Tag Archives: Maths

Principles and practices of effective homework

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

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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…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 and learning.  Next up – maths.

Any unit of work should be planned with end points in mind.  Teachers should start with the relevant National Curriculum statements but more importantly, the types of questions that children will be expected to be able to solve.  Teachers in Y2 and Y6, with their experience of end of key stage testing, may well have internalised the type of questions that would be appropriate for children to solve but resources like the Rising Stars Assessment Bank are invaluable for teachers.  Our expectations of what children will learn are vital. All children are capable of achieving age related expectations given the right support and sufficient time. If we begin a unit with lowered expectations for some children based on perceived ability then we are failing them.

Once the expectations of what children will be able to do at the end of the unit are clarified, teachers should then plan backwards, thinking carefully about what children will need to know and be able to do in order to solve those problems as well as figuring out a sensible conceptual sequence of those things.  For me, these include fluency with number, learning the underlying structure of the problems to be solved, the deliberate teaching of mathematical language and opportunities to reason.  The more I think about this, the more I’m settling on a sequence of development that units of work should be structured around, key parts of the sequence are developed and consolidated over time:
Stages of a unit of work

This model may not work for all topics but I’d suggest that units of work start with pure number.  Contexts can be stripped away to reduce load on working memory and children can get on with learning and practising fluency so that as soon as possible, they are able to recall necessary factual knowledge and manipulate numbers in calculations with little mental effort.  This is by no means rote learning – it should be carefully thought out so that children develop sound conceptual understanding, starting with concrete representations, progressing to more efficient pictorial representations and then on to even more efficient abstract representations.  It is important that teachers remember that the abstract representations are not the maths itself, merely the most efficient way of recording or communicating the thinking.

Fluency

If children can recall number facts and other basic mathematical knowledge within a few seconds, if children can calculate reliably without expending too much mental effort, and if children can recall varied mathematical knowledge, switching between topics, then they will be considerably more able to commit precious working memory capacity to problems that require deeper thinking.  It is for this reason that fluency must come first and continue to be practised in order for that recall to become increasingly accurate and efficient.  Flash cards are very useful here – they provide the opportunity for self-testing and, with a little training, can help the child to become more aware of the what they know and do not know, enabling them to focus their own study.  Teachers figure out what basics are required and deliberately teach those basics if they are not sufficiently internalised already.  If some children are already fluent, they can work on speed and efficiency, for children may well have fluent but inflexible strategies.

Underlying structure

Stories can be said to be a variation or combination of just seven basic plots and expert writers have a sound knowledge of these, enabling them to see stories at a deeper level.  I’d argue that there is a significant similarity in maths; that there are five basic problem structures.  These five structures are aspects of either additive or multiplicative reasoning and are classified based on what is unknown in a problem:

Maths structures

Knowing the structures is not sufficient.  Children must be able to identify the underlying structure from a given problem.  This is no mean feat so sticking to just number problems and avoiding distracting contexts for the time being is important.  Take a problem like this:

Empty box

Hattie and Yates, in Visible Learning and the Science of How We Learn, said that experts see and represent problems at a deeper level.  A novice will only see the surface features in this problem: two numbers and an addition sign.  Consequently, they’ll solve this by adding the two numbers together.  Of course, this is a mistake.  An expert, on the other hand, will know that the whole is made up of parts and that you add the parts together to make the whole.  They may even ignore the numbers at first and read it as part + part = whole, realising that it is a problem where one of the parts is unknown.  They may see it or draw it like this:

Bar model

They’ll use their knowledge of the relationship between the whole and its parts, plus the idea that to find a missing part, you subtract the known part from the whole, therefore calculating 564 – 327 to find the unknown.

It is this kind of thinking that we must get children to do.  Maths lessons should be planned with a sole priority – what will the children be thinking about?  In this first stage of a unit of work, children develop their fluency and then begin to reason about what is unknown in a numerical problem and how to figure out that unknown.

Deliberate teaching of language

The second stage of my model involves building on the number work by adding layers of mathematical language that enable children to talk like mathematicians and understand problems involving ambiguous language. We need to embrace ambiguity because it is in that grey area of language that we can really get children to think hard.  The image below is one I’ve seen many times in many schools and even had up in my classroom in a previous life:

Language.jpg

What this kind of display tells children is that a word equates to an operation.  This is misleading at best and more likely disastrous for understanding.  Words like this only carry meaning in context, for example look at the phrase ‘more than’:

More than

It does not mean that you have to add the numbers together.  It can mean that, but it could also mean finding the difference.  And what of the word difference?

Difference

The ambiguity of language must be deliberately taught and linked back to the underlying structures that children will have been working on.  Teachers model the thinking and ask: In the first  question, is the whole unknown or a part unknown?  What about the second sentence? Draw it…

Wider problem solving

Remembering Hattie and Yates’ assertion that novices see surface features of problems and experts see the same problems at a deeper level, consider this:

Anum and Jay have saved up their pocket money.  Altogether they have £35 and Anum has saved £18.50.  How much has Jay saved?

A novice would read that question and would say it’s a problem about pocket money – the surface feature.  An expert would look at the same problem and say that it’s one where the whole is known and so is one of the parts, but that the other part is unknown.

Once teachers know how experts think, it is perhaps a mistake to simply try and get novices to think like that.  Experts have a vast store of knowledge from which they draw on when analysing problems and so to get novices to eventually think like experts, we must first teach them the underlying patterns.

With this in mind, take a look at this common practice:

Subtraction problems

If we teach children how to subtract, then give them problems that only require subtraction, what are children really thinking about?  Not structure.  Not language.  And yet we can still mislead ourselves when children ‘get the right answers’ that they truly understand what they’re doing.  They may well do but we can’t be sure with tasks like this.  The tasks that we set show what we value.  Perhaps a better task is to ensure that children are thinking like mathematicians, sorting then solving problems based on their underlying structure:

Sorting 1

Sorting 2

Throughout the entire unit

Other considerations during a unit of work include the big ideas in maths.  Coined by Mike Askew, these are concepts that children develop throughout their time at school and are built on year on year:

  • position on a number line
  • estimation
  • equivalence
  • place value
  • numerical reasoning
  • the meaning of symbols
  • classifying
  • sequences

Opportunities should be created throughout a unit for children to think about content in these ways so that they can make connections between ways of thinking and different representations.

Finally, but by no means of least value, teachers must pay careful attention to success criteria. Before this though, objectives need to be sound. Shirley Clarke’s work on formative assessment is important here and there should be clarity between closed and open objectives. Closed objectives are absolute. They have either been achieved or they have not. Procedural success criteria are most appropriate here – steps to follow in order to be successful. Open objectives on the other hand are subjective in that they can be achieved to varying degrees of quality. A selection of tools is most appropriate for success criteria in this instance – strategies to choose from with the goal of efficiency, for example. Year 5 children could be given the calculation 5023 – 3786 and they should be able to, following great teaching, choose a subtraction strategy to solve that calculation in the most efficient way possible. All the procedural work, such as deliberately learning to count up, count back, round and adjust, or carry out column subtraction comes together and all those strategies form a toolkit from which children choose he best tool for the  job.

This model for teaching maths certainly covers National Curriculum aims but more importantly, it strives to get children to think and communicate like mathematicians.

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The teaching of fractions

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

Prior concepts

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.

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:

Covered strips

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.

Mixed number

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.

Complements

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

Complements 2

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.

Comparing

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:

Inaccurate bars

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.

Fraction wall

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.

Number line

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

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

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

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

A fact a day for a term

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

Multiple representations

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

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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