Thursday, January 31, 2013

Now That's What I Call Feedback

I am slowly but surely working my way through James Joyce's Ulysses (my goal is to finish reading it before the summer!). I find the whole stream-of-consciousness technique fascinating (if at times confusing). Today I was in a Grade 6 class and I was suddenly reminded of Ulysses when we used TodaysMeet and saw the stream-of-consciousness of the students.
The students had been working on probability so we asked them this question:

In this can, there are between 10 and 30 cubes. A third of them are blue. What could be in the can?

This is what was in the can:
What followed blew us away. The class had a set of iPads and the students began posting their solutions immediately.
Or their questions.
Or their revised solutions.
Or advice for other students.
Or requests for help.
As a teacher I could see who needed help just be looking at the feed on TodaysMeet. This stream-of-consciousness eventually ran to over 27 pages! Below is just a sample of what was going on. As you read through it, see if you can link all the conversations.

 Isn't all that feedback just wonderful? It was fantastic to see students not being afraid to say if they are stuck and ask questions and other students helping them and sharing ideas. At this point we gave them some additional info. We told them that there were just two colours, blue and green, and that there were between 10 and 20 cubes (not 10 and 30). This really caused problems for some students:

Yes, some students did get stuck but there was so much feedback available that they overcame these difficulties:

Some used blocks to help their thinking:

At this point we asked the class for all the possible solutions. They told us that you could have A) 4 blue and 8 green, B) 5 blue and 10 green, or C) 6 blue and 12 green. What really pleased us is how they were able to reason why these were the only possible solutions (e.g. because these are the only totals between 10 and 20 that are multiples of 3). 
We then gave them one final piece of info: the total number of  cubes was odd. They then had to vote on what was in the can based on A, B, or C as detailed above:

We were pretty chuffed to see someone reason so clearly here!
At this point we wanted to know what students thought of the whole experience. Dave, the classroom teacher, reminded them that we were looking for descriptive feedback. This is what we got. Then picture the smiles on our faces.

So if this the richness in thinking that is potentially there, can you imagine what a disservice is done when students are asked to simply copy a note? In silence?
The final say goes to this student who summed up how we all felt today:
Hooray indeed.
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TodaysMeet can be found here:

Wednesday, January 23, 2013

Representing Patterns

I reckon that the ability to represent ideas in maths is something that most people underestimate or are even unaware of. Yet it is a crucial tool in a mathematician's backpack: it will help him or her gain a deeper understanding by thinking of a concept in a variety of ways. But it is a skill that I saw in action in a Grade 1/2 split class this week.
I showed students the pattern I created below and asked them to create a similar pattern using the shapes at their disposal:

All students came up with something like this. Some where initially concerned that the colours didn't match but convinced themselves that it was OK as it was still square, triangle, square, triangle. A lot of students also extended the pattern without being prompted to. 

They were then asked to represent this pattern without using squares or rectangles and we got something like this from all the groups:

Then we narrowed the attributes and said that the had to represent the pattern again but this time only use one shape. I wondered if they would find this tricky but over half the students managed to get things like:

We allowed students to go on a scouting mission to see other students' solutions. The students who got stuck really benefited from getting the immediate peer feedback.
I then asked them to tell me how they could represent this pattern not with shapes but letters and they quickly gave me some examples.

We then agreed to call these patterns AB patterns.
But I wanted to push them further so I gave them some red counters and asked them to make an AB pattern with the red counters. Sure enough, many groups came up with something like:
We then asked the students to make an AB pattern using themselves. Firstly the organised themselves into boy, girl, boy, girl. Then after a little more thought, came up with stand, crouch, stand, crouch.
I then asked them to represent the AB pattern using sounds (Maths and music are so connected!) and this is something they really enjoyed:
By the way, I like to tell students that I call an AAB pattern 'We Will Rock You'!
And here's the thing which really toasts my crumpet. At the end of the lesson the teacher told me that the students who were the most successful today were her 'weakest' students!
Did I tell you how much I love my job?