeed, Wiles' own proof was only possible because he viewed the problem in a different way to others, and was therefore able to approach it in a way that no one else had tried to (it is worth noting it still took him 8 years!).

I think this message is as important in the mathematics classroom as in the realms of professional mathematics. Many of the different problems we ask pupils to work with in the classroom have different ways of thinking about the underlying mathematics, and if those pupils don't have flexibility in their ways of thinking about the underlying mathematics then clearly they are going to find this difficult. Consider, for example, the three questions below:

1) Cans of pop are sold in packs of 6. If Russell buys 24 cans altogether, how many packs does he buy?

2) Russell walks 24 miles in 6 hours. What is Russell's average speed?

3) The two rectangles shown below are similar. What is the scale factor of the enlargement from the smaller rectangle to the larger rectangle?

All of these problems are solved using the calculation 24 ÷ 6 but importantly the way we interpret the division is different in all three cases. The first question is a classic interpretation of division as grouping; take a dividend and create groups of a certain size (the divisor). In this case take 24 and create groups of 6. The answer to the division (the quotient) is the number of groups created. A key part of this is that the starting number and the number in each group are unit consistent - we start with 24 cans and create groups of 6 cans. However, the quotient is not in the same unit; the answer is 4 packs, not 4 cans.

Contrast this with the second question. In the second question we are not creating groups of 6 miles. Rather we are sharing the 24 miles into 6 shares, with each share being a single hour. The answer now is 4 miles - the dividend and the quotient have the same unit, but the dividend does not. This is clearly a different way of thinking about the division.

The third question is again different from the other two. This time we are not creating groups of 6, or sharing 24 into 6 shares. Rather we are comparing the 6 and the 24 to see how many times bigger 24 is than 6. There are two main types of comparison like this: additive comparison, where we examine how much bigger one value is than the other (this is an interpretation of subtraction often called the "difference" between two numbers) and multiplicative comparison (which is the type used in question 3) where the values are compared to see how many times bigger one value is than the other. This has many different names, probably the most common being a "scale factor".

My point in writing this blog is that if pupils don't understand these different interpretations of division, then they won't be able to answer all of these questions. I regularly encounter (and I am sure other teachers do as well) pupils that do not realise that the way to answer the third question is by using a division. Their concept of division is deficient and although they are capable of answering 24 ÷ 6, they are unable to see that this calculation is the one required. 

Of course, I am not just talking about division here; many concepts in mathematics have multiple interpretations and the more ways of thinking about a concept pupils have the more likely they are to recognise a concept in a certain problem, and therefore be able to work towards solving it.

In my upcoming book (provisionally titled "Representations in Mathematics") I talk about different interpretations of some fundamental mathematical concepts, and how we can use representations to highlight and unite the underlying structures behind them. However, whether you choose to read my book or not I would urge all maths teachers out there to consider the following when approaching teaching new concepts:
  • What are the different ways of thinking about this concept and how it can work?
  • How can I make explicit to pupils the different interpretations of this concept and ensure they are comfortable with each?
  • When solving different problems, how can I ensure my pupils understand the particular interpretation this problem type requires? Note I am not saying that this needs to be explicitly taught to pupils, teachers may choose to ask pupils to explore a number of the possible interpretations to try and make sense of them.
  • Do my pupils have the necessary understanding of this concept to interpret this problem type in the correct way?
Making the different ways of thinking about a concept clear to pupils is going to be increasingly important, not just for pupils' examinations where they will be faced with unfamiliar contexts, but also for beyond school. The evidence is now increasingly convincing that skills such as problem solving and critical thinking are domain specific skills; they depend on strong knowledge of the area to which these skills are expected to be applied. But importantly these skills still need to be developed, and for me part of this strong subject knowledge is a flexibility in interpreting the central concepts of that subject.

* The story in question also appears in the introduction to my upcoming book, so I cannot reproduce it here. I am hoping that now I have finished writing the book that I can get back to blogging more regularly again!

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.