Thursday 26 October 2023

Maps and Maths

Wow, it has been a while since I have written anything other than posts to link sessions to, but I had something in my head today which I just had to write about.

Recently Ofsted have produced a report, "Coordinating mathematical success" where they outline what their inspections have found about what maths teachers are doing well, and not so well. The Association of Teachers of Mathematics (of which I am a member just to make sure any biases are public) have written a response to this which I think is very balanced - supportive in some areas and rightly critical in other. There is one section of this response to highlight:

"It is of course possible to create a journey of ‘small steps’ with the hope that these steps become connected, but we challenge the prominence of these small steps in the report. When undertaking a journey, the primary focus shouldn’t be on each step but rather on an awareness of the landscape and the multiple possibilities within it. "

I have seen these arguments before and also seen people before liken this to a map - with some advocating providing a map of the journey through mathematics, with strict adherence to the planned route and a focus on each "step" of the journey and where it takes us (the "small steps" indicated above), whilst others advocate for just getting out there and exploring, creating our own sense of where things are in the landscape, and looking out for interesting landmarks to go and see. 

The reason this came to mind is that today I was travelling to the lovely High School Leckhampton in Cheltenham to work with the GLOW Maths Hub LLME community on using Cuisenaire rods. As it was a nice day I decided to walk the (little over half an hour) journey from Cheltenham Spa train station to the school. On my way, I used Google Maps walking feature in the way I often do, which is to get it to show me the journey, but not to start the navigation so that I can retain an overview of the whole trip and my progress on it. Now, of course, there is plenty of analogy already there in terms of making sure that pupils retain a sense of the journey and where they are within it, but it wasn't this that prompted my reflection. What did prompt my reflection were three incidents along the journey:

1) I took a wrong turn at a place where the map wasn't complete. I was supposed to follow a footpath that appeared to be the only way I could go on the map. However, when I actually arrived there was a smaller path (that turned out to be the path I should be taking) and a larger path (which I assumed would be the one I would take). I walked for about a minute down this larger path before glancing at my phone and realising that I wasn't following the path laid out for me, and that the path that I was following didn't appear to be marked on the map. Could I have got from where I actually was back on course? Perhaps, but not being familiar with the area I didn't know this (or know how) and so I back tracked the way I had come to get back on the correct path. This cost me a small amount of time, but fortunately I had factored extra time into my journey.

2) A little later I came out of the correct path onto a road on which I was supposed to turn left and follow around an arc to continue my journey when I noticed that, across the road, a new path seemed to continue on in the direction I was headed, potentially cutting out the arc. This path was also absent from the map. In that moment I faced a choice - follow the route as programmed or divert off on this path which I was fairly sure was going in the right direction, and possibly quicker than the route shown on the map, However, I couldn't guarantee that this was the case, that the path wouldn't veer off and take me in an unwanted direction. I made the choice that I suspect quite a few people would make which was to forgo the new path and continue on the route laid out. When I got around the bend a little later, I noticed that there was a path exiting onto the road again, and noted that this was likely the path I had noticed before.

3) On my way back from the school to the station, I didn't really need my map. I had it on, as a safety net, and my have glanced at it once or twice, but I didn't need it to tell me all of the twists and turns I needed to take - I knew the way I had come and was quite comfortable back-tracking along that route. Except I didn't really back-track along that route. I was able to adapt it to make it more comfortable for me. This time I did take the path I alluded to in point 2, and it was indeed quicker than following the road around. I had also noticed on my journey to the school that the small path I had come down in point 1, was connected back to the road I had walked down before joining it at a point further along the road, and that I could probably save myself an uncomfortable and a little muddy walk by cutting out onto the road earlier when I was coming back. So I did that (and this is one of the points where I did glance at the map again to confirm that this would be an acceptable alternative.

Why did this prompt me to think about maths teaching/curriculum (as if the reader couldn't guess!)? Well it suggested to me a few things:

1) The map was ultimately important. Because I was under a time pressure going in both directions (one to get to the session on time, and one to get to my return train on time), I likely wouldn't have got to where I needed to be when I needed to be without the map. Given that school level maths has its own time pressures built in, I think the idea of having the map and a planned journey within it is important, which translates directly to a well-sequenced curriculum that takes learners on a journey.

2) The usefulness of the map, and sticking more rigidly to the planned route on the way to the school was, in part, down to the fact that it allowed me see places where, on my return journey, I could perhaps be a little more flexible. As I mentioned, I didn't use the step by step navigation because I like to retain the overview but I wonder if I had, whether I would have noticed those same places where I could adapt the route as I did in point 3 above. I wonder about whether we are doing a disservice to pupils in situations where we take them through very small steps, particularly for those pupils (like me) who would get frustrated if they couldn't see where that step fell in the journey, and also if this deprives all pupils of the opportunity to notice places where they could take an alternative. However, this is linked to the next reflection/suggestion...

3) It was only because I had followed the route provided on the way there that I felt comfortable making adaptations when I followed the route in reverse. I don't think, it is the reverse thing here that is important - I think it was familiarity with the journey between the two. I feel like I could walk a very similar route again from the station to the school without the map, or only having to reference it occasionally rather than follow it completely (at least whilst it is fresh in my memory). But it was only the familiarity I gained that has given me this confidence. Now I am sure that I could eventually get the same confidence if I took the time to stroll around the space between the two venues, but that time wasn't available to me given the aforementioned time pressures. I also think that the map is useful to help with my explorations were I to have the time to explore, which leads to the next point...

4) The route I took wasn't the only route I could have taken. When I first put my destination into Google Maps, it offered me three possible routes to complete that journey. Being able to look at the map suggested several others (mainly slight deviations or possible shortcuts based on the three main routes). On the route I followed I saw several landmarks - a church, a Texaco garage, street names etc. that were useful checks that I had for when I made the return trip and would be useful were I to make the trip again. But there were lots of things I didn't see because of the route I chose. Some of those things may have been more interesting than the church or the garage. I suspect had I taken a more westerly route my views would have been better (judging by the glimpses of the countryside I saw in that direction whilst following the route I did, and my subsequent scrolling of Google Maps to see what might lie in that direction). 

So what do I take away from all of this?

1) A planned sequence is ultimately useful, but having the step by step navigation would perhaps draw attention away from the journey itself, frustrate those that need to see how their progress fits into the journey as a whole, and might mean those following the journey miss some things that could build their confidence in being adaptable when encountering the same or similar content in the future - so part of our job is making sure that they don't miss these things and that they have the opportunity to recognise the important aspects of the landscape.

2) No route or map is perfect, no matter how expert the map maker/route planner, so it is important to build in time to allow for the occasional wrong turn, even with those who are generally quite good at following the journey - a new journey is still a new journey.

3) For some at least, the confidence to explore and adapt will be enhanced by the security of "the map", the pre-existing knowledge of the landscape that can be accessed, and knowing that they are never too far away from a familiar space.

4) We must build in time for that exploration and adaptation once the map is laid out - if we are only ever moving onto a new journey with every step then, for most, the only thing they will feel like they can do is follow the route exactly.

5) When we plan a journey through a curriculum for pupils there will always be different competing considerations - time, which route allows the "best" experiences, the attitudes of the learners towards possible wrong turns or deviations etc. and we need to evaluate our chosen journey to look at what we sacrifice as much as what we gain.

Of course, the big difference in this scenario is the pupils can't really "see" the map until they have experienced it. I was able to call up a map that someone else had created, but in learning mathematics the "map" exists inside everybody's head, I have my map, you have yours. Learners create their own maps when we take them on a journey through a mathematical idea, and whilst we can communicate something about what that journey looks like, it only becomes real once experienced - I can't simply transfer my "map" into my pupils' heads. I think this reinforces point 4 above about building in time for exploration in areas that pupils have already travelled, revisiting ideas not just to see if they can remember the exact same journey as before, but to give them chances to take shortcuts, amend the route, or explore the consequences of going off on a new path.

My final thought is that I wonder how much of this is specific to maths? It seems like a lot of it might apply to many subject areas but, not being overly familiar with many of their maps, I cannot say for sure.

Sunday 10 September 2023

ResearchEd National conference 2023

 As promised, here is a link to my session from the ResearchEd National conference 2023:

My thanks to everyone who attended, I hope you found it useful!

Saturday 25 March 2023

Sunday 5 September 2021

ResearchEd National Conference

 Yesterday I had the great opportunity to present the ResearchEd National Conference at Harris Chobham Academy in Stratford, London. I would like to thank all those people who attended my talk. The link below will give access to the slides:!AiVD5E48l4WsibZ1ZmiVoGgR2AulPw?e=Dhxz4Q

Thursday 1 October 2020

The Decimal point and Place Value

 My daily twitter browsing showed me the recurring argument about "moving the decimal point" in relation to multiplication or division by 10, 100 , 1000 etc:

The thread is worth a read (you can get to it if you click the image). I read it, and some of the quote tweets around it, and it got me thinking about what I think are two possible misconceptions around the decimal point and place value that seem to exist in the minds of teachers of maths. I am sure some will disagree with one or both of these, and may think that these are misconceptions in my mind instead. But anyway, here goes...

Misconception 1: That the decimal point is a fixed immutable point between the ones and the tenths

There are a number of people saying in that thread that the decimal point cannot move, that it must stay between the ones and the tenths. This for me seems false. The job of the decimal point is to separate or mark the transition between whole values of our unit, and values that represent part of that unit. Of course, in most cases our unit of counting would be ones and the decimal point therefore marks the transition from ones to part of one. But that isn't always the case. Consider for example:

2.6 million

In this, the value to the left of the decimal point does not represent two ones, it represents two millions. The unit we have chosen to count in is millions, and so the decimal point separates the whole millions from the parts of a million. This has obvious parallels to something most mathematicians will be familiar with converting between standard form and our "ordinary" decimal number system:

320000 = 3.2 × 105

In converting to standard form we are literally changing our counting unit from the ones to which column has the highest value in the number we are working with. This means that the decimal point does move, it moves to separate whole values of our new counting unit (in the above example 105) from parts of this counting unit. In converting from standard form we do the reverse; we change our counting unit from the largest valued column back to the ones column, and the decimal point moves concurrently. One could even make the argument that converting units of measure could be viewed in the same way:

3.25 metres = 325 cm

Have we multiplied by 100 to go from left to right here? The physical distance hasn't changed? Would it make more sense to consider that the decimal point has moved due to our change of unit choice from metres to centimetres, and so what was separating whole metres from part metres, is now separating whole centimetres from part centimetres.

Misconception 2: That the decimal point moves when we multiply or divide by a power of 10

Despite what I have written above, and some compelling arguments within the thread, I still come down on the side that it is wrong to teach pupils that the decimal point moves when you multiply or divide by a power of 10. This is not just because of its tendency to be used a trick for teaching without understanding, but more because conceptually it actually doesn't fit with what is happening when we multiply or divide. If we accept that the decimal point moves when we decide to change our unit (perhaps a big if for some), then the decimal point cannot move when we multiply or divide by a power of 10. Consider:

3.2 × 100 = 320

There is no change of counting unit in this situation. In all three numbers, the counting unit is ones, and the decimal point separates the ones for the parts of one. What has changed is the physical size of the numbers that each of the digits represents in the number 3.2. The 3 has become 100 times bigger to become 300, and the 0.2 has become 100 times bigger to become 20. This is synonymous with moving the digits up the place value columns, and definitely not the decimal point down - or even translating the column headings down. Although I can see the argument that says "we can consider 3.2 × 100 as having 3.2 hundreds, and what we are doing is converting that back to a number of ones" I can't make that fit in my own mind with the importance of making sure pupils recognise and appreciate the multiplicative relationships between the place value columns. Even if our learners are completely secure with this, I can't see why we would then use a unitising approach to multiplication to model multiplicative calculations with different powers of 10 - except maybe if we had reached the point where pupils were so secure in this that we were opening them up to another way of making sense of such calculations and perhaps attempting to highlight that moving the decimal point is akin to the equivalent multiplication or division.

In summary, the decimal point definitely can move, but probably shouldn't if we are teaching multiplication or division by powers of 10 unless we are taking a unitising approach to this sort of calculation.

Saturday 15 August 2020

What is a valid mock?

 So Ofqual have released (on a Saturday) the criteria for deciding what a "valid mock" is for the basis of appealing a pupil's A-Level, AS-Level or GCSE grades. If you haven't yet seen them, the criteria can be found here:

One of the criteria in particular caught my eye:

I remember in 2015, when the new GCSE in Maths started, I made myself very familiar with the approach that exam boards take to set their boundaries. It is quite involved to say the least, using national data on prior attainment of the cohort, previous performances of other cohorts, all sorts of data. I doubt any individual school or trust could match those standard - I remember the laughable outcomes when PiXL first tried and the flack they got for the grade boundaries they came up with. My blog on our process at the time is still my most read blog:

Which of course presents a problem. For a mock exam to be graded in line with the exam board's examination standard, I can't see any alternative but to have used the grade boundaries provided by the board, on the same paper(s) provided by the board. Anything else simply doesn't (in reality) live up to examination standards. No exam board would arbitrarily reduce boundaries to reflect that pupils haven't completed a course, which is a common practice in schools. No exam board would simply use the average of a set of grade boundaries, or apply one years grade boundaries to a different set of papers. Any school or Trust who has done this can't sign the form to say their exam was graded in line with the examination standard provided by their exam board. Nor can those schools that use papers they have created themselves, or those that use practice papers that don't have grade boundaries provided.

In my own department we use a practice paper that comes from the exam board, but does not come with boundaries. We set initial boundaries for that years ago, before the first ever exams on the new GCSE and so before their were any past papers in existence. We sat the exam at a similar time to some other local schools, pooled our results together, and applied a similar process that exam boards were applying for their first set of new GCSE papers. We then compared the results that pupils eventually got with their mock exam results, and adjusted the boundaries for the following year. We used the same papers for mocks every year since, continuously refining the grade boundaries as we got more real results to compare to. This is about the most robust system any school could implement to grading, but even this falls short of examination standards. 

Of course, the head of centre only needs to sign a form saying "I confirm this mock exam meets the criteria". No one is going to check. No one is going to ask how they know the boundaries meet the criteria. Unless this is going to be used as a "get-out" clause to deny large numbers of appeals, then I doubt this line will end up meaning much. But it should, and that is the shame of it all.

Wednesday 10 June 2020

Maths as an option?

This tweet caused quite a discussion recently on Twitter.
My answer to this was quite to the point.
However, I was asked to elaborate as to why, and I thought it would make more sense to write it in a blog rather than as a series of tweets that might (would) turn into an overlong thread.
My reasons for opposing Hannah's proposition fall broadly into three camps:

1) There is so much more to maths than what you can do with it (its functionality).
2) The central place that mathematical knowledge serves in our society and history.
3) The inherent biases that exist within education and the minds of maths teachers.

Regular readers of my blog/listeners to my various presentations/interviews will know that I am a big and loud (in both senses) proponent of the view of maths as a collection of connected ideas, and that our job as maths teachers is to support pupils in reaching an understanding of these ideas and how they are connected. This doesn't necessarily mean a "guide at the side" approach to teaching - indeed a wise sage that knows exactly what things to say, questions to ask, examples and representations to show is often exactly what it needed to support in gaining wisdom. What this does mean is a need for teachers to really have a depth of knowledge about each mathematical idea, and its place in the larger concept.

Let us take a couple of examples, in fact the ones mentioned in the thread, namely constructions and histograms. Both are touted as superfluous to requirements in a "modern" mathematical education. However, constructions are intricately and inexorably tied to wider knowledge and understanding of 2-D shapes and their properties. This is often not made explicit because, as Dani Quinn (@danicquinn on Twitter) pointed out, constructions is very often approached as a series of "do this, do that" steps with no effort made to tie it to the larger concept. Getting specific, consider trying to create the perpendicular bisector of a line segment. But rather than thinking about the steps required, think instead about which 2-D shapes have the property that at least one of the diagonals perpendicularly bisects another. The list you might have come up with is:
  1. Square
  2. Rhombus
  3. Kite
  4. Arrowhead (if you class that separately to a kite).
Now think again about the standard image for a perpendicular bisector:
Ask a mathematician: “Where should we live?” | The Aperiodical
Can you see which shape we have created in order to create the perpendicular bisector? It is of course the rhombus. Do we make it clear to pupils that this is the case. Or that alternatives exist? Our list suggested 4 different shapes we might draw that could lead to the same result. Can you see how we would create the others using similar techniques? The images would look something like this:
Except....there is something strange about that last can't be done! Not precisely anyway; you might luck out and get both sets of two arcs to cross on a line that is at 45o to the line connecting the dots, but no way to make sure. So although there are four shapes that could lead to that perpendicular bisector, only three of them are constructable. Can you see why? What is different about the definition of a square compared to the others? Yes, it is because a square is defined both in terms of its sides and angles, whereas rhombi, and kites are defined just in terms of their sides. And although an arrowhead must have a reflex angle, it isn't particular about the size of that reflex angle, whereas the angles in a square must be 90o.

When discussions like this are taking place around these mathematical ideas, then constructions becomes a rich, deep vein in which to explore all manner of aspects of properties of 2-d shapes. Similar discussions can be had around angle bisection, and construction of other shapes. A nice link to make is between the perpendicular and angle bisectors. I will leave these three images as a prompt and say no more...

So what about histograms then? No one uses them in real life! No one creates unequal groups! Whilst this may be true, my thoughts are simply:
  1. They should!
  2. They are still rich links between these and other areas of maths and the social sciences to exploit.
There are plenty of situations where classes naturally arise that aren't equal in size. Many companies wage structures have jobs in salary bands that aren't equal in width. Boxing weight classes aren't equal in width. Months of the year aren't all equally sized. Any situation where frequency is being collected and analysed in these sorts of situations should not be represented in a simple frequency diagram, and kids need to know why. What is it about frequency diagrams that doesn't work if classes are unequal? What can we do if we are in a situation where it makes sense for classes to be unequal, without arbitrarily forcing them to be equal. Take for example this frequency table of salary bands (made up!) for a company:
If you were to draw this as a standard frequency diagram, it might look something like this:

There are several problems with this. The greatest problem is that the 30 to 40 group appears far larger than the 22-25 group, where in fact it only represents one more person. This is of course because the area of the 30-40 bar is far larger than the area of the 22-25 bar, and this is almost entirely down to its width. A similar problem exists in comparing the 18-20 and 20-22 bar with the 22-25 bar. It appears that both of the first two bars would fit almost perfectly inside the third, where in fact there are 4 more people in the first two bars combined compared to the third - a quarter again of the frequency it is supposed to represent. All of these problems arise because we are immediately drawn to the area of the bar as representative of its size rather than its height, particular when bars are different widths.

If we now see the same data plotted using frequency density:

What we have now is a fundamentally different distribution, with the earlier wages much more pronounced. Looking at the first diagram one might thing that there is an abundance of people being paid in the range of 25-40k. No one looking at the second diagram would make the same claim.

This of course comes back to the idea of unit measure. Unit measure is a fundamental concept in geometry and measures. What histograms allow for is a reexamination of unit measure in a new context, and the skilled teacher can use this as a vehicle to re-highlight this concept and show its importance even outside the realms of measuring quantities. The links to the idea of density as being per unit are also important - I always make links to population density and materials density when teaching histograms, and what they have in common around the idea of defining a unit to allow proper comparison. Histograms are also a great place to look at units other than "1"; in the example above the unit is per £1000. Properly taught, histograms allow us to impress upon students that a unit can be whatever we want it to be, as long as it is the same for all applicable circumstances. One of my fondest memories of my teaching pre-lockdown was with my Year 7 class, and defining the unit of a "Mattock" as my height, and then having other pupils estimate how many "Mattocks" tall they were (of course they are all shorter than me so it was a nice way of looking at estimating decimals as well). The joy and understanding that came from that activity were palpable, and if I am teaching those kids in Year 10/11 I will reference it back again when we come to histograms.

Every aspect of the curriculum up to 16 (maybe even 18!) has these connections between and within concepts. Part of the job of a teacher (in my opinion) is to understand these links and use them to create that connected learning experience for every pupil. No pupil should be denied the opportunity to see where and how these ideas resurface and develop - this is at the heart of what it means to teach maths.

This leads me nicely onto my second point - to me maths is universal (in two senses). Mathematics is at the very heart of our society on several levels. Advances in society are inherently tied to advances in mathematics. Our ever increasing technological development is down to advances in mathematics. Mathematics is the language humanity uses to describe the physical processes of our universe. As we develop mathematically, we develop a better language to describe these processes, and this language is what allows us to manipulate these processes to our own betterment. The recent advances in areas such as quantum computing are due in no small part to an increasing sophistication by which some people are able to use mathematics to explore what is possible.

Not only in technology does maths wield its power. The more recent advent and development of statistics has given us a window into human life that simply wasn't possible a few hundred years ago (along with our technology allowing us to connect and share in ways unthought of until recently). We know more about human life as it is now than we ever have done at any point in history, and mathematics goes to the very core of what allows us these insights. Developments in all areas of mathematics are what allows society to develop, and although mathematics has developed to such a point that fewer and fewer people are now capable of advancing the field further, this doesn't mean that we should only be sharing that story with those that might be capable of taking up that mantle. Mathematics has at different points played a central role in our history, our culture and our development as a species. I hope very much that I am not alive in the generation that decides that this birthright should not be passed on to every human being possible, even if relatively few will make use of it in any meaningful sense.

Mathematics is also universal in a different way, in that it is something that literally every human with a functioning mind can advance at. Some people are born without the use of arms and legs that might proclude them from many opportunities that others take for granted. Some are born with medical conditions so severe that they will never walk, never run, never build, never take apart or dissect. Mathematics, as primarily a discipline of the mind, is open to all. Every human with the capacity to think can advance their own understanding of and ability within mathematics, and as such I strongly believe that mathematics has to stay a subject open to all.

This leads me to my third point, the inherent biases of our education system and of maths teachers. Let us take as fact that mathematical attainment correlates well with future earnings (for those that can't take that as fact, this article and this report should be enough to get started with). Let us also imagine that maths, beyond basic functional numeracy, was optional for pupil, post-14 say. Who do we think would be opting for it? And more importantly, who do we think would be "discouraged" from opting for it because schools/teachers are worried about how their success rates in maths might look? I would suggest that those from disadvantaged backgrounds, those from certain BAME backgrounds, those with SEN and other learning difficulties, they would make up a large proportion of those pupils that didn't "opt" to continue their mathematical education. They would be the ones where teachers are having conversations like "maybe functional numeracy would be better for them to support their future aspirations", where of course one of the big things education should be trying to achieve is to broaden kids horizons so that they can continue to re-examine those aspirations in light of what they learn is possible. I can easily imagine the situation where a white middle class family is pushing their child through a mathematical education, supporting with tutors and being ever more demanding of their child's school, where huge swathes of disadvantaged pupils who might actually do well from a mathematical education don't continue it. We all know those pupils that despite everything have come good in maths, and we also know those pupils who haven't. But to deny that opportunity from anyone goes against everything I stand for as a teacher and a human being. Maths as an option is a pretty sure fire way to ensure that the gap widens between the haves and have nots, between the mostly white middle class and those of BAME backgrounds. In a time where, as white people, our relationship with and treatment of other races is under increased scrutiny (and rightly so), the changing of maths to an option would be a huge backwards step in making sure that everyone has the best possible opportunities throughout the rest of their lives. Of course, pupils from disadvantaged and some BAME backgrounds already typically underperform compared to their white middle class peers, and we really do need to look hard at that and take every step to deal with it quickly, but I can't see that part of the solution would be to create a system where large numbers of these pupils can simply be pushed out of it altogether.

The final thing I would say on this matter is this: as a sector we really need to look at what we class as "success" in a mathematical education. If we are talking about anything up to 30% of pupils who reach the age of 14 and are still functionally innumerate, enough that we need a whole separate qualification for them because GCSE isn't suitable, then that is a real problem. I know at this point people will start saying "but the exam system is set up like that" and I understand that (although there is flexibility in it if large numbers of pupils start performing significantly better), but regardless of the grade, I am talking about what these kids actually know and understand. If we are saying that 30% of pupils cant reach half way through a Foundation GCSE paper, after 11 years of maths eduction (bearing in mind that much of the same content also appears on a SATs paper that Year 6s are given) and that it is wrong to even try to support them through that, then I think the problem is not the qualification so much as the previous 11 years. The fact that only 1 in 5 pupils can get over half way through a higher paper is a damning indictment of our education system, and offering an "easier" qualification will not solve that problem. I think our attention needs to be directed much more to making sure that no pupil reaches 14 without already being functionally numerate rather than what we might do for the final two years when they do. On this I have no answers, as my results as a teacher and head of department are no better or worse than many others out there, but as a head of department that is what I want to spend my time thinking about, not which kids I should allow to continue studying maths, and which are going to be denied entry to all of the opportunities that studying maths allows.