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:

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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.

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From formal to concrete
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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?

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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?

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Spot the difference.
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Student explanation

‘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.
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4 students collaborated on this. At times inefficient, but some great conversations.
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Lovely annotations.
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Outstanding clarity and thought

How many right-angles?

After a week of exploring angle sums in polygons, both through interior and exterior angles, we attacked an excellent problem by Don:

I have a shape with 100 sides. What is the maximum number of right-angles the shape can have?

Fascinating to see different reactions to the problem:

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Go straight for the jugular: try and draw a 100-sided shape
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A similar (but digital) attempt
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Overly keen pattern spotting. (The odds/evens pattern breaks for n = 11,12)

One group came up with a hierarchy for solving the problem, that I would struggle to improve on:

  1. Guess
  2. Draw it out
  3. Understand the pattern and apply it

Students worked in trios:

  • Geogebra master (in charge of using technology to quickly sketch ideas)
  • Pattern spotter (in charge of generalising any results)
  • Organiser (ensuring good communication within the group)

 

MathConf9

To a school in Bristol to share and learn with plenty of nerdy Maths teachers. Who knew – not all teachers are under 35! School 21 is a bit of a bubble…

We did speed-dating with people around us, to share ideas and make friends. Here is my favourite:

  • Type a random 3 digit number into your calculator
  • Divide it by 7. Hands up who got a whole number?
  • Divide it by 11. Hands up who got a whole number?
  • Divide it by 13. Hands up who got a whole number?
  • Now, get you 3 digit number and type it in twice (for example becomes 456456). Divide this by 7, then 11, then 13. Hands up who got a whole number?! What number did you get?!

7 x 11 x 13 is 1,001. How could you extend this problem? 101 is prime, 10,001 factorises to 73 x 137, 100,001 factorises to 11 x 9091. Hmm.

 

Talk 1: From Abacus to Zero

By Ed Southall (who shares excellent problems on twitter)

 

We explored trivia about the etymology of mathematical language. Fun, and sometimes useful to hang meaning on, but not much maths. Here are some titbits:

  • Calculus derives from a word for pebbles (maths used to be done by counting pebbles). You can have calculus on your teeth – a term used in dentistry to describe tiny pebbles of plaque.
  • Which words connected to one?
    • Atone
    • Reunion
    • Alone
    • Only
    • None
    • Condone
    • Onion
      • All, apart from condone. Onion means “the big one”. Pearls used to be called onions.
  • If only eleven and twelve were oneteen and twoteen are rules for naming numbers would be almost sane. We are lucky – in Danish the word for 54 translates as “two-and-a-half-of-twenty-and-four”
  • Factor and factory have common root – they both build products (two puns in that sentence).
  • Linear and lingerie have common root – they both derive from the word for thread.
  • Average is an abbreviation of “averie damage”.  When a boat gets broken, how do you work out how much each investor has to pay? You share the total equally between the number of investors, obviously.

Enjoyable, but unsure of how I will use this to motivate students who drag their feet when entering the maths classroom.

 

Talk 2: Two A-Level Topics

Geogebra links:

 

Two friends who did their PHDs together are now writing a textbook for A level, chock-full of proof and links to applications, both at university and in industry. We explored geogebra apps to visualise differentiation from first principles and Newton-Raphson Method for solving messy equations (most equations are horribly messy).

 

Why do we care about Newton-Raphson method? Because it is used in Formula 1 car design, in Google searches, in weather forecasting… This argument left me dissatisfied – either show me explicitly how it is used (although this is presumably horribly complicated) or I won’t believe you.

 

One thing that got me excited was thinking about when the Newton Raphson method would fail (for example if you get stuck in a loop) – for me that is more mathematically profound than tenuous links to the weather.

 

Talk 3: Algebra Tiles!

Mark McCourt, Chief Excec of La Salle Education, ran an excellent session on the use of algebra tiles, from early primary right up to A level. Key thesis: we jump straight to abstract representations of mathematical concepts far too quickly in a topic, at far too young an age.

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Don’t just focus on abstract.

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Claim: they are all the same question.

Bus-stop notation tells you exactly what you are doing. Given the area of a rectangle and its height, what is its length?

Bus-stop division is exactly as hard as dividing algebraic polynomials – you just shift from base 10 to base x. If you understand this deeply enough using algebra tiles then all will be well.

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Bus-stop division with Dienes blocks generalises to algebraic division
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The red side of a tile represent a negative number. To get rid of a positive number you have to add its negative version to both sides.

 

The Maths Diary

Every student in my Yr11 class is making a Maths Diary. In it they will summarise everything they need to know about Mathematics, to help them become better Mathematicians, and to strive to get an excellent grade in their summer exams.

Crudely – it worked for me, so hopefully it will work for them (there are obvious dangers in this approach).

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Notes on 3-dimensional manifolds from my masters. Added notes in margins, colour, quick reminders. Clear but not precious.
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On the inside of each book…

Why summarising?

The act of condensing notes requires the mathematician to identify the essential from ther superfluous.

Why a book?

Cue cards and folders tend to lead to disorganised piles of notes. A book is a self-contained treasure-trove of information.

Some observations

List of every algebraic skill the students need to be able to do. Student plans which topics to focus on and which to ignore, before diving in. Topic list is not high quality (stole from the web)
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Time devoted in lessons and homework to giving each other feedback
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Excellent awareness of what makes a question difficult.
The Drafting Process
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Beautiful presentation, but is there much here beyond facts? Is that a problem?
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Great tips, but hard to scan. Does that matter?

The pros

  • For students who enjoy neat presentation of work, this is a comforting dream.
  • Every mini-test or lesson that students do now ends with a few minutes transferring key ideas in quick note-form back into the diary (key notes jotted in back of book for reference). This is the central store of all knowledge.

The cons

  • What about the student who finds it hard to formally explain their understanding and might benefit more from reams of practice? Is the art of summarising important enough to warrant a short-term loss in their performance?
  • What if the book is lost? (We have already mourned the loss of one diary)
  • It takes a lot of lesson time to maintain and keep momentum

You’re a Wasteman

Reflecting on the past half-term, I have been thinking about my persona as a teacher, and my behaviour management techniques.

Teacher Persona

In my second year I am starting to show more of my character to my students. More singing, silly dances, wall-sits, hammock photos, and bad jokes. I am more comfortable (rather than the artifical “I just do maths” persona of last year), and the students start to see a human being rather than a robot. I am loving it, but must make sure not to go too far, I guess.

Behaviour management

Students often tell me that they just wish I would shout. “It is what I am used to, I need it to stay focussed”. At times, I do doubt my attempts to stay calm and model the lack of anger that I would like the students to learn. I am nervous of students seeing weakness or a pushover teacher – “You just go and get another teacher to sort us out when you can’t cope” a few said last week.

I asked Jess for help with this, who reassured me that, in the long run, my style will work. It is perhaps more difficult and draining than going in with guns blazing, but it is the right way, at least for my personality. Here are some things for me to remember:

  • Positivity – so far at least one student has started copying Mr Judge’s happy dance.
  • Kindness and Calmness – take time to empatise with students, before leaping to conclusions.
  • Consistency and Transparency –  to clearly tell students why I am giving sanctions/praise. This is perhaps a more important use of my time than teaching maths?
  • Following up – with sanctions, conversations, parents, coaches

 

 

Finally, a hilarious and valuable reflection from a student who in the lesson was furious with me:

I also want to apologise for how in the last half hour I started to mess about with others and not focusing on my work and I also want to apologise for calling maths a wasteman lesson even though maths is a great lesson and can help me in the future.

Blaze Outside?

After a 3-day weekend I asked the 15 students in my coaching group how much time they spent outdoors.

The results

(Each row is a separate student. Number of minutes outdoors each day)

Saturday

Sunday

Monday

0

0

0

10

2

45

0

0

0

30

20

45

270

240

240

0

75

0

77

0

64

90

105

45

60

180

60

10

20

0

180

0

10

0

0

0

0

15

90

1

2

60

0

0

0

7 out of the 15 spent less than one hour in total outdoors over the three days. Removing one outlier (an avid footballer), the average time spent outdoors was thirty minutes.

Discussion

It was indeed a cold January weekend, but still, these figures shock me (perhaps a sign of my naivete). I had recently watched a talk from Ken Robinson (the only vaguely positive thing from the BETT conference on edtech, that was otherwise a huge warehouse of salespeople trying to sell ipad cases and wifi systems while completely missing any nuanced discussion about pedagogy). Ken spoke of his latest campaign, with Persil, to encourage children outdoors. Children in the UK are amongst the most housebound in the world.

When I asked the Nehwam children why this might be, they said:

  • It is dangerous outside
  • Their parents think it is dangerous outside (different to the first point!)
  • It is cold
  • Why would I go outside when I can do everything I want online
  • I have family to look after and chores to do
  • I have homework to do

I am so grateful for being given the opportunities and freedom to roam in the wild as a child, and would love to spread this.

A few ideas

I am thinking of how to build a culture of outdoor play with these students. We will play more games outdoors, and I have asked them to take photos of their adventures and send to me to win prizes. Apps such as Wild Explorers and 50 things (by National Trust) give ideas to children for how to get muddy and happy.

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Baby steps for my campaign?

Is there a deep link between screen-time and outdoor-time? The students have all downloaded Moment onto their iPads, an app that tracks how much time you spend on your tablet. Not perfect since does not take into consideration phone/laptop, and does not give breakdown of time (3 hours writing an essay is probably more fulfilling than 3 hours on social media?).

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My iPad-time

Floor Functions

Every Sunday afternoon I spend a happy hour in a cafe with Desmond, a student preparing to study Maths at university. We struggle in the darkness, wading through difficult problems. It is beneficial for both of us. This week, a problem from UKMT (3.5 hours, 4 questions to think about).

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Progression of the problem:

  1. Guess it has something to do with odds and evens, and try that.
  2. Give up
  3. Calculate the first few values by hand
  4. Create a graph of the function on DesmosUntitled picture.png
  5. Realise that the number of factors of n is somehow important
  6. Claim that  Twin Prime Conjecture and Mersenne Prime Conjecture are both true
  7. Get sad when our phone tells us neither has been proved yet
  8. Talk lots about numbers being dragged up or down

 

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Incomprehensible

I really enjoyed applying the skills I had been trying to teach at school – chunking the problem, drawing a picture, making links. We started to understand the problem a bit, but definitely were not near to a solution. See strategies at end of this post.

 

Final stage: ask girlfriend, who happens to be doing a PHD in Maths, to solve it for you. In her words:

“I thought I would want to use numbers that I could understand the factors easily. I realized that if I understand the factors of n, then I don’t know anything about the factors of n+1. So I could do it in a straightforward way.

And your graph showed that it increased on a large scale, but not on a small scale. So I guessed, and knew I would want to approximate above and below”

Knowing to play around with powers of 2 shows great intuition, learnt from many years of practise.

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Problem solving at School 21:

 

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The School 21 Problem-Solving Toolkit
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My attempt at linearising the problem-solving process

Playing with STEP

Thinking about STEP – excellent transition from School to University Mathematics.

Spent an hour in a cafe working with Desmond, a student who is doing STEP this year and is about to go to Oxford to fill his brain with prime numbers.

Here are some links:

We struggled with this question, mostly conceptually, for a pleasant head-scratching hour.

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When R is (2,1), the situation is shown below.

  • The blue line is the scenario that gives the smallest sum of distances OP and OQ
  • The red line is the scenario that gives the smallest distance PQ

We both strongly wanted the blue and red line to be the same line! It turns out they are not… Strange.

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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.