"a dialogue with students which involves meta cognition (thinking about their own thinking) and meta-conceptions (students thinking about their own knowledge and understanding of concepts)"earlier posts:

meta-dialogues are hard to establish

initiating a meta-dialogue

In my last blog I said:

What happens next? Focus Question: What do the other students find hard about fractions?I did ask that question of the class and did get some response (eg. they find ratio harder than fractions) but the response wasn't sufficient to reach the point where my students felt they could design their own fraction question. My students had no intention of approaching this analytically like a teacher might, eg. go through the fractions test, find out which questions the other students got wrong, make generalisations about what they knew and didn't know and devise an intervention to bridge that knowledge gap, etc. No way!

I should explain. I have a relatively small class of year 8 "specials", a mixture of some students with learning disabilities (various), some with behaviour issues and others who are "normal" (whatever that means).

Fortunately, I had thought about this potential road block beforehand and had another question ready to facilitate my goal:

Can anyone see any fractions in the room?A few students could immediately see other fractions in the room and I wrote their responses on the white board. This included what fraction of the class were boys, what fraction were girls. Then one girl came out the front and demanded the white board marker and she wrote on the board, "1/12 is the old guy" (!!!) Of course, the class loved it and this became the exemplar I could use to illustrate how they could write their own fraction problem. We did this collectively on the white board.

The idea of fractions in real life became for me the way out of the dry pages of the maths textbook into something richer and more meaningful.

Next lesson I offered the class a deal. We would go outside for a walk provided that each one student agreed beforehand to find a fraction outside and come and tell me about it. This worked a treat and developed into questions like this:

- What is the ratio between the seats and the bin?
- What is the fraction of red cars?
- What is the ratio between the goalie and the players?
- What fraction of boys is shorter than xxx?
- What fraction of the class is wearing glasses?
- What fraction of the class took off their jackets?, etc.

I have slowed down the amount of content being covered. This is a central point. To go deeper (meta cognition) you have to find a way to go slower.

## 3 comments:

I was inspired by what you said here to come up with a different problem: Figuring out the fractions after something has been constructed. This is commonly referred to as "parts". This might be too tough for your students at this stage, but I was thinking of a recipe. Even as an adult it took a bit of thinking for me to get this. In recipes you see things like "2 parts flour to one part butter". What I realized is this represents a ratio--2:1. What's probably hard is it involves translating one thing into something else:

If you have 2:1, what is the whole? The answer is 2+1=3, or (2/3) + (1/3) = 3/3 = 1. So the "2" part is 2/3rds, and the "1" part is 1/3rd of the whole.

If you put in 2 cups of flour, they now have to have a mental model of "2 cups = 2/3rds of the whole". I've made this easier. It would be more challenging if it was just 1 cup of flour. Then they have to ask the question, "If I have to use the cup measure to fill in the remaining 1/3rd, how many cups, or how much of a cup, should I use?" This leads into what I remember covering in algebra, called proportions: 2 cups/(2/3 whole) = (x cup(s))/(1/3 whole)

It wouldn't have to be represented this way, of course.

What would be a "grand problem" after this is actually going through and making a recipe (like cookies or something), and then analyzing the recipe, like, "Okay, what fraction of the cookies is flour?", "What fraction is butter?", etc. Again, this might be too hard, because you'd have to get into units and conversion of units, but it would really delve into the issue, wouldn't it? I think if they could tackle this then they could really handle fractions.

hi mark,

... and I'm inspired by your comment, which has given me more ideas to pursue with my students

The whole idea of kitchen maths or kitchen fractions is a good path to pursue

btw you reminded me that Seymour Papert has a very interesting section about Kitchen Math in his book,

The Children's MachineOne issue that arises from what you say is the concept of a "whole", what is a whole? It emerges from your 2:1 ratio example that a whole, in this case,consists of 3 parts. That's an issue that I'll need to discuss with my students because the "normal" way is to take a pie shape or some such and then divide it into three parts. But if you start off with a whole of 2 pies then it's a different problem.

I plan to run some lessons now using Kitchen Maths and hope to blog about what happens.

Thanks for the great blog!!! I'm always looking for inspiration. (Even after 20 years of teaching.) I've had students actually make something to eat by dividing a recipe in 2. I still have students who come back with memories of putting in 1/4th a cup of salt because they didn't pay attention to the tsp. Real life makes a difference. It's just really hard to find connections when looking at the standards all the time.

M. Townsley

Rio del Valle J.H.S.

Oxnard, CA

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