US Flags

I’ve been on my travels recently in the United States. I was staying with some friends in Arizona and they had this flag on the wall.  It’s an old flag with 48 stars neatly arranged in 6 x 8 matrix. It was from the days before Hawaii and Alaska joined the Union, sometime before 1959.


Now, I’m sure that all American school children know most of this stuff already, but it got me thinking about the US flag and specifically the arrangement of the stars and it gave me an idea for a lesson.

So, I know that there are now 50 stars for the 50 states of the USA.  But how are they arranged?  And how could we describe this mathematically?  I might start by showing students this zoomed-in portion to give them a hint.

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It is, of course a 5×4 matrix inside a 6×5 matrix.

Was there a 49? Yes, briefly when Alaska joined in 1959. But what did 49 look like? Clearly it’s a square number, but they decided that to keep the aspect ratio looking more like a rectangle they offset the rows like this.


There were actually lots of different versions of the stars and stripes detailed on this wikipedia page:

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It might be better to start with a simpler example from the early days when there were only 13 states (2×2 + 3×3). What other numbers can be represented as a sum of two square numbers in this way?


Is there a better way of doing this one? (It’s 36 stars, maybe 6×4 & 4×3)


You might want to give this table as a handout (or here as a pdf).  Get students to work in pairs to come up with their own designs and compare those to the ones actually used.

And then, what if we go beyond 50? I don’t want to think too much about the political implications of such of move and who might end up as the 51st state, so let’s stick to this as a mathematical exercise!





Quality of Verbal Instruction

I’ve been practicing Bikram Yoga now for about 9 months.  It has utterly transformed my default mental state but that’s the subject of another post for another site somewhere.  During my classes, I sometimes compare my wonderful yoga teachers’ teaching practice to my own and ponder what I can learn from them.  Bikram Yoga is a specific sequence of 26 postures and 2 breathing exercise and is always exactly the same day in, day out, the world over.  The instruction is all done verbally with very little demonstration.  The classes can be 40 or more people and it generally relies on having the more experienced people at the front so those behind can watch and see what they are supposed to be doing.  But is also relies on the the very clear and very well-practiced verbal instructions of the teacher.

Last week I attended the nRich Teacher Inspiration Day  which included a session on Origami led by Fran Watson, the Primary PD Coordinator.  Origami is clearly Fran’s “thing” and she has probably run these sessions many times. One of the things we built, she managed to get us all to do with purely verbal instructions, not even moving her hands – very impressive!

So hopefully you can see where I am going with this… Both my Yoga teacher and my Origami teacher have a well-honed set of verbal instructions.  In neither case, however, is it a script.  If it was they could just load the YouTube video and press play.  They show care for their students by observing closely what they are doing and adapt the script accordingly. It is sometimes more difficult in a classroom to observe your students’ progress but not impossible. Mini Whiteboards are a big help here, whether students hold them up or just work on them on their desks. But also simply looking in your students books at their work. This is something that I used to find really difficult but I know has got a lot better with practice, maybe because I have a better idea of what I am looking for.

So here are the specific techniques that I have seen employed that I think are helpful.

  • Say it once, very carefully and very clearly. These are the pieces of script, the stock phrases that have been practiced and refined through experience.
  • Develop a tone of voice that says “this is the important bit, listen actively now
  • If you need to repeat, use exactly the same words. This makes it clear that it is a “missed instruction” rather than a “new instruction”.  This takes confidence and a self-belief that you have chosen the best form of words in the first place. But that comes with practice and experience.
  • Keep your instructions to the whole class until you make it clear that we are changing “mode” and students are now free to do their own thing. If you need to add clarifying instructions (in response to what you are seeing) you say this to the whole class even though it might be in response to what you see an individual doing. Chances are everyone will benefit.

I’m not saying that there are times in every lesson where this very didactic approach is the right one to use. But when there is a specific task you want students to do, maybe to set them up for the next piece of learning, this is a useful tool in the toolkit. It’s one that I know I’ll be practicing for years to come.

Fundamentals of Area

Area is a topic where students are likely to have remembered a process (you multiply the lengths to get the area) but may not have a secure understanding of what area actually is.  It is worth spending time discussing what we mean by area.  I would prefer my students memorise “Area is the amount of space inside a 2D shape” rather that “Area is length times width”.

Counting the Squares is a good place to start even with students who feel like they know how to calculate area.   The concept of 1cm² as a quantity, an amount of something that bigger shapes have more of is powerful.  Estimation can be useful.  Physical pieces of 1cm² card and a series of rectangles, some with grids, some without is worth doing.

Reinforcing units is important.  Reminding students of the commutative law and rewriting 2cm x 3cm as 2 x 3 x cm x cm can help explain why we write cm² and link it back to algebra.  You could even play with 3cm x 2m.  Does make sense to say that this area would be 6 cmm?

A key step in the process is moving from:

Screen Shot 2016-07-14 at 09.57.31.png…without losing sight of the underlying concept that the rectangle is made up of 12 little squares, each one 1cm².

Moving on to rectilinear shapes, I like this question as a way of challenging thinking.

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The first misconception here would be multiplying 3 x 4 to get 12.  But of course the correct answer is “we don’t have enough information”, an answer which shows depth of understanding and one which I would like my students to be presented with more often.

Maths Hubs: a step in right direction

First I must disclose an interest. Next year, two days of my time will be funded by the Maths Hub programme, albeit a very small slither of the £41m announced today to continue the roll out of the Maths Hub Teaching for Mastery programme in English Primary Schools.  £41m over 4 years is not a lot of money in the context of the 17,000 primary schools in England. I’ve done the Maths – it’s roughly £2400 per school, or £600 p.a. But at this stage, it’s not about trying to reach every school. In fact, the approach that is being taken here is the right one, i.e. schools need to opt in to this programme and at the moment nothing is being forced upon them. Although doesn’t that remind you of some other educational programme which started life in the same guise?

I believe that most teachers of Mathematics would instinctively agree with the core principles of a mastery approach. That is covering concepts over a slightly longer period of time, using the time to go deeper, to consolidate understanding of concepts through carefully constructed practice and problem solving.  To bring the whole class forward together and to provide challenge for higher attainers through depth rather than moving them on to the next topic. The former content-focussed assessment system of levels rewarded pupils and their teachers for ticking as many items as possible on a long list of topics but I reckon most teachers had a strong hunch that this is not how children develop their mathematical abilities.

So it’s a good start in the right direction. But why do the DfE insist on referring to it as the “South Asian method”?

There is nothing wrong with looking beyond our own shores to identify specific areas of good practice that we believe will work well in our country. I just don’t see why it needs to be the cornerstone of an education policy and a justification for a change in approach. There are several things that bother me about the international comparison. Maybe it starts with a sense of national pride. Why must we send 70-odd teachers to Shanghai to learn how to teach Maths when we have so much good practice in our classrooms in the UK, but so little capacity to share it?  How much time does a typical teacher spend learning from best practice in their own school, let alone best practice in surrounding schools. We need more non-contact time broadly across schools to enable this sort of CPD. Secondly is the belief that we can transplant a system of teaching and it will improve results, given the obvious differences in social context and key system differences such as significantly less contact time. And finally this all stems originally from PISA results which are a fairly imprecise measure of the success of an education system and have received calls from a significant number of academics to be stopped.

I was pleased and somewhat intrigued to see that NCTEM’s press release on the same day about the same £41m didn’t once mention Asia.  This is the organisation that is charged with co-ordinating the Maths Hub programme.  I applaud them for focussing on what it actually involves and how the money will be spent rather than trying to justify it by the explaining that we will be copying someone else. It’s a bit like the French admitting that their pop music isn’t very good and so downloading a bunch of English songs then translating them.

I don’t believe the “South Asian method” is in any way helpful as a label, but let’s just see the international comparison as something that has provided an impetus to get some investment in Maths education in the UK. At its heart, that’s what we are talking about here. Providing funding for teachers to leave their classrooms and undertake some CPD and collaborative planing. It is being rolled out gradually from one year to the next.  It can be criticised for not touching enough teachers, but if we want to do it right, we should be learning from one year to the next and taking a long-term view.  Momentum needs to build and short-term success should be judged by the number of schools wanting to join each year.  Once we can demonstrate what can be achieved through reducing contact time and investing in CPD, who knows, maybe some government in a few years time will see that this sort of investment is a no-brainer in order to improve outcomes.

I was at an NCTEM event on Monday launching the programme to the second Cohort of 140 Primary teachers.  There was an explicit focus on Year 1. Obviously the most logical place to start but are politicians going to take such a long-term view and wait 10 years until those students are sitting their PISA tests? Quite rightly they will be looking for continued improvements in educational outcomes before then. I just hope that this programme can be given enough time to build momentum before another change in direction or assessment.  The 4-year commitment is a good start.



nRich games to practice key skills

There is a range of “take turns” dice games like this one on nRich.

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I see this activity as a way of practicing key skills (in this case column addition) but in a much deeper way than repeated practice as you are working backwards to achieve a result.  I would think of this as adding a significant degree of difficulty over simply doing a page of sums; it would be something you would only chose to do once the basic process of column addition is reasonably secure. However, because it is engaging (i.e. competitive) students are more likely to stick with it.

To avoid the need to provide students with 10-dice an interactive dice could be displayed on the board. nRich have this handy spinners tool here and there are lots of other options online.

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All students would then have the same set of numbers and it would be a competition to get closest to 1000.  You might need to note down the numbers as you go to prevent cheating!

Again, of course, the value in the activity comes from the discussion, both in pairs and in whole class.  I like the idea of saying that the target is 1000 but actually rewarding good discussion and reasoning rather than just closest answer (i.e 5 points for a new “noticing”, 5 points for the closest answer)

Another way is to determine all the required random numbers at the outset and the students can fill in the grid with full knowledge of their options.  Less luck involved and so probably less fun!

I would do one game whole-class, where students are playing individually. Then a second game where students are to work together in pairs competing against another pair so they can compare strategies once they have a degree of familiarilty with the problem and get some good discussion going on strategy.

Origami for the end of term

Origami is one of those things that I think I would love to spend more time exploring but rarely do.  I have used the Origami Player with my students  (it works really well as an App within Chrome), which gives excellent visual instructions on making things. The timings have been well thought out and it gives a little timer prompt so you know how long you have got to do that fold before looking up at the screen for the next one.  It’s been an end of term, easy lesson. Nothing wrong with that.

The first session on the nRich Teacher Inspiration Day  last week where we looked at some of the activities here got me thinking about how I could make it a slightly more meaningful learning experience.  

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Still not highly mathematical, but at least it gets students working together and struggling with something.  To build resilience in our students they need see the struggle as a positive and not something to be avoided at all costs.  It was a bit of a metaphor for all learning. A discussion that can be had with students when reflecting on this task might be along the lines of:

  • Did you need help from someone at some point? (yes, good)
  • Did you help someone else at some point? (yes, good)
  • Did you struggle at some point? (yes, good)
  • Did you give up? (hopefully no, good)
  • Did you achieve something you didn’t think you could do before?

This type of discussion can be a powerful motivator and more useful than vague questions like “Did you enjoy it?” or “Did you have fun?”

There are lots of Origami ideas on this page of nRich’s new Wild Maths site. I really like the idea of modular Origami, i.e. each person makes a module and then they come together to create something beautiful.  I have an end-of-term cross curricular session with Year 8 to plan. My Origami paper arrived this morning.  You can use A4 paper and fold down the corner to make a square, but proper origami paper is really lovely and this pack was only £8 for 500 sheets.  It looks great so time to start practising!

I will post an update on this as planning progresses and share some pictures of the final event.  In the meantime, if anyone has an ideas for how to make Origami more mathematical (without spoiling the enjoyment of it!) then let me know.