I ♥ Arrays

Arrays are brilliant.
I'll say that again.
Arrays are brilliant. I love the fact that I can talk about them in a kindergarten class as well as in a Grade 12 class. Arrays allow me as a teacher to help students connect ideas and concepts especially when it comes to multiplicative thinking. Yet I had no real experience of arrays at school but I wish I did; it would have given me a much deeper understanding of some important algebraic concepts.
 
So I was in a Grade 5 class this week. The teacher has been working very hard to address any gaps in the students' additive thinking and was now pretty sure that they were ready for some work on multiplication We wanted to see how they would solve a fairly routine equal groups problem so gave them the following:

A baking tray holds 15 donuts. How many donuts would there be on 13 trays?
 
Now one or two attempted to use an algorithm but were making lots of procedural errors:

Others relied on sketching out the problem:

In the case above the thirteen trays drawn have 10 donuts giving 130. The student then reasoned that the remaining five donuts per tray would give another 65 thus giving 195 in total.
Others used some really neat student-generated procedures:

What was important was that the students sensed that they had to do 15×13. They had also done some work using arrays to represent 6×5, 4×3 etc. So I said that suppose we had to do 16×11 then we would get an array like:

 

By splitting the 16 and 11 into a friendlier 10 and 6, and 10 and 1 the students were able to tell me quickly how much was in each of the four quadrants:

... and from there tell me that the answer was 176 (some added in their head, others wrote their work on paper). We then returned to the donuts question to see how we could use the array to help us here but I pointed out that I didn't want to draw out all the donuts as it would be too time-consuming!

"Wow, that's so much quicker," said one student and many agreed. We consolidated by trying another problem (essentially 17×15) and here is a sample of work:



I love the way that there are no place value lies in this method and that it encourages good number sense. I have seen some students develop this array method into a partial products method:



...which is neat but to me what is even neater is when I can use the array method to explain what happens when you multiply polynomials. When I first came to Canada I learned a new acronym: FOIL. I found a lot of students misunderstood this (it stands for First, Outside, Inside, Last) but they had much greater success when they drew an array:

And when there are more terms in the polynomials it becomes easier to collect the like terms as they lie on the diagonals:

I have shared this method with parents at Math Evenings and it is always neat to see their faces when I give these examples: they finally understand why it works!
I could go on about how arrays make it easier to understand what it means to 'complete the square' or how they can be used to do polynomial division (see James Tanton's excellent video on this).
But for today, it was rewarding to see how quickly the students took to this method and how it fitted in so nicely with their existing knowledge.

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: http://todaysmeet.com/

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?

Super Bear



So it's the last week before the holidays and we're in a Grade 8 class hoping to give them something engaging. Dan Meyer's Superbear task came to mind. We also wanted to take advantage of the fact that this class has a set of iPads at their disposal.
But we were also curious: if we showed the students the above video, what sort of questions would come to mind? Well why not use a backchannel to collate these questions? Having recently seen TodaysMeet in action we decided to give this a whirl.
Now I won't lie to you; I was a touch apprehensive about how the students might use (or misuse) this tool. I have seen adults get a bit silly when using it. Royan Lee (who met with our department last week) gave some good advice: Let them use the tool and let them make the mistakes. And when they do make a mistake then it's an opportunity for learning.
I needn't have worried. Yes, there was a lot of "Wassup?" and "Yo, I'm so hungry I wanna eat those bears" comments but nothing harmful. More importantly, there was a lot of really good questions:

We then generalised these into one question:

How many of the regular bears are equivalent to the superbear?

 

We were careful to avoid all reference to any attributes such as mass as we wondered how the students would approach this. Some chose to use volume:

 Some chose to use mass:

Others used the internet to help them get the relevant information:

 A couple of groups even considered calories:

What was neat was that by giving them no information in terms of measurements they had to be very active in thinking what measurements they actually needed. How they then got these and how they then used these was very impressive especially in terms of showing their proportional reasoning.
We watched the Act 3 from Dan Meyer's site and there was some disappointment from some students that their answers differed from his but we emphasised that he was using his bears whilst we were using ours!
Finally we asked the students to reflect on how they approached the problem today:

What probably made this lesson work so well was the lack of information that we gave the students: it forced them to think what question needed to be answered and from there how to answer it.
After all, part of being a good mathematician is not so much answering questions, but asking the questions in the first place.

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Dan Meyer's Three Act Math can be found here: http://blog.mrmeyer.com/

TodaysMeet can be found here: http://todaysmeet.com/