Lesson Study 3: Towards a Framework

Our question: How to assess collaborative problem-solving?

  • In the first lesson, we thought about assessing before and after collaborative problem-solving.
  • In the second lesson, we explored alternating more systematically between solo and collaboration, maximising the benefits of both.
  • In the third lesson, we shall develop a possible framework for assessing collaborative problem-solving.

Design Principles:

I spoke to Zek, art teacher, about how to assess problem-solving. Assessing a mathematical journey is very similar to assessing a portfolio of artwork?

  • Keep the checklist as short as possible.
  • Don’t always try to prescribe numerical values to everything, you might kill it.
  • Assess a portfolio over time, to clearly evidence improvement


First draft:

  1. 50% through traditional written exam, to measure mastery of standard content
  2. 50% through assessment of collaborative problem-solving. This is split between measuring how the student works within a group, and measuring how the student writes up problems individually, after having struggled on them within a group. Portfolio of work over time.

With pretty colours:


The lesson:

What is the biggest rectangle that can fit in a right-angled triangle?

No help given from teachers at all.

  1. Each of the three teachers observed two groups of 3 students, noting down examples of kindness, self-awareness and spark.
  2. Students write up the problem, individually
  3. Teachers give live feedback on the write-ups, assessing for self-awareness, understanding and rigour.

Examples of work:

Excellently confused struggle – keep on going Fardeen!
On the left, the write-up of the mathematician who drove the thinking in the group. On the right, the write-up of his colleague, who explained the thinking with more clarity. Great example of a person with spark, and a person with rigour.
An example of a teacher’s assessment sheet.


  • Kindness and awareness are too similar? Merge into one, and instead, assess the application of problem-solving tools? For example, W kept on saying “Okay, so how can we make this problem easier?”, deliberately stealing the key question from previous problem-solving sessions.
  • Were the students pressurised by having three teachers strolling around assessing? Is this a bad thing?
  • Should we decide what amazing problem-solving is a priori, or by watching teachers/students/PHD students attack a problem, and note down what they do well?

Next steps:

  • Observe more problem-solving, fighting the temptation to leap in and guide. Watch for generalisable techniques that excellent problem-solvers exhibit.
  • Try in other subjects, not just Maths.
  • Try in the summer exams! Utilise the opportunity that no public exams in Yr12 has given us.

Lesson Study 2: Alternating

Continuing from Lesson Study 1, focussing on alternating between collaborative problem-solving and independent thinking. Collaborating is excellent for blue-sky thinking, for having questions answered and considering the big picture. Independent work is excellent for working through the details, and not getting sucked into group-think.

The Lesson:

Collaborate for five minutes:


Solo for 5 minutes:


The heart of the lesson: collaborate by matching graph to its gradient

Final solo task: extending to gradient of gradient.


  • “I hate it because I have no idea where to start” from students who are uncomfortable, and uncomfortable in the uncomfortable.
  • When all students in a group are confused, how do you ensure that they decrease rather than increase confusion?
  • This was a well-sequenced lesson, but didn’t really answer the question of how to assess problem-solving. What is it that we are looking for in an outstanding problem-solver?

Further resources on how to assess Problem-Solving:

Mathematical Research?

I visited Kings College London, for an afternoon in a stuffy room somewhere in a hospital, to watch a bunch of Yr12 students proudly present to a room of professional scientific researches. They had been working in small groups, an afternoon a week for 12 weeks, on a genuine research question. Using bubbles to MRI-scan tiny blood vessels, using 3D printing to create prototypes, computer programming, applying calculus to biology.

After each group had presented, their supervisor spoke. A common theme running through the observations was that the students had been struggling with the common and real challenges of scientific research, and were helpfully contributing to current understanding. Some of their work will be used as the germ for future Masters Projects – amazing!

Can you work out which are students and which are supervisors?


“You have done proper research”

How to do something like this with research mathematics? Is the abstraction of maths a barrier? You can learn to pipette in a lab without much prior understanding, but in order to understand complex tangent spaces you need years of foundations… Is there a question out there that is approachable and still deep?

Watch this space…

Lesson Study 1: Limits

MG, GD and I are thinking together about how to assess collaborative problem-solving. How can we show and convince others of the efficacy of more open lessons? We used a simple structure:

  1. Independent task 1
  2. Collaborative task
  3. Independent task 2 (similar to task 1)

Teacher then marks both independent tasks, and compares the differences. If the collaborative task were useful, then the second task would show marked improvement.

The Tasks:

Task 1: Area of circle though sectors


Collaborative task: Share understanding of Task 1, using different colour to add further notes

This student found independent work difficult, but was helped effectively by her table. 
Excellent diagram showing limits

When you use a limit it “becomes more true”. What would a philosopher say about this?

Task 2: Area of circle through rings

This is the student who only wrote one line for Task 1 – marked improvement! Interesting limit answer – the circumference is straightened out but never quite reaches a straight line…


  • Some students loved the independent silent work – a chance to think for yourself. Some hated it. What to do about this?
  • “Don’t be afraid of over-explaining” says George, who is more towards LAE on the spectrum between old and new school…
  • Be more precise in the assessment questions I set. It was fun to mark the students’ descriptions, but I couldn’t glean much precise information from them.
  • What is the best ratio of time for Independent : Collaborative : Independent?


Appendix: Limits are Beautiful.

An excellent way to step-up to more sophisticated thinking, grappling with infinity. Here are some more ideas:

Dancing to Rameau

Blog from student here


Took a few students to the Barbican last night, to listen to the LSO, conducted by Simon Rattle.


“Why is there coughing breaks in the music?”  “Why is there so much walking on and off and clapping?”. Classical music does indeed have some bizarre conventions.

Musical highlights:

  • Rattle’s wife, a singer, oscillating like a jelly as she sings Handel Arias
  • Unbelievably quiet start to Schubert’s Unfinished Symphony. We sat at the very back of the hall, and the acoustics were crystal clear!
  • Energy and joy when playing Rameau dances, Rattle jumping around the orchestra while conducting from memory. Contrasts to the stiff upper lip of the percussion player, primly tapping his tambourine.