Farewell Further Mathematicians

I have spent seven hours every week with ten mathematicians this year – and it has been a joy throughout. An outstanding culture, with every student fully engaged in learning just for the hell of it. The students came back in after term had finished for a final party – playing card games and football in the sunshine.

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Iggy and Red-Velvet JJ cake!

Team Photo Time.

  • Shanjey (who was stuck in Luton with family and so appears here in female form)
  • Benyamin with his beloved meticulous notes
  • Suleman, with the rocket we used to model projectiles
  • Ashwin, with his best friend the Rubik’s cube
  • Igoris, blindfolded from when we did Kinky Maths
  • Jezza, in his father’s university gown, playing annoying music
  • Wintana, with her favourite measuring stick
  • Laura, with her Curiosity Box
  • Shirley, with a rude gesture on the bottle of her water bottle
  • Cody, with a stick he brought to the class 9 months ago
  • Ifte, with a boat from Bangladesh and a puzzle box

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W brought maths cookies! 

Some cards: (for personal memory rather than public egotism)

Wintana’s outstanding mathematical metaphors
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Mr Judge Senior

Lovely to welcome Papa into my FM class today. He inspired me when I was a wee lad, with mystical talk of an infinite sum that can equal any number you like. It seemed to run contrary to everything that I thought I knew about the certainty of Mathematics, and intrigued me until I saw a formal presentation of the alternating harmonic series at university. Happy memories of carefully measuring temperature and rainfall over a month in the back garden, or of hand-drawing maps of long Welsh walks, or of painstakingly building giant toy models when Papa joined my primary school class.

The common thread? Not maths, but considered logical thinking. The ability to analyse rationally is the biggest weakness that Papa sees in his colleagues, something that Maths is able to address.

What is the difference between a proof and a demonstration? Why is proof beautiful and important? What transferable skills does proof teach you?

Papa giving an impromptu lecture on Brexit and Equivalence Relations.

Getting joyfully stuck into university admissions questions. Mr Judge Senior was nervous that his maths was rusty, but not a bit of it. Some elegant and left-field ways of tackling problems, with an emphasis on elegance of thinking over churning through calculations.

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Great surprise element in this question! Papa able to wack out trigonometric identities 40 years after he did FM – ridiculous. (Bonus points if you can work out why he was using trig anyway)

Funny moment of Papa’s questioning, when thinking about “How many triangle with integer sides and perimeter 12 are there?”

  • So, what is the key variable?
  • The perimeter?
  • No…, what is the key variable?
  • Er, the angles?
  • No…, what is the key variable?
  • Is it the area?
  • No…, what is the KEY variable?

This continued happily for a few more iterations. The variable in Papa’s head was… THE SMALLEST LENGTH. Funny.

North Downs Way

As always, best to just read the thoughts of the students:

 

Back in March we ran a half-marathon in the snow and frost. Now, we run a half-marathon in the heat and the hills. Only fools do it the easy way. We were off to the North Downs to run along the trails. Training included an Epping Forest race, a fell-race, and plenty of parkruns.

All 13 students safely made it to the race – meeting at 0630 is no mean feat for the teenage circadian rhythm.

Glistening Thames
Nervous pre-race energy
Happy at the start. Will the faces look that fresh in 3 hours time?
Gorgeous

 

The course was out-and-back. This meant that you were constantly bumping into familiar faces. Flash of a smile, and then on your way.

Ifte, focussed.
Gaggle of students, sticking together

Hills are so much kinder on the body – variety of terrain requiring variety of muscle usage. Walking up hills provides welcome respite, and then careering down hill, holding onto control by your fingertips – exhilarating.

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Hills!

Strong friendship!
So strong!
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Daniel, speeding past on his second lap, completing a full marathon. Came third – wow. Great modelling of pushing yourself to the utter limit.

 

Well-needed high five for Rob

James limped to the finish, his left leg covered in bandages (Should he have gracefully dropped out? When does resilience become stubbornness?). Rovish had both ankles covered in gaffa tape, plagued by blisters. Noemi had bloody grazes all down her leg. Zepora powered through, grinning ear to ear, to Box Hill, and then sensibly decided to stop. A courageous decision. All but three of us took proper tumbles, over the treacherous roots in the foresty steep darkness.

Exhaustion at the finish
Picnic overlooking the rolling hills
Tired and happy
Always stretching
Proud finishers

 

Very possibly an overly challenging race for the students – this was in no way beginner-friendly course or conditions. But, better to aim too high and readjust as necessary. Next year, the plan is to hand over the organisation of the races to the students. Where will they go? What adventures will they have? I cannot wait to find out.

 

Following pics are of the students finishing:

Noemi – first student finisher – powerful finish
Ahmet flies through
Rakib’s first half-marathon!
Modestas and Eylin sprint home
Eylin’s grandparents came to support!
Heather screams Ifte home.
Jubril found a friend on the course “I was his mum for the last 2 hours” she proudly said
Aisha floating home
A little boy escorts a crying-in-pain James to the finish
and Noemi helps her friend
What’s that in the distance?! It’s Destinee, Rodi and Rovish! Woohoo!
We have been working for 5 hours, but we still have energy for a sprint finish.
Phew.

Polya: Guess, then Prove

As an examplar of how to collaboratively problem-solve in a lecture setting (rows), this video of Polya is pretty sublime. (MK, you reading?). Long, but so worth a watch. Further discussion here.

Polya’s central themes:

  • Guess, then prove. Guess, then prove. Savages believe or desbelieve blindly. Scientists make a guess, and then think careful about how to decide on the truth of the guess.
  • Inductive reasoning: mathematical problem-solving stealing from science.
    • Observe the situation
    • Spot possible patterns or laws
    • Try to generalise
  • Test your guess. Think by analogy. Consider extreme cases.

(This particular problem is also known at the Cake Problem.)

The fact that Polya seems like a radical and excellent teacher today further proves his point that teaching is an art, not a science.

 

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Maths teacher practising being Mathematician.

As a welcome session for new Further Mathematicians

Such a lovely problem, after watching how Polya did it. The ability to simlpify in two directions (number of dimensions and number of partitions), the relationship between numerical pattern spotting and visual understanding, the lack of technical prerequisites – love it.

Desmond and a bunch of the older FM students popped along to struggle alongside the newbies. Young students were explaining ideas to older, university students were getting mini-tutorials in how to find the nth term of quadratic sequences. Great to illustrate how everyone finds lots of Maths challenging.

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Guess and time. Next time add a third column for percentage certainty.

How to visualise planes?

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(the table is a third plane)
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This student uses their knowledge of Google Sketchup excellently
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Polya’s drawing of four planes is so good – only includes the bare bones. If it included all regions it would be too confusing

Should I use numbers or pictures? Two different approaches here:

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Can I trust the pattern to continue? (Moser Circle Problem). Can I trust myself to visualise all those planes correctly?

Two more ways to generalise:

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The fifth plane intersects the tetrahedron at the four crosses…
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LEI: snazzy new vocab
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Some great quotes from the students.

Really invigorating debate about what makes a great proof. If you are convinced you have a clear picture in your head of the sitaution, but cannot persuade a friend, do you really understand it? If you can’t explain it simply, you don’t understand it well enough says Einstein. If noone understands a proof is it still a proof? Here is my attempt (written after the lesson, since the students were rightly not satisfied with my handwaving). I guessed first, proved second. Does it convince you?

Planes Proof

And, B went away and made the 26 planes on Google Sketchup. Is this a proof? Why not?

Isometric
Can you count to 26?

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.

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: