# Queen Dido’s Problem

Gladly there are no longer any external exams in Yr12. An excellent opportunity to assess what we value. 50% of the final grade will be from a bog-standard exam (we want to prepare the students for next year). 50% will be from a problem-solving portfolio. With the students off lessons to revise and take exams, us teachers have enough time to assess group-work in detail.

The structure:

1. Students are given a prompt
2. Room 1: 25 minutes of independent question generation
3. Room 2: 30 minutes collaboration, with a maths teacher observing and assessing
4. Room 3: Independent write-up (as much time as you need – we are measuring depth of thought rather than speed of writing here)

(Important to have separate rooms for each phase – change of environment signifying change of thought process)

Prompt:

For next time (thanks MK for feedback):

• Find a non-whitewashed image of Queen Dido
• Make the leap from Dido to the Rope clearer
• Think of a less tokenistic way to include female mathematicians

Before we assessed the students, we (well, the rest of the team) had a go at the problem. Lovely to see the deep joy that my colleagues derived from struggling and searching for a route through the Maths Maze.

Room 1: Question Generation

Students enjoyed the narrative – “That Queen was a BOSS” – and came up with some innovative questions, including:

1. Does the layout of the land matter?
2. If I have a rope of length x, and I coil it up in a spiral, what area will it occupy?
3. If the rope traces out a graph (for example boundary of negative quadratic and the x-axis), what length will it have? (Excellently natural motivation for arc-length integrals)
4. Are regular shapes best? Why?
5. What would a graph of area against number of sides look like?

Some students didn’t use the 25 minutes well – came up with a few questions and then waited for the next stage. How to train them to develop more questions, or explore existing questions?

Questions I now want to ask:

1. What if the sides can be curvy?
2. What if the polygon had 3.5 sides?
3. Does a circle have 1 side or infinitely many sides?

Room 2: Collaboration

• It was a privilege to be able to focus exclusively on one group. You can see so much more! From the student who quietly checks her phone to plan use of time, to the dominating student who absent-mindedly rubs out a team-mate’s work. A depth of observation that is not possible in a standard classroom setting.
• Watching students decide on a question to tackle together was fascinating. The sharing varied, from haphazard mumbling to formal sharing protocols and blind voting. Often the sparkier questions were ignored and the safer questions (e.g. if the perimeter of a regular hexagon is 10, what is its area?) were explored. Is this an issue? Some groups focussed on the precise wording of their question, all writing it down separately. Others found their problem-solving lacked direction due a lack of tautness of the question.
• Maybe it was the lack of buzz from surrounding groups, or the fact that this was a formal assessment. But, often the groups lacked zest, quite flat and nervous. A shame that we created this atmosphere – problem-solving should be joyful and energetic. As the students get used to formal collaborative assessments they will thaw, but how can we help to prevent the initial freezing?
• As well as the four members of the A level Maths team, 14 other teachers came to help moderate the assessments (self-awareness is not subject-specific, any teacher can observe for that?). It was an excellent way to spread awareness of our work, and to trigger similar structures in other disciplines? Close analysis of poetry in English? Philosophical debate on an unseen prompt?

• Some great feedback from non-mathematicians.
• Jess thinks that the students would benefit from a focus on improving their quality of talk – using talk structures would encourage more effective and efficient collaboration. Maybe the only feedback students receive in the first term of Yr12 is on their talk?
• Wendy reflected on her PHD in Chemistry, when her supervisor would give her more constraints than the students were given in the problem. “There are very few instances in the ‘real world’ that have a need for that kind of wildly open thinking, even at research level you are dictated to by current trends and where the funding lies, so blue sky thinking, or dealing with problems that need a totally new rationale not yet developed rarely happens – this the domain of the Einsteins, Democritus’ and Newtons of each generation.” Structure liberates – are we making unfair to the students by giving them such an open problem? “Personally I think it would have been a more useful exercise to give the students a problem to solve that relied on their application”. It was definitely my intention to test an application of A level skills in an unusual context – so I think we agree here.
• George continues to be nervous (see our previous debate) about the amount of time students spent feeling lost – are they wasting their time and ending up feeling dissatisfied? “Thinking like a mathematician or a scientist is not an innate skill.”
• Joe thought that the students were speaking like they were writing a paragraph in a history argument. “Here is my main point. Here is my evidence. This is why the evidence helps”. Collaboration sometimes is all about persuading other people why a fact is true. Great links between mathematical persuasion and persuasion in the humanities (as well as expertise about how to mark nebulous essays).
• If a history student wrote “Yes” as their answer to a history question, they would expect to get few marks. Why do maths students who write down a single number as an answer expect to get full marks?
• One big issue that we noticed was the dominance of the students with a reputation for being smart. These students spoke too much, and held the whiteboard pen too much. Their team-mates rolled over too easily, and rarely stepped up to politely tell them to quiet down. We have given a quiet word in the ear of the dominators during previous sessions. How to solve this? Maybe it isn’t a problem – any group will naturally have leaders? Blindly following someone is never good though, and it is worse if the dominating leader mistakenly believes that everyone had an equal voice…
• Tactile prompts are always fun:
• The quality of the mathematical skills lagged behind the depth of conceptual thought. Often it seemed that the students’ brains were filled up with trying to work out why a circle was the best shape, and so things like their application of the sine rule or memory of using differentiation to optimise, suffered.
• Only one group used tech. To plot a graph of max area against number of sides of polygon. Is the lack of tech a problem?
• Astounding intuition from L: “Imagine you have a rubber sack, and you blow air into it. The pressure will push outwards in all directions, so the shape will become a sphere. Obviously there would be no pointy corners. So, the solution to Dido’s problem must be a circle”. Kind of beautiful, but it took 2000 years to create a formal proof of this claim.
• “We always finish our collaboration by creating a timeline of the process” announced one group. Excellent to create bespoke group norms. How to encourage all groups to glue together this well?
• Plenty of teacher analysis:

Room 3: Independent Write-up

This room was largely unsupervised, yet still silent. Students spent from 1 hour to 3 hours (!) in here writing up their thoughts – testament to their commitment and pride in their work.

I sat in here for an hour and struggled to prove Fardeen’s Triangle Hypothesis. Lovely to work alongside the students, and hopefully will be a useful model for the kind of commentary we are trying to create – balancing diagrams, words and symbols.

Reflections while marking Write-ups

If it is easy to mark, then the teacher should not be marking it.

True? Useful? This marking is definitely challenging, but so much fun. Bog-standard marking can be done by Siri/Student/Peer.

• It is too tempting to continue the avenues of thought that the students started.
• The depth of awareness about what students have learnt is improving!
• “I should have spoken up, but didn’t want to sound like a dumbo” reflected one student. “Looking back at it, I realised that we could have done this a different way”.

• Ahmet finishes every write-up with a pararaph explaining why he loves problem-solving. Here is the latest:

“I think that prob-squads and problem-solving is amazing because it tests creativity and collaboration of students and ability of relating problems to past experiences and build upon that. Exams don’t really test that, because it’s a memory-game. You are very limited to constraints of question, where you get penalised for going off-topic. You stay on auto-pilot, which pretty much defines your future”

Nicely put, Ahmet. Am I being irresponsible if I am encouraging students to snootily look down on exams?

• What is the difference between a conjecture and an assumption? (Damien’s excellent question, from marking the Mechanics Problem.) Both involve students somehow using guesswork or less precise thinking, to make a statement whose truth is currently unknown. I assume that friction doesn’t exist in this scenario. I conjecture that every even number can be written as a sum of two primes.. Can you assume that “as n increases, area increases?”. Can you conjecture that “air-resistance is negligible?”. What would a venn diagram of all posible statements look like?
• A general over-reliance of algebra, and under-appreciation of geometric analysis. For example, if I know the centres of two tangential circles, how can I find their intersection point? Immediate response is to attempt to create two equations and solve simultaneously. Thinking geometrically about key properties of the intersection point would have been so much more elegant…