Reality –> Graphs

Use video-tracking to:

  • Build intuition
  • Motivate the use of graphs to describe and explain reality
  • Celebrate creativity



Practical help:

  • Students can make their own videos on phones/tablets, or you could pre-record before lesson.
  • To screencast from your laptop, use this software

Lesson Study 1: Limits

MG, GD and I are thinking together about how to assess collaborative problem-solving. How can we show and convince others of the efficacy of more open lessons? We used a simple structure:

  1. Independent task 1
  2. Collaborative task
  3. Independent task 2 (similar to task 1)

Teacher then marks both independent tasks, and compares the differences. If the collaborative task were useful, then the second task would show marked improvement.

The Tasks:

Task 1: Area of circle though sectors


Collaborative task: Share understanding of Task 1, using different colour to add further notes

This student found independent work difficult, but was helped effectively by her table. 
Excellent diagram showing limits

When you use a limit it “becomes more true”. What would a philosopher say about this?

Task 2: Area of circle through rings

This is the student who only wrote one line for Task 1 – marked improvement! Interesting limit answer – the circumference is straightened out but never quite reaches a straight line…


  • Some students loved the independent silent work – a chance to think for yourself. Some hated it. What to do about this?
  • “Don’t be afraid of over-explaining” says George, who is more towards LAE on the spectrum between old and new school…
  • Be more precise in the assessment questions I set. It was fun to mark the students’ descriptions, but I couldn’t glean much precise information from them.
  • What is the best ratio of time for Independent : Collaborative : Independent?


Appendix: Limits are Beautiful.

An excellent way to step-up to more sophisticated thinking, grappling with infinity. Here are some more ideas:

Dancing to Rameau

Blog from student here


Took a few students to the Barbican last night, to listen to the LSO, conducted by Simon Rattle.


“Why is there coughing breaks in the music?”  “Why is there so much walking on and off and clapping?”. Classical music does indeed have some bizarre conventions.

Musical highlights:

  • Rattle’s wife, a singer, oscillating like a jelly as she sings Handel Arias
  • Unbelievably quiet start to Schubert’s Unfinished Symphony. We sat at the very back of the hall, and the acoustics were crystal clear!
  • Energy and joy when playing Rameau dances, Rattle jumping around the orchestra while conducting from memory. Contrasts to the stiff upper lip of the percussion player, primly tapping his tambourine.

MK visits

MK visits School 21 wearing her three hats: as mathematician, as musician, and as Women-in-Maths activist.

Chapter 1: Women in Maths


According to FMSP, in the UK:

  • 50% of GCSE students are female
  • 38% of A level students are female
  • 28% of A level Further Maths students are female
  • 19% of Research Mathematicians are female
  • 6% of Maths Professors are female

Can you name a female mathematician? The students couldn’t (but then, they didn’t know many male mathematicians either…) MK reeled off a bunch, including Maryam Mirzakhani, the first female winner of the Fields Medal. Gasps of shock when students learnt that this took place only last year.

Good turnout of Yr11 and Yr12 students, and equal gender split – this is everybody’s problem.

We learnt about the following concepts:

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  • Stereotype threat – the risk of confirming negative stereotypes about an individual’s racial, ethnic, gender, or cultural group.
  • Attributional ambiguity – “not knowing where negative responses come from. Is it because my work is not good enough, or because the teacher is prejudiced? Two groups of black students handed in work. Group A had their photos attached to their work. Group B did not. Group B responded more to feedback than Group A, since they knew that the feedback was not related to their race.

Often these biases and effects are subconscious. We therefore need to talk about them, to realise what we need to change about our beliefs.

Chapter 2: Maths and Logic

When Tom came in a few weeks ago, we noticed that the students were tying themselves up in knots about logical arguments. MK taught a session on logic – short lecture followed by problems. She has been honing these sessions back at UChicago, to consciously teach undergraduates the core mathematical skills that they would otherwise be expected to learn by osmosis.

Logic through venn diagrams


Mark, philosophy teacher, gets involved.
  • MK impressed with how well the students work well together. The adults in the room, when given a difficult problem, instinctively go into a nest and thrash out the details independently. The students instinctively work together through talk. Maybe we should now start to also focus on how to go it alone? But, given that most students will have experienced Maths as a solitary sport in their previous schools, it is definitely correct to go strong on collaboration initially
  • Mark noticed that each of the three groups was using a different method (venn diagram, chains of arrows, big tables). When is it okay to let many methods flourish, and when should you just teach one killer method?
  • Students really loved the logic problems – satisfying quick wins in comparison to STEP problems. We did some written by Lewis Carrol. More here

    Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised.

What do you do when you get stuck? MK says:

  • Draw a picture! I drew the same picture, in different ways, over and over for about a year. I will never forget the picture.
  • Stop. Deliberately forget everything you know and do it again, trying to come from a fresh angle.

What is the longest you have spent on a problem? MK thought this question made no sense – you start with a big problem and break it into sub-parts. Solve the sub-parts, except one which is tricky, which needs to be broken down into further sections… Continue constantly. So it all depends on scale.

Chapter 3: O Magnum Mysterium

The Yr10 students are singing Lauridsen, with something ridiculous like 8 parts. It is a great challenge, given that the majority of them cannot read sheet music and are therefore learning by ear. MK came and sang with the sopranos. Great to break into the private-school world of classical music.

Final Parkrun of 2017

J’s times: 28, 25, 23 minutes – ridiculous improvement in three weeks:

Brother running in with sister, charging to the finish:

Z had arrived late, and so was behind the trail-walker (volunteer who walks at the back to ensure everyone safely in). Nerves when she failed to appear, hop on bike to search for her. The rest of the gang all loped back to meet her, met her with whoops and christmas music blasting from their phones. Such excellent and willing support from tired and cold runners. Great great first 5k from Z:

E volunteered this week rather than running – had a lovely time giving back to the Parkrun community. The happy group:

The work that Daniel and I are doing with these students is what I am most proud of, from this term. Getting young people outdoors and active kind of seems far more important than forcing them to understand the graph of the cosine function?

Can’t wait for the Hampton Court Half-Marathon in 2018!!

… Which is true

Steve, PHD student from Warwick, and our old friend Desmond, one term into undergrad Maths at Oxford, came and worked with the FM class today. The students now all copy Desmond’s catch-phrase, “… which is true” at regular intervals.

Feedback to problem write-ups

Tips when stuck:

  • Talk to yourself! Sometimes you find yourself just staring at the problem and not doing much thinking. Talking forces thinking.
  • “Numerical analysis is good. Conceptual understanding is better” says Desmond, on number-crunching your way to a solution.
  • Write down absolutely everything you know

When understanding a maths paper, Steve is happy to decode one sentence per day. Amazing grit.

Steve showed the class a hilarious application of Fermat’s last theorem (more examples here:

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And, the chocolate problem. m by n chocolate bar. How many cuts to reduce the bar to mn individual blocks? (You cannot do multiple cuts at once). Hint: it is a bit of a trick question.


Note in the video the final group are completely silent. When is silence productive and when is it a hindrance?

Cultural Capital Day

Every sixth former spent a day visiting museums across London – the museum capital of the world.

I asked my group of 22 to give me their phones for the day, to stay focussed. 7 people point-blank refused. Some reasons:

  • I need my phone to take photos and notes in the museum
  • I need my phone for maps and the time
  • I feel more secure with my phone
  • I need to monitor my bitcoin investment
  • I need to buy trainers in the sale at 3pm

No-one spoke about snapchat, but there definitely was an underlying desire to stay connected. Students were hurriedly sending snapchats to their closest friends in the bustle of handing their phones in – last few seconds with their beloved black mirror.

The stash
Old-school entertainment: cards.

Anyway. We went to the Science Museum. Sometimes a dated celebration of the British Industrial Revolution, like the unending displays of steam engines. Sometimes overly shallow interactive exhibits, like when you use a joystick to guide a cyclist, to understand the energy crisis. But, also some brilliant things:

  • Found the first ever 0, from maybe 200 AD, in a manuscript found by a farmer near the village of Bakhshali in modern Pakistan. Compare to Roman Numerals, to show efficiency of 0, as something to measure the void.
  • The inventor of the forceps, for childbirth, kept it secret for three generations by blindfolding all pregnant women.
  • The inventor of the x-ray gave an x-ray of his wife’s hand to her, in an attempt of a present. She took it as a horrifying reminder of her mortality.
  • 100,000 homes in the UK are powered by chicken poo
  • Tobacco smoke, at one point in history, was blown into the rectum of drowned humans, to resuscitate them.

By far the highlight was the Wonderlab Gallery (see former blogpost). Normally for much smaller children, but the 17 year olds had an amazing time. Instinctive scientists – conjecturing, experimenting, practicing.

The prisoner’s dilemma in water form: sip or spray?
Thinking analytically



Lessons often start with a headache, a problem that the students will struggle to solve with their existing tools. It serves as motivation for learning new things, and mirrors the development of the tools in the history of Maths.

My students are now so well-versed in the excellent Dan Meyer question, “If this is the aspirin, then what is the headache?“, that they have begun pleading for the aspirin, repeating my language back at me. “Please, just give me the aspirin, I am stuck”. Ha!



Students get stuck in with good humour, playing on desmos, trying to sketch graphs. But, they don’t get anywhere, and begin asking for the aspirin.

Aspirin 1: Play with sticks and string


Aspirin 2: Learn the simplex algorithm, as a way to generalise from the concrete 2D geometric model to the abstract N-D algebraic model.

Wrong and Strong: Exploring Two Extremes

In music rehearsals, there is a phrase, “Wrong and Strong”. If you are unsure of the notes, is better to sing/play confidently. That way, you will be more likely to notice mistakes, and others will be more likely to notice and help you. From music to education – maybe it is best to have a deliberately extreme vision, to use yourself as a guinea pig. School 21 and London Academy of Excellence both have strong (but different) visions, and are conveniently close (geographically at least) to each other.

One morning Tolly (Head of Maths at LAE) and I did an observation swap – we both watched the other teach a Mechanics lesson, before spending an enjoyable hour discussing and trading notes. Throughout the process we remembered not to make value judgements. It is not about working out which model is better, but about working out how each model can incorporate elements from the other.

Here is an analogy:


The Conservatives detest the poor, and exist to ensure money stays in the hands of a wealthy few. Labour detest the clever or hardworking, and exist to celebrate laziness and to destroy our economy.

LAE is a ruthless exam factory, churning out identical model students with excellent pieces of paper but no imagination. Six21 is wishy-washy chaos, sacrificing powerful knowledge and replacing it with hot air.


Conservatives and Labourites (?) are all generally moral people, working hard to improve society. They agree on the end goal, but disagree on the methods.

LAE and Six21 are generally moral schools, working hard to do what is best for their students. They agree on the end goal, but disagree on the methods.

The point? Be careful not to caricature schools. LAE have a healthy timetable of extracurricular activities, and Six21 students sit traditional assessments. While it is useful to highlight differences, we must remember that there are also many similarities.

Lesson 1: Proving SUVAT at Six21

10 minutes exploring the Fundamental Theorem of Calculus, through Mechanics. Why stop at acceleration?

20 minutes group-task practising moving up and down the ladder (this would be the heart of a traditional lesson)

70 minutes working at whiteboards, attempting to prove the equations for constant acceleration (SUVAT equations)

 suvat 1suvat 2

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Tolly observing from behind

Tolly gave me some excellently thoughtful feedback:

  • I focussed on pace in the fluency task, constantly praising for fast pace. I didn’t mention pace at all during the exploratory task. This is something that I hadn’t realised, but would stand by – pace is useful in purposeful practice but might be counter-productive in more open tasks? Thinking deeply is better than thinking fast.
  • Proof is on the abstract end of the Mechanics spectrum. Most lessons lie somewhere in the murky middle. Could there have been more focus on the extreme concrete side? Thinking about what jerk and snap feel like? (See an earlier example using the software Tracker, here.
  • The collaboration between students was productive yet humble. Even in large groups, all were getting involved. Excellent that an outsider could pick up on something that we had consciously been trying to cultivate.

Lesson 2: Understanding Newton’s Second Law of Motion at LAE

40 minutes interactive lecturing. Students, as a group, complete a “Topic Sheet”, which is the bare-bones of key definitions and examples

20 minutes independent practice of skill

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Tolly leading from the front
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Gorgeously clear notes

We spoke about:

  • The importance of interactive lecturing. Tolly values student involvement, and constantly thinks about ensuring that they are actively learning. First guesses at definitions, or ideas on how to start a problem are collected from the group, before being refined. Tolly only speaks for 2 or 3 minutes at a time, always releasing the students to apply the new knowledge on their own. Very fast pace – 8 minutes into the lesson the students had already completed 3 examples.
  • The students felt able to seek further help when struggling, something which can get lost in a more lecturing format.
  • The mini-whiteboards are used for initial thinking, which is then written up on the topic sheet. It is important to both have the rough independent thought, and the crystal-clear final notes.
  • The students’ views on their lessons. Those who I spoke to massively appreciated the clarity and structure of the lessons (topic sheet –> independent practice –> regular quiz). “The topic sheets are amazing for revising”, “It is like the notes are provided for us, we just add on”. Finding the right balance between teacher-led clarity and pupil-led active learning.

I met Vasile, the “Academic in Residence” at LAE, who straddles university and school thinking. He teaches in a traditional lecture format, “Theorem, Proof, Note, Observation. Repeat”. Rigour and precision. Useful preparation for the top students who will move on to a full-time diet of this at university, and a fascinating idea. Vasile has a deliberately light teaching load, allowing him to curate extension tasks for the department.

Next, a table. I love a table.





… is about question generation, creating unique ideas

… is about mastering key skills outside of the classroom

We want students to be confident…

… in exploring difficult problems

… in executing techniques accurately

Problem-solving is best used…

… as a motivation to learn new skills (as the aspirin for a headache)

… as an extension, built off an excellent grasp of existing skills

Notes …

… are flexible (paper/whiteboard/photo) and are not a priority.

… are everything. “It all comes from the notes”. Creating excellent habits leads to excellent understanding

Homework …

… is for mastery

… is for mastery


… is sometimes most effectively learnt when invented afresh

… is handed down clearly and helpfully


… teach best when delivering content they have designed themselves

… teach best when delivering consistent content that all students in the school will receive in the same way

EDIT: this is me over-stereotyping, and is not accurate. In reality, teachers are free to use existing resources or create/adapt their own, so long as every student has excellent notes and is regularly quizzed.


… should be varied, depending on what is being learnt

… should fit the same structure, to allow students to understand procedures quickly and move on to deeper thinking

What are the next steps?

  • To build pairs of Maths teachers between the schools, who observe each other and play-act at each other’s styles.
  • To bring the students together, possibly for problem-solving sessions
  • As heads of departments, to trickle down key strengths from each other. I will focus more on recording high quality notes, and Tolly will focus more on problem-solving.

Tolly, is that a fair summary? I look forward to the next steps in our collaboration!

Hackney Marshes Parkrun

As part of Run21’s training for the Palace Half in March, we assembled at Hackney Marshes on a bitterly cold December morning for a Parkrun.

Nervous before the race, unsure of the course or a good pace, in unfamiliar surroundings. Despite only being a 10 minute bus-journey from Stratford, “it feels like we aren’t in London any more”. Great escape from the grey.

Some beautiful encouragement from the runners, sticking together and supporting all the way. Rosie’s race here.

“I am 73 and I am overtaking you!” crowed a hilarious lady as she plodded past Mustafa on the final section. Great support at all times.

The happy team will definitely be back for more. James travelled for 2 hours to get here, to soak in the companionship.