‘Tis but a Scratch

Inspired by this Monty Python sketch, I asked students to find the area of a triangle, if they know the lengths of the three sides. Gradually I chopped off a mathematical limb, removing ipads, calculators and compasses, before only a brain and a pencil remained.

Some great and unexpected learning:

  • The compass was too small to draw an arc of 13cm. So we enlarged the shape by a factor of 0.5, and thought hard about how that would affect area
  • We thought about how to accurately drop a perpendicular from a point to a line, when measuring the perpendicular height of a triangle
  • We wanted to work out the square root of 21 x 8 x 7 x 6, but without using a calculator. Prime factors to the rescue!

I had given the class 5 minutes to try and find a way of calculating the area of the triangle, using only a pencil and a brain. Some wildly inventive methods, that involved inventing an adapted version of Pythagoras got students close to the correct area. Great creativity, but lacking in precision.

The start of understanding the proof:

Reflections:

  •  If a student knows every constituent skill then they can understand the proof. Whole is greater than sum of the parts – the multiple steps are difficult to keep in your working memory
  • Students have well-trained instinct to expand brackets. Sometimes this isn’t helpful!
  • One student spent a long time trying to prove that s = 1/2(a + b + c). The distinction between something you can define and something you have to prove.
IMG_2237
4 students collaborated on this. At times inefficient, but some great conversations.
IMG_2236
Lovely annotations.
IMG_2235
Outstanding clarity and thought

A levels – opening doors or enlarging brains?

I am thinking of which type of exam to work towards in the new sixth form. In simplistic terms, the one that students probably will do better at, or the one that is more mathematically genuine. Here are some thoughts:

  • Secondary school teachers often are frustrated by the crazy grade inflation they perceive in students Yr6 SATS marks. “No way is ____ a Level 5” we mutter. The accusation is not of cheating, but of coaching to the test. It is unhelpful in secondary school, giving an inaccurate picture of the students and providing unrealistic GCSE targets. If we want accurate marks from primary school, then surely we should give accurate marks from secondary school.
  • Inaccurate grades are not only unhelpful for teachers/universities, they are unhelpful for the student. If a university course requires an A grade, what it really requires is a student capable of A-grade level thinking (if such a thing exists…).
  • I asked my mum (a research scientist) about this. She cannot see any conflict – “Of course you should take the intellectually honest and challenging route”. Set high expectations and students will meet them.
  • I asked MK (an American) about this. In America standardised tests are… standardised. No choice of exam board. Individual teachers/schools don’t have to make this horrible choice, between honesty and what they might perceive as helping their students by opening doors with good grades.

In summary – if a student does not deserve an A grade (whatever that might mean), then playing the system to get them an A grade is delaying failure. Choose the more challenging course, make brains bigger, dream big!

OCR (B) Exams

In conjunction with MEI, OCR have devised a new spec. Features I like:

  • The large data set is comparative information about countries over time. Great for highlighting social inequalities and getting students debating.
  • The third paper includes mathematical comprehension – read a new bit of Maths and then answer questions based on it. Excellent for encouraging mathematical independence.
  • Problem-solving questions abound, with lots of interpretations (that are not hoop-jumping but ensure deep understanding)

 

Edit: KS says “Surely we can teach deep problem-solving in lessons while also using a less mathematically challenging exam board?” A response: our systems should nudge everyone to always keep our mathematical integrity. GCSE interventions often lose sight of the beauty of Maths, because they can. If GCSE exam questions all required deep problem-solving then rote-learning how to rationalise a denominator would become less tempting.

 

Learning from Kings

Karenann and I were excellently hosted by Dan, headteacher at Kings College Maths School. Every student studies Maths, Further Maths and Physics at A level.

  • Every lesson is designed around the principle “Don’t tell the students what to do”. We teach through discussion and questioning. We only break this principle in interventions. Interventions are timetabled in, happen after assessments, and are there to quickly support students.
  • While every lesson will involve problem-solving, we also dedicate a session a week specifically to problem-solving. PHD students come in and help run them, we give far less guidance, and focus on encouraging students to fail, to keep on failing, to tenaciously strive for a solution. No lesson objectives, no rush.
  • Students are assessed by their teachers on core study skills – collaboration, communication, independence and organisation. Interventions (for example go to board-games club to improve collaboration) are put in place.
  • Teachers have a weekly planning meeting, to skill-up those who are new to the Maths, and to ensure teaching is consistent and high-quality.

How do we support the transition from GCSE to A level?

  • It is not true that all our students, even though we are a specialist Maths school, are ready for A level. They might lack study skills (see above), might think of maths as a subject where they can easily find “the right answer”, or might have some subject-gaps.
  • We start with a topic that is new and impressive, but that also will enable the basics (algebraic manipulation) to be covered. Complex numbers works well – requiring expanding brackets, collecting like terms, while also being something that none of the students will have seen before. Recapping completing the square pales in comparison.
    IMG_5664.JPG
    Deliberately hide snazzy methods (multiplying by complex conjugate) at first to encourage excellent algebraic manipulation
  • Students find mechanics particularly hard. This is due to a difficulty separating intuition about forces from the formal modelling. For example, reaction force is equal and opposite to the weight of an object lying on the floor, but this is nothing to do with Newton’s Third Law of Motion.
  • Early interventions are key

Lesson Observation: Yr13 Mechanics

Essential Question: “When I push a block, will it slide or topple?”

  • 30 minutes of teacher-led exploration of the question. 20 minutes of applying the broad techniques to unfamiliar contexts. The students could have been given the question and nothing else. However, in this lesson the content had to be covered quickly, so more teacher-leading was necessary. Dan guided us through, with constant questioning and time to reflect, talk to each other, and predict. Pacy but still involving students and ensuring we all thought deeply.
  • Repeated links to intuition and the physical example (we all had blocks to play with).  “What do you think it might depend on?” , “Intuitvely, should it depend on mu?”, “Translate this into english please”,  “I feel that…”, “Think about the point when it is just starting to happen”. Conscious effort to hone and improve intuition.
  • The most difficult part about the lesson was the logical structure.
    • If I assume that the block slides, then…
    • If I assume that the block topples, then..
  • Teacher quickly assesses work on whiteboards. Nothing really written down formally, no huge emphasis on taking good notes. Focus is on the group collectively thinking deeply together.

IMG_5666.JPG

Counterintuitively, whether the object will slide or topple is not dependent on height of block or mass of block.

IMG_5658
Futher problems
Lesson Observation: Yr12 Matrices.

Lesson focus: to use matrices to solve simultaneous equations.

  • Excellently clear link between prior knowledge and new method. Students unconvinced for the need for matrices to solve equations that they already have a method for. A possible opportunity for technology here: matrices can solve simultaneous equations in three variables, and an app can split out the inverse (useful if you don’t know how to invert big matrices yet).
  • Teacher completes problems on board while students complete on A3 “mini”-whiteboards. Deliberately supportive culture for a class that finds maths hard.
  • Usefully uncovered misconceptions – you can divide by a matrix, and matrices commute.

IMG_5663
Example of student work

IMG_5662
Two geometric interpretations of M= a. How are they linked? I don’t yet know. 
Thank you so much to Dan and Kings Maths School for hosting our visit! We are excited to think how to use some of the exciting things we saw here next year at Six21.

Rekindling Beauty

After a few weeks of mocks and many lessons of focussed revision, it was high time in my Yr11 class for a double lesson of open-ended problem solving. Enough time silently in rows, get rowdy at the whiteboards please.

Untitled picture
Problem-solving helps you become a better mathematician AND pass exams

 We struggled with an excellent problem from Underground Maths, designed as a transition between GCSE and A level mathematics thinking.

Untitled picture1
Stage 1
Untitled picture2
Stage 2

What went well?

  • So much surprising maths covered here. The difference between proving an identity and solving an equation. The equation of a circle. The meaning of a variable. The properties of a quadrilateral. How and when to shift from using a compass to using algebraic variables. (The temptation to correct methods that I hadn’t thought of had to be quashed, to explore surprising links with other topics.)

    IMG_5588.JPG
    Lots of excellent thinking here
  • Outstanding buzz in the room – stealing from other whiteboards, keen to solve the problem, naturally on-task. See video
  • The shift from constructing a circle given a triangle (very tangible) to finding a triangle given a circle (quite abstract) was effective – well done Underground Maths team!
  • I was surprised by who made the best insights into the problems. Not the same as those smashing out the top marks in the exams…
  • Extensions into special types of Pythagorean triples and beyond into Fermat’s Last Theorem (Numberphile video)

IMG_5589
Example of student work (one of the weaker students in the class, or so I thought…)

 Some questions

  • Students flagging towards the end of the afternoon – how do I keep up the pace without providing too much structure?
  • This process took 100 minutes. How to teach students to do a miniature version of this when in an exam, and without any collaboration?
  • How could we have used technology better? A useful visualisation here, which we did not make much use of

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.

Underground Maths

An engaging day thinking about rich A level resources, with Underground Maths. The website is crammed full of rich tasks with plenty of teacher support. A useful reference.

We did this Starter activity. A many-ways problem – either very algebraic or very graphical.

IMG_5332.JPG
My pre-work. I enjoyed putting into practice the techniques I ask my students to use.

A problem that doesn’t look hard, but actually is challenging. How does this affect motivation? Would I prefer a problem that looks really scary but actually is pretty simple?

This would be a great problem to throw at students before they know how to differentiate – if this is the headache then what is the aspirin?

Mathematical Mindsets

We discussed things that we would like our students to think about mathematics, using these cards. Some key thing that I want my students to think:

  • It is good to get stuck. (Teachers need to model this). Also, it is okay to get stuck-in!
  • There are no right answers in maths.
  • Problem-solving is the most important thing in Maths.
  • Good problem-solvers do well in exams
  • Understanding > memory (within reason)
  • If you pass an exam using death by practice paper then you might get a good grade but you will not do well in life. You are merely delaying your failure.

Get stuck-in. Get stuck.

Timing a problem

There seem to be three times to use problems:

  1. Before you know the techniques. Motivation for learning techniques
  2. Just after you have learnt the technique. Application of fresh knowledge
  3. A long time after you have learnt the technique. To encourage connections, recall, re-deriving knowledge.

From GCSE to A level

There is an excellent transition problems bundle, using GCSE knowledge but requiring deeper reasoning.

  • Between the lines (why is there an actual answer for this!)
  • A tangent is
  • A perfect fit (with an excellent extention question: how many pythagorean triples make a triangle with inscribed circle of radius 6? We found 6 integer solutions. What about a circle of radius 7? We found 3 integer solutions. What about a circle of radius n? Lovely opportunity to use excel or python)
  • Name that graph
  • Quadratic solving sorter

You cannot do this task without knowing what an asymptote is. This video clip shows students struggling to use the equation of a circle to find an asymptote (thus showing a deep misconception), without any teacher intervention. When do we leave students to struggle and when do we step in? Does the teacher need to know about every mistake students make?

 

What is 0 the power 0? You need to know the answer to properly understand this problem. Outstanding problem generation.

 

We believe that there should not be a right method or a right answer. This problem fails this criteria.

 

Modelling