Monday, February 12, 2018

Creating Thinking Classrooms (1)

I have been reflecting a lot on Peter Liljedahl's work in the past few months and have been more intentional about implementing his ideas in any of the classrooms I go into. In particular, I am trying to gauge the impact of using three of his optimal practices:

  • start with good questions;
  • use vertical non-permanent surfaces (VNPSs);
  • and use visible random groups.

This week, I went into a grade 8 class who had been working on measurement. I began the lesson by showing them a game called 'Prism or No Prism!' which involves me holding up a shape and the class deciding if it is a prism or not. For the most part, they were correct but about half the class said that a cube was not a prism. When I asked why, they said because it is a cube! As they weren't too clear about what a prism is, I shared with them my 'loaf of bread' analogy:
If a shape can be sliced like a loaf of bread from front to back and give exactly the same size and shape slice, then it is a prism.
"So that would make a cube a square-based prism then!" said one student.
Next I wanted to ascertain that they knew how to get the volume of a prism. The 'loaf of bread' analogy works well here to as we can connect it to layers (or slices) that can be made thinner which leads us to develop the idea that the volume of a prism is the area of one 'slice' multiplied by the 'number of slices' into the more generalised formula, V=Axh
In all of these discussions, we did not look at cylinders.
I then showed them the opening act of Dan Meyer's Popcorn Picker:
I asked "What do you notice? What do you wonder?" Some wondered if one cylinder would give more popcorn than the other cylinder. Others reckoned that the cylinders would give equal amounts of popcorn. So the task was set:
Decide which way you want to make your cylinder. It will then be filled with popcorn!
I used playing cards to create visibly random groups of three students each and then gave each group one marker each and had them work at VNPSs.
The students got stuck into the task immediately, even though they have never been shown the formula for the volume of a cylinder. Whilst there was the occasional dead end (one group got stuck on using V=lxwxh before realising that this wouldn't work with a cylinder!), the students soon reasoned that since the cylinder is a prism, they could work out the area of the base circle and multiply this by the height for each cylinder. Getting the area of the circle requires knowing the radius and some did this by direct measurement whilst others measured (more easily I'd suggest) the circumference of the circle (that is, of course, one of the sides of the rectangle and then divided this by 2π to get the radius.

One of the great things about VNPSs is that as a teacher, it makes it easier for me to see what students are thinking and any errors that they might make.
When we were satisfied that the students had reached a conclusion, we noticed that seven groups opted for the shorter, wider cylinder and one group opted for the taller, narrower one. I filled one of each of these cylinders and, by then emptying the popcorn on the table, we could see that visually most groups had got it correct. It turned out that the group that didn't had the right idea but made a calculation error.
With the students merrily munching on popcorn, I was able to summarise the lesson by using their work on the VNPSs around the room and got them to tell me the formula of a cylinder:
The use of good questions, VNPSs, and visible random groups certainly proved effective in getting these grade 8s thinking. I wondered how it would be for younger students.

Thursday, January 11, 2018

Precious Pentominoes

Here is a task that I made up that I have recently tried which provoked a lot of thinking with students and adults alike.

From a set of 12 pentominoes, choose two tiles to make a closed shape that has symmetry. For example these are NOT allowed:
Once you have a design, then work out the perimeter:
the area:
and the number of sides:
Now calculate a 'cost' for your design as follows:
What is the most 'expensive' design you can make?
The students got stuck into this immediately using the pentominoes to create a variety of designs:
They recorded their thinking as they went:


After a while, we recorded the costs that students had found on the board. Now the students were really keen on finding a more expensive design. There were shouts and screams of delight as groups found new, more expensive designs. After a while, all groups had reached a consensus of what the more expensive design was (I won't spoil it for you by revealing the answer!)

As there was still time left, we extended this by allowing three pentominoes and removing the symmetry constraint. 

As we wrapped up, we asked them what strategies they used. They noticed that the area was always the same but that they had to take care when calculating the perimeter and the number of sides to avoid 'double' counting:

They also noticed that it was important to maximise the number of sides as this was the multiplying factor. So long, 'straight' pentominoes were not as good as the pentomino shaped as a cross.
But perhaps the best comment was as I was leaving:
"Please come again soon and please bring some more questions that will give me a mathematical headache!"

We also tried this with our educators at a Capacity Building session today. Again, the level of engagement was high. We tried also bringing some of the ideas from Peter Liljedahl's research about using vertical non-permanent surfaces, giving verbal instructions, creating visible random groups, and only answering 'keep thinking questions'.

This is when one knows that it is a good task: that adults and students alike are captivated by it.

Wednesday, December 20, 2017

A Nice Number Sense Building Puzzle

Today I was at a school and before going to work with some kindergarten classes, I had a spare 20 minutes to fill. I went into a Grade 4 class to challenge them with this puzzle that we shared on our @DCDSBMath Twitter feed from a few weeks ago:
The students were given whiteboards and told to get cracking! 
And get cracking they did. What I found interesting at first was that they all worked from the top down: they split the 20 into two numbers and worked from there. When I have given this puzzle to adults, I have seen them work from the bottom row upwards. I'm not sure what to make of this!
Initially, some students hadn't noticed that they need to use four different numbers on the bottom row:

A quick reminder of what was required on the bottom row got them back on track. Excitement was evident as the students eagerly showed me a solution. I often found myself saying "Great! I've not seen that one yet. Now find a different one."

There were some mistakes on the way, but these were mainly caught by the students and, because we were using the whiteboards, these were corrected with minimum fuss. I was particularly impressed with these solutions:

I wasn't expecting to see negative numbers being used without prompting!
What I liked about this task is how quickly the students took to it and how much it provoked their thinking. At the same time it got them to use their number sense especially to decompose numbers in different ways. Next time, I would use a larger target value in the top brick to allow them to practice their mental arithmetic for adding and subtracting two-digit numbers.

Monday, November 20, 2017

Using Lego to Build Fractions

Having seen the occasional tweet about using Lego to teach fractions, I was curious as to how effective this might be. Today I tried this for the first time with some Grade 4 students. These are students who have not yet learned about equivalence but have learned about unit fractions and how we can use this knowledge to label fractions.
I worked in the school's Learning Commons and made use of the Lego board that was placed on the wall.
Each pair of students had there own board and a tub of Lego blocks to work with. I started by asking them to find a 2-by-4 block and to fix that to their board. Then I asked students to find a block, or blocks, that was one-half of the original block. Some chose a 2-by-2 block, others chose two 2-by-2 blocks. We also had students use a 1-by-4 block or four 1-by-1 blocks.
In a similar way, I asked students to show me what one-fourth (or one-quarter) might look like. As I listened to the students think there way through this, it became clear that even though they might know the fraction words 'half' or 'fourth', that they did not necessarily know how this connected to the number of parts needed to make a whole. Indeed, when we tried to do some skip counting by fourths as a class, I could tell that there was some uncertainty. To clarify this, I gave them the following challenge:

If a 2-by-4 block is one whole, build seven-halves and tell me another way to say this.

As they did this, I moved from group to group and modelled some skip counting aloud with them:
"One half, two halves, three halves, four halves, five halves, six halves, seven halves."
Now we could use a 2-by-4 block to show that this was equivalent to 3 and one-half:
The students seemed to like this visual 'proof'.
The next challenge was to build twelve-halves and to find what this was equivalent to. The students were able to do this more quickly now and tell me the correct answer. If some students put all their blocks together like this:
we suggested that that their work might be more clear if they leave a gap between each whole:
Finally, we asked them to build six-fourths and to find out what this is equivalent to. Again, they were able to build this quickly:
Some students argued that this was one-and-one-half. Others argued that it was the same as one-and-two-quarters. Then one student suggested that it this meant that one-half was the same as two-quarters.
I seized on this idea: "Who agrees with this suggestion?" 
Everyone did. Unfortunately, the bell rang for recess but it gave me plenty to think about what happened in the lesson and what the students are now ready for. I liked working with the Lego mainly because it was easy for the students to organise their thinking: once the blocks were placed, they stayed put and didn't get knocked all over the place. In fact, I could pick up one student's work and easily show the whole group. 
One drawback of using Lego is that the fractions that you can use are somewhat limited, so I would have to think more carefully about what models and blocks I could use to show thirds, fifths, sixths, tenths etc.

I actually did a similar activity at a Parent Council meeting for one our elementary schools last week but with pattern blocks. I showed the parents a hexagon and asked them to show me what one-third of this was. After they confirmed it was the blue rhombus, I challenged them to build eight-thirds and to then tell me another way of saying this. Once they had lined up eight of these rhombii, they carefully arranged six of these into two hexagons and were able to tell me that eight-thirds is equivalent to 2 and two-thirds. A number of parents actually said "That's why it works!" It was a perfect moment to show them the power of the Concrete-Diagrammatic-Symbolic continuum and how we can use this to develop students' understanding of fractions:

My overall learning of this is then as follows:
1) Equivalence is a key concept in fractional understanding. Without it, fractional computations are built on shaky foundations. Students need to develop this knowledge of equivalence initially through concrete activities before moving on to diagrammatic and thence symbolic activities.

2) Skip counting with fractions is an important prerequisite for developing an understanding of equivalence. Again, this should be developed concretely first before moving on to diagrammatically (number lines) and then symbolically.

Wednesday, October 25, 2017

Exploding Dots and Math Bumps

Last week saw the end of the first ever Global Math Week and what a successful week it was: over 2 million teachers and students took part! I am proud to have been an ambassador for this and to have brought the joy of exploding dots to many teachers and students. If you haven't heard about James Tanton's Exploding Dots then you need to check out this site. Basically, exploding dots are a way of visualising a journey of mathematical ideas from primary to senior grades.
In preparation for Global Math Week, my colleague, Dan Allen, and I held a number of sessions for interested teachers to introduce them to the idea of exploding dots and how they could incorporate this into one of their lessons that week:
For Global Math Week itself, I went into Grade 2, 3, 4, and 7 classes. The students really liked the idea that they were part of a worldwide event and that they were solving the same problems as students in Australia, China, India, Germany, Tanzania or wherever they had friends and relatives.
I started with simple 2 to 1 and 3 to 1 machines. Here, a grade 4 student is writing 7 using a 2 to 1 machine. Listen to all the kabooms happening in this clip:
We quickly learned that we can't use our normal number words to describe our results so instead of saying "One hundred eleven" we said "one, one, one". Then we used a 10 to 1 machine and found out that twenty-three could be written as...23! Here, I could explain to students that nearly all of the math they have learned so far has been in a 10 to 1 machine so in this case we could use our number words 'twenty three'. 
For the Grade 4 and 7 students, I then tried a 3 to 2 machine. 
Kabooms galore! It was neat to see the students taking care to make sure that they did the explosions correctly and checking with each other to see if they got the same result. Where they were discrepancies, they sought to convince each other of the correct answer. 
Finally for the Grade 7s, I tried an operation with them using a 3 to 1 machine:
This they did with no further instructions from me:
From a personal point of view, there is something about Exploding Dots that brings out a beauty I'd never considered in polynomial division. When Sunil Singh first introduced us to exploding dots last year, he challenged us to do 1÷(1–x) and 1÷(1-x^2) using this method. Even though I knew how to do these using more conventional methods, I was gobsmacked by the visuals produced:
I tweeted my excitement to James Tanton who then sent me another challenge. When I got stuck into this, something so surprising and wonderful happened, that I experienced what can only be described as 'Math Bumps':
Joyous maths indeed. 
My sincere thanks go to James Tanton and all at the Global Math Project for helping to spread joyous math everywhere.

Monday, September 25, 2017

Fun for All the Family

I'm always on the look out for math problems that can be used with people of any age. These puzzles are usually easy to explain but not so easy (or obvious) to solve. Here is one that I recently created and posted on our board's Math Twitter account (@DCDSBMath) that has proved popular with kids and adults alike.
To clarify: you can use any number of pieces to make a rectangle so right from the get go it has multiple entry points.
I gave this task to our principals and Math Lead teachers this week and as they worked through it, I was struck by the energy in the room. First two pieces were put together to make rectangles

then three:

and then the question was asked, "Is it possible to use all six pieces?" I put on my best enigmatic smile and said, "Perhaps!" With no immediate solution obvious, these adults had to persist at trying different arrangements.  Snippets of conversations from each table showed some common ground to the thinking going on:
"Are we allowed to flip the shapes?"
"How big must the rectangle be?"
"Either 6 by 4 or 3 by 8."
Then, around the room, shouts of delight went up and high fives abounded as different solutions were found.

When I made this puzzle, I did so knowing that this solution could yield a second solution.

As we moved from table to table airplaying the solutions, I was pleasantly surprised to see some I had not thought about. These two are a nice variation of each other:

As are these two:

But this one really toasted my crumpet:

Which of course begs the question: Can we find all the solutions? I'll leave that up to the reader to figure out.
A question arose during the session as to how to make this task more accessible to students who might have a visual-processing LD. One possible accommodation might be to colour-code the pieces and provide a template for them like this:

What I like about this problem is that not only is it great for developing spatial reasoning, it is one that can be attempted by adults and children alike: it is a problem that would be great for families to do together! And if you like problems such as this, then you will also love games such as Blokus and Kanoodle.
Or for those of you of a certain age, Tetris!