Saturday, 7 October 2017

Rounding Numbers Inquiry


Rounding numbers is an important maths skills that oughtn't be overlooked each year of primary school.


When children round numbers, they are gaining a deeper sense of the value of a number and how each number can relate or connect with other numbers.

It's also an important skill for children to estimate.

Today's generation of children will, like us today, do most of the calculating using calculators on their mobiles.  They won't be the generation scratching out sums butcher's paper.

To use a calculator effectively, we need to be able to round / estimate numbers. Once we have punched in a sum into our calculator, we should think: does this answer make sense? If we don't, we could wrongly assume an answer is correct even if we accidentally typed in a wrong number.

Valuing the skill of rounding / estimating numbers is key for a learner's number sense to deepen. 



We started our inquiry sharing what we already knew about rounding numbers with our table partners. From this, theories, wonderings and reasoning skills were shared.


Which key concepts might help us the most to think deeply about rounding numbers?

We decided to use FORM,   FUNCTION and CONNECTION.


We started with FORM and used the think-pair-share routine and came up with these thoughts and remembered the importance of visualising in maths:





We then decided to use FUNCTION and again used the think-pair-share routine to come up with these thoughts:





We repeated using CONNECTION:





This lead to interesting discussions 



The big picture:






Using this understanding, we our now more appreciative of the learning experiences we will undertake to improve our rounding / estimating skills.





Related Links to Rounding:





















Mental Addition Strategy Experiment



One of our pre-assessment questions was to show the strategy we would use to mentally add the following:



                                   13 + 6 + 7 + 8 + 4





The question was designed to see which of us look for number bonds that add to 10 as an easier strategy.

Not many of us employed that strategy, so I used that for the following investigation into this strategy.


I explained we were going to do an experiment to help us understand our central idea.

We discuss regularly that doing maths quickly does not mean you are good at maths and we often discuss how it is more important to think deeply about the maths we are doing.  However, for this experiment  we will time to see how long it takes us to answer these 3 questions. We shouldn't feel stressed or pressed for time though; we are doing an experiment into strategies. 




Partner A was given the following 3 questions to solve.

The only strategy they could use though was to add each number in sequence.

Just add each number in the order that they are like so:




Partner B monitored to make sure they only used this strategy and they timed how long it took to answer them using a stopwatch.



Experiment result sample:



We discussed our thoughts and feelings about adding the numbers like this. 

We then looked with our partners at the numbers to see if there was an easier strategy to use to solve them.  

One pair shared how they could see some numbers added to 10. If we add those numbers first, it could make make adding them easier and probably faster:



We thought that was an interesting approach.

So, we continued with our experiment.

The same partner answered the same questions, but this time by first finding number bonds to 10.

Partner B had the same role- monitoring they used the strategy and timing them with a stopwatch.

A few felt it wasn't fair that the same person added again.

But, others pointed out in for the experiment to have a fair result, it needs to be the same person.



We then compared how long it took us with this strategy.

Most of us were surprised to see that it took a lot less time to answer them with this strategy.



They shared how they could see this strategy makes adding easier for us.



Some of us though found it took longer to solve using this strategy.

I asked if we thought scientists doing experiments might also be surprised by the results.  We thought they probably were.  I shared how my hypothesis was that all of us would have done this faster with this strategy, but that didn't happen.  Why do we think it took some of us longer with this strategy and for others it was much faster?


One theory shared was that it might take some of us a longer time to look for the numbers that add to 10, but others could find those more easily.

We thought that was a plausible theory.  

(It also told me how some us need some extra help in reviewing number bonds that add to 10, 100, 1 000 etc.) 


Experiment result 2 sample:




We then spent some time reflecting in our Maths Reflection Diaries and were encouraged to think about our central idea.

After writing our reflections, we shared our thoughts with our table.


Quite a lot of us reflected how we need to look for connections and relationships between numbers first so we can then decide which would be the easiest strategy to mentally add.  


We wondered if we could apply this strategy to subtracting numbers and others wondered if it would work with decimal numbers too. Others wondered what other strategies exist for mentally adding and subtracting.

So we partnered up with others and experimented with numbers to find out and then we shared our discoveries with the whole class. 



I think this was a pretty successful way for us to inquire into our central idea. It reminded us of a key mental strategy that most of us had forgotten over the years and it sparked good wonderings to help make the learning student-led.





Related links:



















Wednesday, 4 October 2017

Decimal Numbers

Using the Key Concepts to Think About Decimal Numbers......

We have been inquiring into our base 10 number system and there has been a lot of wonderings happening related to decimal numbers on our class wonder wall (post it note wonderings the children are encouraged to add to when they want). 

From listening into groups and pair number discussions I have been able to determine that whilst the majority of us have a pretty good basic understanding of decimal numbers, there is a significant handful that show that decimals is a concept they still only just beginning to grasp.

In situations like this, the PYP key concepts can be a really useful learning tool. They can help assist those learners in foundation levels of understanding to develop more solid conceptual understandings whilst at the same time deepening the understandings of those learners who already have good understandings.

I explained how we can see that decimal numbers appear on our class wonder wall a lot and so we should try to address some of them.

I asked, 'To think about decimal numbers which key concepts do we think would be the most useful?'

We decided upon:

° Form
° Connection
° Causation

We then used the think-pair-share routine to help us deepen our understandings.


This is what we came up with as we shared together:















Whilst sharing, the children were encouraged to share with us examples of ideas they had when the need arose.  Lots of turn-and-talks took place to help build stronger understandings and communication skills as we thought about ideas.


Doing think-pair-share routines like this is a very useful formative assessment tool and it greatly helps all learners regardless of where they are at conceptually to expand and deepen their understandings.

This also helps develop our sense of being a community of mathematicians sharing theories, wonderings and understandings.

From this, I felt more confident in us moving forward with using decimal numbers and in helping each learner feel more confident in inquiring into the other wonderings we have created.








Friday, 29 September 2017

Maths Reflection Diaries

Maths Reflection Diaries

I have been uber excited about this idea I thought I'd trial with my class this year.  I wanted to find a way that would honour the learner's need to reflect and process their thinking in a free way that works best for them and at the same time could become a useful assessment tool for me to gain a glimpse of what is going on in their mathematical minds so I can better help them in their journeys.

As we all know, as teachers, we take risks all the time in trying new ideas- lots of flops and sometimes amongst them some great successes.  At the moment, I am feeling this 'maths reflection diary' is going to be a wonderful success.

The idea is pretty simple, I cut in half some exercise books ( I figure if the space the children reflect in is small, then it is less intimidating- there is that sense of satisfaction when we feel we use up a whole space) and on the front taped some thinking symbols we could trial:


We have discussed the John Dewey quote, but as reflecting in maths is new to these learners, we didn't go into so much depth about the meaning behind the quote, but in a month or so after they have gathered more diary entries, we will revisit the quote and see how our thinking about it has changed.


After a maths learning experience a few times a week, we spend 5 minutes reflecting about the learning and thinking we have done.

We choose symbols from the table to help visualise the reflecting we are doing. We are encouraged each time to remember that visualising and creativity in maths are key and some children are picking up on that are drawing their thinking, but most at this stage are still more comfortable in writing sentences with the symbol beside each.  That is where they are at and that is perfectly fine.


From this simple routine, I am gaining so much insight into each child's mind. I am able to see where they feel they are being successful and where they may be struggling.  But more importantly, this routine gives each child the valuable time to actually reflect on their learning and it also gives the a voice that they might not otherwise have in the classroom if we are doing whole class discussions. Those of us who may be less courageous or even introverted have a platform to also share what is going on in their mathematical mind. additionally, it gives them a valuable opportunity to think about who they are as a learner and to process the concepts being explored.

Furthermore, it allows me as the teacher to create a dialogue with each child.

We have only started or diary reflections this week and yet there is so much rich thinking and future potential they could take them.

NB: The fold their paper in half so each time they reflect in their diary is half a page.




















This student's creativity and need to visualise in mathematical thinking shines in his entries:
















Sometimes it can really valuable to see misconceptions some might be nurturing:

















I feel I am building a stronger relationship with each child by giving them the daily written feedback and I can tell that because they know I will respond to their diary entry on the same day, they are progressively putting in more thought and some who are reluctant and taking risks to experiment with visually representing their thinking. 


In  a few weeks, we will discuss ways we could improve our reflection diary- perhaps we could add or improve on the symbols or any other ideas they feel might help deepen their learning through reflection. I'm sure they will come up with even better ways than I have.

Reading and responding has instantly become one of my favourite parts of the day. :)

Here is the link to the cover page if you'd like to have and print:















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.



Ohhhhhhh.........



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......