Tom, study-partner-in-crime, kindly came in for a day to work with the sixth form students, 300 intense minutes of A level maths. This is the FIFTH time he has worked in schools with me – such kindness (including last summer in Cambridge and last term in London).
When Tom is graph-sketching, he draws in multiple possible versions of what the line could be. Thinks a while (often fairly loosely – “it becomes very very very big” rather than bothering to use any actual numbers), decides on the correct version, and discards the others. Excellent drafting process.
When talking about problem-solving, L said “when you started with us in Yr10 I thought it was trash. Now it’s amazing”. I wonder what changed? Is this an inevitable slow conversion to a new way of thinking about Maths? Something to explore more.
I briefly mentioned how you could use the power series expansion of e to define how to do “e to the power of a matrix”. W exploded – “Here is circle of nonsense that I accept” she shouts as she draws a small circle on the table. “Square root of minus 1, that’s in here. Infinite series, that’s in here. But a number to the power of a box?! That is way over here! You need to get your ideas and drag them into my circle!”. Hilarious, and so fascinating that she now is completely used to such ridiculous things as complex numbers. The ideas don’t change – it is her circle that simply expands as nonsense slowly becomes nonsense that she accepts.
In the afternoon we explored Kepler’s model of the solar system – spheres within polyhedra within spheres within polyhedra… The kids loved struggling through the three-dimensional thinking. Found the problem in an excellent collection of problems from Stanford University.
15 minutes after the end of the day, the students still refused to leave. A good sign?
It is two days after the Hampton Court Half Marathon, and I am still in shock, walking around in a daze. The culmination of a long, arduous, powerful journey from October to March, including parkruns, longer runs, and weekly training at the Olympic Park.
Bitterly cold weekend. “So cold that I couldn’t leave the house” complained one teacher the next week. Not for 21 brave students, who all turned up at 645am at Stratford station, carb-loaded up, nervous, swaddled in layers, ready. We met a gaggle of 19 teachers at Waterloo, joined a train crammed full of lycra-clad athletes (and my dad), and were off out to Hampton Court.
Mark and I ran with James for the whole course. See his excellent blog here. Throughout, strangers were remarkably generous to James. Is he okay? Should I call a medic? Does he want the last of my food? Does he want my coat? Questions and concern and help came flooding in as James slowly sunk more and more into pain.
After passing the six 21 cheer squad, I felt really good and threw my hands upwards like I just won a 12 round fight in a boxing match.
Story Two: Zepora and Grit
A flash of red hair from a lower level of the gardens, and Heather and I gave chase, jumping over barriers and climbing a wall to meet Zepora, just after Mile 8. She had been running for 1 minute, walking for 1 minute, and continued to do so, like a machine, for an astonishing 4 hours 44 minutes and 22 seconds. What incredible self-discipline. Never stopping the flow, never stopping moving. Smiling throughout.
Students and teachers had waited out in the cold, well after their race had finished, to cheer Zepora in at the finish. What a sight to see 10 students sprinting towards us to bring her home, screaming their hearts out.
Story Three: Wintana and Beauty
Wintana (supporter) was bowled over by the beauty of the palace (built 503 years ago!). She pinkie-promised her friends that they would return in the summer to explore the world-famous maze. Daniel and I had deliberately chosen a race in a beautiful and relatively distant location, to encourage the students to get out of the East London bubble.
What can I learn from those that ran and supported?
When I meet the harpies in the land of the dead, what true story will I tell them? What new stories can I create in my future? How can we make the most out of living in the best city in the world? I love Philip Pullman.
If you spend your life doing nothing but watching television and playing computer games, you will have nothing to tell to harpies in the world of the dead, and there you will stay.
The highlight was the 10 minutes of students popping up, in front of 150 of their peers, and recognising other people for kindness on race-day. A pregnant silence at the beginning eventually gave way to outpourings of gratitude. Thanks for never giving up on me, thanks for always encouraging me, thanks for all the support, thanks for dragging me over the line. We are deliberately creating a culture of recognition in the sixth form, and the authentic and eloquent tributes were testament to the hard work invested from the start.
Thank you to Daniel for training and collaboration, Steve for expertise, Heather and Debs for supporting the girls, Karenann for leading the supporters, Martin for mopping up late students, Rachael for cheering at the finish in the freezing cold, Mark for caring for James, Papa for taking photos, and many more kind and lovely people. Such a great team effort from the teachers.
The next challenge: Run21 takeover Hackney Marshes Parkrun, 200 students from school run it. 19th May. Game on.
In a sequence of lesson-studies, MG, GD and I developed a first draft of a framework with which to assess collaborative problem-solving. In this post I shall reflect on the first time we carried out such an assessment with all 60 A level Mathematicians.
A reminder of the rationale:
We want to measure what we value, rather than value what we measure. Collaborative problem-solving on challenging open questions is an essential skill, both for professional mathematicians and in the modern workplace. Therefore we shall measure collaborative problem-solving.
This problem is excellent because it encourages question-generation, is open enough to play to the group’s strengths, and lends itself to visual thinking.
Record and Reflection of the lessons
I was lucky enough to be able to be in all three lessons. Every teacher embodied the vision of the maths department vividly – emphasising the joy of the hunt. All 3 classes showed outstanding grit, focussing on a difficult problem for 100 minutes without any need for the teacher to monitor behaviour. This wouldn’t have been possible at the beginning of the year. In my group, half the class said they were motivated by the beauty of the problem and half by the fact that their work counted towards their final grade.
After 10 minutes silently inventing questions on their own, the students shared within their groups.
Next, up to the whiteboards to explore a few questions in more detail. A snapshot of one of the lessons, showing how every student is naturally engaged with the maths:
The moment that M makes an astounding link between the spiral and concentric circles:
She is talking about this diagram, and is later able to prove that each ring has the same area:
Notice how MG is quietly stood in the centre of her web, assessing how the students are working, their backs turned to the teacher. Is it right to transform the teacher from the centre of learning to someone quietly filling in observation templates? Very different spatially to a traditional classroom:
In the video below, everyone is contributing to the group. Despite the fact that the mathematical concepts aren’t spot-on, the warmth with which they collaborate is sublime. They made another fundamental mistake later on in their journey, and I just couldn’t decide whether to point it out or let them continue. Am I doing them a dis-service by valuing deep mathematical thinking over mathematical accuracy? If we value more than just the right answer then why should we value the right answer at all?
Finally, the students begun the process of writing up their record of the journey. The journey was collaborative, the write-up will be assessed individually. Is there a tension here?
Students were very quick to pick up a calculator to test conjectures, but slower to pick up Desmos/Geogebra/Excel.
To use Alberto’s language, should the squads work as a team (there is a leader, each person has a clearly defined role, everyone working towards common aim in different ways) or as collaborators (everyone thinks together, everyone is equal). Which is more efficient? Is efficiency the only thing we care about?
I made an effort to welcome other teachers into these sessions – everything we are doing at A level can be translated lower down the school?
Students got a huge satisfying kick out of being able to answer questions that they had invented themselves, even if the questions were a bit trivial from my perspective. For example, one group spent a few minutes proving algebraically that a line with positive x-intercept and positive y-intercept must have a negative gradient. Moral: intervene less, stand back and watch and let the students explore things they find challenging.
In the next post I will reflect on the assessment of the write-up. So excited to read the students’ work, given how much effort they were clearly putting into it…
Planning through thinking, rather than resourcing. No slides provided. No slides needed?
Feedback so welcome.
The journey we will take:
Notation: what does that squiggly line mean? Note, there is a unique integral (if it exists). Signed area. If I know this area, what else do I know?
Estimation. Gaining intuition through rough eye-balling and some good geometric thinking. Is this area finite, infinite or nonsense? If it is finite, guess the area? How about for these curves? (See below for some questions)
Approximation. Explore various methods of methodically chopping up an area into smaller nicer shapes. Play here. Why do we not use sigma notation? Because integration is not sum, but limit of sums.
Formal derivation of integral of small polynomials. The doppelganger to differentiation from first principles (this is more technical and therefore non-examined).
Pause any discussion of area, as the anti-derivative strolls in – “if this is the gradient function, then what could the mother-function be?”. An appreciation that if one anti-derivative exists, then in fact a whole family exists. An appreciation that some functions have no anti-derivative at all. Karenann’s excellent question: “Find me curves whose tangents make a square” is useful practice for finding an anti-derivative.
Finally, the mind-bogglingly convenient Fundamental Theorem of Calculus. “Integral = anti-derivative” WTF. Area is the opposite of slope? Think of this through displacement and velocity. If all you can see is the speedometer, how can you calculate the distance you have travelled? 3blue1brown is the video to watch. Approximate velocity as collection of constant-height lines, since displacement easy to calculate for each segment. Signed area = displacement. Total area = distance. Also, the integral of a function is determined completely by the value of the anti-derivative at just the two endpoints?! How is that enough information? (p.s. might also mention Fundamental Theorems of Arithmetic and Algebra here too?)
Woah this theorem is powerful. Let’s use it to find the average of a function (How does that even make sense?!)
Wait a second, sometimes this theorem fails… And this theorem has a history.
For an utterly awful introduction to integration with not a single mention of area, see here:
A challenge to finish a half-term’s focus on differentiation – can you find a function whose height is equal to its slope at every point?
Start by brain-dumping everything we have learnt in 2018
Can you find a function where the height = slope at x = 1? y = nx was quickly found. So was y = 0, but several groups dismissed this as “too boring” – excellent mathematical taste.
What about at two points? Three points? At n points? One group found “Igoris’ Conjecture” – that a polynomial of degree n will have height = slope at n points or fewer. Igoris, father of the conjecture, then went off to explore the Fundamental Theorem of Algebra, using youtube+notebook excellently.
What about at all points? Desmos + whiteboards. Aha! y = x^x seems to work (bring power down, take away 1…). Really excellent idea, and challenging to refute. What is the difference between n and x in the equation y = x^n? Both can take any value? Something to do with the hierarchy of variables, but it is subtle.
Two people found a solution. W used her excellent wider knowledge and recall, dredging up a memory of a Mathologer video she had watched a few months ago. Amazing! S built on Igoris’ conjecture – if a polynomial satisfies the condition for only a finite number of points, then the function we are looking for must not be a polynomial. Maybe it is an exponential? Let’s use desmos to find which expontential? She created a slider, and played. Amazing intuition. Hear her talk it through here:
6. Talk through the ridiculous properties of Euler’s Number (first discovered by Bernoulli…)
What a great way to finish the term! Great buzz in the classroom – others genuinely excited at S’s breakthrough, spontaneous round of applause when she finished her explanation…
We could have started this lesson with a recap of exponential functions, to nudge students. How much more impressive that S made that link completely cold!
I made zero resources for this lesson. No slides, printouts… Thinking is better than resourcing. I am only starting to learn this.
Beware the danger of getting carried away when something brilliant happens. A few groups were sat outside in the dark, huddled around a quadratic function that was giving them no warmth. Don’t forget them.
Can you prove that there is only one (non-zero) function whose height is equal to its slope? Why must it never have a turning point? What about in higher dimensions, does the question even make sense?
This is a fairly good example of a naked lecturer (based on the essay by Tom Korner). No resources ensures that the lesson is alive, and if teacher can do with no notes then students are encouraged to see that it can’t be that tricky…
“A mathematics lecture is not like a classical symphony but like a jazz improvisation starting from a small number of themes”
“Most mathematicians would prefer someone with something to say but who says it badly to someone with nothing to say who says it brilliantly”
Second trip to the Barbican with Six21 students (first here). This time to hear two pieces by composers on their deathbeds, and Elgar’s first symphony. The students spent their time crying at the beautiful bits, imagining stormtroopers marching across the stage, giving the musicians names (“and then Izzy looked at Gerald scornfully”), getting angry about lack of ethnic diversity (in audience and on stage) and napping on each other’s shoulders to soothing Elgar. Ma and Pa came too, which was lovely. Second movement of Bartok’s piano concerto was my highlight, gorgeous solo chords, unexpected and jazzy.
Inspired by Ben Sparks’ amazing visualisations (see previous blogpost), a few students from the class got a bit obsessed with circles and lines.
Ashwin made another. Here is a draft. He thought about multiplying trigonometric functions together.
Cody’s final piece. Here is a draft. He worked backwards, working out what translation within the trig functions would give him the desired result, using a combination of clear recording and guesswork. Made a great link to the mean of two functions.
So great to see very different working out on the same problem. We deliberately didn’t speak to each other.
4. The Maths dept bonded using hula hoops and tape:
5. The Further Maths class bonded using string and chalk. It is the first time I have “felt” what it is like to be a sin curve – we had to rush when near the centre of the circle, and slow down when towards the edges. Moving as a pendulum. Can we make this out of pendulums?
How can you prove that each dot travels on one line? Can you extend to other shapes? (oval? square?) Can you extend to higher dimensions (sphere!)
The Further Maths gang escaped the clutches of school, migrating to the heart of the West End to join 1,000 other maths dweebs, to celebrate and revel in the joys of Maths. We swarmed into Piccadilly Theatre, up to the Gods, to be entertained by big names in Youtube Maths… Throughout, the emphasis was on stories and jokes, using emotional connections and humour to communicate mathematical ideas to a theatre-full of students.
Watching James Grime (numberphile) encode and decode a message on an actual enigma machine was special. Did you know 10,000 people worked at Bletchley Park during WW2?
Maths Slam – students have three minutes to give a presentation about a topic in Maths. Something to explore in the future…
Ben Sparks: Musician and Mathematician. Everyone accepts that music is played for the good of the soul, so why not accept that for Maths too? Circle or lines? (I programmed this in desmos using mostly trial and error). An astoundingly beautiful and natural progression of Geogebra applets to reach the Mandelbrot set had the whole room gasping in glee. Similarly for the spiral of Theodorus (the rays never intersect), and configuration of sunflower (“which number is the most irrational?”)
Shuffle pack of cards. If you had to guess where the second black queen was, what would be the best guess? (Answer: the 52nd card. Think of where the most likely spot is for the first black queen is. Flip the pack over to consider the second black queen. Counter-intuitive answer, based on elegant shift of thinking)
A4 – only rectangle that when halved gives two copies of itself. Greatest piece of design of the 20th century? Made photocopying so much more efficient…
BF Skinner’s amazing study, investigating how humans and pigeons both create patterns in randomness, looking for meaning where there is none.
Randomness is very hard to create. Best way is to measure natural processes. For example, last two digits of the reading on a decibel-meter (is that a word?). Benford’s law.
Some of the students surrounding us were surprisingly obnoxious and individualistic. A helpful reminder that all the students in this group are consistently kind, always ready to share ideas and support each other. This is linked, maybe, to constant collaboration when problem-solving.
Thirteen students and three teachers met in a sodden Victoria Park, to join in the local running club’s 10 mile race. Z was an hour and a half early, just to make sure. Snapchat was essential in navigating each other through the park to get to the clubhouse.
After weekly sessions at the track, and a good few Parkruns under their belts, we were ready to attack a longer distance. 5 laps of 2 miles was the full course. Excellent to do laps – everyone stays close to each other and people can stop without fuss. 4 6-milers, 5 8-milers, 4 10-milers. Outstanding effort from everyone. Particularly lovely to watch students encouraging each other, and runners/parkgoers cheering them on too. I think that, without exception, this is the longest they have run so far in their lives.
D: “That’s the most alive I have felt in my life – I am normally in my room!”. Perfectly encapsulates the voyaging spirit of running – getting out and exploring the world. Talking to dogs, ogling exercise freaks, taking in the trees and the weird bandstands that dot Victoria Park, working out how to get here, hot tea and cake in the clubhouse – it is all an excellent experience to learn from.
Onwards, towards the mighty half-marathon in March!
Merged the top set Yr11 with the Further Mathematicians, for:
Problem-solving! To get a flavour of how to explore, destroy, generalise, conjecture…
Q & A between the two groups of students. The teachers left the room for this, to ensure maximum honesty. Who knows what was said!
The problem: Crossing the river…
Excellent intuitive introduction to graph theory (which the Yr12 students had seen but the Yr11 students hadn’t). Groups instinctively drawing graphs to represent the relationships between different animals. Some great digressions – what if there were three islands rather than two riverbanks? What if there was a crossing-time associated to each animal? What are all the different situations with exactly four animals?
What if there are more animals?
What if there are different animals?
What if there are lots of rivers?
What if there are lots of boats?
How could a diagram help?
How could a table help?
Hopefully this will encourage a few more students to take Further Maths next year? “It is the hardest A level, but the best one” said W proudly.