Monthly Archives: October 2015

Creating the conditions for mastery

Bart

In one episode of the Simpsons, Bart moves school and is immediately put into a remedial class. He joins a lesson where they are continuing their work from the previous day – the letter ‘a’.  Bart’s observation is: ‘Let me get this straight.  We’re going to catch up to the other kids by going slower than them?’ – See more at: http://www.risingstars-uk.com/blog/creating-conditions-mastery#sthash.OWw90KlV.dpuf

 

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Before, then, now – modelling additive reasoning

One of the parts of the NCETM’s Calculation Guidance for Primary Schools is the ‘Before, Then, Now’ structure for contextualising maths problems for additive reasoning.  This is a very useful structure as by using it, children could develop deep understanding of mathematical problems, fluency of number and also language patterns and comprehension.

The first stage is to model telling the story.  We cannot take for granted that children, particularly vulnerable children in Key Stage 1, will know or can read the words ‘before’, ‘then’ and ‘now’.  Some work needs to be done to explain that this is the order in which events happened.  Using a toy bus, or failing that, an appropriate picture of a bus, we would talk through each part of the structure, moving the bus from left to right and modelling the story with small figures:

FullSizeRender

Before, there were four people on the bus. Then, three people got on the bus. Now there are seven people on the bus.

 

The child could then retell the story themselves, manipulating the people and the bus to show what is happening.  For the first few attempts, the child should get used to the structure but before long we should insist on them using full, accurate sentences, including the correct tense, when they are telling the story.

I have chosen a ten frame to represent the windows on the bus, which enables plenty of opportunity to talk about each stage of the problem in greater depth and to practise manipulating numbers.  For example, in the ‘Before’ stage, there were four people on the bus: if the child could manage it, it would be interesting to talk about the number of seats on the bus altogether and the number of empty seats.  By doing so, they are practising thinking about number facts to ten and building their fluency with recall of those facts.   The task could easily be adapted to use a five frame or a twenty frame.

The next stage could be to tell children a story and while they are listening, they model what is happening with the people and the bus.  After each stage, or once we have modelled the whole story, they could retell it themselves.  Of course, the adult would only tell the ‘Before’ and the ‘Then’ parts of the story as the child should be expected to finish the story having solved the problem.

When the child is more fluent with the language and they understand the structure of the problems, we can show them how it looks abstractly.  For the ‘Before’ part, the child would only record a number – how many on the bus.  For the ‘Then’ part, we would need to show the child how to record not only the number of people that got on or off the bus but the appropriate sign too – if three people got on they would write +3 and if two people got off they would write -2.  Finally, for the ‘Now’ part, they would need not only the number of people on the bus but the ‘is equal to’ sign before the number.  Cue lots of practise telling and listening to stories whilst modelling it and writing the calculation.

10

A more subtle level of abstraction might be to repeat the same problems but rather than the child modelling them using the bus and people, they could use another manipulative such as multi-link cubes or Numicon.  They could also draw a picture of each stage – multiple representations of the same problem provide the opportunity for deeper conceptual understanding.

The scaffolding that the structure and the multiple representations provide allows for some deeper thinking too.  In the problems described so far, the unknown has always been the ‘Now’ stage or the whole (as opposed to one of the parts). It is fairly straight forward to make the ‘Then’ stage unknown with a story like this:

Before, there were ten people were on the bus.

Then, some people got off the bus.

9

Now, seven people are on the bus.

This could be modelled by the teacher, who asks the child to look away at the ‘Then’ stage.  Starting with ten people on the bus and using a ten frame is a deliberate scaffold – deducing how many people got off the bus is a matter of looking at how many ‘empty seats’ are represented by the empty boxes on the ten frame in the ‘Now’ stage.  A progression is to not use a full bus in the ‘Before’ stage – it is another level of difficulty to keep that number in mind and calculate how many got on or off the bus.

Another progression is to make the ‘Before’ stage unknown.  The child will need a different strategy to those already explained in order to solve this kind of problem.  Then story would have to be started with: ‘Before, there were some people on the bus.’  Of course, the adult would not show the child this with the bus and toy people, but they would show the completed ‘Then’ stage: ‘Then, four people got on the bus.’  Finally, the adult would model moving the bus to the ‘Now’ stage and completing the story: ‘Now, there are eleven people on the bus.’  The child would have to keep in mind that four people had got on and now there are eleven, before working backwards.  They would have to be shown that if four had got on, then working out how the story started would mean four people getting off the bus.  They could be shown to run the story in reverse, ending up with seven people on the bus in the ‘Before’ stage.

This task has the potential to take children from a poor understanding of number facts, calculating and knowledge of problem structures to a much deeper understanding.  The familiar context can be used as a scaffold to build fluency and think hard about complex problems with varied unknowns.

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Filed under CPD, Curriculum, Maths

Four add three is equal to two add something

Could Reception and Year 1 children solve this problem?

4 + 3 = 2 + □

Of course they could.  Here’s how.  First children will need to work on their understanding of 7.  Using a manipulative for 1:1 correspondence such as multi-link cubes, we can show how the whole of 7 can be made up of two parts (in the first instance, 1 and 6):

R1

It is important to model the language that will help children think clearly when manipulating the cubes: ‘One add six is equal to seven.  The parts are one and six and the whole is seven.’  It is equally important to talk about the cubes saying the whole first: ‘ Seven is equal to one add six.’  This will help to prevent the misconception developing that the equals sign means ‘the answer is next’.  Then show them how to systematically make seven with other sized parts, talking about the parts and the whole in the same way:

R2

R3

Children should also use the cubes to write calculations.  A little modelling of turning the language of ‘Three add four is equal to seven’ into 3 + 4 = 7, followed by plenty of practice, will be exactly what is needed.

Lots of quality talking, as well as using pictorial representations, will develop children’s fluency with number facts.  Showing different representations, for example Numicon, could strengthen their conceptual understanding:

R4

Some children will grasp this idea quickly, and some will need more practice to internalise the number facts and recall them more fluently.  Those quick graspers can be challenged to think more deeply about the number facts that they are working with.  We can start by returning to the multi-link cubes and looking at two facts:

R5

Here, we can model the talk required to think more deeply: ‘Three add four is equal to five add two.’  Children could repeat that task with different facts to 7 before we show them how to write that as 3 + 4 = 5 + 2.  When children have practised this and can do it reliably with manipulatives, they could draw a bar model of what is happening:

R Bar

A further challenge is to present cubes where there is an unknown:

R6

We could model how to talk about this as: ‘One add six is equal to three add something.’  To model how to work out what ‘something’ is equal to, we simply fill the gap with cubes to make the second row equal to seven, then counting the cubes to figure out what ‘something’ is equal to.  When children have practised and are becoming more fluent, the cubes could be replaced with bars, at first presented in that way but moving on to children drawing it themselves:

R Bar 2

All the while, children could be shown how this looks written down: 1 + 6 = 3 = □.  When they have seen the abstract alongside the pictorial and the concrete, we can try starting with the abstract and asking children to represent the problem with cubes or by drawing bars.

The sequence described, over time, should be enough of a scaffold for the vast majority of children to end up being able to solve such problems and in doing so, develop a deep understanding of early number.

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Filed under CPD, Maths