Patterns of Prague

I was in Prague for a great weekend course on play-based learning this weekend, with Estelle as we're both moving to Kindergarten in September. We also got to explore the beautiful city a bit in the evenings, and Estelle was very indulgent when I kept suddenly stopping in my tracks to snap the amazing paving patterns all around.

I was looking forward to trying these out with my Year 4s. I was not disappointed.

This one first, with us all in front of the whiteboard: What mathematical questions could you ask about it?

So, look for a while at it. How many squares in the black cross?

OK, 33. Look again, if you didn't get 33, and work it out to see that it's 33.

How did you see that?

A good crop of answers. So now on to some individual work. How many squares in the black star? And communicate how you work this out.

There were a few slips here and there, but the good thing is, everyone had a clear idea about the task, and everyone was trying to cut their own path through. It's what our quick image tasks are really good for - "it's over to you - find your own way through!"

And there were so many different ways! This seems so much healthier for students' adaptability and independence than the One-Ring-To-Rule-Them-All approach to calculation and algorithms. 

Martin Joyce made me aware that the Patterns of Prague were already a maths thing:

@Simon_Gregg @estelleash awesome. There's a Mars task called patterns of Prague. https://t.co/5P6XJmJc8j

— Martin Joyce (@martinsean) June 5, 2016

And Danny Brown, when he saw what we were up to, tweeted:

@Simon_Gregg brings to mind this from 'approaching arithmetic algebraically" MT163 @ATMMathematics @journaleditor6 pic.twitter.com/cEmqd5ttpt

— danny brown (@dannytybrown) June 7, 2016

Here's more of that article:

There is a difference between counting and watching yourself counting. It is observing how you count, rather than just counting, which leads to statements about counting, such as 6 X 5 + 6 X 5, or l(w+ l)+w(l+l). An algebraic statement about the number of matches used to create an n by n square, or an I by w rectangle comes from finding a way of counting. A certain awareness is required to be able to count the matches, but a second level of awareness is needed to observe and articulate how that counting is being carried out. It is a double level of awareness - awareness of awareness which is required for you to be in a position to write an algebraic statement. John Mason has talked on several occasions about the following from the Rig Veda: 

     Two birds, dosed yoked companions,
     both clasp the same tree.
     One eats of the sweetfruit,
     the other looks on without eating. 

It is awareness of awareness which is involved in working algebraically. Arithmetic is concerned with the result there are 60 matches. Algebra is concerned with organising the counting, finding a structured way to get the result. To be able to count requires a way of counting, a way of structuring and organising the counting. To be able to count requires you to work algebraically. 

 Approaching Arithmetic Algebraically, Dave Hewitt,
Mathematics Teaching 163

So, these quick images foster independence, a focus on contemplation and communication as much as calculation, and algebraic thinking. Give one a try!

Can students ask - and answer - vast abstract questions, without being taught? Madeleine Goutard on Free and Conquering Minds and Cuisenaire rods

The Cuisenaire Company has republished Madeleine Goutard's Mathematics and Children, and I've been reading my copy.

Here's something. Back in the 1960s she was training teachers in the Province of Quebec, promoting the use of Cuisenaire rods. And yet her first chapter begins with a warning not to use them too much:

“It is generally agreed that concrete experience must be the foundation of mathematics learning. When children find it difficult to understand arithmetic it is at once suggested that this is because it is too abstract; for small children the study is then simply reduced to the counting of objects. It seems to me that there has perhaps been too great a tendency to make things concrete and that perhaps the difficulties children experience spring from the fact that they are kept too much at the concrete level and are forced to use too empirical a mode of thought.” (p2, my emphasis)

What kind of abstraction is she looking for then? Exactly the kind that Connecting Arithmetic to Algebra is recommending: looking for general patterns in the way simple arithmetic works.

"I find it of limited value to ask children a large number of definite, restricted questions whose answers they obtain through manipulation of the rods. On the other hand, I find it most profitable to start with vast questions which can be seen in a number of ways and which permit a continuous analysis of the dynamics involved. This is why I shall consider here families of equivalent additions, of equivalent subtractions, and of equivalent products and quotients." (p3)

And as well as having a clear idea of the kinds of areas that are fruitful to investigate, Goutard had a very strong view of the role of the teacher.

"The teacher is not the person who teaches him what he does not know. He is the one who reveals the child to himself by making him more conscious of, and more creative with his own mind. The parents of the little girl of six who was using the Cuisenaire rods at school marveled at  her knowledge and asked her: 'Tell us how the teacher teaches you all this', to which the little girl replied: The teacher teaches us nothing. We find everything out for ourselves.' 

It is evidently very difficult to give the child so complete an impression of non-presence, and to convince him that he alone is the artisan of his own education, but this is the way in which free and conquering minds are formed." (p184)

You can see that Goutard has an abstract way of writing, quite philosophical and psychological. I find I want to ask her, "How did the teacher teach her?? Yes, I know she stood back and gave her space. But how did she set up her sessions? How do you reconcile the seeming oxymoron of having the student genuinely following their own way, and at the same time having the teacher directing them towards the vast questions? How, not just in general terms, but how does a lesson go? What do you say? What do you do?" I want to see dialogues, with just a little bit of analysis, like you find in Connecting Arithmetic to Algebra. I want to get a feel for actual lessons.

But we haven't got that.

Luckily, all is not lost. If we don't have a clear roadmap, we've got a clear destination and a bearing. And I'm getting a more and more clear idea about the details. It helps to have colleagues like Caroline Ainsworth bringing Goutard's ideas to life. And #MTBoS collagues like  Tracy Johnston Zager,  Kristin Gray,  Mike Flynn, Elham Kazemi and Kassia Wedekind and many others who while they understand the vital importance of the agency of students, also seek ways for them to connect with the 'vast questions'.

I think the square can be circled.

Our beginnings are small:

After which we go off to write our own ones in our journals.

The next step is to take one of these, and focus on it. (For example, I find the relation between the five threes and Jinmin's triangle number way of expressing it worth following. Worth getting the rods out and investigating.) And then you're listening out for the generalisation, the claim. Up on the claims board. Do you agree with it? Then how would you explain it?  Show it with pictures, words, equations, a story...

Year 4 arrange Cuisenaire rods to match written equations #istlive pic.twitter.com/T8edXvL49O

— Simon Gregg (@Simon_Gregg) November 3, 2015

I hope we can do our bit to reanimate Goutard's brilliant philosophy. I hope too to capture a few of the moments while we're at it, and perhaps share them with you.

_____________________

Today we got our "Mathematical Claims" board started. To get the ball rolling I showed an image and asked for remarks. After the obvious features, I asked if anyone could say something more general. "What does general mean?" someone asked. We talked about that a little and then Tibo said:

to which Jinmin adedd:

These ones came a bit too quick and easily; they'd talked about them with teachers before. But they went up on the board to get the idea going. I'm looking forward to claims that come out of a more immersive experience of trying things out, reflection, and talking things through, ideas that are a little more hard-won, (and hard-defended).

37 + 25 = ■ + ●

I've posted about Connecting Arithmetic to Algebra before. This slide, based on the book, is one of a number of useful ones from Kristin Gray's blog:

Today, before the main part of the maths lesson, we looked at an equation on the board. I asked them not to calculate it (although some couldn't resist of course);

37 + 25 = ■ + ●

I asked, "What do you notice?"

Aditi started us off by saying it was balanced. Brilliant! We'd done some work on balanced equations earlier in the year, so I was really pleased to hear this as the first thing.

There were some other good points, and then Justus said the two numbers could be instead:

38 + 25

We looked slowly at what he'd done. Justus said that you could always do this, take it from one addend and give it to the other. Most of the class agreed. 

That's going up on the Claims Board.

"What other pairs of numbers could we have?" I asked.

And then there were a flood of answers.

Alonso gave us:

-1 + 63

and then there were lots of sums with minus numbers in too!

James gave us:

-38 + 100

And Annie:

It's another reason to start a lot more writing in their maths journals next year. There is a lot of great thinking, and it's getting wiped out, rather than pondered over for a bit longer and looked back on. The class were evidently enjoying the freedom of the exploration and talk, but not everyone joined in the discussion. Writing would give more time for everyone to get their thoughts together and put them into words. In September...

On other occasions (like this), I've used Cuisenaire rods to help in representing the pattern. Today the ideas seemed to flow so well without this, but another time I might use them again for something like this.

I like how lots of people have different ideas

I first came across this book in Kristin Gray's blog. The subtitle Strategies for Building Algebraic Thinking in the Elementary Grades gives more of a picture of what the book is about.

Chapter 1 is available as a PDF, so I read it - and liked it,. And bought it. I'm reading it now.

The idea is, we teach arithmetic, sure, but what about noticing the patterns in what we do, making generalisations, or claims, about the ways numbers behave?

It's not about using algebraic notation - far from it - it's about noticing patterns.

An example from our Y4 classes this year: we asked children to look at patterns if you add consecutive numbers. Quite a few children noticed and made the claim that you always get an odd number if you add two consecutive numbers. Some of them could justify the claim. We'd already had a lesson about the addition of odd and even numbers, so they could refer back to that. Eventually we had a whole load of claims, which I tweeted:

the patterns in adding consecutive numbers: add two & you get an odd sum... pic.twitter.com/1oU21aKYSr

— Simon Gregg (@Simon_Gregg) March 30, 2015

There's more going on here too. It's not just about meta-thinking as subject matter, but the emphasis of the book - and often my approach - is the way this thinking is done. Here's an excerpt of one of the many conversations in the book:

Ms. Diaz: ... Have you ever spent time thinking about how you participate in discussions? Like what do you do when something is hard for you to think about? Or when you don't get what someone is saying? 

Kathryn: Well, no one has ever asked me to think about this before. Usually, it is like we just have to have silence. 

Ms Diaz: When I was in school, we didn't spend time talking. Only the teacher did the thinking. But I want us to be a team so that we can all contribute. 

Will: I like how we show our thinking, you know like we come up to the overhead and show our thinking. I like how lots of people have different ideas. 

Brent: It is like there are lots of teachers.

What do you see in this snatch of conversation?

I'm reminded of Freemont's four freedoms mentioned in my last post: freedom to make mistakes, to think for yourself, to ask questions and to choose a method of solution. Conversations like this are places to find these freedoms. Where the teacher asks questions, and genuinely wants to hear what the students think about the question, rather than something that they've told them to learn. It's worth reading Ilana Horn's post on Asking the Right Question.

How do we do that in math class, the place with the most deep-seated rituals of recitation and mindless calculation? How do we move from what English mathematician and philosopher Alfred North Whitehead called ‘inert ideas’ — those which have been received and not utilized or tested or ‘thrown into fresh combinations’?

(I should add, I also love Ralph Pantozzi's coin-tossing activity in the video in the last post, even though, it's not about reflective conversation. It's a teacher-orchestrated activity, and I do plenty of those. One like this can clearly be really enjoyable and memorable and can lead on to opportunities for reflection and conversation. Nicole Louie however mentions "good task worship" in her comment on Ilana's post, and I've been thinking about that lots. You see, I'm always looking for good tasks. Am I a worshiper? I'm still pondering that...)