Proving Pythagoras

After the time-limited Vivas, it was important to find sufficient time for students to get really immersed in a problem, to feel what 8-month Tom was feeling when he pushed through the mud to the light beyond.

The problem: Proving Pythagoras, using Mathologer’s excellent video, and Cut-the-Knot’s incredible bank of proofs.

300 minutes of lesson time to work together + homework and independent study. Outcome – 5 minute group presentation, and individual write-up. Useful to discuss the similarities and differences of presentations and write-ups. What is the correct level of rigour?

Assessment criteria for presentation:

  1. Did you attempt an original proof (either of Pythagoras’ Theorem or a related conjecture)
  2. Did you think about multiple representations of Mathematics?
  3. Did you convey a sense of wonder and awe at the things you were exploring?
  4. Did you present well?

Examples of student exploration:

If I just look at the diagrams on the cut-the-knot website, can I work backwards to what the proof must be?

 

How many 60-triples are there? (These are called Eisenstein triples by other Mathematicians…). This required some really excellent coding. A few patterns began to emerge but nothing concrete – and that is absolutely okay.

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Can I prove the following theorem for a right-angled triangle? For any triangle? What if I pretend that I don’t know the cosine rule, can I still prove it?

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What if I continue the pattern on and on? Can I prove that there are some parallel lines in this crazy diagram? (Geogebra suggests there are parallel lines, but the group couldn’t prove it…)

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Outstanding generalisation:

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Great to encourage students to talk precisely and entertainingly about their mathematical journey, and to provide a space for kind, specific and helpful feedback.

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Preparing for University

Thanks to Kings Maths School for another excellent twilight session.

How to prepare students for admissions tests?

They need the confidence to dive in to challenging questions, to wade through intimidating notation and lengthy algebraic calculations, to be happy with 50% rather than 95%. The dream is to give this confidence to all students in normal lessons, rather than saving it for a select few. Teaching to the top benefits everybody, provided there is sufficient support in place…

Some good resources:

How to prepare students for university mathematics?

Dan made the claim that there are two main diffrerneces from school to uni:

  1. A different type of challenge
  2. The axiomatic approach

Challenge

At school, students learn a method and then apply a method.

At university, students learn a proof, and then either:

  1. Prove a related theorem using an unrelated method
  2. Prove an unrelated theorem using a related method

Axioms

Distilled, rigorous lectures. No examples, no motivation, often proving the obvious (e.g. IVT). Most extreme example: Analysis.

Possible solution – teach a university-level topic in university-style, with school students. Ensure it is a topic that is not on university syllabus – don’t want to spoil the fun.  Example: continued fractions.

Fun fact: Margaret Brown, one of the founders of the school, found that the biggest predictor of student success at university maths was the quality of their peer group. More important than quality of lectures, or past grades. Anecdotally, I completely agree with this. Therefore:

  1. Train students in the art of collaboration at school
  2. Train students in the art of seeking out and nurturing the right peers

 

Final question – which maths faculties out there in the university world are pushing at the pedagogical boundaries? I want to go learn from them!

(w)Restling with Rubik

Cody’s father, Robin, very kindly came in for a sunny session of solving and talking about his huge collection of Rubik’s cubes.

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Engrossed

Some thoughts:

  • Solving the cube is a great metaphor for maths learning – when do you learn from first principles and when do you just learn a method? See below – B just got too frustrated and googled the algorithm. Will doing the algorithm sufficiently many times eventually result in understanding?
  • Some of the cubes looked completely different, but were in fact secretly the same. A great application of isomorphisms. (Some of the isomorphic cubes did not have the same number of colours, or faces – very weird).
  • It was excellent to welcome in a student’s family into the group. Beneficial if they are experts in something relevant to Maths, but also, just beneficial in general. A reminder to more consciously open the doors of the Maths Temple to parents.
  • All the students had learnt how to solve the cube, back in the days of learning about Algorithms. I was the only person who couldn’t. It was a new experience to be the weakest in the group – I was frustrated, lost motivation easily, was constantly asking for help. When I completed the first layer of the cube, I had to go around seeking praise from all the students. Ha!
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Just give me the method!

Parkrun 200

For a few months now, Daniel and I have been (trying to) light fires across three of the small schools – persuading students that running is for them.

Some excellent Tuesday afternoon sessions to boost confidence and build stamina. Sixth form student were especially supportive here – often sacrificing the quality of their sessions to run alongside struggling younger students

To Hackney Marshes, bright and early on a Saturday morning. Shout-out to Joe, Steven and Kazza who all helped ferry students to and from the run. Every volunteer position was taken by a sixth-form student – giving something back to running after their half-marathon experience. James gave an excellent speech as run-director to the runners, welcoming newbie runners to their first 5K.

Briefing by man and baby:

Students strengthening friendships through sport:

Spot the students and teachers:

And we’re off! Beautiful summer’s morning for a run. Jon (13yrs!), who I always used to be able to beat, creamed past me at the start and stayed out in front. What great improvement. Here now follow many excellent photos of students and staff putting everything into their running. For more, click here. Scroll down for a great group pic. 50 runners and volunteers – excellent!

L chases S hard:

Peter, eye on the prize:

C, armed with friendly stick, races to finish:

Support across year groups:

Happy:

Lovely nature:

Flying:

Running back from the finish to support friends to the end:

Who will win, F or her dad?!

Coaching A to the finish line. There is a problem with his ankle, but did that stop him from joining in?! No chance.

Supporting A to the finish:

D, to massive applause, bombs down the finishing straight…

… and finishes, much to the delight of the Six21 volunteers. He was joined by his mum and sister on the run – excellent.

Most of the runners and volunteers. Sorry if you weren’t in this pic!

Next steps: keep the momentum up. Lots of parents came to spectate.

  • Next time they can run.
  • Next time, runners can bring their friends along.
  • Next time, think even more about how to persuade students who think running is not for them, for whatever reason. Jess ran a session for girls and running, we went big on building relationships and routines, but still could have got more turnout.

Anyway. Parkrun is the best.

Midsummer Fell-Running

To Epping Forest, for a 3 mile fell race (only Cat A fell race within the M25, whatever that means.).

“Sorry we are half an hour late sir!” Silly boys.

We started way behind the race – which was a bit confusing for the kind wardens as we ran backwards along the course to try and catch up. Start up by the obelisk, with glimpses of London skyline through the trees.

After scurrying up a steep densely forested hill you suddenly and unexpectedly emerge onto this meadow. Glorious sun, expanse of water, London winking in the distance. It was an incredible moment – yelling out for joy. Picture does not do it justice in any way.

Ahmet powers up the final hill

Modestas sprints to the end

Ifte and the golden light

Look at that light.

OMG. That light.

Rovish and a warden complete the race (Rovish added half a mile or so to the course after getting a bit lost…)

The light catches on the leaves as it trickles down to the forest floor. Mmm.

Lost, but not alone.

At the end of the year, every teacher gives a presentation on how their craft has developed. Excellent idea Jess. Thoughts on other teachers’ talks here.

This post consists of:

  1. My aims
  2. My practice
  3. Further questions

Chapter 1: My Aims

The three myths I have been trying to dispel this year:

  1. If you are good at Maths, then it is easy
  2. Maths is best done alone
  3. Maths must be useful

Fave quote, from Andrew Wiles:

Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of—and couldn’t exist without—the many months of stumbling around in the dark that proceed them.

My express purpose as a teacher is to enable students to feel the joy of this process, for its own sake.

Let’s try and live that metaphor… (I am blindfolded, my partner can see but cannot talk or move. Which team can collect the most beanbags?)

Killer example of professional mathematicians being lost, but not alone, is the polymath project (see for example their work on bounded gaps of primes). Fields medalists working alongside high-school math teachers on an online forum, to create new knowledge.

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Collaboration gets results

Great analysis of this collaboration, by Ursula Martin. What was once hidden away at blackboards during conferences is transparently available for all the world to see. Most of the comments by the collaborators concentrate on examples and conjectures rather than formal proof – people are wandering through the dark room together.

Chapter 2: My practice.

How am I dispelling the three myths?

A culture of problem-solving hits all three.

Teachers modelling being lost alongside students:

Professional mathematicians work alongside students, celebrating the uselessness of maths:

Formally assessing problem-solving tells students we actually value it:

Ask: “If this is the aspirin, then what is the headache?” (Thanks Dan). Example – “I want a function whose height is equal to its slope at every point. Does one exist?”

Do silly things, just for the sake of it, and then analyse them

A farewell card from a student, that perfectly summarises my aims:

Chapter 3: Further questions:

Craft Reviews

Things I have learnt from other teachers, from the summer craft reviews…

Dave: Collaborative Making

  • Headcams on students, to monitor collaboration?
  • Keri Smith
  • “I model my classroom on Lord of the Flies” – best quote ever. Dave all about leaving collection of found objects in a pile in the middle of the classroom and letting the students get on with the important business of playing and improvising, getting bored, experimenting. Not dissimilar to the prompts in prob-squad
  • Claim: Most learning happens casually, without teacher necessarily intending it. You cannot predict the learning, but you can create an environment/culture in which learning is more likely to happen.
  • “I am not interested in Art, I am interested in collaboration and thinking

 

Amy: Slow down; do more

  • Some reading here and here, on how there is more to a good lesson than pace.
  • Long-term planning: focus on the handful of key things the students really need to learn. Ditch the rest – frees up time to focus on what is actually important. Soup analogy from my soup-buddy: focus on the basic ingredients (stock, onions, garlic…), and add further details later on.
  • Weekly routines: structures liberate, students feel safe, transitions smooth, less planning. (This definitely does not mean that lessons are boring, or students don’t need to be flexible…)
  • Now that the teacher is well-rested (not having planned and laminated until late at night), and is utterly present in the lesson (not rushing around the classroom “doing teaching”, hunting for the next card-sort…), she can respond much more effectively to the needs of the students.
  • Utterly pointless to read for five minutes. Thinking and immersion take time.

Zek: Nurturing Creativity

  • Replace “thinking outside the box” with “connecting boxes” – a more helpful analogy
  • Teacher acts as surrogate thinker for the student – student chooses the direction but teacher helps them reach the destination.

Bertie: Building a culture

  • Students and teacher should all feel safe from unkindness, safe to make mistakes, and safe to try things new. Explicit focus on ensuring teacher feels safe in NQT year is great.
  • When is it best to pre-plan how to build a culture, and when is it best to allow it grow organically or accidentally? Often Bertie would say that he only realised why he did something 6 months later, with the benefit of hindsight.
  • Routines build relationships. For example, always eating lunch with the students in the canteen. If it feels like a chore, then stop. It should be a pleasure.
  • Classical self-deprectation:  “Luckily for me the relationships were strong enough that the students accepted the sanctions”. You make your own luck.
  • I think of the students in a similar way to how I think of my children.
  • “You play like you train”

Mark: Is the collective mind more powerful than the individual?

A quote that Mark constantly refers to to remind him of purpose, Russell:

Philosophy is to be studied, not for the sake of any definite answers to its questions since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind also is rendered great, and becomes capable of that union with the universe which constitutes its highest good.

Find and replace Philosophy for Maths…

Great design principles:

  • Philosophy is thinking in slow motion. We prioritise depth over breadth, and precision over generality.  
  • Probing, exploration and dialogue is the mode of the philosopher. We understand that philosophical progress can be slow and that certainty is not always possible.  
  • We are a community of enquiry, collectively exploring big issues. We respect the views and knowledge that people bring to the table. Philosophers develop their own voice and their own stance on key issues.
  • No one is born a great philosopher – wisdom, skill and passion come with application and effort over time.
  • Philosophy isn’t just for rich white males from Western Europe!

Further thoguhts:

  • The teacher’s role is to pick great problems, carefully sequence questions, formulating student groupings.
  • Assessing students should be a joy, not a chore. With Vivas, this becomes possible.

A possible framework for collaborative problem-solving:

  1. Chaos (wait longer than it feels comfortable before teacher to intervene if going hopelessly wrong)
  2. Clarification (start to pin down what we are talking about)
  3. Progress (teacher puts questions on post-it notes on whiteboards, and expects the group to answer them specifically)
  4. Genuine dialogue (and finally now we start to get somewhere)
  5. Silent Reflection (students pulling weird deep-thought faces, when writing up)

 

 

Heath Parkrun

Great hill-training for the upcoming half-marathon. Lovely to see Freddy at his first race – he said he hated every minute of it but will come back for more. Aisha and Ahmet up early after Eid celebrations. Noemi’s family out in force. Modestas ignoring an injury – brave or stupid?

Afterwards, the students headed to the Mixed Pool to splash about. Intrigued to find out who actually jumped in… Only half the students in the sixth form (based on a rough poll) can swim. Is swimming still a necessary skill?

Heather and MK chat at the back and enjoy the views.

Kinky Maths

Jonty went to play squash with a friend for the first time. Jonty had never played before, and his friend was an experienced player. After only a few rallies the friend, almost without Jonty noticing, switched to playing with his weaker hand. They then played furiously, well-matched.

The crucial feature:

The superior sportsman, having been handicapped, now needed to use all of his skill to effectively compete.

How does this relate to collaboration? I have been thinking about the dynamic in a group, where one student appears stronger than the others and tends to dominate. What if they were blindfolded? (We first came up with this idea when Tom and MK, two professional mathematicians, came in to work with the students).

There are alternatives to ensure an equitable experience for all members of the group, such as talk cards (you pay a card into the bank every time you talk to ensure everyone shares the air), or group roles (you can only act as the summariser, for example). The issue with these is that it is easy to forget the rules – you have to actively prevent yourself from flying away. If you are physically blindfolded, then your brain can work as hard as it likes, with no artificial constraints.

Draft 0: Practice wearing blindfolds and giving precise instructions by playing games (Marco Polo…)

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Draft 1: Try in front of a bunch of teachers from around the country.

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The teacher tackled this problem, which merged visualisation with number sense, extending to algebra. Reflections:

  • Communicating to a blindfolded person stimulated precise use of vocabulary, clear descriptions, simplifying language for understanding.
  • Possibly switch between blindfolded and seeing throughout the lesson, for gifted students who don’t quite have the confidence of a teacher?
  • Sometimes it was frustrating to explain to the blindfolded person?

Cody, a student, said that working with me while I was blindfolded “took Mr Judge down a peg or two so that we could work on the same level”. I was thinking flat-out, but was not over-dominating the group. Excellent.

Draft 2: Spend 100 minutes blindfolded, working with students.

The problem, again from Creative Mathematics:

“If you know the midpoints of a polygon, can you work backwards to find the original polygon?”

 

Reflections:

  • Igoris, the only student to be blindfolded, found it a bit too challenging, and eventually took it off to collaborate more effectively. Should only teachers be blindfolded?
  • Providing a constant commentary to the blindfolded mathematician was mostly helpful for all, to aid understanding. Occasionally annoying, when people wanted quiet time to digest.
  • The Seeing Mathematicians found that they were much more conscious of their level of understanding, after having to explain so clearly. Mistakes found quicker.
  • Some students found it really hard –  “I communicate in diagrams, not words”.
  • Even more respect for thinkers like Stephen Hawking, ideas contained within their own mind.
  • If the teacher works flat-out and is not handicapped, then it might be as fun, but it wouldn’t be as educational – students predict they would be able to sit back and watch the expert problem-solve.
  • Would there ever be a situation where choosing to put a blindfold on would improve problem-solving ability of the group?

After 100 minutes, the three groups had markedly different whiteboards. Which is the best? Why?

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p.s.

  1. Here is an article on blind mathematicians.
  2. If you wear a blindfold for 100 minutes then you get surprisingly good at guessing who is not concentrating, and it takes half an hour for your eyes to return to normal…
  3. That evening I went swimming at the reservoir. My goggles broke, so I was blinded in the water. Avoided (mostly) other swimmers by feeling vibrations in the water, and navigated by the feel of the sun on my face. So fun!

Forest Five

I bloody love taking students to Epping Forest. See first draft here. This time, with 10 students, to take part in Orion Harrier’s 5 mile race, as training for our off-road half-marathon!

Bob, organiser, welcomes the gang:

Team warm-up

Daniel trains his son to become the next olympian

Ready!

The start: a line in the field

The course was excellent – gently undulating, mostly on dirt tracks. The course had to be diverted due to cows on the route. What a privilege, still inside the M25! The highlight was when the course dived into the heart of the forest, leaping over tree roots, ducking under branches in the stark shade, fun! The brief moments of panic when the group you are running with loses track of the steady trail of sawdust piles that line the way “which way?!” we muttered to each other frantically. Zigzagging meant our sense of direction was enjoyably destroyed.

James and Rakib navigate around the cows

Sprint finish. Well done Rakib for his first Run21 race!

Strong from Noemi

Modestas storms through

Ifte and Ahmet, both fasting (up at 3am to drink and drink and drink) come home together

Jubril, as always, makes friends to divert himself from the pain

MK, who had a friend until Jubril stole her away

Destinee and Rodi, with dog Lucky and friend Chris, storm down the hill

Zepora and Anne spoke about feminism, NHS, future plans… “We put the world to rights as we ran through the forest.”

The students run to support Zepora – such kindness as ever (see earlier example at Z’s first parkrun)

Sprint finish!!

Thanking canine companion, Lucky.

Look at that forest! And those proud students! Now, to the North Downs for a serious challenge. A lovely note from the two sweepers who ran with the final three students:

We wanted to tell you what a fantastic bunch of young people you brought to the race today! They were determined, persevered and were so supportive of each other throughout the run. And great people to talk to too. Thank you so much for bringing them to the forest to run with us. We hope you’ll be back in August and wish you every success in the half marathon next month!