Purple trail. First snow of the season. Big shift since last seminar (we missed one due to the strike, and winter is well on its way). Kids in full animal-mode, leaping about in the snow hurling snowballs at each other, jumping on ice, jousting with huge branches. Great fun, and a well-needed break from the pressures of high school, time to just be a child.
Notice K’s excellent snowball pitch:
Moments before this puddle’s ice-layer broke. Notice S sneaking up with a huge branch.
Fundraising for our trip to China. Please sponsor us!
Quick warm-up before a 5k in Lincoln Park (with 300 other students from around Chicago!). We all ran out our own paces, supportive and happy in the crisp autumn sunshine. Just after the awards and before the official costume contest (!), the choir was invited on stage to sing.
Running and singing are both excellent ways of building friendships, of listening to your body and your breath, and of enjoying being alive. Woohoo!
Excellent fortitude from all of us as we retained sense of joy, even with drastically different weather to the first week. We are getting ready for winter. No such thing as bad weather. Students getting excellent at spotting things off the track and going exploring.
In order to understand my Mathematical story, it is necessary to first zoom out and consider my family. My Mum’s Dad is a biologist who discovered RNA. My Dad’s Dad was a principal and Professor of Education at the University of Oxford. (Note that neither of my grandmothers had prominent careers despite both being amazing and talented people). My Mum is a scientist, who put the first MRI scanner in a British hospital. My Dad almost got a PHD in Math, before dropping out to become the Director of Finance for the Department of Education. My parents met playing violin together at the University of Cambridge. My two brothers and I also all studied at Cambridge. I say this all at the beginning to emphasize how I have stood on the shoulders of giants – my family’s love of education and of Math put me in an amazing position to be good at and enjoy learning Math. I am so grateful.
A good example – the day after writing this I awoke to 63 messages from my family Whatsapp thread, brothers and father excitedly debating the Math of waterfalls. This is weird and amazing, and I am lucky.
Chapter 1: 0 – 11 years
I remember measuring the temperature and rainfall each day for a month, and, with the help of my Dad, drawing the data as a graph. The sense of wonder as the table of numbers, ugly and hard to read, unveiled itself as a diagram. Math at school seemed to be about memorization – two students would stand at the front of the class and compete to see who could answer mental math questions fastest. I was good at it, and so therefore I enjoyed it. A positive feedback loop.
Chapter 2: 11- 18 years
My dad appears many times in my Math autobiography. Explaining how there was an infinite sum (1/2 – 1/3 + 1/4 – 1/5 + …) that could equal whatever number you wanted it to, depending on how you arranged the numbers. What nonsense. 10 years later at university I learnt why this was true. Instantly answering trigonometry problems that I was completely stuck on. Reading popular Math books on holiday. I wonder now why I am inspired by my father more than my mother even though she is a scientist – have I internalized the stereotype that Math is a masculine pursuit?
At school most teachers were boring, and the Math was beige. One teacher, Mr Baker, ignored the syllabus and taught us things that were supposed to be too difficult for us. I loved his infectious energy and his bravery in tackling challenging ideas. All my Math teachers were the same race and class as me, and I felt completely confident in the classroom.
During these years Math was fine, but the things I really loved were Music and Art. Many hours spent after school in the art studios painting with friends, or playing in jazz bands.
Chapter 3: 18 – 23
I obsessively loved Architecture, and went to a university in London to study how to design spaces – the perfect balance between the Arts and the Sciences. I couldn’t just study Math – that would have meant I was just copying my Dad. I lastd three months – not mature enough to live alone, and surprising myself by missing the clarity of Mathematical thinking too much. I worked on a farm in remote Northern Norway for two months, studying the Math entrance exam in the evenings to get into university for Math.
I devoured Math at University – it blossomed from a one-dimensional subject at school to this fascinating forest of ideas that you could explore and struggle with. I loved topology – which is like geometry but where everything is made of squishy bendy material.
I moved to Cambridge to study for a Masters. I went mostly because Cambridge had an impressive reputation, which is not a good reason. This was the first time I properly found Math difficult. The professors seemed oddly proud of the reputation that this was the hardest course in the world. In two of the six courses I just gave up and didn’t go to the final exam (and so scored 0%). It was a real challenge to my identity as a Mathematician, and I am not proud of how I reacted. I wouldn’t have passed the course without the help of a group of four friends – we spent all our time together, alternating between messing about as students and thinking furiously about Math. One of those friends was to become my girlfriend, and was the reason I moved to the US.
I had a few experiences of teaching Math to children that made me realize how fun and intellectually satisfying it was. One in a local high school where a girl quizzically looked at me and said ‘Sir, are you posh?”. The other for a summer in Johannesburg, South Africa, where far more experienced and superior teachers treated me with respect just because I was white. I was awful at teaching, but wanted to get better.
Chapter 4: 23 – Present
I learnt to teach in a huge school in the East of England, where half the students were Pakistani-British. These students would study at school during the day and then study Islam at the Mosque in the evenings. I tried to teach young children the Math that I had learnt at university – convinced that anyone can learn high-level Math. I failed miserably – not because the students couldn’t do it, but because I wasn’t supporting them in the right way.
My first three years of teaching were in East London. In that time only one student that I taught was White-British, but we teachers never spoke about race – it was not deemed relevant. Is this because race is less important in Britain, or because we were blind to reality? Most teachers were liberal idealistic well-educated white people who wanted to change the world. Gradually I learnt that learning to understand the students and their motivation and interests was far more important than anything I could teach them.
Last year I taught Geometry and Pre-Calculus at Payton. I had never met students who are so focused on their future success – something which is both good and bad.
An art teacher should still make Art. A music teacher should still play and compose Music. As a Math teacher I still need to do Math myself – I meet in the evenings with friends or colleagues to tinker with problems or spend time in holidays going to conferences to push myself.
I have spent the summer reading and writing about this course we are about to do together. I am a bit nervous but so excited, and cannot wait to see what you achieve!
Chapter 5: The Future?
I am really excited to think about how different people in different cultural contexts think about and do Math. Maybe I will continue to travel to new countries to teach? Maybe I will go back to university and try to put into practice all the things I tell my students to do when learning?
Us Math teachers have been talking about lesson plans. “If something is worth learning, then it should be in the lesson plan” is one viewpoint. Substantially changing things on the fly is a sign you haven’t thought hard enough about the learning. Small adjustments to deal with student misconceptions are fine, but don’t diverge for a significant chunk of a lesson.
I disagree. I think that every chance to respond to student’s instinctive thoughts should be grabbed with both hands, as a way of mimicking the process of discovering Math and to pivot around student interest. No matter how hard you think before a lesson you cannot predict what will happen in the heat of the moment. I find that lessons that follow the plan lack zest and joy. When totally unexpected things are happening the students get excited, and I get excited.
Here is what I think is a great example of when not following a plan is great.
Mr J: Does anybody have any questions about the Pythagorean theorem, before we move on?”
D: “I just noticed… 5 squared is 12 + 13. 3 squared is 4 + 5”
Note that D did not answer my question at all. I was looking for people who found Pythagoras hard. He just wanted to share an observation.
At this stage, I had two options. Ignore and continue on with the plan, or explore this pattern further. In the end we explored it a tiny bit, then the class voted on which path to go down. They voted unanimously to explore D’s path, which ended up taking the rest of the lesson (30 minutes)
Mr J: “But 8 squared is 64, and that is not 15 + 17”
D: “Yes, but 64 is double 15 + 17”
Mr J: “Why do you suddenly double?!”
L: “Maybe because the difference between 17 and 15 is 2?”
And now, the game was on. I scribbled a plan on the board:
Google a list of Pythagorean triples
Test the conjecture with some more triples
Prove the conjecture
Students frantically finding the largest triples they could find. Navigating the mathematical internet is a good skill to encourage. Testing, finding counterexamples, realizing their mistakes (“I thought the difference was 25, not 15!”), finding more conjectures (“every triple with smallest number odd contains two consecutive numbers”). G moved to an algebraic statement of the theorem, and noticed it reminded her of something from her Algebra days – the difference of two squares.
A final pattern at the end of the lesson, for triples where the smallest number is even…
2 lots of 8 is the average of 15 and 17
3 lots of 12 is the average of 25 and 37
4 lots of 16 is the average of 63 and 65
What is going on here?
This conjecture is approaching one algorithm for generating triples:
Was everyone involved? How do I monitor engagement while also thinking flat-out as a mathematician?
How to persuade my teaching colleagues that this is worthwhile? I am starting from the assumption that the purpose of school Math is to give students a chance to act like Professional Mathematicians. Is that true?
Payton is currently trying to move towards a model where students in different classes get the same experience. How can that mesh with the idea that lessons should be substantially changed by student conjectures?
By the way, we started this lesson with students reflecting on why they study Math:
Why do I study Math?
If I had to guess, what would I say are Mr Judge’s reasons?
For a grade
Focus on things not in outside world
To understand the world around me
Improve decision making
To help financial math in real world
For a career in future
Encourage deep thinking
To understand things e.g. why a circle is a circle
To verbalize thoughts
Foundation of other subjects that we learn
We will always need to use math
To have a different outlook on the world
To challenge yourself
To create new things
To become professional mathematician
Curious thinker and trust nothing
Has endless possibilities to explore
To discover different topics in math
To get critical thinking and problem solving skills
I am re-reading Measurement, by Paul Lockhart. His mathematical values precisely match with mine. I want to be him. Read the book read the book read the book.
I wonder what he would say to the idea that the ability to view Math as an imaginary land where you explore and play is a privilege that only some students are able to access? That, due to systemic oppression, we would be doing students a disservice by skipping past mastery of skills straight to open-ended tasks?
Here is me attempting to recreate Lockhart’s ethos in my geometry class, full of curiosity at wandering in the mathematical jungle.
The dichotomy between traditionalist and progressive education is a false one, created by academics for ease of analysis. Good teachers know that both have their strengths, and a balance is the only way forwards.
Dry slow-paced practice is terrible. Every task has to be meaningful, taught in context, aimed at a real audience. What does this mean in Math?
If you only focus on open-ended rich tasks, thinking about fluency rather than skills, then you widen the achievement gap. Students who come to you with basic skills in place will fly. Students who don’t have this foundation will never be given an opportunity to build them, and will fall further behind.
Don’t be shy about exhibiting the fact that I have knowledge-based power. By pretending the power doesn’t exist I am not helping my students. Instead I should be explicit about what the students need to do to gain power themselves. Simultaneously, we should also work to destroy the power imbalances of society. “Pretending that gatekeeping points don’t exist is to ensure that many students will not pass through them”
Students already have a voice. My job is to not help them to find their voice. My job is to help them to hone their voice, and make it work in the dominant discourse of American society.
The teacher should act primarily as an ethnographer – actively seeking to understand the diverse cultures of my students.
Teachers are experts in their fields. Students are experts in their lives and communities. Relish and use both.
How can I be more active as an ethnographer? Without overstepping the teacher/student professional boundary, and without embarking on poverty porn?
I agree that there are many skills in English that are essential to modern life – the ability to write a letter/email or speak in a “sophisticated” way is essential to have a successful professional life. I disagree that this idea transfers well to Math. If my students are unable to write a written argument well, they will be limited when communicating with colleagues. If they are unable to apply the cosine rule, there is no inherent limitation (apart from in the arbitrary world of assessments). Surely the role of Math is to encourage deeper thinking? It’s not that I naively assume that through rich problem-solving my students will magically pick up an understanding of the cosine rule (that thinking would be putting diverse students at a disadvantage and is something that I thought last year), I just don’t care at all about the cosine rule.
In Geometry, we proved Viviani’s Theorem. Individual groups’ lightbulb moments diffused across the room, ensuring that everybody had a chance to experience that giddy moment of discovery. Collaborative work to share knowledge, individual write-up to assess internal understanding.
I have been giving the students problems that I would find interesting. “Treat them like experts and they will become experts” is an unwritten mantra. Is there any justification for this? Am I prioritizing my intellectual enjoyment over the students’ needs? In any case, there are some outstanding proofs here:
A great idea to use circles, but is it justified?
Excellent attempt at dynamic proof without words:
This student is thinking very precisely, but I have no idea what is going on:
Equilateral triangles within equilateral triangles…
As a starting point, this is ripe for extension. What about other properties of equilateral triangles? What about other triangles? What about other polygons? what about other dimensions?
A South African high-school student stumbled across a different invariant. His name is now immortalized in the Clough Conjecture. Students began this journey the following lesson…