# Tag Archives: modelling

## A place for everything and everything in its place

Place value is very often one of the first units of work for maths in most year groups and is absolutely fundamental to a good understanding of number.  By getting this right and giving children the opportunity for deep conceptual understanding, we can lay solid foundations for the year.

For the purpose of this blog I’m going to assume that children can count reliably and read and write numbers without error. If these things are not yet developed to the appropriate standard then targeted intervention needs to happen without the child missing out on good modelling and explanations of place value.

Children need plenty of practice constructing and deconstructing numbers, first using concrete manipulatives like base ten blocks or Numicon.  This is to show that 10 ones is equivalent to 1 ten etc.  While they’re making these numbers they should be supported to talk articulately about what they are doing, perhaps with speaking frames: ‘This number is 45.  It has 4 tens and 5 ones.  45 is equal to 40 add 5.’

Read the rest of the article on the Rising Stars Blog.

Filed under Maths

Moving schools and with more than an eye on headship is sure to get you reflecting. The following posts are what I think about various things, in no particular order. Previous posts were about displays, learning generally and maths. Next up – reading.

I’m proposing a model for teaching reading grounded in the various books that I’ve read. The examples will be for a fiction text but I think the principles apply to teaching non fiction too.

Some principles

Good readers combine word recognition with language comprehension to be able to decode the print and understand the language it yields. Once fluent in decoding, it is depth and breadth of vocabulary and general knowledge that contribute to comprehension and so the teaching of reading must develop vocabulary and background knowledge.

Poor comprehenders share many similar characteristics which we need to understand and use to drive the teaching of reading.  Poor comprehenders:

• have limited general knowledge
• have a limited knowledge of story structure or don’t relate events in a story to its general structure
• have a narrow vocabulary and don’t know the meaning of important words
• read too slowly, without fluency or enough prosody to understand the content
• focus on word reading without focusing on content
• make incorrect pronoun references
• don’t make links between events in the text
• don’t monitor their own understanding of what they’ve read
• don’t see the wider context in which the text is set
• don’t build up a secure understanding of the main events in a story
• misunderstand figurative language

When it comes to vocabulary, we can’t teach every word or phrase that children might not know and neither should we. If we do, not only would it be incredibly time consuming but we’d also greatly reduce the experience that children have of deciphering meaning from contextual cues. Some words and phrases need to be taught explicitly before or during reading while others can be learned implicitly during reading.  Either way, if children are to master the language, they must think hard over time about its use.  Put the dictionaries away and don’t start off with ‘Who knows what x means?’  These are both particularly inefficient uses of time and are ineffective.  Instead:

• Model the use of the word in its most common form
• Use an image (this post from Phil Stock is excellent)
• Act it out
• Model other common uses
• Explain word partners (for example, if teaching the word announce you often see make an announcement together)
• Show various forms of words including prefixes and suffixes
• Show words that are similar to and different from the focus word

That last bullet point is not the same as using the synonym model for teaching word meaning.  Telling  a child that melancholy means sad robs them of the beauty of shades of meaning because it is similar to, not the same as sad.

Memory is key. We remember what we think about, so part of teaching reading needs to be giving children plenty of spaced practice in remembering word meanings, general knowledge, events from the text and details of the characters that are crucial to developing a sufficient mental model of the text. It could well be the case that a child who has shown poor understanding of a text is not unable to comprehend it, they just can’t remember what’s necessary to comprehend. Regular low stakes testing of key knowledge from the text is a strategy to ensure this retention and readiness to mind.  Joe Kirby’s knowledge organisers are very useful for this and here’s one I made for Philip Pullman’s Northern Lights.

Stage 1 – oral comprehension

Prepared reading, or providing a brief structural overview, ensures that no child hears the story without some prior knowledge.  In the first instance, read aloud or tell children the story. Capture their interest. Retell it, perhaps in different ways.   Lemov, in Reading Reconsideredidentifies different types of reading and here I’d go for what he calls contiguous reading – reading without interruption from start to finish, experiencing the text as a whole.  It may be sensible to teach the meaning of some words that are crucial for overall understanding of the text but not too many at this stage.  I’ve compiled some thoughts on introducing texts and teaching vocabulary here.

What have children understood?

Clearly it is tricky for teachers to know what children have understood and by asking questions all we really know is whether they are capable of comprehending, not whether they actually comprehend independent of us. Before any specific questioning, it would be useful to get an idea of what they have understood by asking them to tell you about what they’ve just read. The decisions they make about what they say (or write)  reveal what they think is important and you can also judge the accuracy of their literal and inferential comprehension. Aidan Chambers’ Tell me gives advice on developing this in a slightly more structured way whilst still retaining the importance of open questioning.

The key to this stage of reading is the focus on oral language comprehension.  Difficulty decoding should not be a barrier to children experiencing and understanding age appropriate texts.  Lemov puts this beautifully:

Low readers are often balkanised to reading only lower level texts, fed on a diet of only what’s accessbile to them – they’re consigned to lower standards from the outset by our very efforts to help them.

This is one of the reasons why I’m in favour of the whole class teaching of reading and not the carousel type ‘guided reading’.  Listening to texts and using open questions to prompt discussions ensures that the focus in on language development in a way that is not restricted by poor decoding.  Having said that, those children who are not decoding to the standard expected will still need some sort of intervention running concurrently to this so that they catch up.  The benefits of focusing on oral language comprehension have been shown in the results of the York Reading for Meaning Project, written about in Developing Reading Comprehension by Clarke, Truelove, Hulme and Snowling and here.

I’d expect children to then read the text independently, drawing on what they’ve heard from the teacher’s modelling and all the oral language work. Children should have the opportunities for multiple readings of at least the tricky bits.  These bouts of reading become iterative: children build layers of understating with each reading.  For those children whose decoding is weak, they can be directed to smaller extracts, practising decoding and fluency with a text that they should have a decent understanding of following all of the language work.  It’s important to continue to get children thinking about new words that were taught in stage 1.  If that vocabulary is to be reliably internalised, they’ll need multiple interactions.

This is also an ideal point to make some links to non-fiction that can supplement understanding of the fiction. Questioning that involves deliberate comparison between the fiction and non fiction complements understanding of both.  For example, if reading Robert Louis Stevenson’s Treasure Island, spending some time on books or extracts such as below will significantly aid comprehension.

Written responses

Writing is thinking, and to paraphrase Lemov in Reading Reconsidered, not being able to record their thoughts about what they’ve read on paper does not make them invalid, but children are at a significant disadvantage if they are unable to craft an articulate, effective sentence explaining what they have understood.  To this end, returning to those original open questions and working with children to refine their responses and write them effectively is a valuable use of time.  The teacher can model scanning the text for the part needed to refine an idea, or to check a detail, and then children should also be expected to behave in that way.  This post by Lemov makes very interesting reading on that topic.

Stage 3 – targeted questioning

It’s standard practice to ask questions of a text after it’s been read but a great deal of care needs to be taken in choosing or discarding already written questions, or in writing them ourselves. Questions need to be text dependent, otherwise what we’re really doing is getting children to activate general knowledge. An example of this, from Understanding and teaching reading comprehension by Oakhill, Cain and Elbro, is:

Where does Linda’s pet hamster live?

1. In a bed
2. In a cage
3. In a bag
4. In a hat

The possibility of guessing the right answer here would tell the teacher very little of the child’s ability to comprehend text and so asking questions where understanding is dependent on what’s written or what must be inferred from the text is a must. Doug Lemov espouses the importance of text dependent questions in Reading Reconsidered.

When designing questions, teachers must also use knowledge of the characteristics of poor comprehenders in order to model corrective thought processes and to ensure children think in a way that helps them to comprehend more reliably.  For example, we should give them plenty of practice in working out to what or whom pronouns refer.

The education system we work within requires examinations to be passed which then provides opportunities.  Preparing children for success is morally imperative. Write questions in the style of SATs questions about the text, model the thinking process behind successful responses and give children practice doing just that.

Stage 4 – fluency and prosody

Don’t misunderstand – children should be supported continually to read fluently with appropriate intonation and expression. It’s just that to do that well, a reader needs to understand the text. At this stage, that should be the case. Reading for fluency and intonation using a text that children know very well should yield great results and not only that, it provides another opportunity to glean previously missed understanding.

So there it is. A model for teaching a text that moves from oral to printed comprehension; general to specific questioning; and oral to written responses, all the while practising fluency and developing language.

## What I think about…professional learning

Moving schools and with more than an eye on headship is sure to get you reflecting. The following posts are what I think about various things, in no particular order. Previous posts were about displays, learning generally, maths and reading. Next up – professional learning.

With likely a range of often conflicting priorities, deciding what to work on is tricky.  Subject leaders will strive to keep their subject’s nose in front of the rest but ultimately, leaders must be able to zero in on what it is that the children need.  Once that is known, leaders can think about what teachers might need to do differently in order for those outcomes for children to be realised.  The list of things that teachers (could) do day to day is endless so leaders must be able to judge, through experience or by leaning on research, which of those things are worth pursuing and which need to be jettisoned because they take up our time and mental effort for no significant impact.  Research such as that by Hattie is useful but are the interventions described in such research too broad?  For example it is obvious that feedback can have a significant impact on learning but only if it’s done well.  Consider the difference between these scenarios:

• training on implementing a new feedback policy
• training on providing feedback on persuasive writing

Or these:

• training on clear teacher explanations
• training on explaining how to add fractions clearly

There is a difference between being research led and research informed.  Research should be considered in combination with the needs of children and teachers so that leaders get teachers thinking about effective ways to teach.

This would go some way to ensuring that teachers’ subject and pedagogical knowledge is developed, in line with the Sutton Trust report into what makes great teaching. It’s relatively straight forward to ensure that the focus is on those things, however ensuring the impact is a lot trickier. It makes sense for leaders to have from the outset a very clear idea of what they want that impact to be. Phil Stock’s post on evaluating impact (based on  Guskey’s hierarchy of five levels of impact) is very useful here in terms of leaders planning what they want to happen as a result of professional learning and the rest of this post details how one might do that.

Intended impact on outcomes for children

The intended outcomes for children should be set out so that there is no misunderstanding of the standard to be achieved. Using resources like Rising Stars Assessment Bank for maths can help teachers to gather the types of questions that all children will be expected to answer.  The same can be done for a unit of work on reading – find or write the questions about a text or texts, including the quality of response that you’d expect in order to demonstrate age related expectations.  Something similar can be done for writing.  Find or write a piece that would exemplify the standard that you’d expect from children.  Whatever the subject, leaders working with teachers to clarify what exactly children will be able to do and what their work will look like is the goal.

Individual questions would serve as criterion based assessment but for reading and maths, these questions could be compiled into an overall unit assessment and a target could be set for all children to achieve in the first phase of a unit of work. Gentile and Lalley, in Standards and Mastery Learning  discuss the idea that forgetting is the inevitable consequence of initial learning even if it is to a high standard of say 80%+ .  The problem is that for the most vulnerable children, who don’t achieve that initial mastery of the content to anywhere near that standard, forgetting happens more quickly and more completely.  If children don’t initially understand to a certain level, their learning over time is far less likely to stick and will make subsequent planned revision not revision at all but a new beginning.  Therefore, the expectation of the impact on children of any professional learning simply must be that all children achieve a good standard of initial understanding, whether that is judged as absolute through criterion referenced assessment or by a percentage on a carefully designed test.

Now of course, meeting the standard set on an assessment means nothing unless it is retained or built upon. This initial assessment would not be at the end of the unit of work but part way through.   I’d expect, on an end of unit test, higher percentages compared to those that children will have achieved on the initial assessment.  This is because that initial assessment will have served to tailor teaching to support those that require further instruction or practice.  And I’d expect that intervention to have worked.

To summarise, teachers and leaders first set the assessment and the standard to be achieved.  The unit of work is taught until all children can attain the standard, then the unit continues, deepening the understanding of all which is then checked upon at the end of the unit and beyond. The DfE’s Standard for Teachers’ Professional Development (July 2016) identifies the importance of continually evaluating the impact on outcomes for children of changes to practice and so assessments of what children have retained weeks and months after the unit of work are crucial – they ‘ll inform at further tweaks to teaching and professional learning.  When there are clear milestones for children’s achievement, the professional learning needs of teachers comes sharply into view.

Intended impact on teachers’ behaviour

Once it has been decided what the intended impact on outcomes for children is, attention needs to be turned what teachers will do in order for children to achieve those outcomes. Such behaviour changes may be desired at the planning stages of a unit of work, for example in the logical sequencing of concepts related to addition and subtraction over a series of lessons. The behaviour changes may be desired during teaching, for example explaining and modelling how to create suspense in a piece of writing. Finally the behaviour changes could be desired after lessons, for example where teachers receive feedback on how children have done by looking at how they have solved addition and subtraction problems in order to amend the sequence of lessons.  Another example could be providing feedback on their writing to make it more persuasive either face to face or by writing comments in their books.  The key here is that behaviour change is specific to the unit of work.  Having said that, leaders must support teachers to think in increasingly principled ways so that over time, principles can be more independently applied to other units of work and subjects.  As such, intended changes to behaviour must be iterative and long term, with opportunities to make connections between topics and subjects through coaching and shared planning.

For any behaviour change, teachers must see the outcome.  They must see someone doing the things that are expected of them.  This live or videoed teaching needs to be deconstructed and then summed up concisely which acts as success criteria for teachers. For example, in a unit of work on place value, desired teachers’ behaviours could include (and this is far from exhaustive; simply to illustrate the point):

• Plan for scaffolds (and their removal) so that all children can partition and recombine numbers fluently and accurately.
• Intervene on the day if a child shows significant misunderstanding of that day’s learning.
• Use concrete manipulatives and pictorial representations to model and explain the concept of place value.
• Co-construct with children success criteria appropriate to the type of leaning objective (open or closed).

Having such success criteria ensures that both leaders and teachers are clear of what is expected in order for the desired impact on children to be realised. It can also be used to focus practices like lesson study and coaching conversations, which are crucial to keep momentum going and embed change.

Intended impact on teachers’ knowledge

If leaders require teachers to develop certain practices, for many there will be a knowledge gap that inhibits such development. The DfE’s Standard for Teachers’ Professional Development identifies the importance of developing theory as well as practice. Subject and pedagogical knowledge, as well as knowledge of curriculum or task design are all vital for teachers to be able to refine aspects of their practice.   This could be as straightforward as analysing the types of questions that could be asked to get children thinking deeply about place value before teachers write their own which are appropriate to the year group that they teach. Or it could be ensuring that teachers understand and can articulate the underlying patterns of addition and subtraction in the maths unit coming up. It could even be knowing the texts that children will be using for reading and writing in depth in order for them to dedicate future thinking capacity to pedagogical concerns. By setting out the intended theoretical knowledge to be learned and by providing opportunities to gain that knowledge in ways that do not overly strain workload, leaders can set teachers up for successful changes to practice.

Organisational evaluation

For children to improve based on teachers’ developing subject and pedagogical knowledge, there must be great systems in place that allow such development to happen.  Leaders need to be very clear about what it is that they will do to ensure that teachers are supported to act on the advice being given.  Some examples include:

• Making senior leaders or subject specialists available for shared planning
• Arranging for staff to access external training
• Ensuring that observations are developmental
• Planning professional learning using Kotter’s change model

These items become success criteria for leaders implementing long term change.  They can be self evaluated, of course, but external validation of school culture is valuable here.

Reaction quality

The final strand of planning for impact concerns how teachers perceive the professional learning in which they’ll engage. It goes without saying that we’d like teachers to find professional learning not just useful but transformative – a vehicle for improving outcomes for children, personal career development and increasing the school’s stock all at the same time.  One can only create the conditions in which another may become motivated and by taking into account what drives people, we can go along way to ensuring a thriving staff culture. Lawrence and Nohria’s 4-Drive model of employee motivation is very useful here, describing four underlying drives:

The drive to acquire and achieve

If staff are confident that the professional learning will lead to them acquiring knowledge, expertise and success, then they are more likely to feel motivated.  Professional learning then must appeal to this drive – spelling out the knowledge and status that can be achieved through the planned work and never underestimate the power of distributed leadership, carefully supported, of course.

The drive to bond and belong

The school’s vision is key in keeping everyone focused and pulling in the same direction and this can certainly be reinforced with a common school improvement aim as the focus of professional learning.  Finding ways to ensure supportive relationships is crucial.  Culture is the result of what we continuously say and do so leading by example in developing good working relationships will go some to making it the social norm.  Leaders must also look for and iron out any pockets of resistance that could threaten the desired culture.

The drive to comprehend and challenge

This refers to providing opportunities for staff to overcome challenges and in doing so grow.  Setting out each individual’s importance in the school and how they contribute to its success is an example. This is often a long game, with external judgments being made in exam years or in external inspections, so leaders must find quick wins to acknowledge the impact of teachers’ work on the development of the school.

The drive to define and defend

By drawing attention to the good that the professional learning will do not just for the children but in turn for the reputation of the school, we can create a fierce loyalty.  If we get our principles right an articulate what we stand for, this momentum can be very beneficial for implementing professional learning.

This is the job of the leader, striving for improvement in outcomes for children whilst developing staff and building a culture of success. Any professional learning has to have clear outcomes and its only then that they can be reliably evaluated and tweaked to inform the next iteration.

Filed under Coaching, CPD

## 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.

Filed under Maths

## 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.

Filed under CPD, Curriculum, Maths

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.

Filed under CPD, Maths

The question was on the screen:

One year 6 child said: ‘The empty box is in the middle so you do the inverse.  You have to add the numbers together’.

This got me thinking about how children build on their early concepts of number to be able deal with problems like this, which I’ll call ‘empty box problems’.

The underlying pattern of additive reasoning is the relationships between the parts and the whole.   Getting children to think and talk about the whole and parts using concrete manipulatives early on should lay the foundations for them to internalise this underlying pattern.  Every time children think and talk about number bonds, they can be practising identifying the whole, breaking it into parts and then recombining to make the whole once more.

Alongside talking about the whole and parts, children should begin to generate worded statements whilst manipulating cubes or Numicon, for example.  At this point it is important to experiment with rearranging the words in the statement.  They should get to know that ‘four add two is equal to six’ and ‘six is equal to four add two’ are statements that are saying the same thing.  Some discussion around what is the same and what is different about these two statements would be worthwhile.

When children are then shown how this looks abstractly with numerals and the equals sign, this would hopefully go some way towards avoiding the misconception that the equals sign means that ‘the answer is next’.

In the examples used so far, the whole and each of the parts have been ‘known’.  Using the same manipulatives and language patterns, children can be introduced to unknowns.  It seems sensible to begin with giving children the parts and using the word ‘something’ to show that the whole is unknown, i.e., four add two is equal to something.  Some modelling alongside a clear explanation followed by plenty of practice should see children get used to the language patterns needed to think about the concept with clarity.  The next step is to show children the whole and one of the parts, using the word ‘something’ to replace the unknown part.  All of this talk and manipulation of objects is intended to support children to develop a concept of additive reasoning where they do not have the misconception that ‘inverse’ means ‘do the opposite’.

More sophisticated additive reasoning is the understanding of the inverse relationship between addition and subtraction.  Children need to fully understand that two or more parts can be equal to the whole.  From this, they need to internalise the underlying patterns: that Part + Part = Whole and that Whole – Part = Part.  From this, they should be able to work out the full range of calculations that represent one bar model.  Again, it is important to vary the placement of the = sign.

One more way to get children to think about the whole and the parts is to use bar models for calculation practice rather than simply writing a calculation for children to work out.  When done like this, children have to decide what calculation to do to work out the unknown.  Children often exhibit misconceptions such as ‘when you subtract, the biggest number goes first’.  These can be addressed using the underlying patterns; adding parts together makes the whole and, when you subtract, you always subtract from the whole.  When unknowns are introduced, they can be substituted into these basic patterns:

Part + Something = Whole           Part + □ = Whole              35 + □ = 72

Something + Part = Whole           □ + Part = Whole              □ + 35 = 72

Whole – Something = Part           Whole – □ = Part               72 – □ = 35

Something – Part = Part                □ – Part = Part                   □ – 35 = 37

Knowing these patterns will help children to able to analyse problem types in order to decide on the calculation needed.  An additive reasoning bar model with one unknown generates both an addition statement and a subtraction statement.  Showing children empty box problems pictorially, they can talk through the calculations that can be read from the bar model, using the word ‘something’ to represent the unknown.  The next step is to show children abstract empty box problems and get them to map it onto a blank bar model.  They should be drawing on their knowledge that the whole is equal to the sum of the parts and that when you subtract, you always start with the whole.  Eventually, the hope is that the language alone should suffice to work out how to solve empty box problems, with children no longer needing the bars.

Which brings us back to that year 6 child.  Of course, children will develop misconceptions as they make sense of what is shown and explained to them.  By expecting them to think and talk about additive reasoning in the ways described above, it should go some way to building sound conceptual understanding.