Understanding written division

Last week we moved from mental strategies and understanding the concept of division to using written methods. I remember last year, going through the motions of teaching chunking - first on a number line and then vertically.

This year I decided to try out something different and to build up to chunking bit by bit. I found a great website https://desertdesigned.me/2015/10/23/teaching-division-with-area-models/ that showed how to use the area model when dividing. This seemed to be a more visual way to begin. Rather than recording horizontally on the side though, we did all of our recording within the rectangles.

I think this was a really clear picture for the children as it carried on from last week's idea that division is about trying to get as close to 0 as possible.  They all seemed to be on board with this which was great.









The following day we looked at using the same method but recording our subtracting on the right hand side, using a vertical method.

The final stage was to not use the rectangles at all, but to
record using the chunking method recording vertically. Some children were happy to move onto using this, but some wanted to stay with the area model. At this stage I am not worried about which method they use, but the fact that they have understood what they are doing - taking away 'chunks' each time to get as near as possible to 0, and using known table facts to help them do this.

They also came across some sums where there was a remainder and it was not possible to get to 0. This led to a great discussion of what we would do with that leftover number.

What does division mean?

As I am aiming to get the children to have a deeper understanding of concepts in Maths I am trying to take a few steps backwards before racing on with the Year 4 objectives. So we started this week by discussing what division means to them and using the Frayer Model to help guide us with our thoughts.

The Frayer Model was introduced to Paul and I at the workshop we went to on Conceptual Maths. It is a graphic organiser that requires learners to understand not only definitions but also the more complex concepts. It will hopefully promote a higher level of thinking and can also be used to assess deep understanding of the mathematical concept. It will take a bit of modelling but once the children get the hang of it, I believe it will be a really useful tool - not just for Maths.

The children came up with ideas that division was about grouping and sharing. We discussed the term 'equally' and trying to make groups in order to get 0 - or as near as possible. The week was taken up with different strategies we can use in order to help us solve division problems (not using a formal written method). That included using manipulatives as well.

At the end of the week I gave them a sheet where they had to solve various 'mastery' questions. This was the first time they had been exposed to these type of questions. Could they apply the strategies and use of manipulatives independently?

A mix response! One student broke down in tears - not quite what I was hoping for! But at least half of them, independently took what manipulatives they needed and started working on the questions. There was one question where they had to round up after dividing, which was a new concept. But by drawing groups they were able to visualise why they needed to round up, and not have a remainder.

They then blogged one explanation of how they solved the problem.

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Maths Rotations

Rotations Update 07/1/17

Have realised that although this is working well I have forgotten all about giving the children some time to share their learning with the class, which is something I always did at the end of a lesson previously. Need to try to make sure that I am giving some time to this when appropriate in the lesson, not just at the end. 

Rotations Update 04/11/16

Have been doing this for just over a week now and it seems to be working quite well. Having the guided time with each small group every day is really beneficial and helps me to see who is understanding the concept taught and who needs more input. I can also see gaps in the review work and know concepts that need to be revisited - such as rounding numbers to the nearest ten and hundred. Will get some feedback from the year 4 team as they are also trialling this system of teaching maths.

In order to really get to grips with what the children in my class really understand, use and apply in Maths I have decided it is time to change the way I approach maths lessons. I have, for years now, begun each session with a whole class intro, carpet work on whiteboards, set differentiated tasks on same concept as independent work, worked with a group (whoever needed help) then had a plenary (when enough time). But does this really give me a clear picture of each individual child...

After a lot of internet research (pinterest lead me to many blogs) I am going to try out what people are calling guided maths. I have used ideas from 
http://mathtechconnections.com/2015/12/20/math-sorts-a-math-workshop-station/
http://thriftyinthirdgrade.blogspot.com/2016/06/align-your-instruction-to-3rd-grade.html
http://mrelementarymath.blogspot.com/2016_08_01_archive.html
http://www.kangaroomaths.com/kenny4.php

So my plan is that there will be 3 groups (ability grouped) and I will have 3 rotational tasks. I still want to give a whole class input, but it will be shorter than normal - just an intro into the concept. At this point I also want to make an 'anchor' chart which we can add to during the week - basically a visual aid that children can use to help them. At the end of the week a photo of this can go into their books or onto blogs to help them next time when they review. 

Then the children will rotate - guided group (teacher led), independent group (working on the key concept/topic) and review group (working on a previous taught skill/concept). I think the review group is of particular importance as I also want to see how much they have retained and what needs further input. 

So hoping this will help me as structure lessons more effectively. Time may be an issue and I am sure it will need adjusting as we go along...

Mastery in Subtraction

After a week of looking at formal written methods of subtraction I wanted to see how deep their understanding really was. So using some of the questions from the Maths Hub I posted a few questions onto Seesaw and asked them to record their answers and explanations.


Here is an example from Andrey which shows his understanding of solving addition and subtraction two step problems in contexts, deciding which operations and methods to use and why.  



Baimonn explains how to work out the missing numbers in a subtraction sum. This shows a deep understanding of using the inverse relationship between addition and subtraction, and also the column method. 




CC explains how to work out a 2 step problem that involves adding and subtracting.




What was great about this activity was it also showed me which children still needed some work on their understanding of a subtraction written method.

Kamilla was so close here. You can hear her brain ticking as she thinks about the first units column, showing her understanding of using inverse and the concept of borrowing. 


Using a formal method in her book was not a problem, but being presented with the problem in a different context proved that we need to take another look at applying skills and understanding. 



Some confusion between the operations. This also shows me that there is not the depth of number sense that I thought there originally way. Taking another look at place value and using the manipulatives is a must when we review this.

Diving deeper into understanding of multiplication.

Thursday 6th October

Diving deeper into understanding of multiplication.

This has been a tricky week. Started off with looking at table facts, where the children were with them and which ones they still had to learn. Need to communicate this further with parents so they are aware of which tables their child need to learn. Will send home a booklet and some games.

I wanted to know what the children thought multiplication was. So I asked them to draw 5 x 2. The answers were interesting. This was the answer from most of them. I think only one child drew an array, and another 5 groups of 2.

So this gave me some idea of what we needed to do. Arrays were revisited - although when I said that word most of them looked at me blankly! I wanted them to first see an array as a multiplication sum and then to use that to understand the commutative property of multiplication. However, I think I moved too quickly and did not make sure of their complete understanding as the following day I went into associative property and ended up with 15 confused children and a frustrated teacher!

Time to rethink my planning...

Split into 2 groups, no whole class teaching this time around.
Commutative property with Ms Nocha - 2 dice, write the sum, draw the array. Change the order of the dice, write the sum, draw the array = what do you notice. They gave verbal feedback, noticing that it made no difference what order you put the numbers in, the product was the same.

















Associative Property with me - 3 dice, write the sum, group them (using brackets), draw the array, work out the answer. Group them in a different way, or change the order, write the sum, draw the array, work it out - what do you notice. They wrote a sentence to explain.
 = success.

Went back to asking them what they would draw for 5 x 2 = an array or 5 groups of 2 were the answers! Hurrah! What properties do you notice about multiplication - again the answers I was looking for.

The best bit was on the way to lunch we looked for arrays around the school. Hayley then made my day...

Using games to help with reasoning

Monday 26th September

Using games to help with reasoning

Great to see lots of reasoning today whilst children were 'playing' games that involved using mental subtraction strategies that had been explicitly taught last week.

For 'Strike It Out' it was interesting to see Kamilla using the strategy of thinking about the outcome of the difference between her two numbers before committing to her answer. 

Nour, who had been trying to use a written formal method for 'mental' subtraction last week could now see that counting up as an effective method last week, was able to use this when she was finding the greatest difference between 2 out of 3 of her playing cards. 

After about 5 minutes of staring at a number fact family, Kimberly was able to see that by changing numbers around you can create 4 different number statements using 3 numbers (add/subtract only). 

Playing games with maths has become a regular Monday morning session and the discussion of maths and the language that was being used - 'I have the biggest difference', 'oh, I can change the numbers around and make...' is proving to be a great insight into their understanding of numbers.