My #VisibleMaths tour!

Over the Leicestershire half term I went on quite a tour around Britain, talking about some of the ideas from my book #VisibleMaths. Starting in Peterborough on the first Saturday for the always fantastic Complete Mathematics Conference (#mathsconf), I was speaking about threading an idea through the curriculum in a coherent manner - in this case the idea of factorisation.

From there I had a couple of days at home before travelling up to North Yorkshire on Tuesday ready for working with teachers from the Esk Valley Alliance. We had a good time looking at addition, subtraction and division using representations. I particularly enjoyed this trip because my fiancee Rowan came with me and we had a lovely night away and a bit of time in York together.

A day off on Thursday was then the lead in to my really busy time. A short hop across to Birmingham on Friday lunch time for a session with maths teachers from Niksham High School, which was followed by jumping straight on a train to get down to Farnham ready for the excellent ResearchEd Surrey. Many people have commented that this ResearchEd was one of the best local versions for planning and organisation, and I must say I agree! I did a morning session there focusing on the idea of addition and how, with just two ways of making sense of what it means to add, all additions from Y1 to Y12 work in the same way (which was also mentioned in my latest TES blog). The session went down very well, and it was great to then spend the day listening to others talking about education.

Having arrived back in Leicestershire at 11pm on Saturday night, one week after my tour of England started, I then had the thrilling experience of my first ever trip to Scotland on Sunday and Monday. I must admit the sights walking out of Edinburgh Waverley station were simply breathtaking - I am not generally one to take photos of my surroundings, but even I had to capture some of those amazing visages.

The tour finished with my biggest ever audience - a little over 200 Scottish teachers of maths as part of the South East Improvement Collaborative joint INSET day. Having looked at the importance of making sense in different ways (using one of my favourite games!) we looked at making sense of addition and subtraction, and useful models for both of these operations.

A lot of people have suggested they would find a copy of the presentations helpful, so here they are:

Complete Maths Conference Peterborough (Saturday 12th October)

Esk Valley Alliance (Wednesday 16th October)

Niksham High School (Friday 18th October)

ResearchEd Surrey (Saturday 19th October)

SEIC INSET day (Monday 21st October)

Teaching Exact Trig values

***Warning - untested idea alert***

With the advent of all GCSE pupils needing to know exact trig values for 30, 45 and 60 as well as 0 and 90, a lot of people have been searching for a way to make these accessible for Foundation pupils. This came to the fore again on Wednesday prompted by this tweet from Drew Foster:

Now this is interesting for me. I have typically taught this using the two standard triangles:

(Image from Don Steward - Median Blog)

However, I have been know to resort to some of the "tricks" contained in the original tweet, particularly for Foundation pupils or pupils closer to the exams. I know that understanding on its own is not enough to lead to memory, and so I justified this in terms of helping remember. This was particularly needed because, even with kids who understood how the triangles worked, is that they often forgot to draw the triangles, or what the triangles looked like. Instead, they would supplement my teaching by endlessly drilling themselves on the values, usually found in table form.

Now I don't necessarily have a huge problem with this, as it will allow them to access most standard questions. Of course, these days, the concern is around the non-standard questions. When one of those pops up, pupils will likely struggle if they don't have the necessary flexible knowledge with working out exact trig values from what they already understand about trigonometry.

I was mulling this over, and a possible approach came to me. Because it literally only came to me on Wednesday, I haven't had time to trial it yet, but I will be teaching Year 10 set 2 next year, and I think I have it straight enough in my head to try it with them (hopefully this blog will help with that, and I welcome feedback).

The idea centres around motivating the drawing of these triangles by linking them back to a unit circle type definition of the two major trig functions. I will start with a type of inquiry prompt based on this triangle:

And the prompt will be, "What angle is needed, with a hypotenuse of 1, to make the vertical side equal to ½?"

I expect the pupils' first wrong suggestion will be 45˚. At least I hope it will be. Because I want to use it to motivate looking at 45˚ later on. For now, when/if it comes up, I would want to deal with this through isosceles triangles, and the triangle inequality - something along the lines of "If θ is 45˚, then the other angle will also be 45˚, which means both would have to be ½. What does that mean for the triangle?" How much of that I will tell pupils, and how much I will prompt/look for pupils to recognise I will decide in the moment.

Having dispensed with this, the problem becomes a bit more interesting. We could do a bit of calculator trialling, I haven't made up my mind yet. Whether we do or not though, and whether pupils find the result or not, I want to move to justification. The justification for this would come from reflecting the triangle in the horizontal side, giving this picture:

This of course leads directly to θ being 30˚ , as the triangle is an equilateral triangle.

Now of course this is very similar to the way that others would introduce the same idea. I think the difference is the focus of the approach. Previously I would have introduced this by first introducing the triangle, and using it to prompt find sine, cosine and tangent of 60˚  and 30˚ . The question of course is "why these angles?". Just because they are the ones that arise from an equilateral triangle? I think this is difficult for pupils because right-angled triangles and equilateral triangles are not well associated prior to this, particularly in the area of trigonometry. Instead, this seems a more natural question to ask - what angle makes the opposite half of the hypotenuse? It means not having to remember to draw an equilateral triangle of side 2, and then generating something useful,but rather actually drawing what you want to find, and then deducing it.

From here it would seem natural to ask about the length of the horizontal side (knowing it can't also be ½), and what that implies for the trigonometry.

But what about the other angles of 45˚ , 0˚ , and 90˚ ? Well I feel like this can lead naturally to those as well. As I alluded to earlier, I suspect the issue of 45˚ , will have come up already, and so it would be there for us to go back to: "What if the hypotenuse stayed 1, but the angle became 45˚ ?"

Which can be approached in the usual way using a bit of Pythagoras to show that sin 45˚  = cos 45˚  = 1/√2, and that tan 45˚  = 1 (because it will be something divided by itself).

As for 0˚  and 90˚ , I honestly can't see a way of arriving at them naturally, unless we first want to explore 15˚  and 75˚ . This I can see being justifiable, to continue the pattern that 30˚ , 45˚ , 60˚  would suggest. Unfortunately, I can't see a nice way of arriving at these without at least knowing some stuff about trig for non-right triangles. However,  I think the same triangle with a hypotenuse of 1 can at least be used to give an intuitive understanding of the exact trig values for 0˚  and 90˚ .

We can ask questions like "What is going to happen to the opposite/adjacent sides as the angle gets smaller?" Pupils should be able to see that the adjacent side will get close to the same length as the hypotenuse (i.e. 1) whilst the opposite will get very small. This of course is a pre-cursor to the formal idea of limits, and this can later become the limit as θ → 0˚ . Similarly, we can then switch it around and ask "What will happen as the angle gets bigger?", the limit as θ → 90˚ . Again, pupils should be able to see that the opposite will happen, and that the opposite will get close to the hypotenuse, but the adjacent will get very small.

Like I say, I haven't tried his yet, so if anyone wants to make suggestions for how I could make this better, do this in a way to maximise success etc then please do give me a shout.

East Midlands Maths Hub Joint Conference

Hi all! Lot of conferences recently! Last week I was lucky enough to present at the joint East Midlands South, West and East Maths Hub conference. I did a session on using representations (surprise, surprise!).

People at the conference suggested they would like to the slides to that session. They can be downloaded here.

Hope it helps!

NW3 Maths Hub Conference

This Wednesday I was lucky enough to deliver the closing keynote to the Wigan NW3 Hub Conference at Haydock Racecourse. I absolutely loved the chance to mirror the development of an operation through a counters game, before exploring the importance of making sense of mathematics through the power of multiple interpretations of a concept.

The slides from my session are here (including the correct formula - well I hope so anyway!).

Thanks to Lindsay Porter for inviting me to speak (and giving my a lift back to the station) and to Jen for picking me up from Bryn station.

SEND Conference from LIME/Maths Hubs

On Monday 24th June I had the privilege of presenting the closing session at an excellent event hosted at the Ashton-on-Mersey school. This was primarily for teachers of pupils with SEND. The main focus was on the use of manipulatives to support mathematical understanding - a personal favourite topic of mine.

Many delegates suggested they would find the slides useful, so I have made them available for download here.

I should give a shout out and offer thanks to Louise Needham for asking me to speak at this fantastic event.

Putting the "Theory" into Cognitive Load Theory

These days we are hearing a lot about Cognitive Load Theory. But what does this actually mean? Well to understand this it is worth reminding ourselves about what it means to be a theory in science.

A lot of people see the idea of a "theory" as something that is somewhat uncertain. This is often the use in everyday language - if someone has a "theory" about something, it often means they have no more than a vaguely plausible explanation for it.

A scientific theory is different though (or at least a good one is). A good scientific theory should broadly aim to do two things:

1) Explain observed phenomena

2) Predict the outcomes of other observed phenomena

This is what Cognitive Load Theory tries to do. It tries to explain phenomena about how/when the brain forms memories that have been observed, and predicts what might happens in certain circumstances. For example, it has been observed that people find it difficult to remember content if they are reading text at the same time as someone is talking. Typically people in this situation will not be able to answer questions about either the text or the content of the speech. CLT explains this by suggesting that the brain processes text in the same way as speech (in a way, you "hear" the words in your head) and that the brain only has one "channel" for processing auditory input. Trying to process two inputs through your "phonological loop" results in cognitive overload.

So what happens when a prediction goes wrong? What happens when CLT predicts a different outcome? Well the same as what happens when any other scientific theory predicts something incorrectly - either the theory is modified to include the new observation, or if it can't be modified sufficiently then it is deemed incorrect. However, incorrect theories can still be useful. A prime example of this is Newton's theory of gravitation.

Newton's theory of gravitation is wrong. It definitely doesn't adequately explain how gravity works in all cases. This was known in the 1800s, as Newton's theory of gravitation was slightly out in predicting the correct orbit of the planet Mercury. Einstein's general relativity is a better model. Its predictions are more accurate, and more applicable. However, in most cases, Newton's theory is still used. Why? Because it is much simpler. The equations that accompany Einstein's general relativity are absurdly complicated. If you are talking about black holes, or getting close to massive bodies in the universe, they are essential. But for most situations, the equations associated with Newton's theory do just fine. They predict to a high level of accuracy the gravitational forces between bodies. Newton's theory was used to put man on the moon.

So what does this mean? Well if we apply the same sort of ideas to Cognitive Load Theory, what it means is that CLT may well make incorrect predictions, particularly in extreme cases, but that doesn't necessarily mean that the other predictions it makes are automatically wrong, or that they can't be useful. But it also means that if you are going to try and apply the ideas within Cognitive Load Theory then it might be useful to remember the following:

1) CLT may well not a complete theory of cognition, and it may well produce incorrect predictions. This doesn't make it worthless.

2) If you are applying CLT, make sure you read information about the studies that supported aspects of the theory. This will give you a greater appreciation for how useful/accurate its predictions might be for your context.

3) Cognitive Load Theory may well support in your pupils converting more of what you teach into long term memory, but that also means you have to make sure that the memories you are getting your pupils to form are the right memories. CLT can't tell you how to teach the content of your subject so that the connections between topics become apparent - that is part of the knowledge/skill (contentious!) of the teacher.

Research Ed Rugby - Mathematics Teaching for Mastery Using Rosenshine's Principles and Cognitive Science

This Saturday I attended the truly excellent ResearchEd Rugby. Despite all sorts of problems with trains from London to Rugby, Jude Hunton (organiser) got a fantastic line up of speakers together for sessions on Research, Leadership, Maths, English, Science among others.

As part of the maths strand I was speaking in the afternoon. The thesis of my talk was that the NCETM approaches outlined in their Teaching for Mastery program, Rosenshine's Principles of Instruction, and some of the effects proposed by Cognitive Load Theory are the same ideas, discussed using different language.

It is worth reminding ourselves about how these three came into being. The NCETM ideas of Teaching for Mastery came from looking at "high performing jurisdictions" and their practice, as well as the research that underpins their approaches, and suggested what might have the most impact in mathematics education in this country.

Barak Rosenshine derived his Principles of Instruction by examining individual high performing teachers, and the common practice that they share. This of course was not specific to mathematics teaching.

Cognitive Load Theory is a theory for how the brain forms memory, and things that support forming of memory, based on experimental data. Of course, the point of a theory is that it explains observed phenomena, and predicts the outcomes of future phenomena. So CTL aims to explain things that have been observed about memory formation, and predict what might help memory formation in the future.

The three "competing" theories can be shown using these three images:

I contend that many of the ideas from these three sources actually significantly overlap, and in some cases are indistinguishable. Take this example from my presentation:

On the left hand side is a picture of the expression x² + 5x + 6, which then shows how this picture can be rearranged into a rectangle, which shows the factorisation. This can be physically shown using concrete or virtual manipulatives. Then there is an expression on the right hand side which is intended for the learner to factorise (given access to the concrete version of the manipulative).

The title for the slide was:

The proposition here is that this could be considered Modelling, if thinking about Rosenshine's principles, or it could be considered Representation if thinking about NCETM Teaching for Mastery  approaches, or could be considered the use of the Worked Example Effect if thinking about Cognitive Load Theory.

I highlighted several other examples throughout the session:

This activity is one that can be used to support developing Fluency, which is also an example of independent practice and also uses the Goal Free Effect.

This activity prompts Mathematical Thinking, it can be seen as using the Expertise Reversal Effect, and can be used to provide Scaffolding for Difficult Tasks. Note; if you are going to use this task, an important point is that learners should aim to make the minimum change possible from the starting shape in the middle.

The point of course is that these are not competing ideas at all. There is something to be gained from all of them, particularly where they actually say the same thing.

*Hat tips to the Learning Scientists and Oliver Caviglioli for the posters, Jonathan Hall and his website Mathsbot.com for the virtual manipulatives, Open Middle for the open box problem, and Professor John Mason for the Area/Perimeter activity.

*The full presentation can be downloaded by clicking this link.

Reasoning, Problem Solving, Interpretation and Fluency

So as part of my holiday I was listening to Craig Barton's most recent podcast with US educator Michael Pershan. In one part of the podcast Craig talks about his belief that reasoning required fluency before it could be conducted, but that this belief was challenged in a session with the eminent Mike Askew at the joint ATM/MA conference held over Easter. From Craigs description Mike posed a problem similar to this one:

45 × 36 = 45 × 35 + 35           True or False? 

The point was that this sort of question is one that can be considered, and the correct result arrived at, even if pupils are not capable of carrying out the calculation 45 × 36 correctly. Clearly this is true, and inspired this quote from Michael:

"Reasoning happens in the absence of fluency"

Craig reflects on this at the end of the podcast, and talks about how if you "hit a wall" with a problem, if you don't have the required tools in your toolkit (or don't recognise you do) that is when you need to reason. Craig also suggests (although he also admits that it doesn't feel right) that a possible implication of this is that teachers may hold pupils back from achieving fluency in order to allow opportunities for reasoning. Craig goes on to describe the idea of "teaching the fluency first" and then the reasoning becomes part of strategy selection - pose problems that pupils have the toolkit to solve, but pupils need to consider which strategies are appropriate. Craig offers his own SSDD problems as an example. 

It struck me whilst listening to this, that it may well be of benefit to make a careful distinction between what me mean when we talk about reasoning and problem solving. Consider this from the National Curriculum document, that says pupils will reason mathematically by:

"reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language"

There isn't really reference in this to strategy selection, or working through a problem here. This is highlighted much more in aim 3:

"can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions"

This seems much more about what Craig is talking about with not having the necessary toolkit, and therefore not learning a lot (or simply not being able to persevere) whilst seeking solutions. This I think is where I agree with Craig - pupils need to have a secure knowledge of the mathematics underpinning the problem. The aim makes that clear - pupils solve problems by applying their mathematics. If they don't have the mathematics, they can't apply it. The purpose here is as Craig describes at the end of the podcast, to identify the mathematics required, to link the problem type to other problems they have previously encountered etc. The purpose is not actually to learn new mathematics at all. Ideally, it should not be obvious what mathematics will be needed to solve the problem, either from the context of the lesson in which the problem is set, or from the content of the question at all. What I think Craig talks about as "reasoning" about the problem is actually interpretation and strategy selection. There comes a point where we want pupils to be able to look at a problem where the mathematics required isn't explicit, and interpret the problem successfully to identify the mathematics required, before applying that mathematics through an appropriate strategy. One of my favourite areas for these is speed problems. Speed problems are a great source of both routine and non-routine problems. Some problems involving speed are solved by multiplication. Others are solved by division. And then even when a pupil can identify whether they are going to use multiplication or division, they need to choose a strategy for the calculation - for the best way to carry out the division or multiplication.

So where does this leave reasoning in comparison then? Well I actually did a session on this at what must have been mathsconf10 (by the dates of the materials) and it included this slide:

For me, the top question involves reasoning because the mathematics required is clear. That question is about highest common factor and lowest common multiple, and in particular their relationship to two numbers. The lower question is a true "Problem Solving" question (if you ignore the 'real life' context) in that it isn't clear what mathematics is going to be required. One could argue is that the last place you might want this question is in a lesson about LCM (particularly if not surrounded by others that aren't). By contrast, the reasoning question doesn't try and obscure the maths required, but at the same token doesn't just require application of a method. The reasoning question is trying to prompt a deeper consideration of the knowledge that the pupil is developing. To make them think about that knowledge in a way that, perhaps, they hadn't considered before. In the podcast Michael is therefore right, reasoning does happen in the absence of fluency, but that is because reasoning is an important part of developing fluency. To become fluent, one has to be able to approach questions like the first one above, and use knowledge flexibly to develop the chain of logic required. This of course means we can't talk about teaching to fluency before we offer opportunities for reasoning. For me, opportunities to reason are an integral part of the journey to fluency.

My High Five

Ben Gordon on Twitter suggested this idea, and started off with his brilliant "Teach Innovate Reflect" blog. I am going to use the same format that Ben suggested, which can be seen below:

Format:

• What you learnt
• What was the source
• Implications on your practice

I really love the idea of educators sharing a few key ideas from their professional learning. Sometimes it can seem really hard just to filter all of the great practice that comes across your timeline. This seems to me to be a great way of giving others a bitesize of the things that have made the biggest impact on them. So here are my 5 main things I have learnt this year:

1) That there are three levels of curriculum planning

What I learnt: The difference between the intended, implemented and enacted curriculum.

Source: Bauersfeld 1979, via Dylan Wiliam's paper on principled curriculum design.

Implications on my practice: I guess this really helped frame many of our conversations in department, particularly around developing material for our new scheme. What was really useful is having a language to discuss the separation between what we plan, what we teach and what kids learn. I wrote more about it in my article for TES here.

2) About the use of blocked practice

What I learnt: That blocked practice is useful in early concept development

Source: I honestly cannot remember

Implications on my practice: First of all, I am sure I am not the only person that reads/hears things, and has near perfect recall of what they read/heard but cannot at all remember where? But anyway, we have heard a lot in the last year or two about the importance of interleaved practice. What emerged from somewhere recently (although like I say I cannot say where) is the idea that when pupils are first developing a new idea, or applying it in a new way, that they need time to just focus on that idea. Later on, interleaved practice is really helpful in promoting far transfer, but this does need to wait otherwise the new concept/application becomes confused. In terms of my practice, this basically means we do a lot of back and forth stuff in class around the main idea, and it is only when the more independent work starts that interleaved practice comes in.

3) The best ways to use examples

What I learnt: How much backwards fading of examples improves learning.

Source: It wasn't the first place I read/heard it (again can't remember where that was), but most recently on Craig Barton's podcast with Mark McCourt.

Implications on my practice: Quite obvious really, I plan example sets rather than two or three full examples, and within those sets I gradually reduce the support until pupils can work through a few from beginning to end.

4) Some important things about Direct Instruction programmes

What I learnt: That DI programmes need to be under 15 pupils and that over 80% of each hour revisiting rather than teaching new material.

Source: Chloe Sanders (@Chloe_jo) courtesy of the @DITrainingHub at St Martins Catholic School, Stoke Golding.

Implication on my practice: We are trialling a small group intervention using Connecting Maths Concepts with a possible view to expand this a little next year. As part of our preparation for this I was lucky enough to have the opportunity to visit St Martins and see DI in action with the amazing Chloe Sanders. It was in discussion with Chloe that I found about these important rules for DI programmes. Admittedly we were going to use the same programmes, but just getting it straight early on, I think, has helped with the implementation and will help it have more impact when we choose to use it.

5) How great Frayer models are

What I learnt: That Frayer models exist, and how great they are for capturing mathematical concepts.

Source: Jo Gledhill (@JoLocke1) at #mathsconf18.

Implication on my practice: I have been openly critical of the use of knowledge organisers for mathematics teaching. Kris Boulton summed up a lot of the problems with them in his blog. However, when Jo showed a Frayer model (like the one below), I couldn't believe I had never seen them before. I immediately saw how useful they could be for summarising ideas in maths. Over this summer I am going to write them into our new scheme, although I do need to give a little more thought as to how.

Maybe one of these things is a new thing for you. Whether it is or it isn't, I hope you will consider sharing the 5 things you have learned this year, so others have the opportunity to learn from you (and make sure you tag me in!)

A great DI day out at St Martin's

Today I had the enormous privilege to visit St Martin's Voluntary Academy in Stoke Golding. A colleague and I were there to see the use of Connecting Maths Concepts, a Direct Instruction Program that was developed in the United States. I am looking to use these materials to support some pupils who have struggled with maths in the past, and if successful to integrate them into the small group intervention work we do with pupils at my school.

I suspect some will be surprised to hear that from me, particularly after my recent podcast with Craig Barton so allow me to clarify. I am 100% of the opinion that developing understanding of mathematical concepts slowly and carefully is the best way to teach maths, both from a pupil outcome point of view and from a "this is what maths is" point of view. For me, this is what maths teaching should like, and this is what the experience of learning maths should be. So why then would I be looking at a program that (at least on the surface) seems to be entirely about developing "procedural fluency" in isolation? Well for two reasons:

1) I believe that developing understanding carefully and slowly is the best way of going about teaching maths, and that most pupils will develop a strong and flexible understanding of maths by working in this way. But I am not naive enough to think that this will work 100% of the time for 100% of the pupils. It would always be my start point, but for some this will not be enough. We already know that understanding on its own is not enough for retention- pupils forget even those things that in the moment they appear to understand. This is why it is important, even when building understanding of concepts, to plan in opportunities to revisit and re-use ideas. I often refer to this as "picking up an idea", pupils need to pick up ideas they have seen before, play with them for a bit, and then put them down again. And some need a lot more of this revisiting than others. The benefit of this program is that it is at least 80% revisiting previous ideas. And they are built on directly. The links between (for example) adding and subtracting decimals and comparing the sizes of decimals are explicitly made. Couple this with the fact that kids were getting stuff right. Lots of stuff. By some estimates kids in these programs answer up to 500 questions in an hour. And they get the vast majority right. Now I know that maths is not about just "getting it right", but imagine being that kid that only ever got things wrong. That barely even did anything compared to their peers and then mostly got it wrong. Perhaps the only time they got it right was when they had an adult supporting them. Would you be minded to explore the depths of that subject? I know I wouldn't. What I saw today was pupils having the opportunity to be successful in what they saw as maths, something they probably hadn't experienced for the first 6 or 7 years of their education, and then being shown how this can lead them to getting other things right. And I definitely don't think that is a bad thing for eventually supporting pupils to have the productive disposition to explore maths further. Coupled with this is the simple lack of mathematics these pupils have encountered relative to their peers. Whilst the value of "30 million" has been challenged in recent years, it is clear that there is a word gap between disadvantaged pupils and their peers when they start school, and this often widens to the detriment of the eventual outcomes of these pupils. I suspect that part of the power of DI programs is simply the amount of mathematics questions that pupils have to engage with, which will seek to redress any gap in the amount of exposure these pupils have had in relation to their peers.

2) I am far from convinced that these programs have to focus on procedural fluency in isolation. Having worked with Rosenshine's principles of instruction, cognitive science and teaching for mastery principles, I have seen how there is much more to connect them than separate them. Infact this will be part of the subject of my talk at ResearchEd Rugby. It may be that some DI programs do focus exclusively on procedural fluency, but that is not what I saw today. I saw pupils using images of tens frames and larger grids to support their making sense of addition. I saw them making sense of what it means to be a quadrilateral, a triangle, a rectangle, through being exposed to and identifying examples and non-examples. I saw pupils being forced to develop their thinking and language through intelligent questioning, both verbally in the display materials. I saw pupils being expertly guided by their teacher, the fantastic Chloe Sanders who took great care of both myself and my colleague all afternoon.

I would have been happy for the visit to have stopped there, as I had everything I wanted at that point. Instead we were treated to what can only be described as a visit fit for royalty. Firstly treated to lunch with the head, Clive Wright where we had the chance to talk about their journey with DI, discuss the progress of the Knowledge Schools Hub and Chloe's exciting upcoming visit to America to the National Institute for Direct Instruction to talk with (among others) Kurt Engelmann, son of the legendary late Siegfried Engelmann. Then whisked on a tour of the school and seeing the fantastic culture that the team at St Martin's have developed. Every lesson had pupils working with expertly designed materials, taught well by teachers whose expertise were recognised and celebrated, and in classrooms where behaviour was utterly impeccable. A big part of this was the utterly ruthless consistency of application in every classroom. All pupils have access to the same material and challenge, with those who need it supported to achieve as well as others. Every classroom has the same routines, but rather than being stifling to pupils these allow pupils a sense of ease - they know what is expected of them and what they can expect from their teachers. This allows for a relaxed atmosphere where pupils and teachers work together seamlessly for the benefit of all. Everything is thought of and planned, from the lesson materials (not all scripted for DI, but all explicitly taught) to the resealable cans of still water that are available that cut down on plastic waste.

I am honestly not sure I can adequately put into words just how impressive our visit was. My heartfelt thanks have to go to Clive and particularly Chloe who took such good care of us, as well as to all the pupils and staff at St Martin's who accommodated us. I would heartily recommend you visit for yourself if you can and see the incredible work going on at this school - make contact with the Direct Instruction Hub and see it for yourself!

Creating non-standard examples/interweaved examples.

I posed my department a question today in my department meeting:

"Which other areas of maths do pupils need to apply the knowledge that the sum of the angles in a planar triangle is 180 degrees?"

There were some great examples of places such as circle theorems and angles in polygons but also places like coordinate geometry. What we then talked about was the idea of creating non-standard examples from images like those we might find in these area. This led to questions/examples like these:

The beauty of these was that the questions needed very little adaptation to make the focus finding a missing angle in the triangle, and the questions then get pupils used to seeing pictures like this and seeing angles in triangles. Furthermore, if a question can be adapted so that it only requires the angles in a triangle it becomes a non-standard example, but if it can't then it becomes interweaving the other topic into angles in a triangle, or interweaving angles in a triangle into another topic.

We moved the discussion onto other areas as well, so I thought I would share some of the favourite ones that my team came up with:

As part of a lesson about finding area of rectangles, calculate the area of the bars in a histogram. A chance to interleave decimal multiplication. No understanding of what a histogram actually is is required here, but when it is time pupils are already used to looking at them.

Find the area of the shaded triangle. If pupils can solve simultaneous equations then pupils can find the base and height, if not then these could be given to get pupils seeing the triangle. The same stimulus could actually be used at different stages:

1) When first encountering area of triangles with all relevant information given.

2) When solving simultaneous equations in order to find base and height before finding area.

3) When graphing inequalities, which could then lead to the others.

Find the missing angle in the triangle. This was actually adapted from a sine rule question, but could be used in a couple of places before getting to the sine rule:

1) With this information given, just to get pupils ignoring the extraneous information about the sides.

2) Pupils could construct the triangle accurately to find the length x once the angle θ has been found.

Sorry for my absence! AND a note on the abstract

Wow, it has been nearly a year since I have blogged. It doesn't feel that long, and yet I know it has been a long time. I have to apologise to anyone who has missed me (I can't imagine why you would), but the work involved in writing and then bringing a book to the point it can be published is quite something. Alongside this, I have been working hard on our curriculum developments. Before anyone asks, no this wasn't something levied on me by my school in response to Ofsted's new focus on curriculum. It was planned development that we instigated as a department, in response to the reading and development work we had done. I am really excited about its potential, but it has been quite a job of work. There have been new scheme documents to write, and new lesson materials to develop. New assessments to write, and new homework booklets to put together. I am planning a big launch of this at some point, but it won't be this academic year as so far we only have completed the materials for Year 7 and I want at least Year 8 done before we make it all public (plus I have to get permission from my school and team as well!) Hopefully it will be worth the wait. But in the meantime a more recent reflection.

A minor disagreement crossed my Twitter feed a couple of weeks back about the nature of the second term in 7 – 3y. The question was posed as to whether the second term is 3y or -3y. To answer this question I want to focus on what "7 – 3y" actually is.

The first thought is probably "its an (algebraic) expression". Totally correct. But still only words. What is 7 – 3y? This is where it gets difficult. A mathematical entity? A thing?

The truth is 7 – 3y is a pure concoction of thought. It is nothing except how it exists in our minds. That isn't to say that there aren't real phenomena that can be related to the expression. But they aren't the expression. The expression is just there, as an abstraction of the mind. And that means it can be whatever I want it to be. Or rather it is as I choose to make sense of it. If I understand that it can be seen as the difference between 7 and 3y, then I can choose to see it like that. If I can make sense of it as 7 and -3y then I am allowed to do that as well.

For me, this is why it is important for us to ensure we support our pupils in developing understanding. So much of maths only exists in the ways that we make sense of it. Even well established concepts such as addition only exist for us in the ways we are able to make sense of them. Addition may have arisen out of practical ideas, such as collecting objects together, but it has far surpassed that since it has been applied to things like irrational or complex numbers. It is now an abstract concept, there to be made of what I can. So long as I don't contradict the results of other ways of making sense I am fine (for example, I can't just decide that 3 + 5 is going to be 9 - any way I have to make sense of addition must result in 3 + 5 being 8).

If we don't support pupils to make sense of these concepts, to have different ways of seeing and manipulating abstract mathematics within its rules and established prior results, then our pupils will never be fluent in mathematics. They will not have a chance to attain the understanding of which they are heirs to. And then they won't get chance to delve into "the best that has been thought and said" in the field of mathematics.