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.

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This function’s Maclaurin series has radius of convergence 1, unless a is less than 1 (because of singularities in the complex plane)
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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.


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