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
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The Origins of Counting

Rather than our usual Tuesday afternoon departmental meeting, the Maths Team left the school walls and decamped to ritzy Central London, for a meeting about the Origins of Counting, at the Royal Society. As always, the cultural disparities between Newham and the Mall were very apparent, from graffiti on walls to oil paintings of old venerable men on walls. An example of number sense in animals other than humans:

From Number Sense to Number Symbols

A talk by Francesco d’Errico, who claimed he would turn everyone in the audience into an archaeologist by the end of his 40 minutes. Many animals have number sense (hyenas can make comparative statements about the size of an enemy pack and use it to make a decision about whether to attack or retreat), but only humans have number symbols. How did this journey happen?

  1. The journey was non-linear. Innovations in symbols happened, were lost, and then re-appeared later on, often in different cultures.
  2. Homo Sapiens was not the first species to creat symbols. Neanderthals got there first, 80,000 years ago.
  3. The appearance of number symbols was not due to genetic modification (since many animals have comparable number sense to humans, and neanderthals were able to make number symbols). Rather it was due to the plasticity of the brain, and to communication through culture. Symbols have only been developed in the last 5,000 years – nothing in evolutionary timescales.
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Progression of number symbols. x-axis is time (larger x-value means further into the past), and blocks show when different techniques were used. Isolated dots on right show how practices were discovered and then forgotten and then discovered again. 
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For a long time we have used symbols to externalise our memory. Outsourcing memory to google is not a new phenomenon.
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A summary.

The Deep History of Number Words

By Mark Pagel

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Excellent essential question

Language years: if a word is spoken in one language for 5 years and in another for 10 years, then that word has been spoken for 15 language years. “Two” (in its various forms) has been spoken for 148,000 language years. Ridiculous.

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Evolution of words, like evolution of genes
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Decay of words, like decay of isotopes
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The more frequently a word is used, the less it changes. Number words change even slower than this regression would suggest.
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The slow evolution of number words is not limited to Indo-European languages.
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The only words that change slowly across all three language groups are number words. Number words are special!
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Nice inforgraphic

Why do number words change so slowly, across languages and time? Possible answers:

  • A link is built between the number words and the region of the brain links to numerosity. Numerosity is very stable, and so therefore the words are too
  • “Two” is a very clearly defined concept, whereas other words (such as “sofa”) are not.
  • If a word is to be used often, it must be short. There are not many short sounds that are still avaible (they are already words!), so there are no candidates for alternatives to number words.

Implications of Innate Numerosity-Processing Mechanism for Education

By Brian Butterworth, heavyweight academic and oganiser of the conference. He coined the term dyscalculia, and presented here an in-depth analysis of the current neuroscientific research into why some people do not have a strong in-built sense of numerosity.

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The speed with which 3 year old children can count dots accurately predicts their numeracy skills at age 11

His rallying cry: dyscalculia is at least as important as dyslexia, and should be treated as such in schools.

Cedric Villani and Marcus du Sautoy in discussion

Two famous popularisers of maths end the conference with their ruminations on what we have learnt. Cedric is a great speaker, his hands wildly gesticulating and accurately representing the mathematical processes that his mouth is talking about. Marcus observes that, contrary to belief, mathematicians do not see the world in numbers – instead they are hypersensitive to structure. We should not be scared of showing children the big meaty ideas in maths, providing them with linguistic structure to tackle deep ideas.

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Cedric speaks of this cartoon:

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Thoughts

  • We learnt nothing that could be immediately applied in the classroom as maths teachers. But, lifelong learners do not need to be immediately able to apply the new knowledge.
  • We were frustrated by the lack of presentational skills by some of the speakers. Death by powerpoint is never excusable – intellectuals have a responsibility to communicate well.

 

From Graphs to Maps?

This is the final post about the Maths behind the Concrete Project. In a previous post I described how we used graphs and words to describe relationship between variables (such as distance to factory) and how dirty the air was.

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Graphing air pollution

The next step was to move from a graph (which gives information as you move along a 1-dimensional line) to a map (which gives you information as you move about a 2-dimensional plane).

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We tried to make something like this.

I, in my role as mathematician, created a whole load of data for the students to plot on a graph. A year ago I would have thought that this is cheating, but now I focus on the purpose of the project. The aim was not for the students to use Excel to efficiently calculate, but to understand how to plot information and draw conclusions from graphs. Leave the boring stuff to the teacher.

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My role as mathematician

We spent a two double lessons on this. The first lesson was a complete success because:

  1. The numbers involved were easy to understand (we started with lots of simple examples, with made-up numbers)
  2. Colouring is fun! The exercise was novelIMG_5005.JPG
  3. I poured a lot of energy into the atmosphere of the class
  4. There were four adults in the room
  5. I modeled the process very clearly on the boardIMG_5007.JPG
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Most focussed lesson of the year for this student. He loved the clearly explained process.

The second lesson was a complete failure because:

  1. The numbers were now very small. (I toyed with multiplying all of them by a factor of 10 to make more manageable, but wanted the students to get a sense of just how small the concentrations in the air were. Maybe this was a mistake, understanding 2.3  x 10^-19 might be too challenging)8ugtmki_a5f8sdohvky8tpu8nqqwxbbzcn_hzcogkx1k3nlyj6bqtsrltaqsiisvqxs_mb6opgxztqb16nkvb4xvkqebmwkkwcgvlypnxnygenizxl3fyimt4_enoxmfeh1qehn2jla
  2. The students were gradually losing motivation in the project. The concrete factories were no longer going ahead – so why should we continue to work on this project? Fascinating issue, which clearly highlights a potential pitfall of PBL – it could encourage students to only learn when they are solving a genuine problem.
  3. There were a number of students who were very vocal that “this wasn’t really maths”. Because the maps are unfamiliar and involve colouring, there surely musn’t be much deep learning going on. Students often surprise me with their very traditional opinions on what learning looks like.

In the end, teachers followed students. There was no genuine need for the maps to be completed, so we didn’t push for them. Instead, we learnt about sequences and finding the nth term – back to the worksheets that the students recognise as “proper maths”.

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

Flipped Learning at Shireland

Whole Education have organised a year-long action research project focussing on Flipped Learning. The launch event was kindly hosted at Shireland, a school in the suburbs of Birmingham with 30% live safeguarding issues and 64% EAL.

Flipped Learning

As defined at the conference, flipped learning is the following process:

  1. Students learn knowledge, before lesson (through videos or other means)
  2. Teacher, also before lesson, assesses their understanding (this could be through apps that check which students have watched videos, through multiple choice questions…)
  3. Teacher plans lesson based on a secure awareness of each child’s understanding
  4. Finally, the lesson happens. Teacher liberated from explaining from the front, students able to leap straight to evaluating, analysing, creating.

How is this different to…

  1. … traditional homework? Traditional homework involves students applying knowledge they learnt in class, after the lesson. This higher-order thinking surely requires teacher support and so should happen in lessons.
  2. … pre-learning? Pre-learning is where students do a bit of learning before the lesson, but the teacher walks into the lesson blind – no assessment of understanding has taken place.

Shireland’s Philosophy

  • Technology should be mundanely clever. Every teacher, not just the geek in the corner, should be able to use it and see immediate benefits in their classroom. Technology works for us, we are not dictated to by technology. Fix existing issues to ensure all teachers are motivated.
  • Flipped learning is not about the videos, but about the pre-assessment. Spend your time designing excellent assessment, not whizzy videos.
  • Replace 15 learning assistants with 10 people on an E-learning team, who will build an incredible online learning environment. Be bold if you think you are right. This ensures teachers have all the admin done for them, and can focus on other things.
  • Before flipped learning, there was an unspoken agreement between teachers and students – homework was actually a bit pointless, bolted on as an afterthought. No longer!
  • Model the model. CPD is now… Flipped CPD!

Questions and Answers

I am part of a wider team at School 21 thinking about Flipped Learning. Here are some questions the team wanted some answers to.

Q: What about those who don’t do the homework?

A: Shireland have developed a whole arsenal of techniques

  • Students are motivated by a feeling of security – they can now come in to a lesson confident with the material. They understand, finally, the point of homework. In other words, if you design the flip correctly, everyone will do their homework anyway.
  • Clever deadlines for homework. Ensure a day in advance of the lesson, so you can chase stragglers and analyse results.
  • Parental engagement is key. Once parents understand the advantages, they will support you
  •  Be completely relentless at the start of the flipped learning programme
  • If students have still not done the homework, put them in a corner to complete it while the rest of the class do the most exciting and amazing task ever. Sneaky.

Q: What do you do with all the extra time freed up in lessons?

A: The fact that you have more time is a positive, not a negative. More time for deep thinking, peer support, student-led learning, any of the good, deep, stuff. You would teach this higher-order thinking as before, but now with more time and more knowledge of the students’ needs. Your lesson becomes a clinic for fixing issues, rather than a lecture.

Q: How do you ensure this is time-efficient for teachers, daunted by the huge task of curating an entire library of videos?

A: Again, a whole range of answers:

  • Before you make a video yourself, think really hard about whether it is absolutely necessary. Is there already one out there that will do?
  • Do not flip every lesson. Students would have too much homework, and you would not sleep
  • Start small: pilot, evaluate, repeat.

Q: Surely by placing an emphasis on work at home, you are widening the achievement gap between rich and poor?

A: In comparison to traditional homework, students now get more support for higher-level thinking. What was once only available to students with present and supportive parents, is now available to all. Homework clubs and subsidised devices help too.

How to conduct excellent action research?

Shireland is a Research School, and referenced EEF’s research framework. Before gleefully diving in, answer these questions:

  1. What is the question? (Keep it narrow)
  2. What are you going to measure? (Qualitative and quantitative)
  3. Who are you comparing to?
  4. Where are you starting from?
  5. What are you actually going to do?
  6. What were the outcomes?
  7. How will you share the outcomes?

Two things that made me sad

  1. “I like being made to think again” a fellow teacher ruefully observed. It is a shame that teaching does not intellectually challenge all teachers in all schools.
  2. We were given a task to do, having been warned beforehand that this task was designed to make us feel uncomfortable. It was a worded maths question, and the facilitator was a maths teacher. She was comparing giving students a surprise question on a new topic, to giving students questions after they have had a chance to watch preparatory videos before the lesson. This made me sad because:
    1. The comparison is a false one. A good non-flipped lesson would consider prerequisites and not throw students in at the deep end.
    2. A maths teacher should not encourage maths anxiety (sore spot for me).

 

Thank you to Shireland and Whole Education for a thoughtful conference. I can’t wait to revolutionise home-learning and spend more time forensically supporting students to think deeper.