Wednesday, 13 December 2017

Estimating How Many Christmas Baubles?

There are lots of benefits for giving children opportunities to estimate.

Estimating is an important mathematical tool that helps children make sense of numbers and helps them to test the reasonableness of answers they create amonsgt other things. Estimating, I think, can also help children feel more confortable and relaxed with numbers.

As Christmas is approaching, we created strategies to try to esimate the number of baubles in the box:

I deliberately put in some large and small baubles to extend the thinking needed:

We have been recording the strategies we used to estimate. Lots of interesting strategies have been emerging and the children have been applying some of these to change their estimates before we reveal the exact amount.

Here are some sample strategies:

The children have been discussing each the different strategies amongst themselves as the week has progressed and without prompting from me, they have been evaluating the effectiveness or ineffectiveness of strategies being tested.

When we can give children opportunities to naturally be engaged and to discuss mathematical strategies, we know we are on to something good.....

Sunday, 10 December 2017

Peer Teaching: Multiplying Decimals

The Power of Peer-Teaching

We have been inquiring into multiplication strategies and analysing which are more effective than others.

We are now at a point where we continue to add wonderings to our wonder wall about how we could apply those strategies to decimal numbers.

So, today we combined our wonderings of multiplying decimals with visualising what we are doing when we multiply them.

We had groups of 4 and we began using the 'chalk talk' visible thinking routine. In this routine, a large paper is placed in the centre of the group. Each member writes their ideas, wonderings and understandings on the paper at the same time. It is a silent discussion. The members are encouraged to read what others have written and respond by helping them understand, ask a question for clarification or to add on their idea etc.  It is a powerful strategy that ensures each child has an equal amount of 'talk time'; it also allows each child to have an opportunity to collect their thoughts and think how to best communicate their understandings.

To tune into our learning today, we did a 'chalk talk' about decimals. Sample below: 

After our chalk talk, as a class we shared some ideas that emerged. We could only share what someone in our group had shared. We use that strategy a lot especially with 'talk and turn'. I like to think that if the children are encouraged to only share the ideas of others, then they start to listen more intently to their partners. Some very interesting discussions about decimals emerged and we added some more wonderings to our wonder wall to investigate later.

I then posed the question:

How is visualising numbers helpful?

Turn and talk.

The children shared some of their understandings with their partner and then we discussed as a whole class some things our partners told us that we thought were interesting.

From these routines, we had already tuned into what our learning will involve today:

1. Decimal numbers

2. Why we should value visualising numbers.

We often peer teach in our class and we constantly reflect on our peer teaching so we are now at a point collectively where we understand how peer teaching is helpful to our own learning and we are using much more effective and creative strategies when we do it. 

I explained how we are going to inquire into different ways we can visualise multiplying decimals. 

Each group member was to watch a YouTube that explains a different strategy for multiplying decimals.  It was their responsibility to understand the strategy well because their group members will be depending upon them. This sense of responsibility to others instantly engages the children. 

I showed the steps on the data screen that we would follow:

Inquiring Into How To Multiply Decimals

Step 1: Watch and learn

Step 2: Prepare: How will you teach the strategy?
What could you do to engage your group?
Be creative in your approach.

Step 3: Practise teaching the strategy by yourself.
Reflect on what you could do to improve.

Step 4: Teach your group the strategy

Step 5: Ask for feedback from each group member on how well you taught
the strategy.
Each group member will use the '2 and 1' feedback routine:

           ° Two things you did well
           ° One thing you should focus more upon next time you peer
              peer teach

Partner A
Partner B
Partner C
Partner D

Base 10 materials

Hundred grids

Hundred grids

Only whiteboard

From the beginning and throughout, every child was completely engaged in their learning. Lots of creative ideas emerged and lots of reflecting happened as they thought about effective ways to teach their group members.

Samples of what the peer teaching looked like:

Listening to the '2 and 1' feedback the children were providing to each other about their effectiveness in teaching was formative to me, but more importantly informative to the children who will be able to expand upon that in future peer-teaching activities.

We grouped together at the end for an oral reflection.

I asked:   What do we think or feel about our learning today?

All the children felt proud in what they had achieved. A lot commented on how they deepened their understandings of what decimals are or what happens to decimal numbers when we multiply them. Others remarked on how they could evaluate the effectiveness of some of the strategies compared to others. Everyone agreed that visualising was very helpful to understand what we are doing when we multiply decimals. A few commented on how they learnt more creative ways to peer teach.

Finally, all the children agreed that they really enjoyed the learning and that is always key to effective learning and mores to help children develop their passion for mathematical thinking.

Monday, 13 November 2017

Lead In to Multiplication Inquiry

To begin our new unit inquiring into multiplication, I wanted to find a way where I could gain an informal glimpse of where each child was already at with their conceptual understanding and at the same time give them an opportunity to rethink what they already know. 

So, I posed this question:

The children wrote their ideas on a shared paper (so that they could be perhaps be inspired by other ideas being generated in their group). 

To help encourage some diverse thinking, we were reminded how visualising in maths is a key element, so how could we visually show what multiplication looks like.

This served as a very useful pre-assessment to help gain an insight of where each child was at with their conceptual understanding of multiplication.

After some time, we then chose two ideas we had and shared those with our group. This generated some interesting discussions and helped with some misconceptions a few of us were harbouring.

We then swapped our paper to the next group. They read through the ideas and then drew a smiley face beside one of the ideas they found interesting.

The children found it interesting to see what their classmate had found interesting about their understandings and found out orally why they thought so.

Our collective understandings:

We then discussed as a class what the 'big ideas' were that we understood about multiplication.  We had a good understanding that multiplication is repeated addition and some shared the connection with division.  When one student theorised that multiplication sums are the same, giving the example of 3 x 7 = 7 x 3, another student questioned that. 

To help with this wondering, I drew the following on the board:

Which does this represent?   3 x 7    or   7 x 3?

Most of us thought it represented 3 x 7 and a few of us thought   7  x 3

Some children were asked to share their reasoning and eventually, as a class we concluded it must represent   7 x 3 because 'the times symbol represents groups of'.  With this situation we are looking at 7 groups of 3, so it must represent 7 x 3. 

We then sketched what  3 x 7 would look like on our paper to help us solidify this understanding.

As a provocation to help raise curiosity and spark wonderings to explore in our unit, the following was posed: 

We used the 'think-pair-share' routine.  We spent about 10 minutes thinking of different possible strategies and were encouraged to try to also create our own. The strategies didn't need to be effective, we were more importantly trying to find as many different strategies as we could. 

We then shared and discussed these with our group. Lots of great discussions took place especially with the more creative strategies they made. 

Our groups then decided which of these strategies worked and if so, they published them and added them to their group poster.

When we completed our posters, we passed them around to see other strategies others in our class came up with and we discussed a few of these together as a whole class.  

For our reflection, we had some time reflecting our learning in our maths diaries.( Link to Maths Reflection Diaries )  

Using our reflection thoughts, we shared these with our group and then as a whole class we came up with the following big ideas:

This has also sparked some interesting wonderings which we will use in the unit to find out about:

Creating provocations to being number units can be a bit challenging to design. It needs to be able to cater to a broad range of different understandings and to also spark lots of wonderings which will form the inquiries.  Looking at our initial wonderings, I think this provocation was quite successful. The children were really engaged and enjoyed the creativity aspect of creating their own strategies. 

Once we have begun exploring different strategies for multiplication and begun evaluating them, I'm sure they will enjoy this creative challenge I did last year with my class: 

Sunday, 5 November 2017

Exploring the Properties of Quadrilaterals

We have been exploring the properties of quadrilaterals. To help us find out what properties exist, we used a Venn diagram with a partner to compare the parallelogram and trapezium.

We then shared our ideas together as a class and came up collectively with the following:

This was a useful learning strategy to help all of us see different properties we can think about when exploring polygons. As children shared their ideas, they helped others to understand by teaching us what they meant.

We then chose two quadrilaterals and created a Venn diagram to compare their properties.  After comparing, we then evaluated whether we thought the two quadrilaterals had a close or a distant relationship and why.

There was, of course, a marked growth in understanding and the children reflected on how proud they felt in being able to find so many connections.

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: