KS suggested great intro to a lesson on why we need to prove, rather than just trust our pattern-spotting skillz.

3blue1brown has made an excellent series of videos on this

Skip to content
# Glimpses of Infinity, Reminders of Reality

Striving for independent mathematicians in London

KS suggested great intro to a lesson on why we need to prove, rather than just trust our pattern-spotting skillz.

3blue1brown has made an excellent series of videos on this

Advertisements

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)

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.

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

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).

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.

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.

For the past two years I have seen 13 students, four times a week. They arrived as newbies in Yr7, I arrived as a newbie teacher. We grew up together in School 21.

I would have spent more time working 1:1 with opinion-forming students who sometimes disrupted the dynamic. Overall however, it was an unbridled joy to work with the group, and I look forward very much to seeing them about the school.

Another heart-warming Parkrun with students from school. I ran with Abdul (24 minutes), encouraging him to push on through a stitch, which he nobly did for the majority of the run. Esther ran with the girls, and tried to explain to them what a PHD was. Heather coached Yasir, and other runners supported us all through our struggles.

Parkrun is an oustanding thing for students to get involved in:

- Achievable (5km is accessible for all – nothing wrong with walking)
- Free
- Weekly
- Friendly
- Minimal admin from teachers – students just turn up and run!

Inspired by the work of Julian Beever (whose work I learnt about at the Chicago Lab Summer School), we spent 6 weeks understanding how to use our knowledge of trigonometry, similar triangles and three-dimensional thinking to make pavement art.

Geogebra links to play with here (Elevation, Plan, and 3D)

A year later, I look back on this project as a turning point with this class – earning their trust and respect. Although some of the mathematical learning was insecure and needed to be revised, we came together as a group and thought hard about how to apply abstract concepts to tangible products.

On a swelteringly hot day in the final week of school, I took my coaching group out of London for the day, to Epping Forest. (There is evidence of human history in the forest from 2,7000 years ago. It is the size of 3,300 football pitches. There are 50,000 ancient trees. Henry VIII hunted here. The ponds dotted around were born of bomb craters from WW2).

Aims | Reflections |

Understand why people use green spaces, by conducting a questionnaire | Interacting with members of the public is always great. Especially when they are from a different walk of life – the old lady who had walked in the Forest every day for 50 years… |

Have fun outdoors! | Once the kids got past their fear of dogpoo and bugs, finished playing games about who fancies who, then the beauty of their environment flourished. Climbing trees, throwing a ball about, excitedly watching birds, running to stroke a dog… |

Spread kindness in the forest, by picking up litter | We became detectives, imagining the past life of the wrapper/can/condom that we were picking up. Who had dropped it? Why? |

By far the highlight (for me at least – the students’ highlight was lunch) was ploughing through the forest, navigating our way to Connaught Water by our shadows alone. Jaydan constantly checked that his shadow was behind and to the left, as Sophie had told us to do. We dived off the path, over streams, under branches, through bushes. Great teamwork, helping each other through difficult patches. Finally shades of blue spotted through the trees, and we emerged, joyous, on the lakeside.

At times I was frustrated at the hilarious fear with which they approached walking through a few brambles. It is useful to remember that over a three day weekend earlier in the year 7 of 15 students in the group had spent less than 1 hour outdoors.

**Edit: wisdom from Kate on how to deal with negativity**

(Kids are strange beasts: those who were most vocally against the idea of tramping around a forest told me two days later that it was the highlight of their two years with the coaching group…)

- Go bold on games, when tiredness is an issue
- Everyone is experiencing the same rain/hills/tiredness. There are two types of people. Drains only are interested in talking about the problems, focussing on the bad stuff. Nobody likes to be around a drain. Radiators focus on the good things, they recognise that some things are difficult but they think of solutions
- Completely ignore negativity
- Quiet words with ringleaders

All good advice for any groupwork. Somehow it becomes more amplified when outdoors. Just as students struggle to transfer learning from science to maths, teachers struggle to transfer techniques from the classroom to the forest?

I spent an enjoyable if busy week thinking with a bunch of prospective A level students about Infinity. We chose the topic of the endless because it is not explicitly A level content, but is beautiful, important, broad and relatively accessible.

**Key reflections**

- The problems we gave the students were challenging, probably slightly too much so. At the beginning of the week they were struggling to cope, but by the end we had noticed a significant improvement in problem-solving skills. Students were able to stick with a problem for more time (2.5 hrs on the second day, on one problem, was maybe too much of an ask), and were more comfortable collaborating with each other.
- I noticed (as did other teachers) a shift in my teacher personna. No longer requiring to behaviour-manage (or maybe that is wishful thinking – a few were misusing their phones for example), I was able to focus more on the mathematical concepts.
- There are two types of problem-solving. The sniffing dog follows his nose, working forwards (sometimes haphazardly), doing what he can do, until a pattern forms. The wise elephant thinks carefully about what she wants to achieve, and then works backwards from this. The students picked up this language from the week, and agreed that excellent problem-solvers were able to use a synthesis of the two approaches
- All maths teachers descended into an afternoon session for the Festival of Problems. We used some outstanding questions (from Westminster School, curated by Kings Maths School), and had a grand time. If the questions are good enough, then minimal structures are needed.

**Feedback from students:**

- Long open problem-solving sessions were amazing (contrary to my perception that the students were burnt out by the end of it, most of them said they loved being able to spend so much time on one thing).
- Don’t ask us to read chapters on things we haven’t learnt about yet (almost all said this)
- Introduce more practical hands-on sessions

A few pics of the week:

We had an entire floor to work with, which was amazing.

After spending the the week at School 21, KS and I took the remaining students (large rate of attrition in the buyer’s market of sixth forms) to Cambridge, to present what they have thought about to professional mathematicians (or, my friends and brother all who studied Part 3 and mostly are in the middle of PHDs). Tabitha, from Underground Maths, then led a problem-solving session, on this problem about the difference of two squares.

The students presented on:

- A problem I grappled with
- A story that resonated with me
- A question for my audience

**Key Reflections**

Observation |
Outcome |

Tabitha would carefully listen to a group, before adding optional questions – “I have an idea. You don’t need to follow it, but you could” |
Look before you leap. Too often I jump straight in, possibly repeating thoughts the students have already have, or derailing their thought process. |

Nina did not know how to use a calculator, one of the students (much to their amazement), had to teach her. |
Professional mathematics is not about calculation. Constantly remind students about this |

Several of the PHD students had to struggle to understand the ideas being spoken about. “Wait, what even is an infinite fraction?” one asked. |
Everyone, when meeting new Maths for the first time, struggles. |

“It feels like it will be finite, because it is not very wiggly” said Tom. |
Professional mathematicians use intuition much more than students. Nurture intuition more explicitly. |

The session was the highlight of the week, because the professional Mathematicians were instinctively and continually asking excellent questions and providing incredible explanations. All students were hooked. |
A reinforcement in the belief that content knowledge should never be underestimated… |

One student had never eaten a raspberry. Another had never seen a tennis court. Several spoke about Hogwarts. |
Do not forget the “side”-effects of trips to contrasting places. |

**Examples of work**

Brains worked so hard that sleep was necessary…