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.

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



W brought maths cookies!

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

Wintana’s outstanding mathematical metaphors

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?


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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Strong friendship!
So strong!
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.

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.


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.

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

How to visualise planes?

(the table is a third plane)
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:

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:

The fifth plane intersects the tetrahedron at the four crosses…
LEI: snazzy new vocab
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?

Can you count to 26?