Learning Together

Over the last 9 months I have been penpalling with James, a student studying for a Masters in Maths (he loves everything from the history of geometry to application of ODEs to engineering, but specialises in Number Theory), while also a prisoner at HMP Grendon. In order to go to Grendon you have to volunteer. Prisoners have the power to ask fellow inmates to leave, if they become too antisocial. Grendon is specifically a therapy prison (art, physco-drama, poetry, music, gym…).
Today I was lucky enough to get the day off school to go celebrate his graduation from Learning Together. Learning Together is a collaboration between the prison and Cambridge University. Students, from both institutions, meet once a week for 9 weeks to discuss, listen to lectures, write essays. They can study Criminology or Literary Criticism.

Note plenty of time for mingling and reflecting
Here are some observations from the day:

  • The room was so supportive of friends as they took to the stage to share their experiences of learning. Learning as messy, difficult, rewarding. No unkindness ever hinted at, such ,warmth and love from everyone. 
  • I watched as groups of learners, from university and prison, bantered away, completely at ease with their friends. When we used to go in for a day’s singing workshop this level of collaboration was never quite reached.
  • Learning or Togetherness – which is the more important? Learning about the academic definition of Legitimacy or being open to people from seemingly different worlds to yours?
  • Ruth and Amy, co-founders of the scheme, are an incredible team. Great vision, drive, humour. They believe completely in what they are doing, so humbling and great to see! (FOFO – “full on or f*&^ off”)
  •  By the end of the day I was unable to accurately play the game “University or Prison?”. Nor did I want to. Everyone was a learner.
  • Ruth spoke about the close relationship between brilliance and brokenness. In order to reach academic brilliance you must first become aware of and accept the ways in which you are broken, and the ways in which our society is broken.

I finally met James! Former professional wrestler, tapestry-artist, number theorist, beard-nurturer. Quiet, kind, fascinating. In a room full of bustling conversation we, the nerds, sat in a corner and worked through some geometry problems, getting confused about scale factors and applications of Pythagoras. 

Our scribbles – we made friends through a problem…

We spoke about education in prisons. Despite the lip-service, there is only funding in prisons for English and Maths to Level 2 (equivalent of a C at GCSE). If you want to o beyond this, you can study through distance-learning with NEC for A levels, or with the Open University for degrees. Four phone conversations with a tutor, and many lonely nights wading through textbooks. 
46% of prisoners have literacy that is below that expected of an 11 year old (three times the proportion in the general population). For Maths, 52-65% (depending on sources) have numeracy that is below that expected of an 11 year old (shockingly for the general population it is still 49%). 80% of prisoners reject education (I couldn’t find the equivalent stat for the general population, or what this statement really means). 

Learning Togehr: Maths?

James and I would like to set up a scheme, similar to the ones for Criminology/Literary Criticism, but for Maths. Should we focus on numerical competency or instil a deep love of the subject? James spoke passionately about this, taking the words out of my mouth – “Give them the love and they will go away and learn the nuts and bolts as a conesequence”. Prisons provide basic numeracy education, lets give something that only we can provide. (Compare this to the excellent One to One Maths Charity, where prisoners teach each other basic numeracy). One idea would be to organise an intense 1 week summer school, during the 2 week slot in the summer when therapy sessions do not run.
Learning Together is so successful because it brings two groups of people together, who would not normally meet. Who would our second group be, given we were thinking of doing this in the summer holidays? Students about to start their first year of uni? Students at local adult education colleges? Old peoples’ homes? Staff at the prison?
James taught me an excellent phrase – it is “quicker to plait fog” than to use the prison computers. No graphing software to be used here then… He taught me about partition theory and the maths of juggling (originally developed for its own sake, and now with applications to computing).

We spoke of primes…

Blogs: Why, How, What?

Why?

  • To work out what you want to say, and say it well. (Jeffrey)
  • To curate professional portfolio of your practice
  • To help formalise being a reflective practitioner
  • To share expertise and ideas with other people
  • To inform your performance management
  • For your CV

How? 

In a 45 minute session, focus on deep thinking rather than presentation/organisation. If you are not confident with writing an actual blog post then either:

  • Ask someone for help (or look at the tips we gave last time)
  • Write your post as an email to yourself, in a google doc, in a note on your ipad. Worry about building the actual blog another time – the thinking is the important thing.

We will share with a partner our thoughts, at the end of the session.

What?

Stuck for something to write about?

  • Put the flesh on Jeffrey’s story-skeleton –  “Basically, you’ll reach a point in teaching when…”, transformative anectode, “Have you ever… ?”, “Picture the scene: You… “,  convert abstract idea to tangible metaphor. Humans are addicted to stories. Patronise your reader. “So the next time you find yourself…”
  • Reflect on your favourite lesson of the last few weeks – why was it a success?
  • Reflect on your performance management targets for the year – how are you achieving them?
  • Share your snappy findings from this term’s CPD modules?

Examples of blogs 

Floor Functions

Every Sunday afternoon I spend a happy hour in a cafe with Desmond, a student preparing to study Maths at university. We struggle in the darkness, wading through difficult problems. It is beneficial for both of us. This week, a problem from UKMT (3.5 hours, 4 questions to think about).

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Progression of the problem:

  1. Guess it has something to do with odds and evens, and try that.
  2. Give up
  3. Calculate the first few values by hand
  4. Create a graph of the function on DesmosUntitled picture.png
  5. Realise that the number of factors of n is somehow important
  6. Claim that  Twin Prime Conjecture and Mersenne Prime Conjecture are both true
  7. Get sad when our phone tells us neither has been proved yet
  8. Talk lots about numbers being dragged up or down

 

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Incomprehensible

I really enjoyed applying the skills I had been trying to teach at school – chunking the problem, drawing a picture, making links. We started to understand the problem a bit, but definitely were not near to a solution. See strategies at end of this post.

 

Final stage: ask girlfriend, who happens to be doing a PHD in Maths, to solve it for you. In her words:

“I thought I would want to use numbers that I could understand the factors easily. I realized that if I understand the factors of n, then I don’t know anything about the factors of n+1. So I could do it in a straightforward way.

And your graph showed that it increased on a large scale, but not on a small scale. So I guessed, and knew I would want to approximate above and below”

Knowing to play around with powers of 2 shows great intuition, learnt from many years of practise.

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Problem solving at School 21:

 

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The School 21 Problem-Solving Toolkit
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My attempt at linearising the problem-solving process

Playing with STEP

Thinking about STEP – excellent transition from School to University Mathematics.

Spent an hour in a cafe working with Desmond, a student who is doing STEP this year and is about to go to Oxford to fill his brain with prime numbers.

Here are some links:

We struggled with this question, mostly conceptually, for a pleasant head-scratching hour.

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When R is (2,1), the situation is shown below.

  • The blue line is the scenario that gives the smallest sum of distances OP and OQ
  • The red line is the scenario that gives the smallest distance PQ

We both strongly wanted the blue and red line to be the same line! It turns out they are not… Strange.

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Spreading the Word

I am talking maths to two people outside of my school life.

Firstly, to J, currently studying Maths while in prison. We communicate via letters – the thrill of receiving an actual physical piece of paper is rare and powerful. Early days at the moment. The current system is that I tell him of rich problems I have found while planning lessons (the advantage of a properly rich problem is that it challenges anyone, from 5 yr old to professor), and he writes about how the men in the education block get excited solving them – “I’ve got to say, that’s really given me a bit of an adrenaline rush” he writes after finding a pattern in the Graceful Tree Conjecture.

Secondly, to D, currently in a gap year and applying to Oxford to read Maths. His school in Hackney does not have any maths teachers that are comfortable enough with their content knowledge to stretch him enough ready for University. We spent a focussed and happy afternoon in a local cafe drawing fractals and sin(1/x) graphs. He muttered under his breath as he worked, rapidly making connections, scribbling down tiny notes as he went. A joyful reminder of calculus and difficult curve-sketching beyond y = mx + c.

Red curve approximates parabola close to origin, but approximates straight line far away

 

It is an honour to be asked to talk about my subject in a variety of contexts – I want to do more of it!

Oracy in Maths

 

Alongside Heather, I delivered a session on Oracy (using talk in a dialogic classroom to aid learning) in Maths and Science. Visitors can often easily see how to embed talk into traditionally “softer” subjects, but what happens when “there is just a right way to do things”?

Ways to use talk in the maths classroom:

  1. Through games
  2. Through physical structures
  3. Through talk structures
  4. Through rich tasks

Through games, such as skribble, just a minute, pictionary, articulate, taboo. Often the most common and easy way to introduce talk. One example I am currently enjoying is “Which one doesn’t belong?” , a quick and easy way to spark debate. Easily adaptable to the current topic.

Through physical structures in your classroom. I am a bit obsessed with my whiteboards, which are useful because:

  • Everyone can see all the work
  • Knowledge spreads quickly across the room
  • The non-permanence means there is less fear of starting
  • Formative assessment is easy, teacher can survey from centre
  • Students are naturally encouraged to talk to each other

More thoughts in a blog here, and example of whiteboards in my classroom below:

Through problem-solving structures. These could be sentence stems, group roles, timed protocols, toolkits… One example, devised by Rachael, is an adaptation of the coaching model.

  1. Work on the problem for 5 minutes in silence
  2. Coachee talks for 3 minutes (with talk prompts and key vocab visible for support) about what they have done
  3. Coach responds for 2 minutes, with further questions and clarifications.

Finally, and most importantly, through rich tasks! This might be a cop-out on my part, but if:

  1. The students desperately want to solve the problem
  2. Any individual student is unable to solve the problem alone

Then talk will arise naturally as the easiest way of communicating ideas quickly and efficiently between thinkers. Our job as teachers is to facilitate this, with a few well-placed structures. Talk for the sake of talk is banished.

Some places I go for rich tasks: