Category Archives: Curriculum

Principles and practices of effective homework

Homework can have quite a negative reputation.  It is often the source of familial tension as parents make sure their little ones have done it, not to mention the effect on teacher workload.  Research organisations like the EEF have not found it to be too effective either.  That said, research can only judge the effectiveness of existing practices so the job of teachers and school leaders is to find better ways of doing it.  When it is done well, homework can undoubtedly have a positive effect on learning.  The EEF states that effective homework is associated with short, focused tasks which relate directly to what is being taught and is built upon in school.  It also recognises the importance of parental involvement.  With these conditions in mind, here is a set of principles and practices for making homework as effective as possible. 
Read the rest of the article on the Rising Stars website…

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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:

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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.

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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.

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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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Enumerating possibilities of combinations of two variables

With Year 6 children expected to work on the objective ‘enumerate possibilities of combinations of two variables’, we should be clear on the difference between the underlying concept and the algebraic representation of it.  

2g + w = 10

For questions such as this, children should first have a secure understanding of the part, part, whole model.  We can show that 2 lots of something add one lot of something else is equal to 10 by using a concrete manipulative such as Numicon.  First, children represent the whole, in this case 10. Then they can speculate on the two equal parts (2g), trying out g=1 before finding the Numicon piece that fills the gap and therefore is equal to w:

  
Having found one solution, they can continue to work systematically to find alternative solutions.  Trying g = 2 is logical:

  
Lining up solutions beneath the whole reinforces the idea that the expressions are equivalent.  Children can continue to work systematically:

  
This also provides a scaffold for questions of greater depth, such as ‘What is the greatest number that g can represent?  Explain…’  
Subtraction?  Not a problem, although in this case, children must know that for subtraction, you always do so from the whole.

10 = 3g – w

In this question, the whole is 3g and the parts are 10 and w:

  
What is not clear from this model is the trial and error that went into it.  Children may well try 3 ones and quickly realise that it is already less than 10, so subtracting from it will not give a valid solution.  There is lots of scope here for discussion about the smallest number that g could represent.  

The use of Numicon leads nicely into children representing problems as bar models.  Here are the two examples used so far:

   
    
 

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Single and multi clause sentences – an analogy

The new national curriculum this year brought with it some changes in grammatical terminology. One of these changes saw simple, compound and complex sentences become single clause, multi clause coordinated and multi clause subordinated sentences. This post explains an analogy to teach this aspect of grammar. 

   
The analogy (and a smattering of storytelling – good for memory, you know) is to do with living arrangements. The first picture is of Serena. She is an adult who lives by herself, earning money, buying food etc. This is a bit like a single clause sentence: one main clause makes up the sentence. 

The second picture is of Mike and Jean, two adults who could live alone if they wanted to but they choose to live together, sharing all the responsibilities that come with having your own home. This is a bit like a multi clause coordinated sentence: multiple main clauses can be joined in a sentence by coordinating conjunctions. The final picture is of Hardeep and her infant son Gurnek. She is an adult who could live alone, but the child cannot. The child needs the adult. This is a bit like a multi clause subordinated sentence: a main clause is joined to (a) subordinate clause(s) by a subordinating conjunction. 
When children get to know these characters and their situations well, they can link the grammatical terminology to that knowledge. Through lots of spaced practice, in the context of a story or text they know well, they can begin to build their concept of sentence types.

One type of practice can be sorting sentences, modelled first:

  A possible scaffold is to text mark important features for some children, then gradually remove the text marking. Main clauses, subordinated clauses or conjunctions can be underlined, italicised etc. 

The challenge can be increased by changing the context. Grammar through well known picture books is a fruitful strategy not just to work on grammar, but to deepen their understanding of story. The task below uses ‘Leon and the Place Between’:

  
The next screenshot is of some development work on multi clause coordinated sentences: children weren’t familiar with the use of the conjunction ‘for’:

  
Many children still needed a lot of explanation and practice thinking about the difference between main and subordinated clauses, again in the context of a well known story:

   
 Children seemed to found it useful, to begin with, to analyse the sentences using the analogy. Which part of the sentence is a like Hardeep, the mum? Why? Which bit is like Gurnek, the baby? Why? Gradually, the analogy makes way for the grammatical terminology. This strategy was also useful to talk about about the use of a comm a as a clause boundary when the subordinate clause starts the sentence. 

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Multiplying fractions and integers

This is an account of a particularly successful topic from Y5 on multiplying fractions and integers. It’s fast becoming one of my favourite concepts to teach.
I showed multiple representations of a calculation, showing how to make it with cubes:

  
In this example, it was important to make four lots of two thirds first before rearranging the coloured cubes: later when they draw it on a number line, they’ll need to be thinking of four jumps of two thirds. The language of number of parts, size of each part and whole was supported by the bar model and came in handy for the empty box problems later.
Children needed to make other examples of calculations, talking about it with a partner:

  
Most children were fine with this and ready to move on to pictorial representation on a number line. Some needed a bit more consolidation with how the cubes represent a calculation. This is of a set of scaffolded questions where children made the calculation with cubes, then drew it:

  
For those ready for a number line, I explained a worked example before modelling how to do one from scratch:

  
The explanation had to be clear on how to construct the number line accurately, which involved making decisions about how many squares to leave between each whole number (using the denominator of the ‘size of each part’), and how many squares each jump needed to be (using the numerator of the ‘size of each part’). The final decision to make was how many jumps to draw, using the number of equal parts. The whole, and the answer to this particular calculation, was where the last jump ended up in the number line.

When children practised doing this, they chose from using a scaffold of minimally different questions or tackling varied questions:
  
Some children needed a few lessons to keep practising this and each time we returned to it, I started by showing multiple representations of the same calculation:

   

 Some children grasped this concept quickly and needed to work on deeper tasks. I made some subtle changes to the language I used to describe what was happening (bottom right corner of the above image) and used the same bar model to introduce inverse operations with aim of learning to solve empty box multiplication of fractions:

  
Bar models helped to explain what was known and what was unknown, using the language of whole, number of parts and size of each part. I showed how it looked on a number line too:

  
Some managed to get on to dividing fractions and whole numbers, using bar models (which they were familiar with for integer multiplication and division) and number lines to show what was happening:

   
 All the while, more children began to move from basic multiplication of fractions to some deeper thinking about multiplicative reasoning and fractions.

A few weeks later, children worked out the area of rectangles involving fractions and had internalised much of the language to be able to represent questions like this:

  

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Mastering maths curriculum design

Recently, Bruno Reddy and Michael Tidd have written about their experiences in designing a mastery maths curriculum. It seems that there are lots of us designing overviews for year groups with the same goals in mind: long term retention of knowledge and concepts with problem solving at the heart.

It is fascinating to see what others have come up with working on the same project, and also a little reassuring. Bruno provides six tips for creating a mastery curriculum, some of which I’d agree with wholeheartedly and some I’d adapt slightly for a primary curriculum. He advocates lots of practice, separating minimally different concepts, teaching concepts in a sensible order, and spending more time teaching fewer things. All of which sound like good advice to me.

Whereas Bruno suggests going back to basics in the first part of Year 7, I’d say that for a primary curriculum, the spotlight on place value, number facts, mental arithmetic and written algorithms needs to be relentless; spaced out and returned to many times over each year. In lower key stage 2, I’d have these units of work at least termly and more likely half termly. Each time the topic is revised, the expectations can be upped, with more and more time devoted to solving problems. This would enable the teacher to spend the time on good modelling and practice, while gradually increasing the expectation in that topic over the year and introducing different styles of problem. I’d say that this is preferable to organising revision through starters, homework etc alone.

Like Michael, I started with the mathematics mastery example overview.

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Those arrows denoting the continued study of a concept over the year do not provide enough guidance for a teacher as to when and what to specifically revise. For the draft that I’m working on, I wanted to change that. While Michael describes spending a whole half term on fractions, I’d suggest splitting fractions into different units and linking back each time, like in the draft year 4 example below:

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I thought that Children would move from comparing pictorial representations of fractions, to looking at comparing fractions and decimals, to finding fractions of quantities throughout year 4. Each time there’s a unit of work related to fractions, the teacher would check up on prior knowledge (with some sort of test – see benefit number 3 on test potentiated learning) and spend an appropriate time remodelling and getting children to practise previous content before linking to the main work in the topic.

A maths overview like this would be fine for maths specialists but for some, and indeed NQTs and teachers new to a year group, a little more guidance might be necessary. To supplement the overview, I’d have some more detail on the topics themselves including the core knowledge that children must have and be able to recall in order to master the topic and a brief idea of what children should be expected to do:

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This pair of topics will be one unit of work within autumn term 1. These topics will be repeated over the year, so that practice will be distributed or spaced, but another layer of spacing can be introduced by switching between these two topics: a couple of days on place value, a couple of days on mental addition and subtraction. See Robert Bjork on desirable difficulties and a recent post of mine on the same ideas.

All of this, of course, is still a draft. The order of concepts could do with a review, as could the expectations for what children should be able to do in each unit. Also, the overview needs to emphasise flexibility. Although topics are arranged to be revised as shown, there will be occasions when some topics don’t need as much revision while others will need more.

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