A New Year!


Woohoo!! Happy New Year!  Hear the party poppers and experience the joy!


I’m imagining that the night before coming back to school wasn’t as raucous as 31st December can be but I also imagine there were a few unsettled bellies.

I always feel nervous before the new term starts – thinking I’ve forgotten how to teach and plan and remember names and all the other things we have to do in a day to day basis.  But, I also know it’s never as scary as I imagine.  To start of this year, here are my 5 New Year’s resolutions!

1. I will stay organised.

I can be quite messy at work and this can distract from my thinking and students learning.  With this in mind, every day will start and end with a clear desk and a tidy classroom.  I’ve been trying to use a digital markbook and so I’ll be curious to see if that is as seamless as my old paper one.

2. I will make the most of every work day.

I can spend a lot of time faffing, even (or maybe especially) if I have lots of work to do.  With this in mind, I will continue my practice from last year and assign every free lesson a job but this time stick to it.  I’ll try to start jobs when I get them and make the effort to discern the important from the urgent.  I’ll try to keep my email inbox low and respond sooner rather than later.

3. I will post here once a week.

I’ve had this blog thing for a while but I’ve had no discipline with keeping it up-to-date – posting at least one idea or thought or reflection every week should be easy and I’m going to try my best to stick to it.

4. I will try to follow a marking is planning routine.

David Didau has been talking about this for a while and I’m leading a research group at school based around it.  I’m determined to shorten the feedback cycle and make meaningful marking an integral part of all I do.  I imagine I’ll be posting about this soon!

5. I will keep on top of my phone calls.

Since I first started teaching I’ve always made positive phone calls home and I know they make a difference.  But, they often slip away as time becomes more pressured – so I’ll aim to phone home to one student from each class, each week.  From experience, Friday afternoons are a great time for this – awesome way to start the weekend.

Those are my work resolutions – I hope they don’t fail to materialise like so many others or end in disappointment a week or a month from now.  I think each of them are important and am keen to give them the time they need.

How about you?  Any resolutions as you start the new year?


What-if-not – how to extend any problem, no matter how closed


I’ve been continuing my summer holiday reading in between, you know, having a holiday. The Art of Problem Posing is a book I’ve been dipping in and out off since it was recommended to me by Andrew Blair over at http://www.inquirymaths.co.uk and I finally got round to finishing it. The first edition was published in 1983 (the year I was born!) and it is still relevant and useful to my classroom today. This is a really readable book that presents a good model for posing problems both for teachers and students. It has lots of places to stop and engage meaningful with the content and, even though I probably skipped over those bits too much, these really helped to make the points more meaningful.

I was going to summarise and review the book but instead I thought I’d focus on the big take-away that I am going to be using a lot. I think I was already doing this but it is definitely going to feature more and more in my practise – namely, what-if-not.

What if hoax

When we solve problems, our first stage is to accept the problem as given – find the area of the triangle, discover the connection between Pythagorean’ triples, etc – and solve it. We can use the habits of mind and toolkit that we have developed (will blog on this but I think John has done a much better job than I will over here). But, what happens then? Or what if we can’t get started?

Well, we could think about what made that problem up (it was about the geometry of a triangle or it was a list of numbers where a connection seemed to exist) and start to play with it. We can extend any problem by asking the what-if-not question. What if it wasn’t a triangle but a square? What if one of the sides was curved? What if we were interested in perimeter instead of area? What if there wasn’t an obvious perpendicular height? How many other triangles will have the same area? What if the numbers were surds? Immediately, a whole vista of Mathematics opens up and any problem that was quite closed can become open and rich. We can either make the problem smaller and more manageable or larger and more all encompassing.

The book then looks at a couple of examples in detail and details some action of problem-solvers at work.

I’m going to make a poster of what-if-not (that I’ll share here) and encourage students to think about ways we could alter a problem – adding it to the toolkit. This could work really well as a plenary or extension with students of all abilities as we encourage Mathematical thinking. I think it could also help broaden the depth of questions I get from Inquiry lessons – what if something from the prompt wasn’t true or was different? If we are thinking about students having agency and ownership of the Mathematics being taught, I like helping them recognise that any problem that I present to them is one I have found to be interesting and useful but it is not sacrosanct. This could be another way to encourage students to engage Mathematically.

But then again, what if I didn’t do that?


Image from http://www.transitionnetwork.org/stories/mandy-meikle/2012-09/if-not-transition-then-wat


What do Mathematicians look like?

Today, I ran two one hour sessions as part of a transition summer school being run with pupil premium money for upcoming Year 7s (11 year olds).  I wanted to get stuck into some problem solving activities first though, I wanted to dig into attitudes about Maths and Mathematicians.  So, I pulled out an old classic I’ve been using for years that I got from the amazing @PaulineMGaston.

I had students draw a picture of a Mathematician. I was really careful with my language here – I said “Draw a picture of a Mathematician and try to label it … for example your Mathematician may have a furrowed brow because they think really hard.”  (the labelling didn’t really happen this time)

We stuck them up with commentary from the artists and looked at them together:


Some interesting things:  All these people are well dressed, wear glasses and are slim (when I asked one student whether all mathematicians were slim, she responded “Well, you are”).  I wasn’t wearing a suit – I was wearing jeans and t-shirt because it’s the summer holidays – but the other aspects could have been from me.

There was only one girl – @missradders and @JusSumChick reckon the second to last one is a girl, the student assured me it wasn’t but it is quite ambiguous.

Having chatted about the pictures, I asked them a series of questions and hit them with a cut up version of this so I could reveal it one at a time:


We referred back to this at the end of the two problem solving activities to check which of these skills and attributes we used – helping them to realise that, yes, they are Mathematicians!

I really like this activity and do it at the start of the year with some of my classes.  I already have added @Rathematician ‘s suggestion of ‘Persist people’.

What else did I miss?  Mathematicians are ….


Orchestrating Better Discussions

41nct0yO-YLI can’t really understate how important I think discussion is in a Maths classroom, which is why the first teaching book I wanted to read this summer was this one – “5 Practices for Orchestrating Productive Mathematics Discussions” produced by the American NCTM.  I want to summarise briefly, give three good quotes and two things I’m going to take away from this.


This book looks at the key moves that a teacher can make to create a conversation in a classroom that has the potential to achieve meaningful learning.  I like the word orchestrate because that is what is involved.  After reading this book, I think I too often have students feedback after a rich task or investigation but don’t think enough about how I’m doing it.  This book gives a great structure to scaffold discussions: anticipate how the students are going to respond and answer, monitor how different students work and record this, intentionally select the approaches and students you want to share, sequence the presentations to be able to tell a meaningful story and connect the approaches to the underlying Maths.  The name of this blog is ‘joining up the maths’ and so this last point was really exciting to see.  I feel it is really important for students’ to see the links between concepts and subject areas and I think this book will help me facilitate more meaningful and productive conversations.

The authors also point out that for meaningful conversations there needs to be a meaningful task.  We can’t expect students to think deeply if we don’t provoke them and rather spoon-feed them algorithmic approaches.  As we more toward connecting our Maths curriculum more and more, we need to make sure we have good tasks that can help us do some of the heavy lifting (I’ll book more about that soon!).

Three quotes

“…the teacher’s role in discussions is critical.  Without expert guidance, discussions in mathematics classrooms can easily devolve into the teacher taking over the lesson and providing a ‘lecture’ on the one hand, or, on the other, the students presenting an unconnected series of show-and-tell demonstrations all of which are treated equally and together illuminate little about the mathematical ideas that are the goal of the lesson.” (p2)

“Sequencing is the process of determining the order in which the students will present their solutions.  The key is to order the work in such a way as to make the mathematics accessible to all students and to build a mathematically coherent story line.”(p44) Yes!! That!

“Although it is the thinking that goes into the preparation of a lesson that is important, creating some record of the decisions about the lesson is critical for two reasons.  First, the written plan serves as a reminder of key decisions so that teachers don’t have to keep all of the details in their heads.  It supports the teachers as they enact the lesson, reminding them of the course of action that they have set.  Second, the written plan serves as a record of the lesson that teachers can store for future use, revise, and share with colleagues.” (p82)

Take aways

I found this book through Fawn (@fawnpnguyen) and have really enjoyed her posts on deconstructing a lesson activity (part 1 and part 2). This book helped me think again about the moves I make while students are working that can help create more meaningful plenary and discussion.  I love the idea that by selecting and sequencing intelligently the students can tell the story and spot the links – I’m hoping this will reduce the need for me to tie together the threads in a deus ex machina.  I want to make sure I’m selecting and sequencing well and the anticipating and monitoring bit is vitally important if that is going to improve.  There was a suggestion of having a sheet that has the anticipated strategies and logging names and groups on it, so you could be prepared before hand knowing where the story is going to go.  While, I want students to be able to discover for themselves, there are particular threads I want to emphasise at various points depending on the learning goal.

The second takeaway is creating a future record.  I don’t keep anything meaningful in this way.  I have the student prompt sheet and the SMART notebook files but I don’t record any of my own thinking – so year on year I start from scratch.  As far as I know, this is the same for most Maths teachers in the England – maybe I’m just a slow learner!  But, as the rich tasks and investigations get added to, I want to record my anticipations and the student strategies that I didn’t think of to help develop my thinking and that of my team.


This is an expensive book, at almost £19, but I will be dipping into it and using it over the course of the school year.  I think it will make a difference to how discussions in my classroom will be used to more effectively achieve our learning intentions and gives me a framework to assess myself against.


Strategies to help all learners think Mathematically

This is a response to #blogsync topic of “A Teaching and Learning strategy intended to elicit the highest levels of student motivation in my subject”. See more at http://share.edutronic.net/

I’ve always worked in comprehensive schools, mostly 11-18, which means I can move in one hour from supporting a student who has difficult with number bonds up to 10 to helping support a small group of students for Cambridge entrance exams. I love this variety and I love the opportunities it gives me to see links that I might otherwise miss. I think this has helped improve my thinking as a Mathematician and has forced me to find and use strategies to allow all students to develop these skills as well.

I’ve drawn a lot on the work of Malcolm Swan, David Wright, Sally Taverner, Jo Boaler and Andrew Blair.

I’ve split this into two parts – strategies that help thinking and tasks that helps thinking. I’m working on gathering together a more comprehensive list – if you have any obvious ones I’m missing, please let me know!



This is a strategy that is used so much that it almost isn’t worth talking about. Having students think about an answer, share with their partner and then share more broadly helps everyone have a chance to articulate their answers. I find it is a good way of emphasising the use of Mathematical language and the importance of quality explanations.

Self-regulation Cards (inspired by Andrew Blair of http://www.inquirymaths.com)

I’ve recently had some of Andrew’s cards made up that students have on their desks – these provide prompts to help them self-direct their learning. They give ideas like ‘Draw a picture’, ‘What easier problem could help?’, ‘What other information might you need?’, ‘Ask a partner’ and ‘Ask the teacher’. Now, when I drop in to chat to a student I’ll ask them which strategy they are using. This has helped them become more independent and self-motivated.

Mini whiteboards

I don’t know what I’d do without these – how else can I quickly ask a question and process 32 responses instantly? I can quickly identify which students need more support and which concepts need more explanation. My only problem is, I keep running out of pens – do others have better organisational strategies for these?


Odd One Out

This is one of the easiest tasks to implement. Students are given a set of 3 numbers, shapes, functions or whatever and explain why each one might be considered the odd one out.

Using two-way tables or Venn Diagrams

For this task, students are given a large set of objects and then have to categorise them. This helps to pull out similarities and differences, developing Mathematical language and explanations. They then can design a two-way table or be given on to complete. I’ve used this for topics as varied as trigonometric graphs and rotational symmetry of 2D objects.

Multiple representations

To try to avoid focusing on technique rather than understanding, interpreting different representations of the same information can help make links that otherwise go unnoticed. This might be matching numbers (fractions, decimals, percentages, number lines), algebraic expressions (words, symbols, area diagrams) or statistical diagrams (frequency table, cumulative frequency curve).

Always, Sometimes, Never

When helping students to evaluate the veracity of their claims, it is helpful to ask is that always true? Sometimes? Providing provocative statements and having students develop their arguments is a really helpful task to identify key errors and misconceptions (eg multiplying always makes a number bigger, continuous graphs are differentiable, if you double the lengths of the sides you double the area).

These are some of my key moves – what are yours?


Filing cabinets and Toothpicks

I’m a big fan of 3 Act lessons and exploring how story telling can help to engage learners in asking deeper and more meaningful questions, as well as engaging in tasks which are more interesting and engaging.

This week I’ve had a go at two new (to me) activities with different groups. One went great, the other not so much.

First, was the filing cabinet activity from @mr_stadel. I’ve seen this before but always at a moment when it wasn’t useful. I’d tagged it this time and used it as a way into talk about different types of units. The kids developed questions:


and then answered the questions


and we engaged in some really meaningful discussions. Perfect! Really set the scene for alternate forms of measurement.

The second activity arrived on my blog reader last night. I’m in the middle of a unit on sequences and graphs with my Year 9s and so @ddmeyer’s new toothpick activity seemed perfect. But, whether it was because I misjudged the activity or the group or if it was last lesson on a Friday, it flopped. We had two methods going on and we got close to an answer but there was too much angsty frustration for that stage of the week and not enough engagement for my liking. I need to rethink how we run with this one next time.

What is your favourite 3 Act task?


Space and Breakages

I have the privilege of teaching in an 11-18 school and, since I started there, I have had a class in every year group.  I can go from teaching remedial Year 7 (11 year olds) to high flying, Cambridge-bound Year 13s (18 year olds) – sometimes within the same hour!

When I teach A-Level, I teach mainly Pure Mathematics and Mechanics – I don’t really like Statistics beyond GCSE and find Decision a bit dull (though we do normally teach D1 as one of our applied options).  I have a lovely Year 13 group that happens to be mostly Physicists, so they’ve covered lots of the content in Mechanics already.  Today, we were starting to look at momentum and impulse.  I found two great things to stimulate our conversations which I share with you now.

Firstly, this:

A video about momentum FROM SPACE!!! I’m a bit of a space nut – I watch live streams of Mars rovers landing and the like.  I was so excited when I found this and loved the clear explanations and awesome visuals.    We talked about this for ages and then applied it to some problems.

The second hook was an idea I got to demonstrate impulse.  You know the trick where the guy pulls the tablecloth from under the glasses and they all stay standing …. well, that!  It works because the force acts for a very short period of time and so the momentum doesn’t change much.  Well, that’s what is supposed to happen.  I don’t think this is supposed to happen:

Failed Impulse Demo from Kevin Cunningham on Vimeo.

Needless to say, there was much hilarity which you can hear in the video.  When we watched the video on the projector screen we were able to discuss why it didn’t work and troubleshoot it for *next* time.  I don’t think I’d use stiff card or have the glasses quite as close to the edge or give up like I seemed to half way through.  Watch it again … you know you want to!  What else did I do wrong?

Ever had a really good idea go wrong like this?  What is the most memorable hook you’ve used?


Tarsia Jigsaw Revision

My Year 10s are having an assessment next week on everything they have done since September.  I’ve been trying to help them organise their notes, work out what they need to revise, help to address misconceptions through the year and we’ll see how they have retained that information when I mark their test.

Today, I wanted to help them revise so I prepared a Tarsia on fractions – everything from converting between mixed numbers to improper fractions, the four operations and the like – and got them into groups to do it.

image-2As well as the Tarsia, I produced an A5 list of the topics that were being covered.  The students were expected to complete the puzzle as a group and make any notes that they needed to help with their revision.

There was a good buzz in the room – if you haven’t used these puzzles before there is free software to make your own here and there are a ton of them on TES and Mr Barton Maths.

Do you use these?

Are there better ways to use them?  I’ve tried printing them out big and using them as a whole class activity – I’ve tried each person having 6 of the triangles and then putting them together (also as a jigsaw activity) – I’ve tried smaller ones that are individually completed.

What do you do with them?  I decided to do these ones on cards and keep them – other times I have students stick parts of them into their books or construct some equilateral triangles and make up their own or take a photo with their phone as evidence of their work.

What other revision strategies can I be rolling out?

As an aside, I cut these out with a guillotine and so had some nice regular cut-offs – some trapeziums and triangles – I’ve put those aside for another lesson.


Probability Statements & Scales


I was thinking about probability with my lovely lower set Year 7s today (trying to think how to describe them makes me want to write a post about setting). They each had a statement and had to get into order from least to most likely – I couldn’t find a washing line. This worked really well. We then introduced the concept of a scale using percentages and words (0%, 25%, 50%, 75% and 100%) and tried to come up with our own events. This caused uproar as always!

I preempted the conversation by reminding people to stay calm, even if you disagreed, and to indicate that you want to agree or disagree rather than yell at the person making the claim.

One girl, “It is impossible that I will grow a moustache.” [Lots of disagreement!]

One boy, “It is certain that I will come to school tomorrow.” [Even more disagreement with some really good arguments]

We dealt with those (and other) misconceptions and then they began to construct their own probability scales – my go to activity for this since my PGCE year thanks to @PaulineMGaston. Next lesson we will talk about fraction and decimal equivalence some more before we talk about how we work out the probability of simple fair events.




Stumbling into term

Today was hard work. Having a new born at home has really messed with my sleep patterns – last night I couldn’t sleep to near 1 and then I was up at 6.

I taught three hours and had two PPA lessons – didn’t really achieve much in those but started surd unit with my Year 11s. Pitched the start too low but pulled it up quickly.

My Year 8s had fun with the start of a ratio unit. We used @numberloving’s Ratio Fan and Pick (check out the website for top resources!) though we picked up 2 things we thought might be mistakes.

Hoping to sleep tonight and feel a bit more creative and on it tomorrow. I meant to take a picture of some of the things we did today! Boo!