## Problem-Solving: A debate

Organised excellently by Philosopher-Mark, in the style of Moral Maze, George and I debated the following motion, in front of an audience of 30 Yr12 students and a handful of teachers:

The best way to learn problem-solving is to problem-solve

For the motion, my argument. Some of these points are not really that relevant, but might be useful:

• I think that a more general statement is true: “The best way to learn x is to do x”. It is true of learning an instrument, learning to swim, learning to write, to draw, to play sport… Of course there is a place for learning from experts and standing on the shoulders of giants, but the majority of your time should be spent doing. Why should problem-solving somehow break this rule?
• I would rather inspire students to love maths than to train them to be excellent at it. If they are motivated, then they can learn skills from any number of sources (the internet and friends are both amazing things).
• One counter-argument might be that “First, they have to master the basics”. In other words, it is a waste of time to dedicate valuable lesson-time to problem-solving if the students can’t even add numbers correctly. I have three responses. Firstly, what constitutes “the basics” is relative and therefore meaningless. Is multivariable differentiation a basic skill? Maybe at university. Is knowing about place value a basic skill? Maybe at primary school. At any stage there will always be new skills to learn. If you waited for a student to master the basics, you would be waiting for a long time. Secondly, order of delivery implies order of importance. If you start with building skills, and tack a problem-solving lesson on at the end of term, you are sending a clear message to students. “If I had to choose, I would rather you can correctly use the cosine rule than can think independently and flexibly in novel situations”. Thirdly, it would be ridiculous and cruel to expect a student of music to only play scales as a child, waiting till they were 18 before allowing them to play actual music (stolen from Lockhart). Similarly, surely, for Maths.
• Another question might be “What is the role of the teacher then?” If the best way to learn is to do, then does the teacher become irrelevant? No. The teacher carefully chooses problems at the correct level and of the correct flavour to challenge and inspire particular students. The teacher observes first, and models second to correct key misconceptions. The teacher spots mistakes and decides whether to intervene or not – some mistakes are worth stopping instantly and some should be left to flourish.
• My three grounding texts are “A Mathematician’s Lament“, “If this is the headache…” and “Mathematical Etudes“.

Against the motion, George’s argument (correct me if I got it wrong!):

• I know the paths, streams and hills of my childhood intimately. If you wanted to go for a run there, wouldn’t you want to ask me for a favourite route – one that is enjoyable and doesn’t end in muddy fields and barbed wire? By analogy, I know physics intimately. There is a place for getting lost, to ensure you remain aware of the vastness and complexity of science. Most of the time, however, you just “want to get home” without getting lost. My role as a teacher is to guide you, to teach you how to read the map of Physics. I will use my expertise to help you avoid the muddy field of negative signs and to navigate over the barbed wire of resolving forces.
• If you are allowed to get lost too often, you lose joy. As an example, a student wrote “There is a proof, but we’re not smart enough 😦 ” at the end of summary of a problem-solving session. Getting home feels good, and is therefore motivating.

Reflections:

• George and I both view ourselves as experts. I got the impression that George was able to model, clearly and helpfully, the solution to any problem that his students were working on. On the other hand, I have been giving my students problems that I cannot solve (mostly because they are so open as to have one particular solution). Is it empowering to give students a problem their teacher can’t answer, or does it inevitably lead to everyone being confused?
• When the students leave the realm of George’s map, how will they find their way in the wide world?
• George was far more successful in capturing the imagination of the audience with his “muddy fields” image, and by restricting himself to one clear position. Consequently, there was a large swing from the beginning to the end of the debate, from “agree” to “disagree”. Well done!
• If I was to restrict my argument to a story, as George did so effectively, it would be from when I was 16, learning to become a sailing instructor. The sky was grey, the sea was choppy, I was cold and tired. I was trying to tow 5 boats behind a powerboat, but first had to tie them on. My fingers were fumbling with a bowline, trying desperately to get some slack on the rope. It took me 10 frustrating minutes to complete the knot. Only at that stage did my assessor, who had been watching me throughout, step in and tell me “You were trying to use the wrong knot. Next time use a round-turn and two half-hitches”. Apologies for the arcane sailing references, but I have never forgotten when to use each knot – it was absorbed into my gut through being allowed to make mistakes and fail.
• It is excellent that we both agree on the importance of problem-solving, and are passionate enough to organise a successful debate.
• A number of students asked excellent and eloquent questions. Training students in the art of teaching is important, since it gives them the tools to give constructive feedback to their teachers. If either George or I ever stray too far away from the happy middle-ground, hopefully our students can bring us back.
• Some students said “Maybe it is good to get lost in Maths, but good to be guided in Physics.” Is this due to a genuine difference in disciplines, or because they have been brainwashed by their teachers?
• George mentioned that he starts to have a horrible time when lost out in the wilderness. I specifically seek out the feeling of being lost when out running – I love the unexpected discoveries. Is this just a case of different personalities giving birth to different pedagogies?
• There are two ways you can love a subject. You can love the rush of getting lots of things right and being good at it. Or, you can love the feeling of challenge and exploration. Only one of these is sustainable – eventually everyone stops being good at their subject, once they get to a high enough level.

See here for notes on the process. Students disappeared off home to write up the journey they went on. Huge amounts of care put into this – half say because they enjoyed the problem and half because they know ti counts towards final mark. Here are some reflections on the write-up:

In general, students should use diagrams more, read their work aloud to check it makes sense, and don’t make the mistake of thinking that including reflections means you should exclude actual Maths.

The next assessment of problem-solving… (see first here, and student write-up here)

1. For homework, complete this excellent task from Underground Maths. (I tried it last year)
2. Begin the lesson by drawing a Map of Maths. Some hilarious responses.
3. In the meat of the lesson, use the two-way table as a prompt. What other headings could you use? How could you create links to other parts of Maths?

4.  Maybe less nitty-gritty mathematical thinking this time, but more higher-level thinking about how to create links between disparate parts of Maths? Unsure, let’s wait until the write-ups come flooding in…

## Seeing Squares: The Art of Question Generation

A first draft at assessing problem-solving with Further Mathematicians.

1. Find a problem, written in 1950 for university applicants to Stanford University
2. Work on it with Tom, and then work on it a bit more alone.
3. Distil the question to its bare bones

4. Students work, at first alone, and then in front of 30 visiting teachers, on the problem.

5. After noticing how hard the students found the task of generating questions from the prompts, I asked PHD students to do it too…
6. Students analyse all questions that were asked, and start to analyse what makes a good question. Could you guess which were written by professionals and which by students?

We came up with the following rules for question generation:

 Do Don’t Ask “How” questions Have an end goal in mind Look at interaction between two variables Make links to other topics Examine the scenario from multiple viewpoints Ensure your question is comprehensible to a stranger Be trivial Be vague Ask questions with a yes/no answer Make assumptions in your questions – include them as sub-questions instead

Examples of good questions:

• What are the conditions required to create a 45 degree angle from the lines going through a square’s vertices? What would the collection of possible “45 degree points” look like?
• Suppose the squares have side length 1 – what is the largest possible perimeter for the triangles formed from the straight lines and the square (with the 90 degree angle still assumed)

Examples of poor questions:

• Is it drawn to scale?
• How can I link this to trigonometry?
• Are those squares?

7. After all of this warm-up, when the students saw the original question they were remarkably quick at finding the answer (once they had parsed the word-heavy question). Great recall of circle theorems.  I made a toy on Geogebra to illustrate the problem

## London is the best city in the world.

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.

A quotation from hero Philip Pullman. See previous blog-post for an outstanding story that 21 students created

You live in London. How are you exploiting the riches and joys that the city has to offer? Which stories are you creating?

Here are some links to explore. Lots of places are desperate to encourage young people to explore, so there are plenty of excellent offers.

What have I missed?

## 300 minutes of Tom

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).

Some thoughts:

• 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?

## Grit, Kindness, Beauty

If you do three things, you should click on:

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.

The Journey

The morning

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. This was the only race across the whole country to not be cancelled – so lucky!

The race (complete pics here)

Story One: James and Kindness

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.

The Aftermath

On Tuesday in assembly, we reflected:

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.

Post-script:

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.

A reminder of the framework:

The problem: The Spiral of Theodorus

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?

General reflections:

• 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…

## A journey through Area

Planning through thinking, rather than resourcing. No slides provided. No slides needed?

Feedback so welcome.

The journey we will take:

1. 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?
2. 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)
3. 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.
4. Formal derivation of integral of small polynomials. The doppelganger to differentiation from first principles (this is more technical and therefore non-examined).
5. 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.
6. 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?)
7. Woah this theorem is powerful. Let’s use it to find the average of a function (How does that even make sense?!)
8. 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:

## Height = Slope

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?

1. Start by brain-dumping everything we have learnt in 2018
2. 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.
3. 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.
4. 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.
5. 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…

Some notes:

• 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”