Wednesday, 7 June 2017

Form: What is a cubic centimetre like?

Using the PYP key concepts are a fabulous tool for helping children delve deeper into mathematical concepts.

Today we explored the cubic centimetre.

Looking at the PYP key concepts, we thought about which might be the most useful to use first to help us gain a deep understanding of cubic centimetres. Some of is thought 'connection' and explained why they thought so. Others suggested we use 'function' and a few felt perhaps 'causation'  Most of those thought 'form' would be the most useful.  After hearing everyone's reasons, we decided to use 'form' today.

Looking at a plastic cubic centimetre, we used the think-pair-share routine to explore it.

Eventually, our class sharing looked like this:

Though a simple learning experience, it generated a lot of interesting discussions. There was some debate over whether it has to be a cube in shape. To help, we made a cube that was 1 cubic centimetre.  We then squished it down.  Is it taking up the same amount of space? - Yes.  So, what does this tell us? - It is still 1 cubic centimetre because it is the same amount of space; therfore a cubic centimetre does not need to be a cube in shape.

We then found objects in the room and estimated and then measured their volume using cubic centimetre cubes.

We can sometimes overlook the key concept 'form' as being a bit basic. However, when we encourage children to really stretch their minds whilst using it, great understandings and wonderings can be explored. 

To read more about using the key concepts, click the link below:

The Power of Using the PYP Key Concepts in Maths

Tuesday, 6 June 2017

Creative Volume Problem Solving

Giving children opportunities to problem solve creatively is a key element of mathematical thinking we need to be constantly valuing in our classrooms rather than 'getting answers'.

To help with this, partners were given a thesaurus and an atlas to examine and were asked: 

Which do we estimate takes up the most amount of space? 

(I didn't want to use the term 'volume' just yet to allow the children to gain a better understanding of what volume actually is and I had chosen two books which looked like they had a pretty similar volume)

Without using any resources, rich discussions took place as partners discussed and created estimating strategies.  They then individually recorded their thoughts using the sentence starter:

I estimate that the thesaurus / atlas takes up more space because.......

Giving children sentence starters like this, I think, helps them develop stronger communication skills and also gives them an opportunity to deepen their own thinking without being influenced by others.

After they wrote their thoughts, we discussed together.

Firstly, we showed hands how many of us thought the thesaurus took up more space (4), the atlas (11) and the same amount (5).


Students were invited to share why they thought so and some very creative thinking and strategies emerged.

We first heard from those we thought the thesaurus takes up more space:

- I visualised smooshing the book down as if it was playdough and I could see it taking up more space than the atlas.

- I can see the thesaurus is much thicker so I can sense it takes up more space.

- I visualised taking out all the pages of both books and lining those pages up side by side. I could see that the pages of the thesaurus would be much longer so that makes me think it takes up more space.

We then heard from some students who thought the atlas takes up more space:

- I imagined cutting the atlas in half and stacking both halves on top of each other. By doing that I could visualise it being thicker and so taking up more space. 

- I drew a dot at the halfway mark of pages on the side of the thesaurus. I then placed the atlas beside it and imagined doubling it. By doing that I could visualise it taking up more space.

- I opened the thesaurus up at exactly halfway and placed it spread out on top of the atlas.  I then looked at them at table level and could see clearly that the atlas took up more space.  

- When I placed the thesaurauas on top of the atlas, I could see it is almost half the area size of it. So, I visualised doubling that and then tried to estimate the thickness of both to see which might take more space. I'm not exactly sure, but I think I can see the atlas taking up more space.

We then heard from those who thought they took up an equal amount of space.
They had used similar estimating strategies and some had thought about estimating the width and length to mentally calculate which took up more space. That was interesting to me because up till then no one had suggested the books being rectangular prisms nor formulas for measuring the volume of those shapes.

I then asked, what mathematical concept are we using when we measure how much space an object takes up?

A few of us shared how it is volume.

What things in our room have volume?

- chairs, tables, our bodies etc were shared.

What about the air in our room? Does it have volume? 

That question raised a lot of different opinions. When asked why we thought so, someone shared how a balloon gives us evidence that air has volume. 

Another shared how fire 'breathes in' oxygen and when there is no oxygen left in a room a fire goes out so that must prove air has volume because when there is no more of it, it has an impact on fire. Amazing thinking! :)

Another wondered, but what about in space? There is no oxygen there so does space have no volume?

We wondered about that..........

We then began investigating:

How many different strategies can you create to measure the volume of each book?

The 'how many' part of investigations really opens up possibilities for children to stretch their minds.  We are not valuing an answer; instead we are are valuing creative thinking.

We discussed how creative thinking is a key element in maths and that it doesn't matter in this learning experience if the strategy is effective or not. What we are doing is thinking creatively.

Partners were given string, MAB units, longs and flats as well as being able to use rulers or any other object in our room to explore the measuring.

Loads of very impressive creative strategies emerged.....

Some students counted the number of pages inside both books and did calculations to try to find out which had the most number of pages and then doubled / halved the number to see if it would help them measure their volumes.

One student found a connection between numbers and others were discussing the difference in volume they felt they had measured between the two books. 

Some cut the string to use as a measuring tool when comparing.

 This student thought about whether if he rolled a bottle once, then placed the measurement using a pencil and then rolled it again to measure whether it would help him measure the volume of both books or not.  He wasn't sure if it would help, but his idea intrigued many of his classmates and me.

Lots of very rich mathematical theories, reasonings and knowledge were being shared whilst partners created different strategies.

We did a gallery walk hearing the different strategies and were amazed that the incredibly creative ideas that emerged.

I think this was a pretty successful investigation and introduction to volume which we will expand upon in further learning experiences.  

Monday, 13 March 2017

Inquiring into the Square Metre

Inquiring into the Square Metre

There is a common misconception that a square metre has to be square in shaped.

When children are introduced to square centimetres and square metres, its easy to understand how this misconception grows.I wonder though if it prevents children from being able to deepen their conceptual understanding of surface area.

I wonder if by constantly having children use square cm and square m as a unit of measurement, are we denying them from really understanding how surface area is a measurement of space rather than a measurement of number of square cm etc.

To help address this misconception and to deepen our conceptual, understanding of surface area, the children entered and saw a piece of paper on the floor surrounding by metre rulers:

We did a 'silent gallery observation' where children came up close and observed what they saw and then returned to their see-think-wonder paper to record their thinking. They could keep returning to the display as much as they wanted:
After recording, the children shared their ideas with their table partners and then we shared and discussed some of these as a class:

We thought the first wondering was interesting, so in pairs we found ways to try to find out how many square cm equals 1 square metre.

We had quite a few different ideas and strategies.

When students shared, one in particular stood out where she showed how we measure the area of a rectangle:

She then used this explanation to explain how we can find out using the same strategy:

We thought 10 000 was a lot, but also interestingly, when we sat around the square metre paper again, a few of those commented that it seemed there were a lot more than 10 000 fitting inside it.  I had to agree.

I then posed the question: Does a square metre have to be square in shape?

We wrote our theories on a post it and then posted them on the line (above).

We noticed how it was almost 50-50.

We heard from some people who thought it had to be a square and their reasonings.

We then heard from others who were in the middle and finally we heard from those who thought it doesn't have to be square in shape.

To help with the debate that took place, I then cut part of the square metre paper.

I then moved that cut part and joined it to another part of the paper.

What shape have I created?

- A decagon.

Is this area still 1 square metre?

We had a stand off again with a 50-50 opinion.

We heard some ideas and their reasonings.

One student then remarked how it is still 10 000 square centimetres and that's why it is still 1 square metre.


Light bulbs started flashing all around.

Another then remarked,so if think whether the surface area is 10 000 square cm, then we will know if it is a square metre.

We all agreed.

Estimating is a key skill we should always encourage children to do in maths. When we estimate numbers etc, we are gaining a deeper sense of the concept.

We then did a hunt for objects in our room that we estimated had an area less than, approximately equal to, and greater than a square metre.

After gaining a better sense of the size of a square metre, we then thought about how we could create different shapes that had a surface area of 1 square metre.

Children partnered up with others and sketched possible ideas.  Loads of rich reasoning discussions and creative ideas took place.

Some samples:

We partners felt confident in their measurements, they then used masking tape to create their 1 square metre shapes.  We didn't have time to finish these, but when we do, we will then examine the shapes created and see if they are or are not square metre in area.

This pair had sketched a square metre and then split it in half. The then joined the two triangles and worked out the lengths:

Some amazingly creative shapes are been explored and we are all excited about making and sharing them tomorrow......

Sunday, 5 March 2017

Valuing Creative Thinking in Maths

What message do we want to impress upon children of what maths is about?

When our children leave our year level, what would we like them to feel mathematical thinking is about?  How would we like their perspectives on maths to have grown, deepened and changed by the time they move up a year level at the end of the academic year?

This year, one of my goals is to help the children in my class to appreciate the wonders of number relationships and connections. That our number system isn't something to fear, but to play and make discoveries about. I also hope to impress the idea that mathematical thinking isn't about getting an answer. An answer is merely a byproduct of the rich process we can take our minds when creating and evaluating strategies. 

To help deepen our understanding that maths is about creative thinking, today we looked at the following context problem:

Stuffed with PizzaFile:Supreme pizza.png

Andrés and Margarita are stuffed with pizza!

Andrés ate one-quarter of a cheese pizza. He then ate three-eighths of a pepperoni pizza  and finally ate one-half of a mushroom pizza.

Margarita ate five-eighths of a cheese pizza followed by another half of the mushroom pizza.

All the pizzas were the same size. Andrés says he ate more pizza than Margarita because Margarita did not
eat any pepperoni pizza. Margarita says they each ate the same amount of pizza.

Who is correct?   /    How many different strategies can you create to solve it?

The key to opening up an investigation like this, I think, is changing the last part to state 'how many different strategies.........'

If we present children with closed problems where the answer is deemed what we are valuing, then we can only expect children to think the correct answer is what is important.

When presented with this problem, I asked my class, "What do we think we are valuing in this investigation?"

As we have been doing learning experiences like this a lot of the year, happily, we were able to identify that we are valuing creative thinking. Others thought we are valuing how there are many different ways to solve a problem and that they are all good. Another shared how in maths we should try to evaluate different strategies to see which are more effective and why that is so.

When I hear the children sharing these understandings, I feel like the big messages of the year are settling in their minds and this is also observable by their much more positive perspectives on maths.

If we as teachers, feed children the perpetual message that maths is about answering closed questions, then it is no wonder why so many children develop negative attitudes towards it. If we help children see how maths is about being creative, they more freely take risks with their thinking and begin enjoying where their minds take them.

Children buddied up with their table partner to create different strategies to solve the problem. Loads of rich and deep discussions took place. Loads of trial and error and lots of peer teaching of some misconceptions being harboured also took place. Listening in to the diverse discussions, it made me realise how all this amazingthinking and learning could never be taking place with a traditional chalk-n-talk maths lesson. Real mathematical thinking was taking place and children were feeling passionate about their creativity.

After some time, partners then published some of their strategies on paper and these were then shared around the room. As others read their strategies, they used post it notes to give constructive feedback. This strategy further helped us to see even more possible ways we can visualise and problem solve in maths.

Some of our ideas:

After sharing our strategies, we then used Padlet to write a short reflection about this learning experience. Here are some responses:

When we read reflections like this, we have to know we are doing something right in our classrooms......

Tuesday, 31 January 2017

Maths is Creative Thinking: Measuring Time

Maths involves creative thinking.

When kids are given opportunities to discover this and why creative thinking is important to mathematical thinking, even those who might harbour negative feelings, become much more open-minded towards maths. They start to rethink and understand that maths is not about getting an answer.  It helps them to understand that mathematical thinking is about coming up with your own ways to try to solve problems. It is about creating your own strategies and then reflecting on their effectiveness. It involves analysing strategies to find their pros and cons. 

Often, in traditional maths learning classrooms, children are spoon-fed strategies to replicate and then repeat. That robs them of rich opportunities to create their own strategies and to really delve into the concepts being explored. When we give children opportunities to creatively come up with their own strategies (and allow those strategies to be mad or ineffective) we help tear down those misconceptions that there is only one way to solve problems. We also help tear down feelings of apprehension some children harbour fearing they are not doing the 'right' thing. 

When we ask children to be creative in maths, we are giving them the much needed freedom and opportunity to explore the way their mind works. 

That's important.

We all think differently and we should be encouraging children to develop their own thinking processes, not programming their brains to think a certain way.

To begin our investigation into ways we can measure elapsed time, we used a Padlet and shared our initial understandings by answering the question: 

How and why is creative thinking important in maths?

Padlet is a wonderful way to give a voice to everyone in the class, not just those courageous enough to share their thinking.

After ideas were shared, we were asked to find someone else's thought that made is think about it differently then we had. 

I like this strategy for a few reasons:

1. It encourages children to actually read and think about others' thinking.
2. It helps validate the thinking of class members and helps them feel their thinking is appreciated.
3. It helps generate deeper levels of discussion. When a student shares another students' idea, we all become more involved in analysing it.
4. It is a more active way of sharing thinking. When we discuss orally, some children miss out of the initial and then struggle to catch up with what is being discussed. Using the Padlet, they can read and see what thought is being discussed and this helps them to think about it more.

We then used another Padlet to share and discuss our understandings of:

How does visualising help us with mathematical thinking?

These thinking and discussions helped 'set the scene' for the sort of thinking and investigation were about to do.

We then looked at the following problem:

Using the phrase 'How many different strategies.........' instantly makes the enquiry open-ended. There are now many possibilities and that is what we are valuing.

To help us understand what we are valuing, I explained how the strategies we create do not have to be the most effective. In fact, we should try to challenge our thinking by also creating mad, ineffective strategies because they will help us with our thinking. We can create strategies that will take a long time to help save. That's alright.  Let's just explore different possibilities.

We used the think-pair-share routine.

Children had 10 minutes creating different strategies on the paper.

After 10 minutes, we discussed what we thought about creative thinking.

Some very in depth and interesting reflections emerged such as:

° Before I thought creative thinking was just about doing something in a fun way. Now I understand that you actually need to think really hard and deeply about something. 

Everyone agreed that creative thinking does require 'hard' and deep thinking.

° I think it's interesting that creative thinking is a balance between hard thinking and fun thinking. 

° I get different answers with the different strategies I created and I'm trying to find out why.

This last comment is exactly where we want children to be: without being prompted, find reasons why a strategy is working or not. We all agreed this was a great thing she was doing.

I then explained we were going to do a 'silent gallery walk' to see what others were experimenting with and exploring as possible strategies.

Why do we think we are going to do this?

- So we can see how creative we can be?

- We could be inspired by other people's ideas and use them?

That's interesting.  Do we think it is alright to use other people's ideas?

- Yes, because we might see an idea and be able to change it to make it even better.

- I don't think we should just copy someone else's idea though. It's the same as stealing their thoughts and pretending its yours.

- But we can be inspired by someone else's idea and see if we can be even more creative with it.

-Its similar to what we did last week with creating logos. We learnt that designers often collaborate and share all their ideas to make a great idea together. If we see someone's strategy we like, we aren't speaking it. I think we are becoming a team with that person. Even if they don't know they are part of the team.

With those ideas we agreed that the purpose is to be inspired. If we find an idea interesting, we should try to experiment and play with it.

Some of our creative strategies:

So many unique and interesting thoughts can emerge when we give children the freedom......

We then silently walked around examine the strategies being created. When children found something inspiring, the returned to their seat and continued creating different strategies to solve the problem.  

After another 10 minutes had passed, table partners then shared their strategies with each other. Some rich discussions took place as they analysed the strategies and the reasoning behind them. The children naturally evaluated the effectiveness and visual creativity without being encouraged to.

To expand our creative thinking further, we then looked at our next task:

The new strategy created needed to have an element from each person. Some initially felt this was too tricky, but after time we all found possible solutions to do this. 

We then shared our new strategies on the data screen.  We explained each person's strategy and then how they were combined.  Their classmates were encouraged to ask them questions about the thinking process rather than the product and that generated some interesting perceptions on what creative thinking entails.

Partner Sample:

Partner A's:

Partner B's:


New Combined Strategy:

Another Pair showing each's strategy and then combining it: 

We then wrote a quick reflection using the following prompt:

Here are some excerpts from our reflections:

° I feel like this is the first time I've ever really understood what creative thinking is REALLY about. I used to think it was just fun thinking, but I know now that it involves hard and complex thinking too. 

° ...we can even take the easiest word problem and make it super complex when finding ways to solve it. 

°......visualising helps me make sense of what I am trying to work out in my mind.

° This helped me discover that I can create many different ideas and I feel more successful as a mathematician. 

°......helps me make more sense of what I'm doing.

° .....When I kept hitting road blocks with the strategies I was trying to make, it made me think of ways I could solve them. Actually, so much thinking went on in my mind doing this. I was amazed that my mind could handle them all so well. 

°I like how there isn't any pressure when we are creating different strategies. Its fun and challenging at the same time.