Manipulatives

The #elemmathchat conversation is always at the slightly difficult hour of 3 am. here in France. I did set my alarm this Thursday, and did manage to wake up to turn it off, but that's as far as it went. So I caught up with it on Storify and favourited and commented a little. Here's one I liked.

@MarkChubb3 Everyone should have a ladder of abstraction handy. https://t.co/9j0XlN5xMD

— Simon Gregg (@Simon_Gregg) October 23, 2015

and then it went like this:

@MarkChubb3 Can you give me a particular instance of that? ;-)

— Simon Gregg (@Simon_Gregg) October 23, 2015

and

@Simon_Gregg Dividing fractions... How many understand what's going on here? Some don't see need for concrete if Ss can get an answer

— Mark Chubb (@MarkChubb3) October 24, 2015

Tracy quoted this from Making Sense in her blog post recently:

“In traditional systems of instruction, teachers are asked to provide feedback on students’ responses, to tell them whether or not they are right…this is almost always unnecessary and usually inappropriate. Mathematics is a unique subject because…correctness is not a matter of opinion; it is build into the logic and structure of the subject…There is no need for the teacher to have the final word on correctness. The final word is provided by the logic of the subject and the students’ explanations and justifications that are built on this logic” (Hiebert et al. 1997, 40).

It's a bit like Gattegno had it in his picture I've shown before:

Gattegno was saying, the knowledge (K) doesn't get poured into the student (S) by the teacher (T); the teacher communicates what they want to communicate by pointing them towards something that will give them the knowledge directly.

This is especially true when the affordances of a manipulative and the way the student has been asked to explore it give the student instant feedback.

27 allows you to fit three nines exactly along next to it. They fit really neatly!

Although Gattegno cuts a slightly odd figure to us now, and his "lesson" is evidently a kind of performance that is at the end of a series of lessons, because he's at the root of how (and that!) we use Cuisenaire rods, we owe him a lot. This second clip is where things really start to get going:


(I wonder, how did Gattegno make that link between "one third of" and x 1/3? Also, how did he get the children comfortable with thirds? I find that's often puzzling for children.)

I feel that his diagram isn't quite right. I want to put conversation into it. Gattegno is talking with the students a lot. But, for me, it's especially student to student conversation - which is notably absent in this video, but needn't be for Cuisenaire rods to be used to give students access to the logic of maths. My diagram would look more like:

My elaboration of Gattegno's picture

I love how Gattegno goes off from the rods into writing equations about 27. Again, this can be done with students making their own equations.

Caroline Ainsworth, following Madeleine Goutard's lead, gets students to write lots of equations about a number. You can see how this could follow on from some version of that 27 discussion:

And more Year 2 maths free writing that they are bringing in from home. @stocklandschool #YesUCan pic.twitter.com/HDvP6SLPUR

— stocklandmaths (@stocklandmaths) October 14, 2015

Here's a page of a child's writing from Goutard:

This seems a really fruitful direction, that I'd like to make my own. I've headed off that way before, but there's a lot further to go.

And have I answered Mark's question? I'm not sure. But I was struck recently, how at Toulouse's "Nuit des Chercheurs", how even a University Professor, Arnaud Chéritat, and his students are using 3D printed models to understand something that's too illusive without something to handle and look at:

& these are the models Arnaud Chéritat made to understand how to turn a sphere inside out! (https://t.co/havucDig4Z) pic.twitter.com/EpfSdioxjn

— Simon Gregg (@Simon_Gregg) September 26, 2015

____________________________________________________________________________

Illustrations added for my reply to Joe's comment below:

Illustration A: What can you say about this picture?

Illustration B: What can you say about this picture?

The ladder of abstraction

I'm interrupting my list to write about this. It's sort of in reply to a tweet from Tracy Johnston Zager, though I suspect I haven't answered the question in exactly the way she intended:

@mpershan @MathMinds @Zakchamp Maybe @Simon_Gregg could chime in about how he handles/introduces manipulatives?
— Tracy Johnston Zager (@TracyZager) September 12, 2014

No, actually this chime can be part of the list.

7. Climb up and down the ladder of abstraction

I've been reading Roy Peter Clark's entertaining and instructive book Writing Tools: 50 Essential Strategies for Every Writer over the summer (link to free podcasts of this on iTunes). One of the 50 strategies is "climb up and down the ladder of abstraction". Here's a snippet:

The ladder of abstraction remains one of the most useful models of thinking and writing ever invented. Popularized by S. I. Hayakawa in his 1939 book Language in Action, the ladder has been adopted and adapted in hundreds of ways to help people ponder language and express meaning.
The easiest way to make sense of this tool is to begin with its name: the ladder of abstraction. That name contains two nouns. The first is ladder, a specific tool you can see, hold with your hands, and climb. It involves the senses. You can do things with it. Put it against a tree to rescue your cat Voodoo.
The bottom of the ladder rests on concrete language. Concrete is hard, which is why when you fall off the ladder from a high place, you might break your foot. Your right foot. The one with the spider tattoo.
The second noun is abstraction. You can’t eat it or smell it or measure it. It is not easy to use as a case study. It appeals not to the senses, but to the intellect. It is an idea that cries out for exemplification.

Here's Hayakawa's illustration of his ladder:

This fits very well with the way I like to teach. Say in maths, I want the kids to use stuff they can touch or be physically part of. To take an example that I think works really well, I'll get them using the Cuisenaire rods to make their own patterns, and then I want them to take the numbers inherent in those patterns, then beyond that, the relationship between the numbers, and if possible to abstract that relationship into algebraic form. I'm quite prescriptive about what I want them to do, but hopefully the limitations are creative, because I want them to be creative.

If you haven't read Richard Feynman on why there is no science education in Brazil, do. He was astonished at how little teachers went down the ladder again from abstract to concrete.

"I didn't see how they were going to learn anything from that. Here he was talking about moments of inertia, but there was no discussion about how hard it is to push a door open when you put heavy weights on the outside, compared to when you put them near the hinge – nothing!"

bench by Zaha Zahid

The architect Zaha Hadid is designing the new maths gallery for the London Science Museum, a building I spent a lot of time in as a kid. She has a maths degree, and she uses maths as a place to find new abstract forms for her buildings. She was also inspired by the abstract mathematical paintings of Kazimir Malevich.

Black square by Kazimir Malevich

So, she's using abstract mathematical forms and ideas, and, quite literally, making them concrete:

"When I came to do architecture people said you must know how to add. There is that aspect to maths, of course. But there is another that was of interest to me and that was abstract thinking, and that was when I realised how important that degree was."

Zaha Hadid's design for the Mathematics Gallery

To me, that's thought-provoking.