Category Archives: pedagogy

Some principles on classroom routines in the new year

Before getting into the nitty gritty of planning some actual maths lessons for the new school year, my mind has been swirling around some principles for how I manage my classes next year.  Routines don’t come naturally to me, I used to actively avoid them in my previous life before teaching.  After the last few years, I feel I have learnt the hard way and have now come to truly appreciate the power of a good routine established with a class.

Many of these routines are whole-school and I intend to embrace these as much as I can next year, having previously maybe been a bit sneery of such things.  For example, the lesson-entry routine in our school is great.  All pupils stand behind chairs on entering. They can get their equipment books out, etc.  They can have a look at the activity that will be on the board. But they don’t sit down until we have a chorus of “Good morning, Mr Horley”.  They do this in every lesson (up to and including Year 11) and it is well-established school-wide. It gives a nice, clean positive start to the lesson and encourages oracy right from the start.

However there is only so far whole-school approaches can define routines, individual teachers need to fill in the gaps with the minutiae of the detail in the routines of their own classrooms that build upon the whole-school principles.

At some point I will attempt to write the details of my routines for each class.  At this stage, here are my principles:

  1. Say it simply.   I want my pupils (and my!) mental energy to be devoted to doing maths not remembering routines.  Whatever I do has to be easy to explain to the most overloaded Year 7.  I need to do it in small chunks.  It’s important to get routines established early but let’s aim for the first week rather than everything in the first lesson.
  2. Say why.  I have a mantra for next year: as far as possible every “rule” will come with “and this is why we do it this way”.  Hopefully they will agree it’s a good idea, be more likely to remember it and be more likely to follow it!
  3. Reiterate, constantly.  If they don’t get it all in the first week, not a problem because I will be reiterating consistently throughout the year.  I need to avoid veering “off message” to try to liven things up, which I am prone to doing.  A bit of boring repetition is not actually all that boring for kids if it is done with positivity and becomes “the thing we do”.
  4. Do it collaboratively with the department.  Back to my initial paragraph, my classroom does not exist as an island in my pupils’ lives.  As a team we need to constantly discuss what we do consistently across the department, and what we have autonomy on.  Even where we have autonomy, we should aim to share ideas, especially with the new members of the team. It’s difficult to find time to meet formally so it is more likely to be the informal chats in the office where this gets done.

Next is to get into the detail for each class…




Knowledge Organisers in Maths – The Journey Part 3

Read Part 1 and Part 2.

As our final versions of the Knowledge Organisers are being printed over the summer holidays, I have been preparing some of the resources for how we test our pupils on the knowledge contained therein.  As I wrote before, the real value in actually creating these resources is in the discussion that it prompts amongst the teachers in the department.  So it’s not ideal that I am doing this on my own in the holidays. I hope to carve out some INSET time at the beginning of term to run through this with the department, especially important for our new trainees.

Towards the end of summer term we did manage to have a quick discussion and agree on a couple of principles for testing.  As a school, across all subjects, we have decided to make the Knowledge Organiser tests high-stakes.  They are high-stakes in that pupils will get a detention if they fail to achieve 80%.  Also, most departments are doing Multiple Choice tests, mainly to keep the marking burden light.  An implication of making the tests high-stakes is that they have to be marked by teachers.  We need to ensure that pupils can achieve high success on their tests by learning purely what is in the Knowledge Organiser.  We are not testing application of knowledge, just the knowledge itself.  In Maths, this means we are not doing a standard maths test where the pupil would be working out answers, but we are testing that they have learned the definitions and examples as given.  We will do this using multiple choice questions where there may be more than one correct answer, maybe we will call these Multi-Multi Choice Questions (MMCQs).  Below are some examples showing the extract from the Knowledge Organiser and the associated MMCQ.

Screen Shot 2018-08-01 at 06.49.36.png

Screen Shot 2018-08-01 at 06.47.57.png


Screen Shot 2018-08-01 at 06.53.05.png

Screen Shot 2018-08-01 at 06.48.11.png


Screen Shot 2018-08-01 at 06.49.21.png

Screen Shot 2018-08-01 at 06.48.45.png

This all feels very different from anything I have ever done or seen done in Maths before so it makes me pretty nervous about how it will work.  But I am also excited because if our pupils can start lessons with a firm knowledge of these basic facts and definitions, maybe they will make more progress in doing the maths that uses them.

Please do let me know if you have any thoughts or comments based on your experience.  I am not publishing the whole set of MMCQs yet as we will be working on them next term, but do get in touch either via comments below or on Twitter if you’d like me to e-mail what I have got so far to you.

Visiting Totteridge Academy

As a maths department, what achievement would you feel most proud of?  An outstanding set of GCSE results with a high proportion achieving 9-7? Data showing excellent progress between Year 7 and 11?  Pupils that visibly enjoy maths and actively engage in lessons, showing that they are willing and able to think mathematically?  A team that effectively improves the teaching practice of all of its teachers and manages to continuously improve the teaching in all its classes? Or having the confidence to share these achievements with other maths teachers by inviting them to your school for a day.

Personally, I would like to be pushing towards the bottom of that list and that is exactly what the team at The Totteridge Academy (TTA) are achieving.

I always find it a hugely powerful experience to extract myself for one day from my own familiar routine and setting to see how others are doing it.  I have come away from TTA today buzzing with ideas, an enthusiasm that quite rightly should be tempered by the mantra of “do one thing and do it well”.  It’s great to glean ideas from others, but any change in how we do things back home is worth nothing without the commitment of the team that’s behind it.  I feel that, at least at a department level, we need 100% consensus on implementing new ideas.  If we can’t get that, we shouldn’t do it.

It has been a rapid change at TTA and I’m sure that the things I have seen today are only part and maybe not that big a part of the story.  But I’d like to reflect on 4 things that I found interesting.

1, Standards of oracy, use of domain-specific language.

What struck me here was not just the way that teachers insisted on ‘right is right’, the Doug Lemov principle that I know many teachers strive for, but how pupils were on board with it too. The classroom culture was such that pupils would put up their hands to comment / correct answers given by other pupils.  Not in a smug, you got it wrong kind of way. But to build up the answer so that collectively as a class we can be certain we have it nailed. Maths is seen as something which is precise and there is a satisfaction in completely and correctly answering a question.

2, Use of chants

As an alternative to Knowledge Organisers (see previous post!) the team at TTA have developed an A3 sheet of about 40 “chants” that pupils learn through the year. Examples include:

Comparing fractions…            …find the LCM

Estimation…                              … 1sf

Multiplying fractions…            …top top bottom bottom

Dividing fractions…                  …x by the reciprocal

Factors of a number…             …go into a number

Multiples of a number…         …are the times tables

I loved the way these were used in lessons.  During an explanation from the teacher or from a pupil, they would say the first part, pause and then the class responds with the second part.  This was clearly a well embedded part of the routine that all classes seemed confident with. A really slick way of reinforcing core knowledge whilst keeping pace to the explanations.

3, KS3 5-a-days and parental involvement

The principle of empowering parents to help their children isn’t going to patch all the gaps that you might have with Year 7 and 8 but it makes sense to give it a go.  In a targeted way, parents are given a weekly set of numeracy questions with worked examples to give them the confidence to help their children at home.   I want to find out more about this and how it develops over the course of the year.

4, Group work

The key to success here is group accountability.  The groups are consistent from one lesson to the next and they accumulate points as a group over the term.  Anyone in the group may be called upon to offer an explanation to the whole class.  The lesson I saw started with pupils working individually on a problem presented to them.  It wasn’t a race amongst the group to get the answer first, in fact they seemed to be conscious of each other’s work and would slow down and offer advice if another group member wasn’t getting there.  Once all had agreed upon an answer, they would next rehearse their explanation. Again, there was real collaboration and awareness here.  One would start the explanation, get to a certain point and then pass the baton on to the next member in the group.  They would all practice a piece.  Once they finished, if there was time, they would rehearse the explanation again. It all seemed very natural to them, I don’t think I have ever seen such a high level of collaborative work in a maths classroom!

All in all, an inspiring day. Many thanks to the staff of The Totteridge Academy for hosting so many of us.




Knowledge Organisers in Maths the journey – Part 2

Part 1 is here.

I have somewhat mixed feelings about “making your own stuff”.  On the one hand I feel that there are thousands of teachers across the world spending time making resources for things that have been taught for decades, so why are we wasting time reinventing the wheel?  But on the other hand there is something to be said for using something in our teaching that we know to be exactly the way we want it, that we know inside out because we wrote it ourselves.  And when that something isn’t just written by the individual teacher but by a team collaborating on a project it becomes that much better in quality and hopefully the team feel that much more confident in using it.

So has been my experience of creating our first batch of knowledge organisers in Maths.  I started the process thinking that there must be loads out there and it is surely just a question of picking the one we most like the look of.  Although I have magpied bits, mostly from Andy Coleman’s comprehensive collection posted here, our particular version has been the result of around 2 hours of department time where we have discussed the specific nuances of how we model solving linear equations, the use of specific terms in our teaching (e.g. a heated debate on indices vs. powers), and written the words live on screen as a real collaboration between the 5 of us.  Admittedly a small team helps here and more people would likely have made the process take longer.  I am always super conscious of taking up teachers time in meetings but everyone seemed to be enjoying it and getting value from it. It is something I hope we can spend some more time doing in our department next year with our new staff members.

So I am sharing our work with a bit of hesitation.  Firstly they are far from perfect and definitely far from finished.  We haven’t actually used them with real live pupils yet, they have been created in a vacuum. By this time next year, I expect them to look very different.  But more importantly I wouldn’t want departments to miss out on the experience of collaborating to create something they own as a team.

During our discussions on the maths pedagogy, we had to keep reminding ourselves of the principles of what we were trying to achieve with the knowledge organisers:

  1. They are not trying to achieve everything, just the key “facts” that need to be learned.  The absolute base knowledge that learners at all levels need to access the curriculum.
  2. As soon as we found we were trying to write teaching points we would stop.  It’s not a text book or a revision guide.  It is not aiming to explain how to do things, but be a concise list of key information.
  3. The definitions don’t need to be perfectly mathematically rigourous.  They should be durable, in the sense that they are not contradicted by future learning.  But of paramount importance is that they make sense to our learners and are written in language they can access. Often in maths examples speak louder than definitions.
  4. Low-stake tests and pupil self-quizzing.  This is the next part of the journey but something we need to be mindful of now.  How do we expect our pupils to engage with this? What will our low-stakes tests look like?  Current thoughts are that we give them a key fact, they fill in the definition and an example of their own, ideally different from the one given to them.

So, for what it’s worth, here they are.  By all means copy individual definitions if you like them and they work for you.  Just don’t take the document and print out 30 copies to give to your class as that is unlikely to get anyone very far.

Knowledge Organiser Maths Yr7 Term 1a 1b v1

Knowledge Organiser Maths Yr8 Term 1a 1b v1

Knowledge Organiser Maths Yr9 Term 1a 1b v1


Part 2 is here.


Visiting Michaela

As with many things, I find myself somewhat behind the curve when visiting Michaela Community School in Brent this morning.  Without doubt, more has been written about this school in the 4 years since it opened than any other school in the country.  I found this recent post from Adam Boxer useful, it contains links below to blogs written by the staff – seemingly a part of the job description for working at Michaela.

The school prides itself on “being different from other schools” which was the message being delivered to Year 8 by the head Katharine Birbalsingh within a few minutes of my arrival. Being “different” inevitably attracts attention in itself, but it is also part of the mission of the leadership to “spread the word”.  They are not content with creating a great school in a corner of London, the bigger goal is to revolutionise the system.  And so, I joined the steady stream of daily visitors wanting to know more.

Having read so much about the place, much of what I saw was what I expected. Impeccable behaviour in classrooms, and seamless, silent transitions between lessons. The chanting of Invictus at lunch time, followed by the daily topic for discussion and then an “appreciation”. What struck me most was the relentless focus on oracy (not that they call it that) with not a single mumble being allowed to pass unchallenged in lessons or elsewhere.

But my main reason for visiting was to see the maths in action and in particular to get an insight into teaching of lower attaining groups at KS3.  The sets are given different names and at no point is anyone allowed to utter the words “bottom set”. I was told in hushed tones which set I was seeing.  The approach makes sense although in my experience it doesn’t really matter what you call them, kids work out pretty quickly which set they are in.

Central to the maths curriculum is a set of booklets that have been created by Dani Quinn and her team. They become the structure and central resource for the lesson.  I saw them being used in the following ways:

  • As a reference for the instruction part of the lesson where sections might be read out aloud by the teacher or the pupils.
  • Some parts were fill-in-the-blanks, scaffolded tasks as a lead-in to independent work.
  • As a source of individual questions to be modelled by the teacher
  • As a source of individual questions for pupils to do on mini-whiteboards
  • As sets of questions for pupils to work on independently
  • And, as I was told by a student, as a revision resource

Although they are used as a core resource, teachers were using the booklets very flexibly, not in any way seeming to be constrained by them and often veering off-piste to do some additional mini-whiteboard questions, for example.

All groups have the same booklet but the lower set(s) would be expected to proceed at a slower pace and therefore not cover everything in there.  I don’t see a way around this. If you are going to set students then of course the lower groups will have to either learn fewer topics or cover them in less depth.  Bart sums it up when he joins the remedial class in a new school:


I have many thoughts on setting in maths still flying around but that will have to wait for another day. In the case of Michaela, what was interesting was that the lower sets are no smaller in size.  I counted 24 in a bottom set Year 10 and was told that there must have been some absent.  There are simply no behaviour issues in these groups and the insistence of seeing every single whiteboard means that nobody is left behind. I realise how much I rely on circulating the room and providing individual instruction with my smaller bottom sets.  I saw little of this. Instructions and corrections were mostly being given from the front of the room even if they were targeted at individuals.

Another component of why this works was the use of self-reporting.  I have all but given up asking questions like “who didn’t understand that?” or “who got that right/wrong?” I would always prefer to see actual evidence of understanding on whiteboards or in books rather than asking students to self-report by putting hands up or by using Red-Amber-Green cards to show level of confidence. But what I saw at Michaela was a classroom culture where pupils seemed to be showing honestly where they were at. There was no stigma attached to being right OR wrong.  “Is there anyone in this room that doesn’t understand that?” Two hands shot up immediately. A short whole-class discussion ensued based on the student articulating clearly what they didn’t understand with another whole-class example. To have this quick verbal check in addition to formative assessment of student work is very efficient. In another lesson, when a student just said,“I don’t understand” this was immediately thrown back with “you need to ask a specific question”.

What I didn’t see was any use of Powerpoint.  Michaela teachers are masters of the visualiser. Everything I saw on screen was skilfully projected snippets of the booklet or some other printed material or their modelling of solutions using pen and paper under the visualiser with the occasional show-call of student work.  PowerPoint is not banned and they would use it for particular images or animations that benefit from it, but the point is, there is no “clicking through” a presentation as the structure of a lesson.

Visiting any school is refreshing and enlightening – I am very grateful that I am being allowed to use some of my Year 11 gain time to do so.  We can learn so much from each other, but mustn’t get carried away with the notion of “it works there so it will work here”.  Contexts vary, and as I discussed with Ms Birbalsingh before I left, the fact that Michaela has started from scratch with both its pupils and its teachers is a unique situation very different from trying to improve an existing school, especially when it comes to expectations on pupil behaviour.  It’s inspiration for making changes at your own school, not an instruction manual for how to do it.

Would I want to work there? Well yes, who wouldn’t want to work in an environment with impeccable behaviour, excellent resources that the team has created and continually refine, and therefore has a strong sense of ownership in. The relatively low contact time (13 hours per week on average, 19 max) is somewhat offset by the high amount of “duty” time – the high presence of staff is key to maintaining the behaviour. But that still leaves quality time for collaborating as a team, especially as time is not spent creating PowerPoints.  My current school won’t be seeing the back of me anytime soon, though – a daily commute from Wandsworth to Wembley Park is certainly not what I’m looking for right now!


Knowledge Organisers in Maths – the journey part 1

I am instinctively cynical about any “latest new thing” in teaching. As a profession, we have been teaching kids things for centuries, and I feel it it is highly unlikely that anyone will suddenly stumble upon something genuinely new that will have a significant impact on learning.  Knowledge Organisers feel a bit like they are the latest new thing, so I have to say my heart sank ever so slightly when I was asked to join a middle leaders meeting this week to discuss our plans for introducing them next year.

In reality, I’m sure there is nothing particularly new here.  If I could go back 30, 40, 50 years, I would probably find examples of these being used in schools even though they may have gone by another name.  And I think the name is part of the problem.  To me, “Knowledge Organiser” doesn’t really describe what they are.  Maybe it’s because I’m old enough to remember the Filofax of the 1980’s which were called Personal Organisers. Filofax The Original Organiser

There was a whole world of wonderful inserts which you could buy and arrange how you desired, clipping them into the 6-ring binder, a wonderful way of whiling away your day! The name “Knowledge Organiser” implies to me that the learner will compile some sort of folder themselves over time and decide how to organise their own knowledge.

From what I’ve seen, that’s not what they are.  They are Fact Books.  I’d like ours to say “Maths facts you need to know in Year 7”.  I feel that is more descriptive of what they are and how they will be used.

I’ve decided to write this because I left my meeting feeling much more positive about them than I was at the start.  This is a whole-school initiative – surely the only way to introduce them so that pupils and parents see a consistency across subjects.  Although I agree with a lot of what Kris Boulton says (Why Maths Teachers Don’t Like Knowledge Organisers) I think it is right and proper that we support the whole-school initiative and don’t fall back on the refrain “but Maths is different” thus risking acquiring (reinforcing?!) a reputation as the awkward squad!

So why am I becoming more convinced of their possible merit? In our meeting we had a good discussion on the principles of what Knowledge Organisers are and how we should use them, namely:

  • They list the base level of facts that pupils need to know to achieve in this subject in this term.  They are not intended to cover everything. They are not to be seen as a syllabus or a revision guide.
  • Their core purpose is to enable pupils to engage in self-quizzing. We will need to explicitly teach pupils how to do this.
  • They will form the basis for fortnightly quizzes.  It is this retrieval practice which creates the change in change in long-term memory.

There are plenty of examples of excellent Knowledge Organisers out there (Jo Morgan has a collection on this page.)  I intend to lean heavily on these when creating our own.  After all, there is nothing new about these facts, although inevitably we will want to tweak and adapt to fit our Scheme of Work and format.

So my starting point is to sit with the team and see what we think.    It’ll be interesting to hear their thoughts..,

Part 2 is here.

Part 3 is here.



I’ve learned to love routines since I started teaching and this term I am particularly pleased with the results I am seeing with a routine I have established around low-stakes testing, homework and lesson starters.   A game-changer for me this year has been Hegarty Maths which my school uses for homework and has been integral to this routine and to encouraging and enabling pupils to take responsibility for their own learning. I have also been reading a lot more about spaced practice and have been sold on the idea of the testing effect and the power of low-stakes testing.

It won’t work for all: context is everything.  My context is 3 x 1.5 hour lessons a week with a low attaining, small (15) but actually quite mixed Year 9 class with a high level of EAL. There are some pretty significant attendance issues with a handful of students who have been in about half of my lessons this year.  And the usual challenges of lack of confidence / self-belief, reluctance to engage, poor learning behaviours, etc. that go with a bottom set.

It’s taken me a while to realise that the main obstacle to overcome for many learners is not so much understanding of new concepts but the retention of knowledge from one week to the next, one month to the next.  It is not something I think I ever experienced at school, so I suffer from expert blindness and I struggle to empathise. Until recently, I used to get frustrated when I occasionally revisited topics with a class and they had “forgotten everything”.  We want our pupils to see the deep connections between topics in mathematics as this goes some way to helping solidifying learning. But that is not sufficient for all pupils.  Some will struggle to see the connections in the first place, for others those connections may have a transient impact and not have much impact on long-term memory.

This is where spaced practice comes in. Basically going over topics that include:

  • Core skills that will have been taught since primary but are still not secure
  • Topics from that last couple of weeks
  • Topics from earlier in the year

So, our routine looks something like this:

Screen Shot 2018-06-01 at 10.21.45

Wednesday: 5-6 starter questions on the board that I write.  Here is an example:

Screen Shot 2018-05-27 at 21.43.16.png

There is overlap between these questions and the tasks that are set as Hegarty Maths homework for the week which is due the following Tuesday. I circulate the class to assess how they get on with these questions, what they have remembered and will maybe spend a bit longer doing some more of those questions on mini-whiteboards if I feel it will be helpful.  This whole process could take anything between 15-30 minutes before reviewing Tuesday’s test (more about this later) and then getting into new learning and the main part of the lesson.

Friday: Knowledge Test 1. A single side of A4 with 12 questions which they do at the start of the lesson.  Here is this week’s example:

Screen Shot 2018-05-27 at 21.53.53.png

I usually give them about 15 minutes which is long enough for highest attainers to attempt every question.  The lower attainers have often stopped before then so I circulate and will give hints on mini-whiteboards and encouragement to them.  We then review these questions immediately, they mark corrections with green pen and they keep the test paper to revise from.  Each question has a Hegarty Task number on it.  3 of these tasks (the underlined ones) were set as the homework. They are encouraged to look up the other tasks if they need to revise them.  The homework is set on Tuesday, due for the following Tuesday but I always make a point to praise pupils that have done the homework early, before the Friday test as they will have more success in the Friday test.

Tuesday: Knowledge Test 2. These questions are exactly the same topics / Hegarty tasks as Test 1.  Here is the example from this week:

Screen Shot 2018-05-27 at 22.00.22.pngSo they know exactly what is coming up and should be able to get over 80% in this test.  I tell them this and about two thirds of the class are now getting 70% or more. They do this one in silence, closed book. I offer minimal support, take them in at the end and mark them after school that day, which takes about 20 minutes.

Wednesday: Feedback on Test 2 after Starter questions as described above.  This is usually pretty quick, around 5 minutes at most.  After looking at their scores they  then make corrections in green pen as I model solutions under the visualiser. With the result looking something like this.

Screen Shot 2018-05-27 at 22.02.41.png

The tests are then added to their folders so they are all kept together (rather than in books).


I toyed with the idea of giving detentions based on the result of Test 2, but I know there are some who would be getting a detention each week and that would be counter-productive.  It also goes against the idea of low-stakes testing, i.e. the threat of a detention immediately makes it high-stakes and I already have some pupils with significant maths anxiety which I don’t want to make any worse.  So I set detentions based on effort on homework which I can see clearly on Hegarty Maths.  But I still give them the message that I expect them to be getting at least 10 out of 12 on Test 2.

The most obvious benefit so far has been completion of homework which has improved considerably.  I’d like to think that I am gradually shifting the attitude of “I need get my Hegarty done to avoid a detention” to “I need to do my Hegarty so I can do well in my test”.  Clearly I’d rather the predominant attitude to be “I want to do my homework as it will help my learning” but I think this is a step in the right direction!

Overall I guess this means that about 30% of lesson time is now being spent reviewing previous topics.  That feels about right to me although it is significantly more than I used to do.

Do you do anything similar with your classes? Anything else you think that I should consider tweaking?  Share ideas, please! Get in touch by leaving a comment or via Twitter.  Please DM me if you’d like the full set of Knowledge Tests.

UPDATE: I have put the first 6 weeks’ test into a folder here. I do think the value is in tailoring these to your class, but also that this can be a bit time consuming. So if these are of use, please feel free to download, adapt, etc.  I will try to add more as I make them.





Just follow these steps and you’ll be OK at Enlargements

Here’s a statement which I don’t think will be too controversial – I would have thought maths teachers the world over would agree with this:

I want my students to gain a deep understanding of the mathematics, not just follow a procedure to get the right answer.

This is our aim. We don’t always get there. We have different ways of getting there. I have recently re-read this seminal paper by Skemp from 1976.

Screen Shot 2017-06-01 at 13.18.19

In it he talks about Relational Understanding, which I have generally thought of as understanding of concepts, and Instrumental Understanding, which I think of as understanding of procedures.  In my teaching I have been inclined to build conceptual understanding first, and then see what methods make sense from there.  However, I’m starting to think that it’s not that simple.  There are some situations where some instrumental understanding might come first and act as a foundation on which to build relational understanding.  Ultimately we want both, but the order in which we achieve this is not always the same.  We should not dismiss a didactic approach that provides a clear sequence of steps and worked examples as a part of the journey to a deeper mathematical understanding.

I recently observed a colleague teaching the topic of Enlargements to a Year 11 revision class, who are entered for the Foundation paper.  He used a method which I don’t think I had thought of before.  It was heavy on the Instrumental Understanding, but it worked and the students were doing some more tricky fractional and negative enlargements with centres of enlargement not at the origin.  About as hard as it gets for these types of exam questions (thanks for Maths Genie for these examples)

Screen Shot 2017-06-01 at 13.36.31.png

So, what was this wonderful method?  Well, it might be nothing new, but it used vectors.  It relied on students being secure with describing the translation between two points using vector notation.  Given that this is how Translations are described and that we typically teach Translations (along with Reflections and Rotations) before Enlargements in a topic called Transformations, this should be a build on / consolidating of what was learned a few lessons ago.

The steps go something like this.

  1. Label the vertices of the shape you’ve been given (say A,B,C,D, something like that)
  2. Circle the centre of enlargement, CoL (helpful to distinguish it from a vertex later)

Screen Shot 2017-06-02 at 14.57.56.png

3. Find the vector that moves you from the CoL to each vertex.
4. Multiply each vector by the scale factor

Screen Shot 2017-06-02 at 14.26.07

5. Now use those vectors to plot the new points, starting from the CoL again. Connect the vertices to form the shape.

6. Finally draw in some ray lines to convince yourself that you have not made any mistakes.

Screen Shot 2017-06-02 at 15.14.58.png

I think that’s quite a neat method that enables us to go straight to what might be seen as the most difficult example. The only real difficulty here, however is multiplying a (pretty simple) negative fraction by an integer.  Something which should be secure by the time this topic is being taught. And if it’s not secure, well here is an opportunity to practise it without detracting too much from the main objective.

Another benefit, is that it might be easier for students to plot points by counting squares rather than draw accurate, extended ray lines, as pointed out by Mr Blachford on Twitter.

Once students have done a few examples, we can draw attention to some things, for example:

  • If the scale factor is >1, it gets bigger, if it’s <1 it gets smaller
  • The scale factor applies to each side length of the shape (but not the area…)
  • Negative scale factors always place the image the “other side of the CoL”
  • All ray lines must go through the CoL. This is how it is constructed.

I would use Geogebra (as in fact I did to create these images) to examine what happens when we move the CoL. (Note: If you are the in the default US language setting, Geogebra uses the terminology Dilation rather than Enlargement – which actually is more descriptive of what we are doing, isn’t it?!)

Jun-02-2017 15-24-44.gif

And from there we can go onto questions that show the enlargement and ask for a description.  I would use Geogebra for that bit as well as it is very easy to create images that can then be used to ask learners to describe what they see.

Just for fun, I had a go at a 3D version in Geogebra.  I’l leave you to decide whether or not it adds anything. You should be able to access it here. This is what it looks like:

3D dilation.gif

This is a specific pedagogy for this topic.  It’s one way of doing it. For other topics, for example fractions, I would prefer to spend a decent chunk of time on building conceptual understanding before focussing on algorithms to get right answers. But that’s for another post…





“I noticed” vs. “I liked”

I am hugely privileged in my role this year as a Teaching for Mastery Lead having had the opportunity to join various Teacher Research Groups (TRGs). This, combined with the work we are doing in Year 7 (see this article), and mentoring trainees has meant I have spent nearly as much time joining other teachers’ classes as I have taught my own classes. Whilst that makes me feel slightly guilty, I hope that this will pay off next year and in years to come –  I am learning so much in the process. It would be great if all teachers had a sabbatical year, 4 or 5 years into their teaching career where they teach a 50% or less timetable and spend time observing others in their own school and other schools nearby including cross-phase (i.e. primary-secondary). A pipe-dream maybe, but it could go some way to alleviating the retention problem at that crucial stage in a teacher’s career when it should be getting more manageable but often doesn’t.  There were some encouraging signs of this in the recent Education Select Committee report on Recruitment and Retention of Teachers.

Sitting and watching (actually I rarely manage to sit still for long, the urge to get up and engage with students is too strong!) someone else’s lesson only gives half the picture, however. Going hand-in-hand with the lesson is the shared reflection on that lesson afterwards between teachers. And this is the point.  It is not a lesson “observation” in the traditional, pre-2014 Ofsted sense.  I am not there to evaluate the teacher in any way. I am a fellow professional who has another perspective on the learning happening in that room.  Because I am not leading the lesson, I should be able to notice things, and I may notice different things than the teacher who is leading the lesson.

A lesson observation is traditionally is followed by “feedback” which is more often than not a one-way conversation between the observer and the observed.  Usually it is a very polite affair which starts with a lot of “I liked…”, “I thought … was lovely” – the WWW.  All nice to hear, but do you ever get that feeling that these are platitudes and really you are waiting for the EBI? The “I thought maybe you could…”, or “In the past, I’ve tried…”  I’m not saying that this style of feedback is not useful, especially when the observer has many more years experience that the observed.  But I would say that anyone with more than a few months’ experience in the classroom has something to offer and that the conversation should start off very differently.

Earlier this year I was invited by Danny Brown to join a lesson of his. It was last thing on a Friday and after the lesson I also joined his staff meeting.  The lesson was fascinating, but it was the staff meeting that has really stuck in my mind since.  It wasn’t a department meeting as such, but a voluntary gathering to reflect on a lesson that had been given by one of the department and observed by others.  The focus was on something that was “noticed”.  It was not an attempt to analyse everything that happened in the lesson, but a focussed discussion on something that was interesting for some reason and that we can all learn from.  There was a high degree of respect and trust between these teachers and the discussion became deep, insightful and .

I have been practising this ever since in discussions following lessons, be they informal “feedback” with colleagues (I dislike this term because it implies a one-way flow of traffic) or more formalised TRGs as part of my Teaching for Mastery work.  It takes some practise.  Commenting on something without evaluating it can be tricky.  You sometimes feel like you aren’t really making a point.  But actually just clarifying what happened at a particular point can then open into useful conjectures as to why that happened.  This is where different insights from different people in the room can become really powerful and is the essence of a fruitful TRG discussion.

We need to see a major culture shift in our schools. For too many years, lesson observations have been about scrutiny and accountability and not about close collaboration of a team of professionals seeking to improve their practice.  This has led to a culture of fear in schools where many teachers still would rather not have someone “observe” them because it causes anxiety as they feel they are being judged.  I would warmly welcome anyone into my classroom at any time and would always want to know what they noticed, but I recognise that is not a common attitude amongst teachers.  We need to practice how we share these noticings with each other so that they are truly supportive, non-judgemental and lead to fruitful discussions.  And we need to be open and receptive to these discussions and realise that they are about mathematics and learning.

The approach can be very time effective.  We don’t need to sit through an entire lesson to notice something interesting, 10 minutes might be enough.  One noticing might spark a couple of useful insights on a short post-lesson conversation.  I might call this “noticings-lite”.  It’s not a huge investment of time, the bigger challenge is the shift in culture.

If we are serious, though, we do need to organise this and having more than one adult seeing the same lesson can generate the range of perspectives. This is what the Teaching for Mastery programme is achieving this year and for once it is coming with funding to enable teachers to be out of class.  In my experience watching each other and carefully analysing lessons is simply the most powerful form of CPD there is, far more beneficial than most whole-school INSET.  I hope it continues to grow in a funded sustainable manner to increase the skill levels of all teachers.