Tag Archives: Deliberate practice

SATs not as hard as it looks!

One of my favourite responses when working with with children on tricky problems is, ‘Oh is that it?  It looked much more difficult!’ As May draws closer, children in Year 2 and Year 6 up and down the country are preparing for end of key stage SATs. Tests often invoke strong opinions among teachers. As adults who have typically done well in the education system, tests may never have been a worry and we may see them as a chance to shine and something to look forward to. Others may hold the view that testing children is barbaric and sucks the life out of curricula as schools teach to the test. Either viewpoint, or any gradation in between, does not change the reality that schools are accountable for the success of children on tests. Perhaps more important than accountability though is ensuring that all children, particularly those who are disadvantaged, are able to graduate from an education system that provides qualifications through examinations and have access to wider opportunities in the future.

Every school will be familiarising children with the upcoming tests, most likely by using practice papers, with the aim of children knowing what to expect and in turn doing the best that they can when the time comes. In my experience, there are a number of strategies to do this well and there are also some strategies that could well do more damage than good.

SATs are the ultimate summative test for primary school children and it can be tempting to recreate these summative conditions when preparing children for them. Practice tests done in exam conditions where they receive an overall score at the end have some value but could well set children up for failure, creating anxiety as the high stakes take their toll. Removing the test conditions gives children a chance to learn how to take tests. If the stakes are made lower still, for example by removing the importance of the score achieved, then we can go some way to normalising test situations and therefore reducing the likelihood of anxiety.

It is very easy to get hold of past papers and although the examples here are maths questions, the principle applies to reading, spelling and grammar tests too. One important first step in teaching test technique is to model what a successful test taker does and verbalise their thoughts. Displaying certain types of question, saying what you’re thinking and showing what is appropriate to record is crucial to encouraging children to do the same. From this modelling and explanation, teachers can co-construct success criteria for how to go about the test. The criteria will be a selection of tools to choose from depending on the question being tackled. Sure, those who are successful in tests know subject content very well but by explicitly showing what it is that successful test takers do, we can unlock the mystery of how to be successful. Looking at the KS2 sample tests, the success criteria for answering those types of questions might be:

paper-1-sc

sc-2

Over time, advice like this can build up and if children can internalise it, they will be equipped to deal with tricky problems. Of course, strategies like this are no use without good content knowledge but when combined, set children up to succeed.

Once strategies have been modelled, children can be set off practising. Again, it’s tempting to give children their own paper and have them complete it as they would have to during the test. However, it becomes a much more valuable exercise if children talk about what they’re doing so getting pairs to complete papers collaboratively gives them an opportunity to talk and hear how someone else goes about tackling a test. A few guidelines help to keep them focused:

  • Both work on the same question.
  • Agree an answer before moving on.
  • If you disagree with your partner, explain why you think you’re right and listen to their explanation too.

One of the stressors of testing is the time constraint. When children are practising test techniques there is no need for such constraints. Over time, they’ll get quicker and the strategies they work on will become more autonomous. At that point, time restraints can be put in. For example, you might set the target of getting to question 6 in 10 minutes or halfway in 15 minutes.

We’ve all experienced that frustration of seeing children answering a question wrong in a test. This doesn’t have to be the case when they are practising and like in any great lesson, teachers react formatively to the information before them. If everyone is struggling with question 4 about fractions of numbers, then stop them and teach them how to do it, give them a few extra practice questions and make a note to return to it soon. If it is just one pair or a handful of children struggling, then a little scaffolding, followed by some more practice will help. The example below comes from the KS1 sample test:

KS1 Maths SATs.png

Having seen that this pair of children did not know how to approach the question, the teacher explained that division can be seen as sharing and that this is asking to share 35 into 5 groups. The teacher, in blue pen, drew five groups and began sharing one at a time before the pair completed the question. Now evidently that won’t be enough for that pair to have understood completely so it can then be followed up with sufficient practice to internalise the idea.

Once children have completed the practice tests, teachers will be keen to know the score they achieved as well as looking for specific detail about which questions and topics children struggled with. The well-worn phrase ‘Check your work’ will I’m sure be repeated countless more times with varying levels of patience but that means nothing unless children are explicitly taught how to do so effectively. The way that test are marked can encourage the habits of checking. The most structured way would be to mark each question with the number of marks awarded:

Mark the page.png

When scripts are marked this way, children can see which questions they were successful in answering and which they got wrong. When the tests are returned, children can look for the questions they got wrong, and if it was a case of making a mistake, can discuss what happened with their partner and make the necessary corrections.

This may be a sensible place to start but of course it makes children reliant on the marking to see where mistakes have been made. A gradual removal of that scaffold could involve marking the score for each page rather than individual questions:

mark-the-page-2

In this example, out of the 3 marks available for the questions on this page, the pair of children scored 1. It is then down to the pair to re-read questions to first of all determine which are incorrect and secondly to work through it again to see what went wrong.

A third option, to remove the support a little further, would be to count up the total marks, only telling children something along the lines of ‘You scored 33 out of 40. Find and correct the mistakes.’ It goes without saying that these marking strategies push for corrections of mistakes and will do no good if the child never knew the content well enough in the first place.

Test papers are valuable resources to use in the classroom, not least because of the teaching opportunities for test technique that they allow. One subtle but significant benefit is the varied practice they provide too. During maths lessons, the focus may be narrowed to one objective or concept, and rightly so to provide focused support and practice. Tests’ varied questions though provide a great opportunity for revision, to interrupt forgetting and to provide teachers with a wealth of information with which to inform future lessons.

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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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