Sticks and Shadows

Paul Lockhart’s excellent book Measurement (a constructive follow-up to his infamous diatribe about the start of maths education, A Mathematician’s Lament), includes this gem:


I adapted for mixed-ability Yr9 class. Some students were thinking deeply about the shape of the curve, others were practicing measuring angles and lengths and plotting coordinates.

From formal to concrete
Multiple ways to measure the shadow…


Geogebra app proved the most popular, since the least brainpower required to use it. Maybe only give this to the students once they have struggled with a physical metre-ruler or the unit circle first?


Lovely conversations about how a shadow can have a negative length, and why we would want to think of the metre-ruler sticking straight into the ground and still casting a shadow.

Some students found four squares representing 30 degrees a challenge. Which graph is correct?

Spot the difference.
Student explanation

Apples and Kings

Four lessons in the morning in Stratford, and then into central London for two conferences.

Apple Teacher

In a luxury hotel just off Trafalgar Square, we were served posh cakes and snazzy presentations from a slick Apple Team. At times the focus on design was almost laughable – we spent forty minutes learning how to remove the background from an image. All style, no substance.

“Learning how to tell compelling stories with data is a skill that we all need in the age of fake news” we were told. This seems questionable. Surely a more important skill is to critically evaluate data, to work out if it can be trusted? Twisting messy data into neat stories is a problem, not a solution.

Some things to take home:

  • “Browse each other’s learning”. Explicitly encourage students to observe other students, learn from each other. This could be through whiteboards, walking around the room, or ensuring that all work is publicly viewable online.
  • The ipad is a tool for getting out of the classroom. You can explore the universe by app, facetime an expert on the other side of the world, ask a large group of people a question on Twitter… (Is there a danger that what was once a field trip now becomes an exercise on Google Earth?)

Walk over Westminster Bridge in blazing sunshine to Lambeth, to the King’s Maths School, wedged tightly in between blocks of flats. A different world. For…

Further Maths Forum

A workshop, led by Michael Davies (head of Westminster Maths Dept for 30 years and writer of STEP questions). Great contrast to previous conference – chalk and blackboard, with occasional computer graphing. Clear delivery of talk, but no question of no substance here.

We grappled with Taylor and Maclaurin Series.

Untitled picture
This function’s Maclaurin series has radius of convergence 1, unless a is less than 1 (because of singularities in the complex plane)
Untitled picture
Every derivative of this function at 0 is 0, so its Maclaurin series is the constant function y = 0. This is an example of a non-analytic function (where analytic unhelpfully means its taylor series converges to the function)

Pedagogical points about this Maths:

  • It is surprising that complicated functions can be approximated by polynomials. No reason why this should be true. Why should the behaviour at x = 0 tell you the behaviour for any other value of x? If I know exactly how you behave today, can I predict what you will do for the rest of your life?
  • Each time I make a better approximation by adding more terms in polynomial, the earlier terms do not change. Again, not obvious. This makes approximation by this iterative process practical.
  • An identity is when coefficients of each power of x is the same on each side. 1 + x + x^2 + …. = 1/1-x is not an identity, it is just pointwise true when you evaluate at any valid point of x.
  • 2+ 3 + 5 is just the sum of the numbers in the set (2,3,5). In an infinite sum you are not doing this – consider the alternating harmonic series as counterexample. Infinite sums are properly weird
  • If gradient and curvature are both 0, not necessarily a point of inflection. (What if the third derivative is 0 too and locally the graph looks like a quartic?)


General points:

  • Think carefully about the example sheets you give students. Constantly interweave previous learning, and make the questions “desirably difficult” – challenging enough for deep thought. Too often we focus on quick wins.
  • For A level you need good technical skills (to be able to do a page of algebraic manipulation, for example). Start this early – don’t allow any coasting through the beginning of the course and relish wading through the working.

Fun with Further Maths (these are the problems we worked on)

I asked Michael for any tips to ease the transition to difficult A Level Mathematics for students who struggled with GCSE. “I don’t have that problem, all my students got an A* and are very strong”. Understandable, but frustratingly unhelpful.

Flipped Learning at Shireland

Whole Education have organised a year-long action research project focussing on Flipped Learning. The launch event was kindly hosted at Shireland, a school in the suburbs of Birmingham with 30% live safeguarding issues and 64% EAL.

Flipped Learning

As defined at the conference, flipped learning is the following process:

  1. Students learn knowledge, before lesson (through videos or other means)
  2. Teacher, also before lesson, assesses their understanding (this could be through apps that check which students have watched videos, through multiple choice questions…)
  3. Teacher plans lesson based on a secure awareness of each child’s understanding
  4. Finally, the lesson happens. Teacher liberated from explaining from the front, students able to leap straight to evaluating, analysing, creating.

How is this different to…

  1. … traditional homework? Traditional homework involves students applying knowledge they learnt in class, after the lesson. This higher-order thinking surely requires teacher support and so should happen in lessons.
  2. … pre-learning? Pre-learning is where students do a bit of learning before the lesson, but the teacher walks into the lesson blind – no assessment of understanding has taken place.

Shireland’s Philosophy

  • Technology should be mundanely clever. Every teacher, not just the geek in the corner, should be able to use it and see immediate benefits in their classroom. Technology works for us, we are not dictated to by technology. Fix existing issues to ensure all teachers are motivated.
  • Flipped learning is not about the videos, but about the pre-assessment. Spend your time designing excellent assessment, not whizzy videos.
  • Replace 15 learning assistants with 10 people on an E-learning team, who will build an incredible online learning environment. Be bold if you think you are right. This ensures teachers have all the admin done for them, and can focus on other things.
  • Before flipped learning, there was an unspoken agreement between teachers and students – homework was actually a bit pointless, bolted on as an afterthought. No longer!
  • Model the model. CPD is now… Flipped CPD!

Questions and Answers

I am part of a wider team at School 21 thinking about Flipped Learning. Here are some questions the team wanted some answers to.

Q: What about those who don’t do the homework?

A: Shireland have developed a whole arsenal of techniques

  • Students are motivated by a feeling of security – they can now come in to a lesson confident with the material. They understand, finally, the point of homework. In other words, if you design the flip correctly, everyone will do their homework anyway.
  • Clever deadlines for homework. Ensure a day in advance of the lesson, so you can chase stragglers and analyse results.
  • Parental engagement is key. Once parents understand the advantages, they will support you
  •  Be completely relentless at the start of the flipped learning programme
  • If students have still not done the homework, put them in a corner to complete it while the rest of the class do the most exciting and amazing task ever. Sneaky.

Q: What do you do with all the extra time freed up in lessons?

A: The fact that you have more time is a positive, not a negative. More time for deep thinking, peer support, student-led learning, any of the good, deep, stuff. You would teach this higher-order thinking as before, but now with more time and more knowledge of the students’ needs. Your lesson becomes a clinic for fixing issues, rather than a lecture.

Q: How do you ensure this is time-efficient for teachers, daunted by the huge task of curating an entire library of videos?

A: Again, a whole range of answers:

  • Before you make a video yourself, think really hard about whether it is absolutely necessary. Is there already one out there that will do?
  • Do not flip every lesson. Students would have too much homework, and you would not sleep
  • Start small: pilot, evaluate, repeat.

Q: Surely by placing an emphasis on work at home, you are widening the achievement gap between rich and poor?

A: In comparison to traditional homework, students now get more support for higher-level thinking. What was once only available to students with present and supportive parents, is now available to all. Homework clubs and subsidised devices help too.

How to conduct excellent action research?

Shireland is a Research School, and referenced EEF’s research framework. Before gleefully diving in, answer these questions:

  1. What is the question? (Keep it narrow)
  2. What are you going to measure? (Qualitative and quantitative)
  3. Who are you comparing to?
  4. Where are you starting from?
  5. What are you actually going to do?
  6. What were the outcomes?
  7. How will you share the outcomes?

Two things that made me sad

  1. “I like being made to think again” a fellow teacher ruefully observed. It is a shame that teaching does not intellectually challenge all teachers in all schools.
  2. We were given a task to do, having been warned beforehand that this task was designed to make us feel uncomfortable. It was a worded maths question, and the facilitator was a maths teacher. She was comparing giving students a surprise question on a new topic, to giving students questions after they have had a chance to watch preparatory videos before the lesson. This made me sad because:
    1. The comparison is a false one. A good non-flipped lesson would consider prerequisites and not throw students in at the deep end.
    2. A maths teacher should not encourage maths anxiety (sore spot for me).


Thank you to Shireland and Whole Education for a thoughtful conference. I can’t wait to revolutionise home-learning and spend more time forensically supporting students to think deeper.

Structured Exploration, Proactive Self-checking

When I began to use Desmos and Geogebra in maths lessons, the tasks were lacking in structure. “Click on this link and have a play” I would say optimistically, hoping the students would be self-motivated enough to wonder and ask questions themselves. Varied success. Here are two first attempts at providing specific purposes for technology in the Maths classroom.

Structured Exploration

Simpy put: “Click on this link, have a play, and then do this subsequent exercise”. Allows teacher assessment, students given some (but not total) direction. Examples:

Proactive Self-checking

Here are three levels of marking:

  1. Teacher holds the answers, and marks herself. Teacher has good knowledge of student’s understanding, but the delay between question and feedback can be long, and this is time-consuming for teacher.
  2. Student is given the answers somehow (back of textbook, on a separate slide…) and passively checks their work against it. Immediate feedback, but often lack of cognitive depth in feedback
  3. Student works out the answer in a different way. This level encourages building of links between separate areas of maths, and allows the student to continually assess their progress. Let’s call it Proactive Self-Checking.

Here are some examples of proactive self-checking (do you think it is important enough to need capitalisation?):

When practising plotting points from an equation, students can compare their hand-drawn graph to a desmos graph.

Desmos doesn’t have that weird kink in the line – what have I done wrong?
The line on Desmos is going up but mine is horizontal – something is wrong!

This is a bit obvious, no real links made. How about encouraging the use of area models to check expanding brackets?

I might use a more efficient method to calculate, but by using this method to check I am constantly reminded of the link

How about starting to understand simultaneous equations,to check that I solved any equation? If I am using bar modelling or the balance method to solve my equation, then being able to link the answer to a picture is deep stuff!

2x + 5 = 3 has a solution at x = -1

How about using pictures to make sure you can simplify expressions?

I don’t know what I have done wrong, but at least now I know I need to ask a peer for help!

Your next steps:

Think about providing structured exploration for angle rules?

Design proactive self-checking for your next lesson?

Use this collection of exercises as an intro to Geogebra with your class?

Search the excellent Geogebra database or Teacher Desmos Site for ideas?

Try this challenge?


Desmos to spot mistakes

Desmos is an excellent graphing calculator. Nick had a really great conversation with himself. The graphing calculator disagreed with the line that he had plotted. Which was correct? He oscillated between faith in himself and faith in his ipad, before spotting that 2 x 2 was not 2. Technology used as a self-marking tool can be powerful. To check at the end of the journey, rather than as an overly helpful crutch from the beginning of the journey.

Who is right??