Sunday, 11 March 2018


I'm thinking about variation.

In Mr Barton's podcast of his conversation with John Mason and Anne Watson, Anne Watson says:
"Comparison is a hugely important idea throughout mathematics. When you have something to compare, you don't focus on the thing itself, you focus on connections, same, difference, relationships between the things, relationships within the things."
I like the way she keeps it simple:
"Sadly, I think variation has become Variation, with a capital V, and some teachers think there's a right way to do it and a wrong way to do it. There isn't: it's actually embedded in mathematics."
Back in 2006, Anne Watson and John Mason wrote a short article, Variation and Mathematical Structure in the Mathematics Teaching Journal.
It seems to me, beginning to think about his, that considering variation is useful right from the start of education.

When we present the classic shapes poster to 3 year olds, we start doing things with variation.
Children get the message, whether we intended it or not, that shapes with straight sides 'sit' with their sides, not their corners, at the bottom. Unless they're 'diamonds'! That rectangles and squares are different things. That triangles and pentagons and octagons are regular. It seems innocent enough, but teachers have some lessons later on undoing these impressions. This is why Christopher Danielson's Which One Doesn't Belong?s are so good. They consider the possible variation, including counter-examples, to help students think about the categories for themselves.
David Butler also has some great quadrilateral posters which reflect the real range of possibilities.
Similarly, when we always write 2+3=5, with the sum on the right, we're not doing justice to the possibilities with this notation. We should  sprinkle in some 5=3+2 versions too.

Often a series of examples leads to the possibility of certain generalisations. For instance, back in this post, I talked about how a set of differences can lead to  noticing about subtraction.
Indeed, it did lead to a conjecture:

The other day I saw this tweet from Duane Habecker:
Putting this series of subtractions next to each other invites noticing - oh look, you can carry on to negative numbers! - and - look! - when you subtract -1, it's just like adding 1. I wonder if it always works like that...?

You could see a 'number string' routine as a way of presenting examples to show the range of  possibilities and asking students to make links and comparisons between them. Take this number string in a 1st Grade (Year 2) lesson with Kristin Gray.
The number of examples is limited; the focus on responding to an unknown in different places in the equation. There's a lot of thought that goes into how the different examples relate to each other.

Another routine that I've seen and used just a little is What's the same? What's different? with two images; there are lots on Twitter with the #samediffmath hashtag. Here's one:

I've thought about variation lots when designing Which one doesn't belong?s, but not calling it variation. I've just had a piece on them, written with Jim Noble, in Mathematics Teaching. and I've got a folder of them.
The design of them is fascinating: how to elicit the most responses, how to allow particular relationships to be noticed. Here, on this Othello board, my theme is half. That's what's the same about them. I could have introduced one where it wasn't a half. So we're mainly looking to arrangement for the differences. Are the two halves 'the same'? How many are there in total? Which colour is 'on top'? And so on.

I want to become more conscious of how we present examples, how we vary them, how we display new relationships in them. I'm now going to be looking out for how variation crops up in various places, trying to make my awareness of it as a theme more systematic.

Friday, 9 March 2018

Être plutôt qu’avoir ?

After Le Maître est l'Enfant, Estelle and I watched another French education documentary tonight, Être plutôt qu’avoir ? -  'To Be rather than to have?' It was a look at education, especially in France, historically. I found the pictures of school before the 19th-century desks-in-rows era interesting. Like this from Pieter Bruegel The Elder.

And then a look at some of the ways that some educators have enriched education for primary-age children: through circle times, philosophy for children, forest school, Montessori classes. Lots to think about again: good to now and again, or even regularly, challenge the way we teach - how much is a product of the relatively short history of public education, and how much is a conscious approach to real education?

We saw Célestin Freinet working the vegetable patch with his students. And some of his 'pedagogical constants', a kind of manifesto:
  1. The child is of the same nature as us [adults].
  2. Being bigger does not necessarily mean being above others.
  3. A child's academic behavior is a function of his constitution, health, and physiological state.
  4. No one - neither the child nor the adult - likes to be commanded by authority.
  5. No one likes to align oneself, because to align oneself is to obey passively an external order.
  6. No one likes to be forced to do a certain job, even if this work does not displease him or her particularly. It is being forced that is paralyzing.
  7. Everyone likes to choose their job, even if this choice is not advantageous.
  8. No one likes to move mindlessly, to act like a robot, that is to do acts, to bend to thoughts that are prescribed in mechanisms in which he does not participate.
  9. We [the teachers] need to motivate the work.
  10. No more scholasticism.
  11. Everyone wants to succeed. Failure is inhibitory, destructive of progress and enthusiasm.
  12. It is not games that are natural to the child, but work.
  13. The normal path of [knowledge] acquisition is not observation, explanation and demonstration, the essential process of the School, but experimental trial and error, a natural and universal process.
  14. Memorization, which the School deals with in so many cases, is applicable and valuable only when it is truly in service of life.
  15. [Knowledge] acquisition does not take place as one sometimes believes, by the study of rules and laws, but by experience. To study these rules and laws in [language], in art, in mathematics, in science, is to place the cart before the horse.
  16. Intelligence is not, as scholasticism teaches, a specific faculty functioning as a closed circuit, independent of the other vital elements of the individual.
  17. The School only cultivates an abstract form of intelligence, which operates outside living reality, by means of words and ideas implanted by memorization.
  18. The child does not like to listen to an ex cathedra lesson.
  19. The child does not tire of doing work that is in line with his life, work which is, so to speak, functional for him.
  20. No one, neither child nor adult, likes control and punishment, which is always considered an attack on one's dignity, especially when exercised in public.
  21. Grades and rankings are always a mistake.
  22. Speak as little as possible.
  23. The child does not like the work of a herd to which the individual has to fold like a robot. He loves individual work or teamwork in a cooperative community.
  24. Order and discipline are needed in class.
  25. Punishments are always a mistake. They are humiliating for all and never achieve the desired goal. They are at best a last resort.
  26. The new life of the School presupposes school cooperation, that is, the management by its users, including the educator, of life and school work.
  27. Class overcrowding is always a pedagogical error.
  28. The current design of large school complexes results in the anonymity of teachers and pupils; It is, therefore, always an error and a hindrance.
  29. The democracy of tomorrow is being prepared by democracy at the School. An authoritarian regime at the School cannot be formative of democratic citizens.
  30. One can only educate in dignity. Respecting children, who must respect their masters, is one of the first conditions for the redemption of the School.
  31. The opposition of the pedagogical reaction, an element of the social and political reaction, is also a constant, with whom we shall have, alas! to reckon unless we are able to avoid or correct it ourselves.
  32. There is also a constant that justifies all our trial and error and authenticates our action: it is the optimistic hope in life.

Friday, 23 February 2018

Becoming the Math Teacher You Wish You'd Had

Tracy Johnston Zager's book, Becoming the Math Teacher You Wish You'd Had: Ideas and Strategies from Vibrant Classrooms is a - perhaps I should say the - book that I'd recommend to all math/maths teachers, and that includes all of us primary/elementary teachers. And for those of us working in inquriy-led IB PYP classrooms, the fit is perfect.

Tracy takes a range of things that real mathematicians do as her starting point. Chapter 7 for instance is titled, Mathematicians ask questions. She starts with a quote from Jo Boaler's book What's Math Got to Do with It?:
Peter Hilton, an algebraic topologist, has said: “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” Such work requires creativity, original thinking, and ingenuity. All the mathematical methods and relationships that are now known and taught to schoolchildren started as questions, yet students do not see the questions. Instead, they are taught content that often appears as a long list of answers to questions that nobody has ever asked. 
She gives examples of resources for encouraging questioning, like 101questions and Notice and Wonder.

My favourite part of the chapter is one of the dips into real (and yes, vibrant) classrooms, this time with Deborah Nichols' first and second grades (p152). The question had come up, 'Are shapes math?'
The first step was to find out what the students wondered about shape.

And here are their questions. What an amazing set:
I want to ask some of these students what they meant by some of these! That first one, 'How big can circles go?' - is that about the practical constraints on us creating circles? Or is it about how circles start to look straight when they're really big? Like us walking on our planet. Or is she asking about circles in space? Or something else? Is the question 'How round can a circle be?' related?

Also, 'Are shapes fragile?' Did the questioner mean can shapes be distorted easily? Like the way a triangle or a tetrahedron model made of edges is quite robust, but a square or cube can be deformed easily?

Anyhow, what the teacher did was to arrange a sequence of experiences, with shapes that allowed for there to be a real dialogue between these questions, the shapes themselves and what the teacher needed to be learnt. Without allowing the inquiry to go off in directions that wouldn't really answer the questions, the students' questions - and the answers that came - stayed to the fore. Bit by bit the students built up the knowledge and vocabulary they needed to answer the more mathematical sides of their questions. And they thoroughly covered the learning that's set down for those grades.

As Tracy writes:
Perhaps knowing that students' inquisitiveness leads to the same ideas that mathematicians study and standard-writers emphasize can help us feel less pressure to tell and cover and explain. If we allow students to ask, we will likely end up in the same place but with much more engaged, empowered students.