Visiting Langdon Park

MG and I visited Langdon Park on “Looking Outwards” day. Every teacher at school has scattered to the winds to learn from other ways of doing things across the country. What a great idea!


As we waited in reception we struggled through a STEP question, getting a bit confused with the hierarchy of variables.


Key learning: think five years into the future, not one half-term into the future. In order to do this you need a stable core of staff – there are three maths teachers who have been working at Langdon Park for the past four years, and love working with each other


On assessment:

  • Only formative in Y7-9.
  • At A level, end-of-unit assessments. 1/3 are difficult problem-solving questions, that have been seen a few days before the test. 2/3 unseen. Less than 90% means you have to re-take in your own time. Homework is self-assessed, and teachers monitor quality of homework through the test scores. All exercises happen at home, leaving time in lessons for activities.
  • Kate questioned why I might want to assess collaborative problem-solving. Why kill it through formal assessment, surely it is enough to do more subtly in every lesson? My answer: my assessing formally you are able to track over time, and you remind everyone that it is something that you value. Who is right? Unsure.


On interventions:

  • School leadership tend to think the more time students spend with their teachers, the better. The teachers disagree – overload leads to exhaustion from teachers and students. “We want to see our students for the minimum amount of time”. It is about independence, and wellbeing. “If they work or don’t work, that is their problem. I am here to help them, not monitor them”.
  • Instead… residentials. Early in the year, take the students for a weekend away at a youth hostel by the sea. Do fun maths, focus on resilience, give the students lots of independence (I don’t care what time you go to bed, as long as you are up by 730 for breakfast). The focus is not to blitz through exam technique, but to build a strong culture in the class and to promote passion.


On the curriculum:

  • Yr7 on number, Yr8 on algebra, Yr9 on geometry. Go deep on each topic and then don’t revisit it. If students properly understand “division” in Yr7, then that will set them up for life. GCSE exams are easy if you understand a few core examples (for example multiplicative reasoning), and can apply them. Yr11 is about preparing for exams, but Yrs7-10 are mixed ability exploration and understanding.
  • The curriculum is then translated into a “Learning Journal”, structured exercise book for students, which contains within it space for notes, pre-printed assessments, pre-printed structures for problem-solving. Rough work done elsewhere. Beautifully effective.




On team-planning:

  • Leave the school! Go to a café, or to someone’s house, and bash out an afternoon’s work.
  • Create printed resources – a snazzy ringbound booklet of the term’s planning is much better than things floating around on Google Drive?


On notes:

  •  Key revision notes are made by students, throughout the year. Not copied from the teacher’s board, or from a textbook. By creating themselves, students take ownership and embed the learning more. Very similar idea to my Maths Diary. Learning note-taking before GCSE gives great autonomy to students when they reach A levels.


Thank you to Kate and Paul for hosting us – we came away brimming with great ideas!




The Three Faces of Fractions

A teacher’s guide to fractions.


Please read this first before looking on to the resources. Think first, resource second.


The three faces of fractions:  Region, Length, Process.

Fractions are confusing for students. There are (at least) three possible ways to think about fractions. Throughout, we are using Geogebra to illustrate core examples:

  1. Region (part of whole) Fractions. This is the standard way to introduce fractions to younger students.
  2. Length (measure)    Fractions on a numberline. Fractions exist as single numbers (not a pair of two numbers), on the number-line. Students often disagree with the statement “fractions are numbers” – thinking of fractions as lengths should help to counteract this misconception.
  3. Process (operation) Fraction and Division. A fraction is a process, encoding division into a single thing. 2/3 means two divided by three.


(Note, there are other ways of thinking about fractions, such as “fraction of  quantity”, or as a ratio. We decided to stick to three for simplicity).


Celebrate the ambiguity, constantly move between these three faces, and discuss with students when each viewpoint is best. Confusion arises when students attempt to use the a way of thinking about fractions (for example region) in the wrong context.


Other comments

Language note: Every time anyone says minus, a kitten dies. (“minus seven” is ambiguous – are you subtracting seven or talking about the number negative seven?). Similarly, any time someone refers to 1/3 as “1 over 3” a puppy dies. “1 over 3”  encourages the misconception that a fraction is about two disconnected numbers. “1 over 3 plus 2 over 5 is just 3 over 8!”. Instead, say “one third” or “three sevenths”, which encourages thinking about fractions as numbers rather than pairs of numbers.


The core concept, we claim, is equivalence of fractions.

  1. 3 and 5 look different, and are therefore on different places on the number line.
  2. 2/4 and 5/10 look different, but still are at exactly the same place on the number line.
  3. If you understand equivalence, then you understand simplifying fractions, ordering fractions, adding fractions.


The lesson sequence:

  1. First we explore student ideas and skills in the first lesson, and introduce the three faces of fractions in the second.
  2. We will then spend a lesson on each of the faces of fractions, exploring them in detail. In each lesson the skills are the same (equivalence –> simplification, improper fractions, and 4 operations), but the perspective has changed.
  3. The final three lessons start to summarise learning and look outwards, creating links to other parts of Mathematics, and to concepts outside of Maths.



Lesson Question Details
1 What is a fraction? Student-led. Pre-assessment, explore student ideas about fractions.
2 Why are fractions confusing? Teacher-led. Introduce the three faces of fractions
3 Why are fractions regions? Equivalence

    • Simplify
    • Improper vs mixed
    • 4 operations
4 Why are fractions lengths? Equivalence

    • Simplify
    • Improper vs mixed
    • 4 operations
5 Why are fractions quantities? Equivalence

    • Simplify
    • Improper vs mixed
    • 4 operations
6 Can I show mastery with fractions? Overall skills. Students choose which face of fractions they use.
7 Why are fractions useful? Explore applications, within maths, science, news, other
8 Have I actually understood fractions? Student summary

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.

MK visits

MK visits School 21 wearing her three hats: as mathematician, as musician, and as Women-in-Maths activist.

Chapter 1: Women in Maths


According to FMSP, in the UK:

  • 50% of GCSE students are female
  • 38% of A level students are female
  • 28% of A level Further Maths students are female
  • 19% of Research Mathematicians are female
  • 6% of Maths Professors are female

Can you name a female mathematician? The students couldn’t (but then, they didn’t know many male mathematicians either…) MK reeled off a bunch, including Maryam Mirzakhani, the first female winner of the Fields Medal. Gasps of shock when students learnt that this took place only last year.

Good turnout of Yr11 and Yr12 students, and equal gender split – this is everybody’s problem.

We learnt about the following concepts:

Untitled picture

  • Stereotype threat – the risk of confirming negative stereotypes about an individual’s racial, ethnic, gender, or cultural group.
  • Attributional ambiguity – “not knowing where negative responses come from. Is it because my work is not good enough, or because the teacher is prejudiced? Two groups of black students handed in work. Group A had their photos attached to their work. Group B did not. Group B responded more to feedback than Group A, since they knew that the feedback was not related to their race.

Often these biases and effects are subconscious. We therefore need to talk about them, to realise what we need to change about our beliefs.

Chapter 2: Maths and Logic

When Tom came in a few weeks ago, we noticed that the students were tying themselves up in knots about logical arguments. MK taught a session on logic – short lecture followed by problems. She has been honing these sessions back at UChicago, to consciously teach undergraduates the core mathematical skills that they would otherwise be expected to learn by osmosis.

Logic through venn diagrams


Mark, philosophy teacher, gets involved.
  • MK impressed with how well the students work well together. The adults in the room, when given a difficult problem, instinctively go into a nest and thrash out the details independently. The students instinctively work together through talk. Maybe we should now start to also focus on how to go it alone? But, given that most students will have experienced Maths as a solitary sport in their previous schools, it is definitely correct to go strong on collaboration initially
  • Mark noticed that each of the three groups was using a different method (venn diagram, chains of arrows, big tables). When is it okay to let many methods flourish, and when should you just teach one killer method?
  • Students really loved the logic problems – satisfying quick wins in comparison to STEP problems. We did some written by Lewis Carrol. More here

    Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised.

What do you do when you get stuck? MK says:

  • Draw a picture! I drew the same picture, in different ways, over and over for about a year. I will never forget the picture.
  • Stop. Deliberately forget everything you know and do it again, trying to come from a fresh angle.

What is the longest you have spent on a problem? MK thought this question made no sense – you start with a big problem and break it into sub-parts. Solve the sub-parts, except one which is tricky, which needs to be broken down into further sections… Continue constantly. So it all depends on scale.

Chapter 3: O Magnum Mysterium

The Yr10 students are singing Lauridsen, with something ridiculous like 8 parts. It is a great challenge, given that the majority of them cannot read sheet music and are therefore learning by ear. MK came and sang with the sopranos. Great to break into the private-school world of classical music.

Final Parkrun of 2017

J’s times: 28, 25, 23 minutes – ridiculous improvement in three weeks:

Brother running in with sister, charging to the finish:

Z had arrived late, and so was behind the trail-walker (volunteer who walks at the back to ensure everyone safely in). Nerves when she failed to appear, hop on bike to search for her. The rest of the gang all loped back to meet her, met her with whoops and christmas music blasting from their phones. Such excellent and willing support from tired and cold runners. Great great first 5k from Z:

E volunteered this week rather than running – had a lovely time giving back to the Parkrun community. The happy group:

The work that Daniel and I are doing with these students is what I am most proud of, from this term. Getting young people outdoors and active kind of seems far more important than forcing them to understand the graph of the cosine function?

Can’t wait for the Hampton Court Half-Marathon in 2018!!