Melting Brains with Infinite Series

My papa sparked a desire to understand wacky series, when he told a 9-year-old me that the alternating harmonic series can sum to whatever you wish it to. Addition commutes, except if you do it lots and lots?!

A happy two hours exploring this with FM class, using Excel without fully trusting Excel.

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The students’ emotions at the end – thoroughly angry at Maths. Mission Accomplished

Best Student Quotes:

  • “If the alternating harmonic series can converge to anything, does that mean 1 – 1 + 1 – 1 … also can?”
  • “1 + 1/4 + 1/9 + 1/16 + … is an infinite sequence of squares. Lined up they are infinitely long, but do they have an infinite area? Oo, isn’t that like Vsauce?”
  • “Eventually the percentage of numbers with a 9 in them is basically 100”
  • “I jump to the right more and more, and then end up … on the left?!” yells W as she hops around the room angrily (-1/12).
  • “Don’t worry sir, it doesn’t take too long to add up the first 50,000 terms”
  • “I feel dumb in Further Maths”. Obvious Warning: if it is too hard, people will drop out. How to ensure students know that this kind of thing is not on the exam, without constantly referencing exams?

Further links:

 

 

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Thing Explainer

With five of my Yr12 FM gang, into the heart of London for a talk on how to explain science and maths simply to a public audience.

Reddish tinge of sunset

 

We were listening to Randal Munroe (of xkcd fame), and Marcus du Sautoy.

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We looked around at the thousand or so people in the audience. “There is not a single Asian person I can see” exclaimed Ashwin. “They are old and white” said Cody. Very different to Stratford station, where we began our journey.

 

Some points from the talk:

  • 500 years ago it was possible for one person to hold in their brain almost all of the scientific knowledge we had. Now, no one person could properly explain how an iphone works.
  • Marcus was given “A Mathematician’s Apology” by a maths teacher, gave him the key to a secret magical garden. He writes to show the magical garden to as many people as possible
  • You sometimes use sophisticated language to convince yourself and others that you do in fact deserve to be here, you are clever enough.
  • Marcus used the first few rows of the audience to show Cantor’s diagonal proof – each row was an infinite decimal, each person a digit
  • Humour is a way of connecting speaker/audience (we are the same!) and to diffuse tension
  • Teacher: “What is the biggest number?” Student: “5246?” T: “What about 5247?” S: “Ah, I was so close!”
  • The hardest problem that humans tackle is working out how they come across to other people.
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Beautiful room!

Inventing Triangles



The advantage of multiple classes is you can refine, compare and contrast. Loving it.

This kind of lesson requires, from the teacher:

  • Mathematical agility, to be able to quickly check if a triangle is worth exploring, or if a conjecture is worth exploring
  • General knowledge, to link in unexpected ways (see below)
  • Love of chaos – got no idea where we were going to travel

One group decided the outer diagonals of the triangle would be 1,3,5,7,9… and then used a Pascal-like rule to construct the rest. They abstracted to just odds and evens, and started noticed nested patterns of triangles. 


Another group started with rt(2) as the diagonal, and multiplied rather than added. They too found the nested triangles…


I got very excited at the parallels with Sierpinski’s triangle. The whole class crowded around, awed into silence, to compare and contrast the two. (One made by continuing down an infinite number of rows, the other chops a finite thing infinitely many times).


Later that afternoon I noticed precisely the same pattern in a textbook. After yelling about being the Neil Armstrong of triangles, boldly going to new mathematical lands, I was disappointed to see that someone got there before the kids. Does it matter? Not really. It does matter that I was ignorant of it existing during the lesson – the excitement would have been less genuine.

My Grandkids’ Lung Cancer

“But the project finished two terms ago!”, Rochelle complained after legging it from her home to meet Cameron and I outside the school gates one weekday evening.
That isn’t how truly authentic projects work – they don’t neatly fit into term-schedules. We were off to summarise our report on the air pollution that would be caused by a proposed set of concrete factories, slap-bang in the olympic park.
In two hours of dense, tightly protocoled discussion, Cameron was the only speaker to get a laugh. Finishing his planned speech, he paused, “I just have two more things I want to say”. I couldn’t decide whether to leap in and move things on, but he spoke, eloquently and from the heart, about intergenerational responsibility. “When you are old we are going to look after you” was the punchline that got the chuckle.
Rochelle, finishing our presentation with the line “Do you really want to be responsible for giving my grandkids lung cancer?” (Which we hastily had drafted while in the waiting outside, scribbling up against the wall), cut through the technical jargon (Section 10.3, 2011 Planning Act, B9 usage) to create a phrase that the room remembered. Several people came up at the end to shake her hand and reference that line. 
The committee voted unanimously against the building of the concrete factories. (My) summary of their position – we accept that all cities require building materials, but we are shocked by the poor quality and thoughtlessness of this specific application. The chairman, in his summary, specifically praised the students – “to hear the two youngsters speak was powerful”. 


“Most adults would have struggled doing what you just did” said Terry Paul, local councillor, to Rochelle


The other protestors went to the pub to celebrate. The kids declined the offer of a beer, and instead went home safe in the knowledge that their learning had truly changed something . “Don’t let anyone tell you that you cannot make a difference to the world” one of the local residents joyously yelled as he shook Cameron’s hand. 

A Tentative Lesson Structure

I have been naturally falling into a rhythm in my Further Maths classes. This structure seems to work well when students are already on board with Maths (and therefore follow you through headaches and open exploration), and when they are meeting lots of new mathematical structures for the first time.

 

 

Table to clarify thinking, and then a record of the lesson below.

 

 

Obviously, any attempt to codify what all good lessons looks like is ridiculous.

Chapter

In general

In particular – Matrices

Why?

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Provide students with a headache to wrestle with

Ask students to solve increasingly complex simultaneous equations

Before providing the aspirin, you need to give the headache. Dan Meyer

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Give students key information that it would not be reasonable for them to discover themselves

Lecture on how to multiply matrices (link back to multiplying vectors, which in turn links back to multiplying numbers)

Minimise teacher-talk while not wasting time on wishy washy discovery-based learning

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Ask students what questions they have. Important to do individually first – no cross-contamination

“What do you want to know about matrices?’

Now that they have a basic idea, questions start flowing. (Without teacher-input, the blank page is too daunting)

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Explore! In groups of three or four students?

Structured comparison of matrices and numbers.

Students need to learn how to behave like a mathematician – conjecture, root around in the darkness, play, prove.

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Big reveal: you now have the aspirin for your headache

After playing around with matrices and working out what the equivalent of division is, you can solve big systems of simultaneous equations

Provide an ending to the story, reminder of why this new bit of Maths is useful.

A record of the lesson

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Headache time
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Initial questions
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Explore time

From Apprentice to Master

A year ago I met Desmond a few times in a cafe, to struggle through some STEP problems, in preparation for his university applications.

Now, he is getting ready to leave London, to study Maths at Oxford. Before he left, he came to Six21 and spent two hours with the 11 Further Mathematicians, leading them down his merry path of algebraic problem-solving.

The explanations were sometimes confusing and fast, but this small sacrifice was easily worth it for the benefits:

  • The students responded so well to a leader only a few years their senior (Desmond called the boys “bro”, and could get away with it).
  • The students felt comfortable enough to repeatedly ask for explanations – “I still don’t get it”.
  • Excellent role-model for high uni aspirations. “You all have the GCSE grades to get into Oxbridge for Maths. All you need now is to work hard”
  • Start the exposure to tricky STEP questions and the associated tricks (add and takeaway something disgusting to an expression to reveal something simple after all). Final thing Desmond said to me: “Make sure you do loads of STEP questions with your students”

All 11 students decided to stay for an extra half-hour after the end of school to try and finish the final problem (prove that any number pq, where p and q are primes greater than 2, can be written as difference of two squares in exactly two ways). Great testament to Desmond’s session.

Best of luck for the future Desmond!

 

Questions: (could also use this)

(20172 – 20162 + 20152 – 20142)/(2017 + 2016 + 2015 + 2014)

 

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First reflections on Further Maths

Originally there were 7 students doing FM, all male. It took one question, literally one question, “Would you like to do Further Maths?”, for four girls to leap at the opportunity to do it. They just hadn’t considered the possibility. Great reminder to ensure everyone knows they are welcome to join the Maths Clique, especially those from groups that have historically been under-represented.

 

Highlight of the first week – exploring complex numbers.

  1. I introduced complex numbers as natural next step in progression (x + 2 = 0, 3x – 2 = 0, x^2 – 2 = 0, x^2 + 1 = 0).
  2. Students generated a whole load of really excellent questions

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    (If students were able to answer their questions by end of lesson, it is ticked)
  3. “Trust your intuition, and follow all the rules you already know”. The group were able to solve quadratic equations with complex solutions, manipulate expressions involving complex numbers, mess about with fractions using their knowledge of surds as analogy.
  4. When looking at powers of i (remember, all I have given the students is the definition of i)
    • “We are trapped in a loop!” yells Wintana
    • “It’s like a circle” chips in Ifte
    • Noemi draws the circle, notices that it looks like the real number line goes through horizontallyIMG_7218.jpg
    • “I guess the y-axis is the imaginary number line” jokes Igoris sarcastically

What beautiful and constructive conversation from the students. Replicating the discovery of the complex plane, inventing it for themselves. Obviously discovery-based learning has its pitfalls (“Now class, you will discover the trigonometric ratios”), but once you give the students the initial seed they can run very far indeed! I am very excited.

BTW, reminds me of a recent podcast by Ben, Ben and Blue about how lecturing is doing worse than doing nothing at all. Inspired by the work of Carl Wieman, physicist turned educationalist. An expert teacher has comfort in chaos, is able to move agilely in response to the class. Lecturing, in comparison is comfortable, easy, reliable.

Central School, Vanuatu

For a morning, we popped into a local school to observe some lessons and learn more about education in Vanuatu. (New resolution: go to a school on every country I visit).

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Classes of 40. Everyone studies maths up to the end of school (age 19). Classes are mixed, and work is the same for all students. External exams throughout secondary education (following a New Zealand exam board).

 

Kids are kids everywhere (we found this with the teenagers, and Gretchen found this with the 7 year olds). Atmosphere relaxed and friendly as we walked through the school. No teachers running for the photocopier, no over-boisterous students. Maybe a lesson starts five minutes late, or the students don’t turn up for lesson at all (as happened with one lesson I waited for, for 40 minutes).

 

 

“you take all the terms in the first bracket, and you multiply them with all the terms in the…”  intones the teacher pleasantly.

“…SECOND BRACKET” choruses the class.

 

The teachers we saw had really excellent subject knowledge (strong calculus for example), and also said the thing they value most is problem-solving and deep thinking. The students were working through rote exercises. Same tension as in London then.

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Does this give the illusion of understanding, or it is a useful way of ensuring the whole class stays focussed? I saw very similar style of delivery in Soweto – is it a fluke, or a consequence of colonial education, or similar for another reason?

 

Teachers train for three or four years, and were shocked to hear that in England you could have a full timetable straight away.