In our number of the day sessions (which we haven't done recently but various students keep asking me that we do again) we evolved to writing the number of tens in the day number as one way of saying something about the number. 94=9x10+4. Most of the students seemed comfortable with this.

When we used letter names for Cuisenaire rods (which we stopped doing about half way through the year), it was natural to write 2r=p for 'two red rods are the same length as a pink rod'. Notice how that hardly seems like multiplication at all; more like counting.

I'm intrigued by how this seems to be easy for young children. Multiplication is meant to come later. What is going on?

The answer seems to be that, after the children have got to know the rods, they work as "units". I've just been reading Christopher Danielson's excellent Teacher's Guide to his wonderful

If we used cubes instead:

there are other units involved, units we could count. There are maybe four lots of counting going on at the same time. Counting how many cubes in the red two. Counting how many cubes in the brown two. Counting how many twos there are. And counting how many cubes there are in the four. It's too crowded; too much counting at the same time. We all know how patting yourself on the head and drawing circles on your stomach at the same time is a lot more than two times harder than doing them separately.

But, when through familiarity we've got used to say how the yellow rod can be seen as "five" (being five of the smallest white rod in length), it begins to behave as a unit. So, if we have four of them:

When we used letter names for Cuisenaire rods (which we stopped doing about half way through the year), it was natural to write 2r=p for 'two red rods are the same length as a pink rod'. Notice how that hardly seems like multiplication at all; more like counting.

2r=p |

I'm intrigued by how this seems to be easy for young children. Multiplication is meant to come later. What is going on?

The answer seems to be that, after the children have got to know the rods, they work as "units". I've just been reading Christopher Danielson's excellent Teacher's Guide to his wonderful

*How Many?*and it's been helping me to think about this idea of a unit.there are other units involved, units we could count. There are maybe four lots of counting going on at the same time. Counting how many cubes in the red two. Counting how many cubes in the brown two. Counting how many twos there are. And counting how many cubes there are in the four. It's too crowded; too much counting at the same time. We all know how patting yourself on the head and drawing circles on your stomach at the same time is a lot more than two times harder than doing them separately.

But, when through familiarity we've got used to say how the yellow rod can be seen as "five" (being five of the smallest white rod in length), it begins to behave as a unit. So, if we have four of them:

it's like having four apples. That the five is composed of five ones is in the background, not interfering with our count.

Which is my guess about why everyone in K3 (5 ad 6 year olds) was able to think about multiplication in this lesson (we call it 'times'; every now and then I throw in a "lots of"):

There's an album of these photos. (The tracks are available from Numicon. They were developed by Tony Wing, starting from the work of Catherine Stern.)

I'd be interested in your thoughts.

I'd be interested in your thoughts.

Love this so much. Some of the benefit has to be exactly what's cumbersome to us. The rapid accumulating length of the multiplicand. Multiplier and multiplicand is so clear, and the rearrangement into arrays is so powerful. Plus I think it makes the properties of multiplication make physical sense. In array, of course axb=bxa. But look how amazing it is when 5 reds is the same as 2 yellows.

ReplyDeleteQuite a few students have observed at various points that axb=bxa, and even suggested it's always true. We haven't put them into an array yet, which for me is a wonderful proof.

DeleteThis seems so close to algebra! I wish I'd had been introduced to maths through cuisinaire rods and discussions about numbers, multiplication and how to think about abstract ideas.

ReplyDeleteIt is a great way to learn - there's no scary cliffs to climb or descend, no getting 'left behind', no 'not understanding'...

DeleteThanks for sharing this Simon. It certainly shows the power of the rod's again. You hit on some really key ideas, the students ability to unitze is really developing quickly in your class. That fact that the rods are already unitized is helpful but they can also choose to verify by using the ones (white) cube. I think what you said about comparing it to using snap cubes is key, on the snap cubes it's easy for them to verify and revert to counting where with the rods they have to think in units or have to actually go and get a white cube and line them up to verify which still gives some students that option that may still need it. The second thing I think you really make clear is the use of the language of "piles of" or "groups of". I always think of Kathy Richardson's book where she speaks to using language like that for younger students and avoiding saying multiplication which lets it develop more organically before they hear that word that often causes setbacks because they think multiplication is so much more difficult! Its the mindset piece!

ReplyDeleteThat thing about language is key isn't it? Using a new term at the start can say, 'Oh, this thing is so different that we're going to need to learn a whole new word for it.' Which, however much some students will enjoy a new word, sets up a barrier for some where there doesn't need to be one. Generally, I go for learn the thing first, and learn the word later. Or I might throw the word in casually here and there without drawing much attention to it, like we would normally.

DeleteIn case other readers of this blog post are interested, https://constantinides.net/2018/06/08/structures-in-arithmetic-teaching-tools/ also discusses algebraic structures present in Cuisenaire - the comments section draws out a further point from Simon's interesting post.

ReplyDelete