# Tag Archives: mathematical modelling

## Teaching Ratio

Novices and experts see problems differently.  Whereas a novice sees superficial features, an expert notices deeper underlying patterns, discarding the often irrelevant and distracting contextual information.  Here’s an example:

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## The teaching of fractions

There are certain prerequisites for children to develop a solid understanding of fractions.  First, they must understand, through work on additive reasoning, that a whole can be split into parts and that the sum of those parts is the whole.  There’s a short step into multiplicative reasoning from here – that a whole can be split into multiple, equal parts and that the whole is the product of the size of each part and the number of parts.  Once this is understood, children can begin to think about the whole being worth one and the parts being fractions of one.  The ideas that follow are broadly sequential in terms of conceptual development.

Children will need to manipulate various representations of fractions, for example making them with fraction tiles (as both bars and circles); taking strips of paper and ripping them in to equal parts; and drawing bars and circles, dividing them into equal parts.  It is worthwhile to get children to do lots of judging by eye and marking equal parts of a whole as well as using squared paper to do so accurately.

Of course, there is a lot of language to work on whilst manipulating these models of fractions.  Children need to be shown clearly the link between the total number of parts and the language (but not yet necessarily the written form) of the denominator: two parts – halves; three parts – thirds; four parts – quarters etc.

With a secure start in the basics of splitting a whole into equal parts, children can work on the idea that fractions always refer to something.  A third, for example, doesn’t stand alone.  It might be a third of an apple or a third of twelve sweets or a third of one whole.  Modelling these full sentences and getting children to speak in this way should solidify their understanding of proportion.  Through the sharing out of objects, even very young children can work on the concept of fractions of numbers – sharing six sweets between three children means that each child has the same number of sweets and that two sweets is one third of six sweets.

Once children are comfortable with the idea that an object or a set of objects or a number can be split into equal parts, and that each of those equal parts can be described as a fraction of something, that object or that set of objects or that number, they can go on to work at greater depth.  By comparing strips of paper or bar models that are the same length yet are split into different fractions, children can look at the relationship between the size of each part and the number of parts.  That is, the greater the number of equal parts, the smaller the size of each part.  Children should be expected to think about how ¼ is smaller than ½ because ¼ of one whole is one of four equal parts whereas ½ of one whole is only one of two equal parts.  Then, questions like this should be relatively straightforward:

The understanding that unit fractions with larger denominators are smaller than unit fractions with smaller denominators will contribute significantly to work in comparing fractions later on.

Children could begin to look at improper fractions and mixed numbers next.  Using ¼ fraction tiles, they could make one whole and then see what happens if you add another ¼.

This lends itself to counting in unit fractions but we should exercise caution.  Children may be able to chant ‘Three quarters, four quarters, five quarters…’ but early conversion to mixed numbers as well should help to secure their understanding of the relationship between them.  Manipulatives like fraction tiles and multi-link cubes are great for representing improper fractions because they can trigger accurate mathematical talk to describe the improper fraction (the total number of cubes as the numerator and how many cubes in each whole as the denominator).  The same can be done to describe the mixed number (the number of wholes, then what is left over as a fraction of a whole).

Returning to additive reasoning, children could generate complements to 1 whole and record them as addition and subtraction statements.

A slight change to the representation used can support children to work with complements where denominators are different:

Placing two bar models of equal length one on top of the other is great scaffold for comparing fractions.  When the denominators of the fractions are the same, the bars should not even be necessary but when they are different, the image can help to structure thinking.

When dealing with fractions with different denominators, the practice that children had earlier of judging by eye to split a whole into equal parts and marking the divisions themselves becomes crucial, otherwise, things like this could happen:

A standard fraction wall is all that is needed to begin work on equivalence and the first step is of course shading one fraction and looking up or down the fraction wall to find fractions of equal size.  When children are comfortable with that, they can begin to look at patterns in the abstract representations, particularly the link between times tables, numerators and denominators.

Using the language of simplifying or cancelling fractions without first talking more generally about the concept is a mistake.  If children are well versed in using a fraction wall to find equivalents to a given fraction, it is only a slight tweak to talk about finding the equivalent fraction that has the fewest total parts.  It would be tempting to talk about finding the equivalent fraction that is ‘closest to the top’ of the fraction wall but this would be a mistake too.  The language of simplifying or cancelling can be used to attach to the concept of finding the equivalent fraction with the fewest total parts to get children thinking conceptually soundly.

One further aspect of thinking of fractions is to consider them as numbers.  To do this, plotting fractions on a number line directly beneath the bar model is a good way of linking the two representations.

Representing fractions as a proportion of one, as a part of a quantity and as a position on a number line significantly supports children’s development of proportional reasoning and ensures that future tricky concepts such as calculating with fractions can be built on a secure foundation.

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## How can a child catch up to learn times tables in one term?

Children should know all times tables by the end of year 4, but there are children that slip through the net, taking much longer to learn them.  There are also children that may seem to have learned times tables by the end of year 4, but forget and have to work into upper key stage 2 to relearn.

This post describes a plan to get children who are in year 3 and 4 and who are not on track to understand times tables by the end of year 4.  The plan is also for children in year 5 and 6 who still do not know their times tables.

A fact a day for a term

The basic structure of the plan is to work on one fact per day.  Working with commutative facts such as 3 x 4 and 4 x 3 together, and taking into account that familiarity with tasks should accelerate the work the longer it goes, a term is a sensible time frame to work in.  This will be systematic, working from x10 to x5, then x2, x4 and x8, then x3, x6 and x9, finishing with x7, x11 and x12.  This is to enable links to be made between times tables.  Within each times tables, we’ll work in increasing order of times tables (i.e., 10 x 1, 10 x 2, 10 x 3 etc.).  Of course, different children will have different starting points, not all starting with 10 x 1.  As days pass, children will consolidate their understanding of a times tables through repetition, multiple representations, counting and low stakes testing.

Multiple representations

For times tables to stick and to be useful in other areas of maths, they need to be rooted in secure understanding.  To allow this to happen, each fact will be represented in different ways, in the first instance by the teacher but increasingly by the child.  The first representation is Numicon, using the example of 4 x 5:

Using this we can explain that 4 x 5 means 5 lots of 4 and that by counting in multiples, we can find out that 4 x 5 = 20.  Children will have done this for 4 x 1, 4 x 2, 4 x 3 and 4 x 4 in the preceding days so they should be able to count in 4s.  However, they may need to do some skip counting, where they whisper or say in their head each number except for the last on each Numicon piece (1, 2, 3, 4; 5, 6, 7, 8; 9, 10, 11, 12…).  The Numicon also helps to lead into other representations:

Repeated addition: 4 + 4 + 4 + 4 + 4 = 20

Bar model:

Number line:

All the while, the child is practising counting in 4s, and thinking about how 4 x 5 = 20.

Commutativity

One more representation can lead the child into working on the related commutative fact.  An array gives a little further practice seeing how 4 x 5 =20:

Rotating the array shows how 5 x 4 has the same product:

This can lead into counting in 5s to get to 20 and showing that 5 + 5 + 5 + 5 = 20.  Then, repeating the representations of Numicon, a bar model and a number line will help to internalise the commutative fact.

Low stakes testing

Having worked on this new fact (and its commutative relative), the child can then work on remembering facts that have been previously worked on in days gone by.  Practising recalling times tables is of course a great way of ensuring that they come to mind immediately when needed.  Quick, effortless recall means that little cognitive effort is required to summon the knowledge, thereby keeping as much working memory as possible freed up to solve a problem that needs the times table fact in the first place.

There are two ways of working on quick recall of times tables.  The first is if the child has a reliably secure understanding of multiplication.  In this case, simple testing such as asking ‘What is 3 x 5?’ or the use of individual flash cards will be fine.  However, if a child is still not quite there with conceptual understanding, testing by using objects or images can help to get them to think mathematically instead of guessing.  The teacher shows any of the pictorial representations already described to prompt thinking about the number of groups, the size of each group and ultimately quick recall of the whole.

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## Multiplicative reasoning

If a child understands additive reasoning and the relationship between the whole and its parts, it is a fairly straightforward conceptual step to understand multiplicative reasoning.  Multiplicative reasoning should be modelled as repeated addition in the first instance.  Adding multiple equal parts (for example 5) might look like this:

5 + 5 + 5+ 5 is equal to twenty.  Children need to understand that multiplication allows for efficient repeated addition.  You  have your thing to be multiplied (5) and the multiplier (4): 5 +5 + 5 + 5 = 5 x 4.  Creating arrays and deliberately connecting repeated addition with multiplication makes for sound understanding.

How children work out the whole should not be taken for granted.  At first, children might count each item in the array.  Counting in multiples can be achieved by first skip counting.  Children might whisper the numbers while counting except for the last in each row, which is said out loud.  Then replace the whispering with counting in their heads and then simply saying the multiples.  Over time, given sufficient practice, children will internalise these times tables.

Commutativity is important here – the array used above shows 5 x 4 but rotated it shows 4 x 5.  Times tables taught systematically and with such conceptual support should be straightforward for children to learn comfortably before the end of year 4, especially when we consider it like this:

Of course, children need time to practise well and multiple representations help children to make connections.  Graham Fletcher’s blog post describes the use of pictorial representations on flash cards – an approach that is a great form of low stakes testing to support the learning of times tables.

This image supports the understanding of having a ‘thing to be multiplied’, a multiplier and a whole.  With practice, children will be able to subitise from glancing at the flash card, becoming fluent and accurate with times tables recall.

Some children will grasp all this quickly and can work at a greater depth while children that need more practice with the basics get it.  Still using the array, children can easily begin to think about distributivity simply by splitting the array into parts:

The part above the line is 5 x 2 and the part below the line is 5 x 2:

5 x 2 + 5 x 2 = 5 x 4.

There is lots of scope for systematic thinking about equivalence with a task like this.

Arrays are perhaps not the most efficient of representation so a progression is to get children to be able to represent multiplication in bar models.  First though, Numicon to work on the language of size of each part, number of equal parts and the whole:

Numicon is a great manipulative to represent multiple parts because of its clarity of the ‘size of each part’.  Multi-link cubes could work too, but children would need to organise the parts into different colours to differentiate between them:

Building worded statements using a manipulative will ensure children practise the language needed to internalise the concept of multiplicative reasoning.  Dropping in some of the  inverse relationship between multiplication and division could be useful here too.  Doing it systematically can also help keep times tables knowledge conceptual and not shallow:

Commutativity could be brought in again – showing that 3 groups of 4 is the same as 4 groups of 3 using manipulatives arranged with intent.  Alongside this, comparing the similarities and differences with the worded statements will get children to think with clarity about equivalence between two multiplicative expressions.

Bar models are a versatile representation that can be used to solve a wide range of problems later on, so getting children to sketch out multiplication and division statements using bars enables them to practice a versatile skill.  We should expect great accuracy in their drawings – they should be representing equal parts.  If children also represent the same expression on a number line beneath the bar model, we can encourage links between representations and lay the foundations for trickier calculations and problem solving as they progress through school.

Update: The NCETM recently published this account of teaching the six times table, with some great ideas for depth.

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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:

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.

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.

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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## 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):

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:

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:

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:

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:

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

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:

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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## Mastering the common problem types in maths

I’ve been wanting to combine some of the thinking I’ve been doing into cognitive overload, worked and partially completed examples, and the bar method as a pictorial representation of mathematical problems.  The lesson described below is what I did, with a substantial eye on the expectations of the new national curriculum and the idea of mastery.

A worked example

A good explanation clearly takes children through the steps needed in order to solve a problem, but these steps should be rooted in the deeper structure of a problem and not the superficial.  As such, the success criteria that the teacher works from and that children refer to should support that expectation of analysis.  In the example, I showed my class how they could solve a comparison problem where both the total of the parts and the difference between them is known, but the value of each part is not.  To solve this type of problem, they need to be able to pick out totals and differences, as well as get an idea for which part is bigger/smaller.  The success criteria I used was this:

Then, I took one question and walked them through each of these three steps:

A further few questions can be prepared and ready if further modelling is necessary.

Partially completed examples: scaffolding for mastery

Children will need to deliberately practise representing the information pictorially.  Work into cogntive overload suggests that when children are overloaded, a number of things can happen:

• They’ll complete the first or last instruction only
• They’ll lose their place in the sequence of instructions
• They’ll abandon the task

In an effort to prevent this things from happening, the work I gave most of the children in my class included partially completed examples.  These aim to reduce the cognitive load while still providing the opportunity for deliberate practice.  The first few questions had some information already transeferred onto the pictorial representation.  Gradually, there was less and less of this until children were solving problems with just the basic structure of the problem given.

Undoubtedly, some children will need more practice with heavy scaffolding before it is removed and some will need much less.  Working in this way makes adapting the scaffolding for different children easy to manage.

Trickier problems

There will be children who are already able to solve this kind of problem, but the process modelled and the success criteria will still be of use to them to solve trickier problems.  In the tasks that I chose to give to my class, I gave them three parts to compare:

Although the basic structure of the problem is sketched out for children, in this example I did not give them any labelled sections, as the children I was intending this work for I felt would not need it.  Were I to do this again, I’d have another variation with some labels provided so that the questions are partially completed.  Hopefully, this would enable some children to work with questions this complex that otherwise might not have been able to.

Sure, this is but one lesson and it will take time for children to master the underlying patterns of problems so that they can solve them efficiently.  So it got me thinking about when different problem types should be introduced or when they should be mastered by.

Mapping out the problem types

With all this is mind, I set out to allocate problem types to phases for when they shoud be introduced.  The idea is that children in KS1 will master the basic additive and multiplictive probem types;  lower KS2 will master more complex additive and multiplicative reasoning problems as well as multi step problems based on the basic additive and multiplicative types; and upper KS2 will master still more complex additive and multiplicative reasoning problems as well as a wider variety of multi step problems.  Of course, children in each phase wlll need to revise older problem types, the goal being that eventually, they can see the undelrying pattern of a problem, thinking their way clearly to an accurate answer.  There are not many problem types for each phase, so they can be practised over time in a variety of contexts.  Each time they are revised, the scaffolding provided can be gradually removed so that towards the end of the phase, children are solving the problems with no scaffolding at all.

Introduction of bar models for additive and multiplicative reasoning Year 1-6

The expectations here are high, but achievable.  Mastering the common problem types by the end of KS2 will set children up very well for the next stage of their education.

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