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.
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:
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?
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.
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.
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.
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)
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
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
… 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
… 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!
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.
Tom, in his fourth year of his PHD, very kindly came to the Big Smoke from leafy Cambridge, for a morning’s problem-solving with my Further Maths Class. STEP questions are just beautiful.
We started with an exploration of being stuck. Tom told us the story of his struggles:
He was stuck for 8 months. 8-month-Tom is his new name.
His teacher (supervisor) did not understand what he was doing, but was still able to re-motivate him at their weekly meetings.
He took a 2 month break from the problem, returning with fresh vigour and a new perspective
When he finally broke through the problem in a café, he went for a walk in a park, dancing and singing for joy. What a perfect image.
Only 3 people in the world understand what he did.
Top tips for students:
Cultivate the art of being bored. Take breaks from doing Maths, and from doing anything. Checking your phone or playing basketball are not acceptable, you need to let your mind wander. We went outside and each student wandered around the freezing playground in silence. Only Tom had a breakthrough in that time – fluke or because he has trained his mind so well?
When you are stuck, just keep on working. It is tough, but guess, plug in numbers, try easy cases. Failing is still learning. Cultivate the art of being stuck.
Don’t leave your homework till last evening. Give yourself time to get stuck and for the answer to come to you when your mind is empty.
Tom noticed that students would continue down a dead-end for longer than he would. Constantly change tack, attack from new directions, to maximise your chances of success.
Observations from the lesson:
Tom and I were both completely stuck at times, on the problems. The kids loved this.
“Sir can I have a two month holiday so I can solve this STEP problem?”
I need to help the students with their formal understanding of logic.
Some groups stayed just on the first part of the question, and dived into the rabbit-hole. Generating questions is such an important art.
Highlights from video: Tom pacing up and down waving his hands “Yeah it is true, I think it is true … … Yup, I think it is true” (that pause lasted 30 seconds). “O so now we are proving, not disproving! Saucy” exclaimed Wintana.
Implications for university:
Problem-solving does take place in student kitchens or in communal study areas – students struggling through example sheets, going over lecture notes together.
What is lacking is a sense of autonomy. Students in general are not learning to generate questions, to learn which questions are useful/beautiful/interesting. They tackle questions given to them, for which they know there is an achievable answer. There is no real ownership of problems. If you continue to a PHD, then this skill is essential. If you leave the world of academic Maths, then this skill is essential. Therefore, this skills is essential for all.
Maybe I should go and do a PHD, to reflect on whether my problem-solving skills have improved through helping others develop theirs?
Ifte and I went for an afternoon’s explore at the newly created Dyson Institute – an offshoot of Warwick University nestled within the Dyson HQ in the middle of nowhere in Wiltshire. 33 undergrads in the first year of the experiment. 1 day of intensive lectures, 1 day of self-study, and 3 days working as an engineer for the company. The students/employees get paid, rather than paying, for the experience. A fascinating model, always a privilege to visit somewhere with a strong vision.
Here are some thoughts from the day:
4% of students study engineering in the UK. In Singapore it is 40%. What is the ideal number?
Despite it being an Open Day, the closest we got to seeing any engineering was being shown the terrifyingly secure doors, controlled by fingerprint scanners, that let in the engineers to the inner sanctum. A shame.
Is learning only worthwhile if it has a direct impact to industry? What is the link between university and employment? Is this a commercialisation of learning? A necessary evil or something to be actively celebrated?
Universities still have a role to play. It is just that the more varied the options for students, the better
Currently the institute had a standard A level offer. Given that students’ experiences at school will be so varied, why bother? With a cohort of 30 students you can assess in more personal ways, and support those who need it at the beginning of their journey.
Only 1 in 8 engineers are female. Ask a child to draw an engineer and they will draw a dirty man fixing something, rather than a woman creating something beautiful.
We heard of teachers transforming their practice with the help of inspiration from the Dyson Foundation. A class of quiet students methodically filing away at their acrylic torches was transformed into a class of joyful chaos, students all pursuing products that have a real need.
Dyson is a deliberately youthful organisation. James Dyson, at our swanky meal, said “I crave naïveté and hate experience”.
Speaking of James, his name was dropped constantly by everyone throughout the day. I asked about this, and the employees spoke of him as a centralised point of inspiration, rather than as a weight crushing down on them. Fine balance for a figurehead to strike.
One of the students said “At school, the teacher knows the right answer. Here, they have no idea. If they already knew, then there would be no point in asking”. This is not the distinction between school and university, but between bad and good teaching?
I overheard a Dyson student complaining to his teacher – “You made us de-bug a computer program just using pen and paper. When would we never need to do that?” The tyranny of relevance rears its head here – learning for learning’s sake has scarpered?
Thank you to the Dyson Institute for a fascinating insight into a really exciting new journey. Best of luck!! It has reminded me to get stuck into the world of university teaching.
Twice in November, Yr12 students expanded their brains while thinking hard about UK Maths Challenges.
First, every Mathematician in the year-group sat a 2 hour Individual exam. A few were visibly nervous the day before – “How can I revise? What if I fail?”. The idea of an exam (anything done individually in silence in protected time) that is done for fun, and means nothing, is utterly alien to them. This is a bit of a shame.
My predictions of who would do best in the challenge were inaccurate. Students who are thoughtful and quiet did excellently, but tend to fall of my radar in wizzy interactive lessons. A useful reminder. Some students who don’t seem to be putting much effort into normal lessons and are struggling, still did excellently. I wonder why? The questions are assessing something different to a usual Maths exam…
A few weeks later, after enjoyable Friday afternoon training sessions, four Further Maths students traipsed across London, to Pimlico Academy, for the team challenge. 18 teams, with disproportionate number of boys, and of private schools.
It is a rare event where warm-up maths questions on the tables are enough to reduce a room of 72 teenagers to silence, just for the joy of solving. A shame that there was not more talk – either within teams when trying to solve group problems, or between teams (I love a bit of structured socialising between different schools, but my team definitely did not appreciate my forcing them into meeting others)
“Demoralisingly hard, but fun” summarised S at the end of the afternoon. Private schools dominated the top positions, which would be a hard thing to fix, even if you thought it was a problem… We came 14th out of 18th, very respectable. There are no winners or losers in the Maths Temple, only worshippers at the altar of truth.
I wandered around asking the other teachers “what is the vision of your school”, and was baffled by how baffled they were at the question. “What do you mean? It’s just a school…” their confused faces said. I take for granted the strong vision of School 21.
Afterwards, organised fun – walk along the Thames Path (beautiful reflections of skyscrapers on the river) to Turner Collection at the Tate Britain. Heated argument, when S took a photo of a detail of one of Turner’s sea paintings and turned it into a meme that A had “invented”, for snapchat dissemination. ‘Delete that! That’s MY meme!”. Ha!
Teach yourself to solve a rubik’s cube (Every student can do this, and the craze is slowly spreading across the year-group. I cannot solve it still.)
Intro sessions, to understand the journey from words, to pseudo-code/flowchart, to formal python code.
Collaboration with Computer Scientists:
We have joined up with the Computer Science class. We send them algorithms, they make them sexy and efficient. They send us algorithms, we prove they always work.
Example: the above pseudo-code for corridor greetings became this code. A joke death-predictor became this
A student who studies both maths and Computing went away and revised his knowledge of matrices to find the inverse of a matrix
We debated, inspired by the moral arguments of Cathy O’Neil in Weapons of Math Destruction. The Wright brothers used to swap sides when they couldn’t agree, to empathise with the other point of view. In that vein, the computer scientists had to argue that “Algorithms do more harm than good”.
Task: collect as many blue dots as possible, as few reds as possible.
Learning: inequalities in the plane.
Outcomes were very surprising. The assumption was that students would fiddle around with straight lines. Instead, they leapt ahead to bizarre curves, often using trial-and-error, and the power of online graph-sketching.
Is it okay that these curves would never have been dreamt up if the students didn’t have access to technology?
Would there have been more profitable learning if I had restricted the game to straight lines?