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.

Should trainee teachers that "fail" be allowed to continue working to QTS?

I have been teaching for nearly 14 years, and spent over a decade involved in training maths teachers in some guise or another. Like anyone involved in teacher training, I have seen trainees that have developed really quickly and by the end of the course have looked like they have been teaching for years. I have seen trainees that are badly struggling and clearly not suitable - most of them have deferred or left before the end of the course. But I have seen some that just need a little more time. They get on well with the assignments, they are professional in their approach, but for one reason or another they are not quite on track. Often it is due to struggling with behaviour, occasionally for other things, but they aren't far away. These trainees face a difficult choice - defer and return later, having perhaps done a bit more work in schools, give up and write off the year, or try and extend their training. But some are not in a position to defer, and it can be a bit of a stigma for having had to defer for anything other than a family situation. The costs involved in extra time at a University can also be prohibitive for some, and not all school based training can easily accommodate extension beyond a year.

Given our recruitment and retention problems in maths education at the minute, I wonder if it is not time to revisit trainees like this. Trainees that are not quite going to get there in a year, but could well get there. Trainees that have completed the theory work, but need more time on the practical. People that are invested in teaching maths, that really want to make it happen, but just need a bit longer to work on it.

I wonder if it should be possible for a trainee in this position to get a job in a school, perhaps initially as an unqualified teacher, but to be able to continue working towards QTS through their training establishment. They could then achieve QTS at any point in the year, at which time they could move to the qualified pay scale.

This is by no mean a fully fleshed out thought. What would happen if a "failed" trainee couldn't find a job? How long before they have to start from scratch? What would happen if they did get a job, but didn't achieve QTS within a year? Should they be be allowed to keep going? And again, if so, for how long? But I still wonder if we could find ways of answering these (and other) questions that we might not be able to bring more committed people into the profession. The idea also speaks to my values as a teacher and trainer, I would always much rather work with and support someone who is struggling than dismiss them.

I would welcome thoughts on this idea, how we might answer the questions, and what other issues would need tackling, or even just whether people agree or disagree that this idea has merit.

Time to revisit...Teaching for Mastery

In two weeks time on October 13th I will be delivering a session that shares the title of this blog. The blog is meant to act as a preview to the session.

“Mastery”. Some people see it as the latest buzz-word to be shunned until we wait for the next “big thing”. For others it is central to teaching. For some it is a confusing term with no clear idea of what it actually means. And I can sympathise with all of these views…

The idea of “mastery” has been around for a long time. People much more knowledgeable have written about its provenance, its history and its progress to the modern day. Neither this blog nor my mathsconf session will be trying to reinforce or reinterpret any of this. I will not be attempting to explain the structure of a mastery curriculum (which is not exclusive to a mathematics curriculum). Better men than me have already done this, not the least of which is the LaSalle CEO Mark McCourt (if you haven’t read his blogs on mastery then you must). Saying that, it is important to understand certain aspects of its structure to understand where I hope my session fits in.

One of the central aspects of a mastery curriculum is teaching in a way that all pupils can access from their starting point, and then carefully assessing their understanding throughout the teaching process. A second is the use of correctives where the initial teaching isn’t successful – having different ways of approaching concepts when the first way falls short. The biggest aim of my session is to try and showcase some of the ways that teachers can approach this. Starting with what I see as important ideas to consider when thinking about structuring learning, I then aim to share practical examples of approaches that could be used either as part of the initial teaching or as a corrective approach. For those that know me, it won’t be surprising to hear that much (but not all) of this focuses on the use of representations to reveal the underlying structure of an idea (given that my book “Visible Maths” is entirely concerned with the use of representations and manipulatives to reveal underlying structure).

As an example, but not one I am using in the session, consider the “rule” that one negative number divided by another negative number results in a positive answer. Consider -15 ÷ -3:

One way of representing this is to use double sided counters; these usually appear with a yellow side (positive) and a red side (negative). Two different coloured counters can also work, and in fact to model this calculation we only need to consider negatives so a single colour of counter will suffice. The image above shows -15, and now we have to think about how we divide that by -3. One way of thinking about division is to think about creating groups, so a possible way of looking at this calculation is, “Start with -15 and create groups of -3.” These groups can be seen below:

When we think about division like this, the result of the division is “How many groups can we create?”. We can see that this process creates 5 groups, which means that -15 ÷ -3 = 5.

  

Often this “rule” is taught as an arbitrary rule, without any attempt to show where it comes from. In many classrooms, one could be forgiven if kids believed that the only reason this is a “rule” of maths is because teachers says so. But this rule is a necessary rule – if division works in the way we know it does then the answer to -15 ÷ -3 cannot be anything but 5. I finish my session with a discussion around other “rules” of maths, how appropriate representations can show where these rules actually come from, and also discuss how we can manage the transition from using representations/manipulatives to the abstract calculations. Hopefully I have whetted your appetite to hear more about teaching approaches that can support mastery in mathematics, and I look forward to seeing you (whether in my session or not) in Birmingham. Don’t forget to join us for the pre-drinks and networking the night before as well!