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!