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…

Underground Maths

An engaging day thinking about rich A level resources, with Underground Maths. The website is crammed full of rich tasks with plenty of teacher support. A useful reference.

We did this Starter activity. A many-ways problem – either very algebraic or very graphical.

IMG_5332.JPG
My pre-work. I enjoyed putting into practice the techniques I ask my students to use.

A problem that doesn’t look hard, but actually is challenging. How does this affect motivation? Would I prefer a problem that looks really scary but actually is pretty simple?

This would be a great problem to throw at students before they know how to differentiate – if this is the headache then what is the aspirin?

Mathematical Mindsets

We discussed things that we would like our students to think about mathematics, using these cards. Some key thing that I want my students to think:

  • It is good to get stuck. (Teachers need to model this). Also, it is okay to get stuck-in!
  • There are no right answers in maths.
  • Problem-solving is the most important thing in Maths.
  • Good problem-solvers do well in exams
  • Understanding > memory (within reason)
  • If you pass an exam using death by practice paper then you might get a good grade but you will not do well in life. You are merely delaying your failure.

Get stuck-in. Get stuck.

Timing a problem

There seem to be three times to use problems:

  1. Before you know the techniques. Motivation for learning techniques
  2. Just after you have learnt the technique. Application of fresh knowledge
  3. A long time after you have learnt the technique. To encourage connections, recall, re-deriving knowledge.

From GCSE to A level

There is an excellent transition problems bundle, using GCSE knowledge but requiring deeper reasoning.

  • Between the lines (why is there an actual answer for this!)
  • A tangent is
  • A perfect fit (with an excellent extention question: how many pythagorean triples make a triangle with inscribed circle of radius 6? We found 6 integer solutions. What about a circle of radius 7? We found 3 integer solutions. What about a circle of radius n? Lovely opportunity to use excel or python)
  • Name that graph
  • Quadratic solving sorter

You cannot do this task without knowing what an asymptote is. This video clip shows students struggling to use the equation of a circle to find an asymptote (thus showing a deep misconception), without any teacher intervention. When do we leave students to struggle and when do we step in? Does the teacher need to know about every mistake students make?

 

What is 0 the power 0? You need to know the answer to properly understand this problem. Outstanding problem generation.

 

We believe that there should not be a right method or a right answer. This problem fails this criteria.

 

Modelling

MathConf9

To a school in Bristol to share and learn with plenty of nerdy Maths teachers. Who knew – not all teachers are under 35! School 21 is a bit of a bubble…

We did speed-dating with people around us, to share ideas and make friends. Here is my favourite:

  • Type a random 3 digit number into your calculator
  • Divide it by 7. Hands up who got a whole number?
  • Divide it by 11. Hands up who got a whole number?
  • Divide it by 13. Hands up who got a whole number?
  • Now, get you 3 digit number and type it in twice (for example becomes 456456). Divide this by 7, then 11, then 13. Hands up who got a whole number?! What number did you get?!

7 x 11 x 13 is 1,001. How could you extend this problem? 101 is prime, 10,001 factorises to 73 x 137, 100,001 factorises to 11 x 9091. Hmm.

 

Talk 1: From Abacus to Zero

By Ed Southall (who shares excellent problems on twitter)

 

We explored trivia about the etymology of mathematical language. Fun, and sometimes useful to hang meaning on, but not much maths. Here are some titbits:

  • Calculus derives from a word for pebbles (maths used to be done by counting pebbles). You can have calculus on your teeth – a term used in dentistry to describe tiny pebbles of plaque.
  • Which words connected to one?
    • Atone
    • Reunion
    • Alone
    • Only
    • None
    • Condone
    • Onion
      • All, apart from condone. Onion means “the big one”. Pearls used to be called onions.
  • If only eleven and twelve were oneteen and twoteen are rules for naming numbers would be almost sane. We are lucky – in Danish the word for 54 translates as “two-and-a-half-of-twenty-and-four”
  • Factor and factory have common root – they both build products (two puns in that sentence).
  • Linear and lingerie have common root – they both derive from the word for thread.
  • Average is an abbreviation of “averie damage”.  When a boat gets broken, how do you work out how much each investor has to pay? You share the total equally between the number of investors, obviously.

Enjoyable, but unsure of how I will use this to motivate students who drag their feet when entering the maths classroom.

 

Talk 2: Two A-Level Topics

Geogebra links:

 

Two friends who did their PHDs together are now writing a textbook for A level, chock-full of proof and links to applications, both at university and in industry. We explored geogebra apps to visualise differentiation from first principles and Newton-Raphson Method for solving messy equations (most equations are horribly messy).

 

Why do we care about Newton-Raphson method? Because it is used in Formula 1 car design, in Google searches, in weather forecasting… This argument left me dissatisfied – either show me explicitly how it is used (although this is presumably horribly complicated) or I won’t believe you.

 

One thing that got me excited was thinking about when the Newton Raphson method would fail (for example if you get stuck in a loop) – for me that is more mathematically profound than tenuous links to the weather.

 

Talk 3: Algebra Tiles!

Mark McCourt, Chief Excec of La Salle Education, ran an excellent session on the use of algebra tiles, from early primary right up to A level. Key thesis: we jump straight to abstract representations of mathematical concepts far too quickly in a topic, at far too young an age.

1
Don’t just focus on abstract.

2

Claim: they are all the same question.

Bus-stop notation tells you exactly what you are doing. Given the area of a rectangle and its height, what is its length?

Bus-stop division is exactly as hard as dividing algebraic polynomials – you just shift from base 10 to base x. If you understand this deeply enough using algebra tiles then all will be well.

3
Bus-stop division with Dienes blocks generalises to algebraic division
4
The red side of a tile represent a negative number. To get rid of a positive number you have to add its negative version to both sides.

 

Flipped Learning at Shireland

Whole Education have organised a year-long action research project focussing on Flipped Learning. The launch event was kindly hosted at Shireland, a school in the suburbs of Birmingham with 30% live safeguarding issues and 64% EAL.

Flipped Learning

As defined at the conference, flipped learning is the following process:

  1. Students learn knowledge, before lesson (through videos or other means)
  2. Teacher, also before lesson, assesses their understanding (this could be through apps that check which students have watched videos, through multiple choice questions…)
  3. Teacher plans lesson based on a secure awareness of each child’s understanding
  4. Finally, the lesson happens. Teacher liberated from explaining from the front, students able to leap straight to evaluating, analysing, creating.

How is this different to…

  1. … traditional homework? Traditional homework involves students applying knowledge they learnt in class, after the lesson. This higher-order thinking surely requires teacher support and so should happen in lessons.
  2. … pre-learning? Pre-learning is where students do a bit of learning before the lesson, but the teacher walks into the lesson blind – no assessment of understanding has taken place.

Shireland’s Philosophy

  • Technology should be mundanely clever. Every teacher, not just the geek in the corner, should be able to use it and see immediate benefits in their classroom. Technology works for us, we are not dictated to by technology. Fix existing issues to ensure all teachers are motivated.
  • Flipped learning is not about the videos, but about the pre-assessment. Spend your time designing excellent assessment, not whizzy videos.
  • Replace 15 learning assistants with 10 people on an E-learning team, who will build an incredible online learning environment. Be bold if you think you are right. This ensures teachers have all the admin done for them, and can focus on other things.
  • Before flipped learning, there was an unspoken agreement between teachers and students – homework was actually a bit pointless, bolted on as an afterthought. No longer!
  • Model the model. CPD is now… Flipped CPD!

Questions and Answers

I am part of a wider team at School 21 thinking about Flipped Learning. Here are some questions the team wanted some answers to.

Q: What about those who don’t do the homework?

A: Shireland have developed a whole arsenal of techniques

  • Students are motivated by a feeling of security – they can now come in to a lesson confident with the material. They understand, finally, the point of homework. In other words, if you design the flip correctly, everyone will do their homework anyway.
  • Clever deadlines for homework. Ensure a day in advance of the lesson, so you can chase stragglers and analyse results.
  • Parental engagement is key. Once parents understand the advantages, they will support you
  •  Be completely relentless at the start of the flipped learning programme
  • If students have still not done the homework, put them in a corner to complete it while the rest of the class do the most exciting and amazing task ever. Sneaky.

Q: What do you do with all the extra time freed up in lessons?

A: The fact that you have more time is a positive, not a negative. More time for deep thinking, peer support, student-led learning, any of the good, deep, stuff. You would teach this higher-order thinking as before, but now with more time and more knowledge of the students’ needs. Your lesson becomes a clinic for fixing issues, rather than a lecture.

Q: How do you ensure this is time-efficient for teachers, daunted by the huge task of curating an entire library of videos?

A: Again, a whole range of answers:

  • Before you make a video yourself, think really hard about whether it is absolutely necessary. Is there already one out there that will do?
  • Do not flip every lesson. Students would have too much homework, and you would not sleep
  • Start small: pilot, evaluate, repeat.

Q: Surely by placing an emphasis on work at home, you are widening the achievement gap between rich and poor?

A: In comparison to traditional homework, students now get more support for higher-level thinking. What was once only available to students with present and supportive parents, is now available to all. Homework clubs and subsidised devices help too.

How to conduct excellent action research?

Shireland is a Research School, and referenced EEF’s research framework. Before gleefully diving in, answer these questions:

  1. What is the question? (Keep it narrow)
  2. What are you going to measure? (Qualitative and quantitative)
  3. Who are you comparing to?
  4. Where are you starting from?
  5. What are you actually going to do?
  6. What were the outcomes?
  7. How will you share the outcomes?

Two things that made me sad

  1. “I like being made to think again” a fellow teacher ruefully observed. It is a shame that teaching does not intellectually challenge all teachers in all schools.
  2. We were given a task to do, having been warned beforehand that this task was designed to make us feel uncomfortable. It was a worded maths question, and the facilitator was a maths teacher. She was comparing giving students a surprise question on a new topic, to giving students questions after they have had a chance to watch preparatory videos before the lesson. This made me sad because:
    1. The comparison is a false one. A good non-flipped lesson would consider prerequisites and not throw students in at the deep end.
    2. A maths teacher should not encourage maths anxiety (sore spot for me).

 

Thank you to Shireland and Whole Education for a thoughtful conference. I can’t wait to revolutionise home-learning and spend more time forensically supporting students to think deeper.

Why are chimneys like party poppers?

We are nearing the final stages of the English and Maths project, where a bunch of Yr9 students are using their understanding of how pollution spreads to convince the local residents and the planning officials that building 4 factories in the heart of the Olympic Park is a bad idea. Jess wrote about the project planning here, and the press attention here.

The mathematical foundation for the argument against the factories has constantly shifted over the past term. Both frustrating and rewarding. Here is the journey, starting in October.

  1. Initially I assumed that any theoretical understanding of pollution would be difficult for a nurture group, probably delving into multi-variable differential equations.
  2. Therefore, we at first planned to record levels of pollution in the local area, and use this data to predict what the pollution might be like after the construction of concrete factories. We were very kindly given a class set of tags that paired with phones to give data. Unfortunately the data was not detailed enough, logistically it was difficult to go out and record high quality data, and not every student had a phone. The dream of students independently going out and using their phone to change the world was somewhat dashed. That said, a few students do continue to use their tags and record the pollution in their daily lives – small win.
  3.  If measuring air pollution was not going to work, would we have to go back and understand the theoretical models? A few weeks of emailing around pollution scientists in London and treading water in lessons ensued. Elsa, from Southbank University, came and spoke to the class, and taught me about the Gaussian Plume Model. We had found an equation that could map out spread of pollution.untitled-picture
  4. First attempt: students to understand the equation and substitute in the relevant variables (for example, distance away from concrete factory and speed of wind). I made a beautiful but hopelessly complicated flowchart to aid this process.untitled-picture
  5. The equation was too unwieldy. Next step: I plug the numbers into the equation (or sometimes students use a ready-built calculator to do it themselves, and the students plot the graphs and analyse the results. Plotting graphs with very small and very big numbers involved revealed great teaching points about rounding and scale.untitled-picture

unnamed (3).jpg

We discovered that the students were having difficulty understanding the variables affecting how dirty the air that you breath in is. For example, what does “distance from chimney to my location”? Building on an idea from Jess I made a double-lesson analogy of Chimney as party popper, with the exploding paper representing the pollution. untitled-picture

unnamed-1
Distance to factory, distance to side of factory, and height of chimney, become real.

 

unnamed (2).jpg

Conclusions 

  • Through this tortuous journey of dead-ends and frustrated ideas we (accidentally?) exposed the students to a variety of mathematical ideas. Had we gone straight to Stage Five: plotting graphs and analysing them, then there would have been no data collection, no understanding of equations, no algebraic substitutions.
  • The fact that we teachers genuinely had no idea of how to solve the problem gave us intellectual excitement throughout the project, useful in pumping up the students and our lessons.

Snooker Table: The Journey of a Problem

  1. Find a really great problem on the internet (Thank you Colin Foster!)
  2. Share it with maths teachers on a meeting I facilitated about scaffolding rich tasks
  3. Ask for feedback from teachers who then taught it
    Outstanding meta-cognitive questions by Rosie
  4. Spend two hours on evening excitedly making resources rather than marking

5. Refine after working on it with one class

Spot the student’s mistake?
6. Post the idea on Facebook, and have a friend improve your app (even though you have been working with Geogebra for 3 years and he only found about it a week ago)7.Teach it to another class…

Top Set Yr11 with brains switched off at end of term…

8. Repeat
Latest draft of resources:

Maths on Youtube

Youtube can be used to give students information:

 

Youtube can be used for students to give information:

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:

 

Barging into the Semi-Private World

Graham Nuthall, in the Hidden Lives of Learners, talks of three worlds of a student:

  1. The Public World. The world of the school, dominated by the teacher. If you walk into a classroom this is the world you usually see.
  2. The Semi-Private World. The world where children manage their status in their social group, with often different rules and customs to that of the Public World.
  3. The Private World. The world inside a child’s head, rarely seen and difficult to access?

Each Thursday evening I take a few Yr11 students to Starbucks to work on Maths. The aims are to:

  • Increase independence (students in theory get on with their work and I get on with mine)
  • Deliberately barge into the semi-private world, showing students that it is acceptable and profitable to work in spaces other than school
Plates and iPads work as mini whiteboards…