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.
- Students are given a prompt
- Room 1: 25 minutes of independent question generation
- Room 2: 30 minutes collaboration, with a maths teacher observing and assessing
- 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)
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:
- Does the layout of the land matter?
- If I have a rope of length x, and I coil it up in a spiral, what area will it occupy?
- 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)
- Are regular shapes best? Why?
- 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:
- What if the sides can be curvy?
- What if the polygon had 3.5 sides?
- Does a circle have 1 side or infinitely many sides?
Room 2: Collaboration
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…