Two Quartic Polynomials

Problem 1, from Don

What is the side length of the equilateral triangle?
  •  Thought about this with Rosie on the last day of term. Ended, through three simultaneous equations using Pythagoras with a quartic polynomial to solve – eurgh, disgusting. Is there a more elegant solution?
  • Initially I worked out what the answer was using Geogebra – something I would never have done a few years ago when I was not familiar with dynamic geometry software 
  •  How does it link to the construction problem: given three parallel lines, how can you build an equilateral triangle with vertices on the lines, using straight edge and compass only?

Problem 2, from 3Blue1Brown

13 dots on a circle. How many regions?
  •  Waking up at 4am from jet-lag I thought about this problem to pass the time.
  •  I love colours.
  •  At one stage I worked out I had a quartic polynomial sequence, since the fourth difference was constant. I typed in the sequence into google and out popped Pentatope numbers, hidden in Pascal’s triangle. Pleasing!
  •  After an hour or so I arrived at a description of the solution, without the faintest idea of how to explain. I shallowly searched for patterns rather than stopping to think about why. We then watched the elegant solution by 3Blue1Brown.
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Snooker Table: The Journey of a Problem

  1. Find a really great problem on the internet (Thank you Colin Foster!)
  2. Share it with maths teachers on a meeting I facilitated about scaffolding rich tasks
  3. Ask for feedback from teachers who then taught it
    Outstanding meta-cognitive questions by Rosie
  4. Spend two hours on evening excitedly making resources rather than marking

5. Refine after working on it with one class

Spot the student’s mistake?
6. Post the idea on Facebook, and have a friend improve your app (even though you have been working with Geogebra for 3 years and he only found about it a week ago)7.Teach it to another class…

Top Set Yr11 with brains switched off at end of term…

8. Repeat
Latest draft of resources:

Maths at the Science Museum

Tom, third year of his PHD, and me, second year of teaching, met to investigate the much-publicised Mathematics Gallery that the Sceince Museum. Designed by Zaha Hadid but opened after her death, cool purples and huge sinuous forms inspired by the fluid dynamics of flight. Beautiful space.

The gallery website states: “Our bold and thought-provoking new gallery, designed by Zaha Hadid Architects, examines the fundamental role mathematicians, their tools and ideas have played in building the world we live in.” It does this by exhibiting objects, such as an electricity pylon or a lottery machine, that have been influenced by mathematical ideas.
There are a number of things that drove us slightly crazy:

  •  There was only one meaningful mathematical statement in the entire exhibiton (or if there were more they were too hidden for a 2 hour visit to find). It was a diagram showing the angle between lines from a boat to the sun and the moon. Alex Bellos is happy in his review to compromise knowledge of the maths involved – for him revelling in the beauty of the objects on show is enough. We found the absence of link between maths and the objects in front of us annoying.
  •  There was not a single object that is less than 20 years old on show. The most modern thing was a Mathematica textbook. This is sending out the message that Maths has finished its contribution to the world around us. A historical curiosity rather than a pillar of the modern environment.
  •  There was only one interactive you could play around with, but the sliders lagged and froze, and the maths was far from obvious. If the only thing you can do on an interactive touch screen is scroll through text, then don’t bother buying an interactive touch screen, buy bigger cardboard.
  •  The curators were surely aiming to win the World Record for number of variants on the phrase “…which could only have been understood using complex mathematics”. The number of synonyms for “complex mathematics” was mindboggling. The idea that “it is too hard to explain this” either patronises the audience as flitty idiots or reveals a lack of imagination in the curation.
  •  A child would come away from this exhibition thinking “If I wanted to be a ____ (insert jobs such as shipbuilder, radar mechanic, architect), ___ years ago (insert number that is at least 50) then maybe I would have to know maths. I don’t, so as I suspected, learning about congruent triangles is BORING”.
  •  There was not a single piece of pure mathematics on show. (The previous gallery has racks of beautifully crafted glass Klein bottles, for example.)

 

I have been loving Dan Meyer’s hilarious Pseudo-Context Game (every Saturday he posts a photo from a textbook and asks his readership to guess the maths the textbook author wants the students to learn). Inspired by this, one of us would only look at the object and the other would give three options for how maths links to it.

Was this model here to handwave about the mathematics of the bow, about the most efficient journey a ship should take on a sphere, or the best way to pack objects into a ship?

Here are some ideas to improve the exhibition (for my reference when I open a Maths Museum in 15 years time)

  •  Include pure maths – some people enjoy maths for its own sake
  •  Include any maths – don’t patronise the audience
  •  Include playing with maths – situations where viewers can predict, hypothesise, change variables and consider how that affects outcomes

WonderLab

Megan, after a school trip with Yr2 students, recommended a visit to the new interactive gallery, WonderLab. Torrents of young families crowded in to play with air currents and magnetism, with two unabashed men running round too. Ridiculously more fun than the Maths Gallery, so why would anyone ever choose Maths over Science? Things we thought about:

  •  How many pulleys should I use to pull my weight up? (Think toddlers in car seats delightedly pulling themselves up high into the air)
  •  Why is my nose cold? (Infra-red camera)

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    Beard insulates
  •  How much background radiation is there in this room? (An INCREDIBLE display, just an analogue particle detector, in which you could see the traces of electrons muons, alpha particles)
  •  How can I create standing waves? (You could control oscillations at both ends of a 3m bit of rope, to create standing waves with up to 6 nodes – mesmerising)

    IMG_6835.JPG
    Tom conducting ropes
  •  How can I make things float in space? (Magnets, magnets, magnets)
  •  Can I hear through my teeth? (Bite down on a metal rod, fingers in your ears, and the music vibrates through your skull to your eardrums)

We were thoroughly immersed in what the curator calls “The serious nature of play” – if I just tweak this a tiny bit more than maybe… What if we… I never thought that…
The maths section here mostly consisted of puzzles (Tangram and Rush Hour Game). All games that could have been played on an ipad but were presented in pleasing wooden blocks. Fun but not deeply generalisable to fundamentals of maths in the same way that some of the scientific wondergasms were.

Deepening Habits, Reclaiming Evenings

I mistakenly believed that planning until 10 each night would be confined to my first year of teaching. This year, my second, I am trying to break out of a cycle of late nights and planning one day at a time. Burnout beckons otherwise.

How can I achieve this?

  • A blisteringly tight timetable. Excellent structure, but if I fall behind then how to catch up?
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My weekly timetable. Can you work out what the colours mean?
  • Shared planning across the department. How to ensure that all lessons are high quality, in a consistent house style, and that all teachers can deliver them with autonomy and passion?
  • Changing my marking practices.
    • Plan excellent reviews that, when marked, can tell me exactly what the student does and does not understand. A good idea but I rarely use it.
    • More peer assessment and self-checking exercises, to ensure students get quicker feedback on their work. A good idea but I rarely use it.
    • Class Act, an app, allows the teacher to quickly note down in-class assessment. A good idea but I rarely use it.
    • Seating plan on a clipboard, that I jot down notes on as I walk around. A good idea but I rarely use it.
    • (Spot a pattern?)
  • Redrafting (lowering) my expectations – if every lesson is planned in detail then I will be too tired to deliver them with any verve
  • Building up a bank of resources and expertise to draw on. I use OneNote to store my resources.Untitled picture.png
  • Using pre-existing resources more.
  • Carefully assess what I should spend my time on. I might enjoy searching for and scaffolding an open problem in number theory, but is it self-indulgent or genuinely most effective for student learning?

I have too many solutions to the problem, and currently forget to use any of them. My next goal is to work out how to build them into the structure of my daily habits, to nudge myself into always doing them, instinctively.

Transforming Functions

A happy afternoon spent analysing how functions move as you change coefficients. If I know the shape of f(x), what do I know about the shape of cf(ax + b) + d? Use Desmos to see things quickly (but maybe also skate over deeper thought as to why?)

 

I love a table to organise thoughts:

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With lots of words
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With explanatory colours

Next time:

  • Go beyond “if it is in the brackets then it is the opposite of what you would expect” to a deeper understanding of how the graph moves
  • Tweak the activity to ensure that no graph can be drawn without deep thinking. An over-reliance on graphing software is a dangerous thing.

 

 

Maths on Youtube

Youtube can be used to give students information:

 

Youtube can be used for students to give information:

Adding

Assessment week at school. In the moments when I wandered around the room rather than frantically marking at the front of the class, I was absolutely fascinated by the journeys the students were going on. 
For example, for this question, I watched a student for five minutes

  1. First, stare at the question and do nothing for two minutes
  2. Suddenly, without warning, start to write out the numbers from 36 to 90 (missing out a few by accident on the way)
  3. Start to count how many numbers there are
  4. Get confused, so start from the beginning, this time labelling as you go
  5. Get confused, so add 36 to 49, just in case
  6. Write 50 as your answer, because 50 is close to 49

The student was completeley focussed, attacking the problem from multiple perspectives, and persevering. 

Addition has no inverse.

Another student used trial and improvement. Built his own subtraction algorithm.  If you don’t understand that subtraction is the inverse of addition, then you learn increasingly ingenious ways of using other tools. Look closely at the sequence he chose – it reveals a lack of place value understanding too.

Spreading the Word

I am talking maths to two people outside of my school life.

Firstly, to J, currently studying Maths while in prison. We communicate via letters – the thrill of receiving an actual physical piece of paper is rare and powerful. Early days at the moment. The current system is that I tell him of rich problems I have found while planning lessons (the advantage of a properly rich problem is that it challenges anyone, from 5 yr old to professor), and he writes about how the men in the education block get excited solving them – “I’ve got to say, that’s really given me a bit of an adrenaline rush” he writes after finding a pattern in the Graceful Tree Conjecture.

Secondly, to D, currently in a gap year and applying to Oxford to read Maths. His school in Hackney does not have any maths teachers that are comfortable enough with their content knowledge to stretch him enough ready for University. We spent a focussed and happy afternoon in a local cafe drawing fractals and sin(1/x) graphs. He muttered under his breath as he worked, rapidly making connections, scribbling down tiny notes as he went. A joyful reminder of calculus and difficult curve-sketching beyond y = mx + c.

Red curve approximates parabola close to origin, but approximates straight line far away

 

It is an honour to be asked to talk about my subject in a variety of contexts – I want to do more of it!

Oracy in Maths

 

Alongside Heather, I delivered a session on Oracy (using talk in a dialogic classroom to aid learning) in Maths and Science. Visitors can often easily see how to embed talk into traditionally “softer” subjects, but what happens when “there is just a right way to do things”?

Ways to use talk in the maths classroom:

  1. Through games
  2. Through physical structures
  3. Through talk structures
  4. Through rich tasks

Through games, such as skribble, just a minute, pictionary, articulate, taboo. Often the most common and easy way to introduce talk. One example I am currently enjoying is “Which one doesn’t belong?” , a quick and easy way to spark debate. Easily adaptable to the current topic.

Through physical structures in your classroom. I am a bit obsessed with my whiteboards, which are useful because:

  • Everyone can see all the work
  • Knowledge spreads quickly across the room
  • The non-permanence means there is less fear of starting
  • Formative assessment is easy, teacher can survey from centre
  • Students are naturally encouraged to talk to each other

More thoughts in a blog here, and example of whiteboards in my classroom below:

Through problem-solving structures. These could be sentence stems, group roles, timed protocols, toolkits… One example, devised by Rachael, is an adaptation of the coaching model.

  1. Work on the problem for 5 minutes in silence
  2. Coachee talks for 3 minutes (with talk prompts and key vocab visible for support) about what they have done
  3. Coach responds for 2 minutes, with further questions and clarifications.

Finally, and most importantly, through rich tasks! This might be a cop-out on my part, but if:

  1. The students desperately want to solve the problem
  2. Any individual student is unable to solve the problem alone

Then talk will arise naturally as the easiest way of communicating ideas quickly and efficiently between thinkers. Our job as teachers is to facilitate this, with a few well-placed structures. Talk for the sake of talk is banished.

Some places I go for rich tasks: