Tuesday, 16 February 2016

Investigating Division Without Death-By-Worksheets

Developing strong mental strategies is one of the most important elements of mathematical thinking children need constant help in developing.

In real everyday life, we use mental strategies far more than pen and paper strategies. 

In real everyday life, we use calculators on our mobiles far, far more than pen and paper strategies.  

Even when using our calculators, we are using mental strategies to quickly assess whether the answer we get is a plausible or not. 

So, it stands to reason that our maths curriculums should give far more importance to mental strategies.

Today we discussed what division is.

We first did a 'talk n turn' and then shared our thoughts together: 

- grouping numbers or objects
- separating numbers
- finding out how many will be in a group from a whole group
- finding out how many might be in a group and how many remaining
- it's the opposite of multiplication

These were some or our ideas. 

We hadn't yet thought of a key basic conceptual understanding though yet.

So I suggested: Let's look at multiplication is and that might help us think of other ways we can describe what division is. 

This eventually helped us to add how division is repeated subtraction.

We were able to determine this once we remembered that multiplication is repeated addition.

That in itself was an interesting revelation of understanding amongst students. 

We looked at 4 x 5

How is it repeated addition?

Most of us felt it meant 4 + 4 + 4 + 4 + 4

Does it?

4 x 5 means =   4 groups of 5

Does that mean + 4 + 4 + 4 + 4?

Quite a few confused looks.

Someone then suggested that if 4 x 5 means 4 groups of 5 then as repeated addition it must mean 5 + 5 + 5 + 5.


When things like this emerge with the majority of my class, it indicates to me that we need to spend some time later revisiting multiplication by using manipulatives and visuals to help us really solidify our foundation understandings. 

So we looked at 20 ÷ 5. 

On our mini whiteboards we thought about how it looks as repeated subtraction.

20 - 5 = 15 - 5 etc

To lighten up our mood, a drew what they were to imagine themselves standing on a cliff face with sharks below.  A man was holding a gun to your head and demanded you tell them what 84 ÷ 4 is mentally.  

What's an easy mental strategy to solve this.

The children recorded their ideas on their mini whiteboards.  Then we shared our strategies with those close by.

Students were then invited to come and share the strategy they thought of:


A lot of us had thought of using the split strategy and we discussed why that made it easy to do mentally.  

A few thought of the compensation strategy by making 84. a friendly number by changing it to 80.   We thought that was interesting and gave it a try with 96 ÷ 4 to see how it might look with other numbers.  Changing 96 to a friendly or easy number like 100 made it easier to mentally divide.

A few thought of using the half and half again strategy which we agreed was another useful mental strategy for those numbers.

Then one student showed a short division method.

We had recently discussed doing written algorithms in our heads when we looked at multiplication mental strategies.  I didn't need to say anything.

Some children explained how it isn't the most effective way.  We need to look at the number relationships and think of best mental strategies to use for those particular numbers.

Another student added that if we do that mentally we are making life difficult for ourselves.

It's great seeing those messages making impacts in children's minds.

We talked again about building neuron pathways like we always do when we explore mental strategies.  We are understanding now that if we practise mental strategies, they will form string neuron pathways in our minds so we will use them when we need them.

What could we do to build pathways for these division mental strategies?

- Make our own questions and try to solve them.

- Make questions for our partners to solve.

To great ideas so we spent about 5 minutes with either of those two approaches we wanted to make to help us with any of those mental strategies we just saw. Our introverted classmates had the option for their alone time to process their thinking and those extroverted could apply their personalities too.  It's a win-win catering with options given like this. 

An additional and important benefit from this is that we needed to be able to create questions that could be solved with the mental strategy we wanted to practise.  

When children create their own questions, they need to really deepen their thinking about the strategy.  Which numbers will work for that strategy and which won't- Why?  These were discussed amongst some of us during our enquiry practise.

This type of thinking is far more beneficial than simply being given a worksheet with set questions to practise a strategy.

To then add some more thought- and hopefully some creative thinking we first shared different symbols for division that we knew.

We then looked at this division symbol:

and had some thinking time to wonder why that symbol was actually created to represent division.

We shared our theories with those around us and then as a whole class.

Some great ideas emerged:

I particularly liked one student telling a story how the two dots were once best friends. They had a fight and separated (division) and built a wall between them forever.

Despite is liking all the suggestions, it did still leave us wondering what the real reason was.  Had we thought of it?

We liked to think even if we hadn't, we enjoyed creating our own and that is what creative thinking really should be about. 

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