#MTBoS Mission 1: Rich Task – Painted Cube

I really wanted to be involved in the MTBOS (Matho Twitto Blogo Sphere) challenges this year – I was a bit lame last year and never got off my bum to do anything about it.  I’ve been a bit late on last week’s Exploring the MTBOS challenge but I thought I could catch up today.

I wanted to talk about a problem that my Year 8s have been working on this past week – The Painted Cube.  Now, this used to be one option for GCSE coursework and I’ve used it for that in the past.  I think it is a pretty good example of a ‘Low Threshold, High Ceiling’ task – the problem is easy to pose, everyone can do something and the students can run with it, extending it to the limit of the their Mathematical ability.

Here’s the problem:

3x3x3Imagine a large cube made up from 27 small black cubes.

Imagine dipping the large cube into a pot of yellow paint so the whole outer surface is covered, and then breaking the cube up into its small cubes.

How many of the small cubes will have yellow paint on their faces?
Will they all look the same?

nRich have a nice animation to help with the launch here.

There were a number of skills that I wanted to be used during the few hours that we were looking at this task.  I wanted students to collect results systematically, I wanted them to look for and spot patterns, to come up with general rules, draw isometrically, consider the volume of cubes and cuboids and to relate their rules to the physical reality of the problem.  I knew I could encourage some to look at graphs to summarise and present their results, to begin thinking about algebraic proof, expansion of quadratic and cubic brackets and, potentially, propose results in 4D (sadly, none of my 12 year olds got that far this time round).

I had students working in pairs or individually – I gave them access to some cubes but encouraged them to move towards visualising using diagrams because we didn’t have enough cubes for a 10x10x10 example. Lots of discussion and lots of excellent thinking ensured.  The quality of the questions was superb and the student support for each other was excellent.  Some students weren’t clear on drawing isometrically, so a student expert demonstrated with the visualiser while another narrated and answered questions.  There was some tussling over the multilink cubes but that calmed down after a while.

We spent two hours of lesson time looking at this problem.  We looked at finding nth terms earlier in the term and are about the begin a section on volume, so I thought this was a good place to drop this problem in.

I’ve asked every student to spend another hour on their own for homework to take the problem further, so I’m looking forward to see what they bring back after the weekend.

I love that from a simple problem we can get so much Maths that connects to everything else – when they come back on Monday we’re going to mindmap all of the Maths we used in this one simple problem and marvel at the wonder of Mathematics.

Using the Moodle Lesson Module

I do like Moodle – not many other teachers in my school use it at the moment and so I do have students asking me why I’m obsessed with it. I think that might be stretching the truth but I worked out a way to use the Lesson module today which really excited me. 
I had my Year 11s period 5 today and I was trying to get them to investigate what happens to the area and volume of cans of drink when you double and triple the lengths. Unfortunately, this was severely hampered by  their inability to calculate the volume and surface area of a cylinder. I thought that post-mock this group of C+ targeted students might have been able to do that – not so. 
There were a lot of mental blocks going on and there were several students missing – I worriedly scratched my head and asked if anyone would find it helpful having a video explaining this. Flipped classrooms eat your heart out. A few did, so I put it on my to do list and stated to do this after school on my lovely ipthen school did not provide this) and my new stylus that I got for Christmas. Then I remembered Moodle lessons. 
The lesson module lets you set up a series of pages that students can navigate by making choices – think choose your own adventure. So, I knew I wanted to assess all my students on their ability to work ut hte volume and surface area of cylinders in various orientations and with varying information given.  But, now  before they answer some questions they have the choice of watching a screencast from me, a page from or just jump in and do it. It finishes with an evaluation into how useful the whole exercise was. If they like it I’ll do a few more and then get them to start designing their own. 
It was quite confusing to set this up. I had to draw a map of what I was doing and refer to it quite a lot (see below). As with everything on Moodle there were options galore – I kept it as simple as I could. All told, it took about an hour to do the screencasts, set up the pages and write the questions. I can’t give guest access to school Moodle but if you want a copy of the module to upload to yours I can do that. (you’ll probably want to change the videos though – my students can understand my Northern Irish accent but I’ve taught them for a while now!). I think I’d be able to do another one a little bit quicker but not by a lot. 
(the plan is on my desk at work at the moment but I’ll upload it here soon – then this sentence won’t be here). 
Have you used the Moodle lesson module? What has worked? Did I make it too complicated? Are there easier ways to achieve what I did?

Remembering formula – teaching for recall


So, today was the first day back to school after the Christmas holidays – back to the grind in 2012. Today was a pretty busy day – like most UK schools every day of the week is different for me. I thought I’d try to think of one thing I could record on here each day for a while, both as an exercise to encourage me to keep thinking about my practice in deepest, darkest January and to remember things I have done that have worked. 
I had my Year 9s today and we started to think about circles.  We spent some time drawing them, labelling the parts and then measuring to try to find any pattern. Most were on board and, even with the dodgy measurements, we could see and explain that the diameter was twice the radius and also that the circumference was about three times the diameter. We had a conversation (maybe it was more a lecture, but they were interested, honest!) about Greeks, irrationality and pi. I introduced the formula C=pi*d and we thought about that and saw how that might help with any circle we might meet for the rest of our lives. 
Over the years, I have tried a number of ways to help students remember this formula and the area formula. Today, we did my favourite – hotter/colder!  Here’s how it works:

  • Pick an object to hide
  • Send a student out of the room
  • Hide object
  • As student enters whole class chants “c equals pi d” on loop
  • Closer = louder, Further = quieter

Simples. I liked it and the kids seemed to get it. I bumped into one later in the day and said “C equals” to be met with “pi d”. I know on its own it’s not teaching for understanding but it was fun, they understood it and normally they remember it. I see them tomorrow, so let’s see how that goes!

Do you have any innovative ways to remember formula? Any classic songs you sing or play for your class? 




Tac Tiles

I’ve been inspired by something I saw on another blog. Simply take a picture of something I’m working on and upload with a comment or two.

This is a Tac-Tile tray and I’m currently using these to help with algebra in KS3. They link geometry to algebra and have helped my students lot. I’m able to use them well for collecting like terms but I’m trying to think how to expand on this. The kids love them.

They were originally developed by DIME in the early 90s but they don’t seem to be in production any more. I had my set laser cut by a local school design department at £3 a pop. Pretty good! I’ll put more up about these as I develop resources for them.

It wasn’t like this in my day …

Sometimes it is embarrassing admitting that I teach Maths.  I’ll be at a party, my job will come up in conversation and then people will tell me about how much they hated/loved Maths – how they are terrible/great at it.  Sometimes, that is where the conversation will move on to more interesting things and I’ll sigh with relieve.  But, every once in a while, I’ll be treated to the erudite insights that my new friend has to share on the nature of the educational system and the failings of kids these days.

What I always find uncomfortable about these conversations is the subject of expertise.  After all, everyone has been a student for at least 13 years, surely everyone has a valid opinion about what makes a good school, a good teacher and a good set of qualifications.  Well, I’m not sure I agree.  I’ve trained to do this job, I’m studying for my MA and I work on engaging and enthusing 200 students a week – maybe my opinion should carry a bit more weight in this conversation?

I’m still digesting Carol Vorderman’s report and I’ll likely be posting about it for a while.  I was really interested to see that some things never change:

Employers and their organisations also complain about the low level of mathematical competence of their new employees and now many hold numeracy courses in mathematics to allow them to ‘catch up’. p4

This is the same reasoning that was given for the commissioning of Cockcroft in 1983.  Maybe things were better some mystical day in the past or maybe, just maybe, the raising of the leaving age of compulsory education is of importance here.

We have had four different qualification systems since the Second World War:

1) School Certificate

2) O Level

3) O Level and CSE


The report talks about each of these and points out that none of the systems worked for every student or every employer.

Major problems have been associated with all of them.

There never was a Golden Age. There have been times when we have met the needs of our more able children at the age of 16, but we have never had a satisfactory provision for the whole cohort. p51

I think this is really sad.  We do need to be designing curricula and qualifications that are engaging and relevant to all of our students.  I agree we can’t continue to watch 300,000 students a year be classed as ‘failures’ when they don’t achieve a C grade at GCSE.  We need appropriate and worthwhile qualifications that allow for success and progress in a meaningful way that employers will respect and take on board.  Why should the foundation curriculum be a cut down version of the higher?

nor is it good enough for the system to gear itself to them [the top 15%] and for everyone else to receive a trickled down version of their requirements. p3

We need a better system.  I want to be part of that.

In the past, end-users [students] and teachers had much greater influence over school mathematics and we advocate a return to that situation. p5

Bring it on!

Maths Curriculum Design in the UK (England in particular)

When the National Curriculum was published there was much furor the impact this might have the individual classrooms around England.  Teachers were furious to be forced to teach a given set of topics, in a given order and to be have to use the level descriptors imposed from outside.

[Update: Adam points out I made a mistake here.  While the NC set out things that needed to be taught, it wasn’t until the introduction of the National Numeracy Strategy that order was felt to be imposed.  Of course, this was non-statutory but often wasn’t treated as such.]

That was long before I joined the teaching profession.  I’m about to start my 7th year as a qualified Maths teacher.  All my career I have worked in the state sector, working in different roles in 4 different schools.  All of those have used the Sample Medium Term plans from the National Numeracy Strategy as the basis of their schemes of work.  There was some tweaking and developing, but the underlying structure was the same.

With the ongoing curriculum review and the assurances from Whitehall that teachers are going to have more autonomy, I’m worried – are we ready for it?

In the recent report, “A world class mathematics education for all our young people”, this comment is made:

In this country we have a long tradition of successful innovation in curriculum and pedagogy, particularly in mathematics, science and technology. This has come almost entirely from the grass roots, from individual teachers, small groups or charitable organisations, and has often been done on a shoestring. By contrast, many large government initiatives have not been particularly successful but have cost a lot of money. p24

In my experience, this long tradition was largely suspended with the advent of the National Strategies.  There is still work going on (Bowland and Nuffield come to mind) but this is only brought into most classrooms when it fits neatly into the pre-existing schema.  I agree with the report when it says:

There has been a culture of policies which are non-statutory being almost universally viewed as obligatory by teachers and schools, due to the government agencies’ reliance on them for their tick-box style of assessment. This has not helped the mathematical education of children. p8

With the dismantling of the strategies and the support that has provided for years being removed, I’m concerned we aren’t ready to take on the challenges of developing innovative curricula in our own departments.  I’m not convinced that the spiral delivery of the Sample Medium Term (SMT) plans has been effective but equally I don’t know how many teachers of my generation have been equipped to be able to design something better.  While I welcome the opportunity to unfurl our wings and allow innovation back into the development and design of the mathematics curricula on a local and institutional level, I fear the supports are being removed too soon and that rather than innovating we are at risk of stagnating.

Am I wrong?  Is your department already using an innovative curricula that bears little resemblance to the SMT plans? Is the spiral curriculum a keeper?

Brain Rules – John Medina

I’ve really enjoyed reading Medina’s book. I bought it three years ago, started a difficult job and put it down having only read a few pages. Picking it back up to engage with and think about has been really helpful.

Summer is that time in which lots of teachers move into dry-dock and tweak those obviously failings, reflect on what worked and what didn’t. Other colleagues forget entirely about the classroom and don’t think about school again until Sept 1st. What has been interesting this summer for me, is that I have been excited to keep the learning going, supported by Twitter, my Google reader, my to-do list and my notepad.  Another sign that the bailing of water has stopped, the storm has stopped and the refit of the teaching boat can continue unabated.

In his book, Medina sets out 12 rules that he derives from recent advances in neuroscience and fully supports all of his claims with peer-reviewed research.

The links this makes with everything I’m thinking about – assessment, the importance of the visual, the resetting of the ‘ten minute’ attention window – is great.  A real sense of everything being connected.  When I think about my learning as a teacher I see lots of important parallels.  I don’t learn well when under continual social stress, I don’t learn well when I don’t sleep or exercise enough and so on.  Medina gives some suggestions for further research and ways his findings might be implemented in the workplace and school.  As I think about how these findings might change my living and working, reflecting on how I can use this information to help my students then I feel the book I’ve read has been worthwhile.

Highly recommended.

Standards Based Grading (SBG) in the UK

This has been a fun year. I went from being a Head of Department in a “national challenge” school to working as a normal member of a Maths Department in a school that isn’t doing too badly. It took a while to get used to that shift in gear but I’m there and ready to start changing things again in my classroom.

Term finishes here on Wednesday and so all meaningful teaching is finished. I’ve been reading a lot about Standard Based Grading and seeing similarities with some of the work I have been trying around Assessing Pupils’ Progress. The focus on that has been using dialogue as an assessment tool, discussing with students and using probing questions in meaningful ways.

I really like Standards Based Grading from what I’ve seen but when I look at the systems that others have implemented, I have a quandary.

Quandary 1: Spiral Systems

Most of the systems I have seen are based on the American model of Algebra I, Algebra II, etc. but I teach in the UK. My school , like most schools in the UK, works closely to the Sample Medium Term plans published to work alongside the Framework. These have been added to and developed, probing questions and rich tasks have been added, but the structure is predominantly the same. For me, this seems to be problematic.


Because this is a spiral based system. In the first year of secondary education kids study bits of Number, Algebra, Geometry and Statistics. In the second year, they study most of that again but only in a bit more depth. And in the third.


I guess I’m a bit daunted in thinking about implementing this system in this context. I have to cover quite major concepts in each year in most areas of mathematics. I have the state published level descriptors and objectives, but I feel like most lists are the very similar for each year. Or at least, I feel like I want evidence of all the previous standards when I am faced with a new class.

Quandary 2: 3 hours a week versus every day

Another thing which seems to be different in the UK is that we don’t see our classes every day. I see a class for 3 hours in a week, at different times on different days. I have seven different classes to prepare for, at 6 different grade levels and across the ability spectrum. Should this make a difference?

Quandary 3: Teacher Assessment

We have two grading systems in the UK dependant on the age of the student. From reception (K) to Year 9 students are given a level from 1 up to 8 for each subject. In Maths, this is broken mainly broken into the 4 areas I listed earlier (Number, Geometry, Data, Algebra) with an extra one called “Using and Applying”. The same assessment criteria is used all the way through their schooling and kids are expected to make levelled progression through each year. So, in a class I might start with level 5 students and be hoping to take them to level 6 or so by the end of the year. I don’t normally get a break-down of the strands but one grouped grade and so have to work out where the students are and work from there.

After that students are externally assessed and are graded at 16 (GCSEs), 17 (AS Levels) and 18 (A-Levels).

So, I train my students to know what level they are at and why. Show them how to make the next steps and we go! But, with five strands going on there is a bit of criteria overload.


I want to do this – I’m willing to put the work in over the next month or so to prepare for it and the next years to tweak or redo it – but I want to know if anyone in the UK has already done some work on this and can give me some advice? Is this something you’ve thought about and discarded? If so, why? Are these problems distinctive to the UK?  Is the UK solution just to have previous level, current level and next level for each strand?

I’ve found it really helpful reading old blog posts from Kate Nowak, Dan Meyer and Jason Buell but would really value some UK based thinking on this. Or someone assuring me that the situation is so similar I should stop hesitating and just get on with it.

Interesting New York Times Article

Interesting article about whether training teachers or paying them more is the best way to go.

Some interesting points about MKT (Mathematical Knowledge for Teaching) – evidently teachers who score higher on this knowledge achieve better results [see Deborah Ball]. Evidently, no closet to knowing how to ‘teach’ this knowledge though, so just another measure to see whether the teachers you already have are able to do their jobs effectively.