Tag Archives: Deliberate practice

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.

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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.

What should leaders prioritise?

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
  • Providing access to a coach (and training for coaches)
  • 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.

 

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What I think about…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.  First was displays.  Next up – learning.

Asking teachers what learning is surely throws up disagreements of varying degree from polite dispute to outright warfare.  What makes sense to me is that learning is a change in long term memory.  Too often, children don’t manage to transfer concepts from working memory to long term memory and without that internalisation, we cannot say that they have learned.  All we can say is that they have done some work.  Now that work might well have been good, but teachers and leaders need to be aware of the difference between short term performance and long term internalisation.

Performance vs learning and the importance of desirable difficulties

The key paradox is that to improve long term retention, learning has to be made more difficult in the short term even to the extent of being unsuccessful.  We remember what we think about and learning happens when we have to think hard about content.  If children are thinking about things other than what we have intended for them to learn (a distracting context, for example) then that’s what they’ll remember.  If they haven’t had to think too hard, they may well produce some decent work but the thinking behind it is less likely to be retained.  So what does this mean?  Units of work and individual lessons need to be planned around what it is that children will be thinking about.  Each decision about what the teacher will do and what the children will do needs to be justified with that question mind and amended accordingly.  We all get better at what we habitually do – we become more efficient – so if we require children to be able to remember knowledge, procedures and concepts, we must give them ample opportunities to practise remembering those things.  The efficacy of the testing effect has robust evidence and seems to work because testing (either yourself or a teacher posing questions) triggers memory retrieval and that retrieval strengthens memories.  Flash cards are a perfect example of this in action.

What’s important is that this testing is low stakes – no grade, no mark at the end of it, just practice in remembering and feedback on responses.  Feedback can take two forms.  Firstly the feedback can be from teacher to child and is as simple as telling the child what they were good at and what they misunderstood, then correcting those misconceptions.  Secondly, feedback can be from child to teacher and involves the teacher using the information to plan what to do next to develop understanding further.

Low stakes testing is a desirable difficulty – one way of making learning difficult (but not too difficult) so that children have to think hard.  Other desirable difficulties apply more to curriculum design:

  • Interleaving (switching between topics)
  • Spacing (leaving some time between sessions on a particular topic)
  • Variation (making things slightly unpredictable to capture attention)

By presenting content to children little and often, with increasingly longer spaces in between, teachers can instill the habit of continual revision rather than only revising when some sort of exam is approaching.  As such, concepts are internalised and retained rather than forgotten.  Robert Bjork’s research on desirable difficulties can be found here:

Knowledge

The idea of knowledge can be divisive.  Recalling knowledge is often described as lower order thinking and many are keen, quite rightly, to get children to do higher order thinking. This can be dangerous because knowledge is necessary but not sufficient.  Higher order thinking skills rely on a sound basis of knowledge and memory so teachers must ensure that these aspects are fully developed before expecting success in higher order thinking.  Knowledge needs to be internalised too.  It’s not enough to be able to Google it.  The more a child knows, the easier it is to assimilate new knowledge because more connections can be made:

Knowledge

Scaffolding

Children are more alike than different in how they learn.  Attempting to teach to a child’s perceived learning style is nonsense.  Everyone, no matter what we are learning, requires three things: knowledge, practice, and feedback on how we’re doing.  It is of course true that children come to a lesson with varying levels of prior knowledge and to a certain extent have different needs in order to be successful.  Teachers may have (and many, I’m sure, still do) differentiated tasks three, four or more ways – an unnecessary burden on time and a practice that reinforces inconsistency of expectations, particularly of the perceived ‘lower ability’ children.   For those children that are behind their peers, if they are not supported to keep up with age related expectations, they will be perennially behind and will never catch up:

Keeping up Differentiation

If we only cater for their next small step in development, we’re failing them.  Instead, all children should be expected to think and work at age related expectations.  Teachers should scaffold tasks appropriately so that all can work at that expectation and we do not have a situation where ‘that’ table are doing something completely different.

Scaffolding

For children that grasp concepts quickly (not our ‘most able’ children – heavy lies the crown…), teachers provide opportunities to deepen their understanding before acceleration into subsequent year groups’ content.  Undoubtedly, there are a small number of exceptions to this.  There are some children that have a lot of catching up to do before we can even think of getting them to keep up with age related expectations.  But if they are removed from lessons to carry out this catch up work, then everything will always be new to them – they’ll miss seeing and hearing how children are expected to think and work.  It is much better to precisely teach, and get them to practise, the basics that are not yet internalised in short bursts and often so that they remain with their peers as much as possible, experiencing what they experience but having the support needed to catch up.  This could be basics such as handwriting and number bonds, for example, and teachers should work closely with parents where there is a need to catch up to set short term, focused homework until the basics internalised.

Intervening

When children misunderstand something, when the work in their books is not to the standard expected, is a crucial time.  Paramedics talk of the golden hour – one hour after an accident – where if the right treatment is given, the chances of recovery are significantly higher.  With children’s learning, if we leave misconceptions to embed or even thrive, we’re failing them.  Even if we mark their books and write some wonderful advice for them to look at and act upon the next day or the day after, we leave holes, holes which children can slip through.  When there is a need, we should intervene on the day so that children are ready for the next day’s lesson and are keeping up.  This of course requires flexible and creative used of TAs and non-class based staff but from experience, it works. Interventions focus on the work done that day.  For some children, pre-teaching may be more beneficial.  Before the school day starts, they are shown the main content of the day’s lesson and carry out a couple of practice examples so that when it comes to the lesson later on, they have some prior knowledge which will improve their chances of success in that lesson.  This concept is in contrast to pre-planned, twelve week intervention programmes where children are removed from other lessons for significant periods of time.

Learning is complex and relies on many interrelating and often unpredictable conditions.  That said, there is much that we can control and doing so greatly increases the likelihood that what we intend to learn is learned – really learned.

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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:

TT numicon

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:

TT bars

Number line:

TT 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:

TT Array 1

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

TT Array 2

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:

Array 1

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:

Times tables facts

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.

Flash card

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:

array 2

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:

Multiplicative reasoning2

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:

MR3

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:

MR4

MR5

MR6

MR7

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.

bar and no line

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

 

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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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