Tag Archive | representation

# Summing Up – Teaching Addition

Addition is perhaps the most straightforward of the four operations to understand, but that doesn’t necessarily mean it’s always easy to teach.  I was speaking to a Year 5 teacher earlier this week who had planned to start this term by spending a couple of weeks revising the four operations with her (lower set) maths group.  She’d planned only a couple of days on maths, but after two days still felt there was a lot the children weren’t understanding and decided to keep going for the rest of the week and possibly into next week too.

Having a good understanding of the progression of skills for addition can help when trying to ‘unpick’ why older primary children are having difficulty with it.  Are they familiar with the vocabulary around it?  Do they have good mental images of what is happening when they add?  Are they being let down by a shaky grasp of basic number bonds which leads to mistakes in some steps of longer methods?  Is there understanding of place value secure?  Have they got a good conceptual understanding of the method they are using rather than trying to remember a ‘trick’ they were taught last term or last year?

Children’s early experiences of addition should be very practical and the idea should be introduced in a meaningful context?  They need to understand addition both as combining two sets of objects and as adding more to an existing set.  Early Years environments tend to be full of opportunities for them to do this.  We have 4 red cars and 3 blue cars, how many cars altogether?  Seth has 3 sweets and Tami gives him two more, how many does he have now?  After lots of experience with concrete objects, children may be ready to move to using representations.  So instead of using actual cars or sweets, cubes or counters might be used to represent them.  Another step beyond this is to use pictorial representations, such as drawing circles or dots to represent objects.  One very powerful representation for addition is the bar model (as above) which clearly shows two parts making one whole.

After lots of experience of combining and augmenting sets, children will begin to learn some of the number facts, beginning with those within ten.  Working with tens frames, Numicon or Cuisenaire can all be helpful when doing this.  Tens frames and Numicon particularly reinforce the idea of ten being two lots of five, helping children to build mental images of numbers.  This program is useful for demonstrating the use of tens frames.  Begin by adding on one more and then two more – if children are used to counting, this should come fairly easily.  After a while, they will need to start learning the number bonds to 10 and within 10 – doubles can often be a good place to start with this.  For some ideas about learning number bonds, one of my blog posts from last year has some ideas.

Before children start to add larger numbers together, they should be familiar with the idea of place value and have experienced making numbers with Dienes or other base ten equipment. As the totals they add go beyond ten, they can use the Dienes units to model this and start to exchange ten unit cubes for a tens rod.  Later, children can start to add single digit numbers to 2-digit numbers using the Deines.  It can be helpful to also model how to do this using hundred squares.

When adding 2 digit numbers, again it is helpful to start by using equipment so that children see the way that partitioning and recombining works in practice.  This can later be recorded using lines and dots to represent the Dienes rods and cubes.  The next step is to record the partitioning and recombining process, eg.

and later still to more formal column addition.

The important thing is not to move children on until they are secure at any particular stage.

The same process can then be applied to increasingly large numbers so that by the time children get to the top end of primary they are able to add numbers with at least 4 digits.  When children are familiar with decimal numbers and their place value, the same methods can be used to add decimals too.

# Order, Order – Reading, Writing and Ordering Numbers

Early in the term is a good time to go over the basics of reading and writing numbers and putting them in order.

Young children need plenty of practice in reading, writing and representing numbers, and this can usefully be part of a ‘Number of the Day’ activity like this one.   It’s important that children start developing their number sense alongside this and representing numbers with practical equipment like Dienes, Numicon or ten frames or by drawing tally marks will help them to do this.  The Gordons program ‘Dienes and Coins’ is useful for these representations and this site has some nice interactive ten frames.  As children get older, they need to learn to understand much bigger numbers.  Children are often fascinated by really large numbers, and once they get the hang of how the number system works, more able KS2 children will enjoy trying to read and write multi-digit numbers.  This Wikipedia page lists the names up to centillions.

Number tracks and number lines can also be useful in helping children get a sense of the relative size and position of numbers.  The Mathsframe site (which I love) has a really useful activity where children put numbers on number lines.  There are lots of different levels at which to use this activity and the option of showing divisions on the line or not.  Older children need to also get a sense of how decimal numbers work and the Decimal Number Line ITP is very useful for this.  The programme allows us to ‘zoom in’ on a portion of the number line and expand it to look at what happens within that portion.  Children in KS1 will also be starting to find numbers on hundred squares.  One useful activity is for children to cut up a hundred square along the horizontal lines and then lay out the rows end to end to make a 0-99 (or 1-100) number line.  This helps them to see the connection between the hundred square and number lines and shows them why we skip to the next row when counting on over a tens barrier.  Putting together ‘jigsaw’ pieces of the hundred square can be useful for children in developing their understanding of how these work, and Nrich has a jigsaw activity which can be used either in its interactive version or in printed form.

Ordering numbers is also an important skill.  When the NNS first came in, washing lines of numbers were standard in nearly every classroom and these are well worth using still.  Having sets of number cards of different sizes means that these can be regularly changed.  Children enjoy putting numbers in order on these or spotting numbers which have mysteriously been switched overnight.  The Gordons ordering program uses this idea and again Mathsframe has some good ordering activities.  Some of the levels on this are only available to subscribers but a subscription is good value in my opinion as there is a wealth of resources on this site.

Reading, writing and ordering numbers links well with work on place value and there are ideas for that at my blog from last year and there are other ideas for teaching number on my Number Pinterest board.

# Picture This – The Importance of Representation in Teaching Maths

Last week, I talked about the great importance of building conceptual understanding in teaching maths and how fluency should build on this understanding rather than be based on teaching procedures without understanding.  One of the most powerful ways of doing this is by using concrete materials and representations and there are a wealth of these available to us.  When I was in school (admittedly rather a long time ago now), the only concrete materials I can remember are some shells and counters we used to help us do ‘sums’.  There was very little in the way of representation either – possibly shapes and fractions might be illustrated by diagrams, but otherwise little comes to mind.  Admittedly, I did manage to learn maths despite this, but even with the benefit of a maths degree, I found that some mathematical concepts became much clearer when I started teaching them and discovered representations that would support me in this.

This week, I have been reading a very helpful book by Tandi Clausen-MayTeaching Mathematics Visually and Actively introduces a whole range of concrete and visual material to support teaching maths in different areas.  Clausen-May argues that visual and practical approaches are vital in teaching children who may have struggled to learn maths in a more abstract way and the book is aimed mainly at teachers of these groups.  However, I believe that these approaches are actually beneficial for children of all abilities.  I want to be upfront and admit to being sent a copy of the book by the publishers for possible review, but I have no hesitation in recommending it.  The book is divided into chapters for several different areas of maths and for each introduces some key ways of using visual and practical approaches.  I am always keen to use this sort of approach in my own teaching, but I found here some useful reminders of approaches I was already familiar with, together with some that were new to me.  As well as key representations and materials for each area, there are also practical ideas about how to use these in the classroom and suggestions for further reading.  Information is also given about online tools and information, or in the case of concrete materials, guidance as to where these can be obtained.  As a bonus, a CD is included with the book, on which can be found useful printable materials and powerpoints.

In schools today, lots of visual and active approaches to teaching mathematical ideas can often be seen in Early Years setting and in KS1, but much less in KS2 and beyond.  Where representations and concrete materials are used it is often with less able children.  Children can then become reluctant to use these because they see them as ‘babyish’.  We need to use these approaches much more routinely, so that this sort of stigma is not be attached to them.  Admittedly, some of the concrete materials will need to be bought, but arguably this is a much better use of our budget than buying text books or photocopying worksheets.  Many can be fairly simply made or printed off and in many cases there are interactive versions available (although caution needs to be taken that these don’t completely replace the ‘hands on’ experience of manipulating objects which is so important in the early stages of learning a new concept).

I have started a pinterest board which includes some of my favourite concrete and representational resources and I hope to be adding to this regularly as I remember and come across others.

# In with the New – The New Primary Maths Curriculum

Like it or loathe it, the time is coming when it will be impossible to ignore the new curriculum (unless of course you teach in an academy).  Year 6 will have another year to continue with the old curriculum but other year groups need to start teaching from it from next September.

I am currently taking the NCETM Professional Development Lead Support course (which I would so far highly recommend) and had my first residential training at the end of last week.  In the main I found this somewhat reassuring.  I am sure that Michael Gove had a heavy influence in determining much of the content and in particular the emphasis on the aim of fluency with recalling facts and using procedures, and generally higher expectations by the end of the primary years.  Despite this, the three overarching aims are difficult to argue with, focusing on fluency, reasoning and problem solving.  The NCETM approach is to emphasise that fluency can only be achieved, and should only be achieved by building on a foundation of good conceptual understanding.  Their training and the training that we in turn will be passing onto schools explores the key role that representation and the use of concrete apparatus has in building up this conceptual understanding.  They are also keen to encourage teachers to make connections between different mathematical ideas in their teaching.

My worry is about how well this message will be conveyed to schools.  I have had two years of training as a Primary Maths Specialist, another year of work towards my masters in primary maths education, training as a Numbers Count teacher and have done lots of reading and research in addition to this.  I understand the importance of representation and of making connections.  I have seen the damage that can be done when children are moved too quickly to working with abstract mathematical procedures before they have been able to build up their conceptual understanding to support this.  I have experienced those wonderful ‘light bulb’ moments with KS2 children who have fallen well behind and lost all confidence in their mathematical ability, but given the chance to step back a little and revisit concepts of place value or calculation using concrete apparatus, suddenly see how it works.  Many of my colleagues however have not had these opportunities.  I’ve learned so much from the high quality professional development I’ve received in the last few years and could probably fill at least a year’s worth of weekly staff meetings by sharing all of this.

In most schools, professional development time is very limited.  Maths has to vie with many other subjects and priorities for staff meeting and Teacher day time.  Courses can be expensive and require teachers to be covered which adds to the expense, and budgets are limited.  In my opinion, however, it is good quality professional development which has the potential to make a huge difference to the quality of teaching and learning in schools.  If even half the time and money which is currently spent on inspecting, monitoring, evaluating, tracking data and gathering evidence was spent instead on good quality CPD, I believe the impact would be incredible.

The introduction of the new curriculum could be a great opportunity for schools to revisit their teaching approaches, to ensure teachers are clear about progression and route ways, to explore the range of concrete apparatus and representations which will support conceptual understanding, to explore the links between different mathematical ideas and to share approaches and ideas.  But this will require significant investment of time and money.  I suspect, however that many schools will not find the resources to do this and instead the new curriculum will be presented as a list of requirements with the result that many teachers will feel under pressure to move children on too quickly, which could lead to even less conceptual understanding.

In his (always helpful) blog yesterday, Derek Haylock also made the very important point that the format of the new assessments (currently being developed) will have a great influence on what is actually taught in schools.  Will these assess children’s understanding of underlying concepts, their ability to reason mathematically, their ability to apply their skills to problems?  Or will they focus on assessing the children’s ability to use mathematical procedures fluently?

# Less is More – Supporting Less Mathematically Able Children

Last week’s #mathscpdchat focused on what we could do to support less mathematically able children.  It’s an important issue for teachers.  Poor numeracy skills put children at a definite disadvantage in life as outlined by National Numeracy here.

In my experience of teaching less able children, I have often found that one of the main problems is that they have been moved onto abstract methods and thinking too quickly, before they have really got a strong sense of number and good mental images to support their understanding.  Pressure to prepare children for assessments contributes to this, but we need to be aware that if we move children on too quickly, we are often trying to build understanding on very shaky foundations and sooner or later the cracks will show.

In last Tuesday’s discussion, we agreed that building up basic number sense was essential.  Ideally, this starts to happen in Early Years Settings and KS1 with lots of use of concrete apparatus and representation, but in KS2 and beyond, the use of manipulatives and images remains an important tool in building up understanding.  The CRA approach to building understanding is a good one to bear in mind.  We start with Concrete apparatus, move to Representation when children are more confident and finally to Abstract when children have a firm grasp of what is happening, linking each step to the previous one.

I’ve found ten frames and Numicon particularly useful for helping children to build their number sense, but Cuisenaire, multilink and Base ten equipment can all be helpful too.  Another way of building this is by regular use of dot talks in the lower years of primary and number talks at higher levels.  In dot talks, children are presented with a pattern of dots and asked to work on their own to calculate how many dots there are in all.  Then the class or group discuss the different ways they worked this out.  This helps children to see different ways of breaking up numbers.  Number talks work similarly.  Children are given a calculation and initially work on their own to solve it.  Then the class discusses the different approaches.  Again this helps children see that there can be multiple approaches to the same problem and that no one way is the ‘right’ way.  They may also start to see connections between the different approaches.

@School-LN reminded us of the importance of making connections, and suggested an interesting way of helping children to do this.  Children are given sets of numbers, shapes or bar charts, for instance, and asked to sort them into groups and then explain their choices.  For less able children, maths can seem to be a lot of disconnected facts and procedures that they have to learn, but if we can help them to make connections, they start to realise that there is much less to learn than they feared.  @PGCE_Maths suggested the report ‘Deep Learning in Mathematics’ which is well worth reading and argues the case for focusing on connections and relationships in maths rather than technical procedures.

@Janettww had some experience of using 3 act maths lessons, where students devise their own questions before attempting the maths and has found it very motivating for students at all levels of ability.  This seems to be something that could really promote mathematical thinking.

@bm332 also raised the important issue of classroom climate.  Many students really lack confidence and it’s important that they feel able to speak up when they don’t know or don’t understand something;  @Maths4ukplc also pointed out that mistakes need to be valued as learning opportunities.

So altogether, lots of food for thought and lots of good ideas.  The complete record of the discussion can be found on the NCETM site here and I’ve also put together a pinterest board which includes some of the resources mentioned together with some other ideas.

# Wonder Walls – Maths Working Walls

Have you ever had the experience of looking for a household object, knowing you’ve seen it somewhere but unable to remember where, then finding it in a place that you walk past several times a day?  If things are there long enough and we don’t make use of them, they become ‘wallpaper’ and we often stop noticing them altogether.

Unfortunately classroom displays can suffer from the same fate.  We can spend hours in the Summer holidays putting up impressive displays, but if we don’t ever refer to them, sooner or later our children will stop noticing they’re there, let alone making use of them.  This is where working walls should come into their own.  The idea of a working wall is that it should be full of things that will support children’s learning and help them to learn more independently.  They should be constantly changing to match our current topic.  I appreciate this can be difficult to achieve in the life of a busy teacher and so my top tips for saving time would be:

• Keep things simple – there’s no need for triple mounting and laminating (unless it’s a resource you will use again and again), as long as it’s legible and clear.
• Keep everything – devise a system for filing away your resources so you can dig them out next time you teach this topic.  I usually keep things in folders labelled by topic.
• Make use of printable resources – lots are available from sites like Teacher’s Pet and Communication4All.
• Get the children to help – independent or homework tasks could include making posters about your current topic, showing how to use a method or illustrating some new vocabulary.

What should be included on a working wall?  This might vary according to the age of your children, but might include:

• Vocabulary related to your current topic (the very useful Cheney Agility Toolkit has this editable word wall which you could use)
• Relevant models and images
• Worked examples of methods – these can be screen shots from your whiteboard or photocopies of children’s work
• Problems and challenges – make these interactive if possible, perhaps by children responding on sticky notes (Nrich have some good posters that could be used for this)
• Number lines or washing lines related to your current learning (eg. lines counting in hundreds or in decimals or in multiples of 2)
• Examples of children’s work (What A Good One Looks Like)
• Real life examples of your current topic (again this is a good task to give for homework – ask children to look for eg. examples of circles, or bar charts or timetables and bring them in)
• Photos of children working on practical tasks
• Practical resources that children can use (eg. mirrors, hundred squares, number lines etc.)
• Success criteria

Whatever you include, make sure you refer to it often and wherever possible refer children to it when they need help.

The Educationworks site has several examples of good working walls for both maths and literacy and there are other examples and ideas on my Working Walls Pinterest board.

# Stocking the Toolkit – learning number bonds and tables

My older daughter is now in her second year of teaching.  Just before she began her PGCE course two years ago, I had a lot of fun putting together a ‘teacher toolkit’ for her as a present.  It contained lots of useful teacher tools: sticky notes, staplers, useful teacher books, laminator, highlighters, paper cutter, lolly sticks etc.  She’s found it very useful, but it did nothing to show her how to teach (except for possibly the teacher books).  She needed her PGCE course, and most importantly experience in the classroom and observation of others for that; as a thoughtful and reflective practitioner, I know she’ll be honing and adding to her skills throughout her teaching career.   Similarly, my plumber could loan me his toolkit for a day and I still wouldn’t be any nearer to fixing the dodgy radiator in the bathroom.

Rapid recall of number bonds and tables facts is a very useful tool in any child’s mathematical toolbox.  When tackling word problems, for instance, it reduces the cognitive load for a child if they can focus on visualising the problem and how to solve it without the distraction of having to work out number facts from scratch each time.  However, there’s no point in having these tools available if the child has no idea how to use them or how they relate to the world around them.  So, having learned the facts, it’s vital that we give our children lots of opportunities to use and apply them, doing this in ‘real’ contexts across the curriculum wherever possible.

It’s also crucial that children understand what these facts mean.  When I was in primary school ( a frightening number of years ago), we all learned our tables by rote but I suspect many children who could find the answer to 9×7 in an instant, had no real idea that they could use the answer to work out how many days until Christmas when told they had 9 weeks to go.  So, before trying to memorise any number facts, children should always have plenty of experience of combining objects in different ways, both using concrete objects and visual representations.  This idea, for instance, shows how a multiplication fact can be represented as repeated addition, arrays or groups of objects and also uses the commutative rule to generate a related fact.  For addition facts, children need lots of opportunity to explore numbers and the different ways they can be broken up into different parts.  Activities like this one using number bond bracelets or this one using number spiders should be a staple in KS1 classrooms (and probably in KS2 for children who still haven’t got good mental pictures of numbers).

With these foundations in place, we need to think about how our children are actually going to learn the facts.  For this, there really is no substitute for practice, but we can at least make this practice as painless as possible.  In fact, many children enjoy the feeling of mastery as they see their mental stock of number facts increasing and become increasingly fluent and rapid in their recall.  There are lots of games and activities, both concrete and online, for reinforcing number bonds and tables and my pinterest board has lots of ideas for this.  One proviso I’d make though is that whilst many children respond well to working against the clock, some definitely don’t and for them activities which don’t involve time pressure will probably be best.

To make the task more manageable too, we need to explicitly teach children that lots of facts are related which cuts down significantly on the number of facts that need to be learned.  Using fact family triangles and generating fact families so children learn that with number facts it’s ‘Buy One Get Three Free’ should be a regular part of the classroom routine.

If you’ve read my other posts, you might know that I’m a big fan of daily counting using a counting stick.  This video shows how Jill Mansergh used a counting stick to teach a group of teachers at an ATM conference the 17 times table.  Even Mr Gove doesn’t advocate us teaching the 17 times table in primary school (although give him time), but the basic process that Jill uses here could of course be used for teaching any times table and has the added benefit of linking nicely to counting along the number line in steps, which might be useful when it comes to teaching division too.

In my experience, most children are able to learn their number facts with fairly rapid recall, given sufficient practice.  However, there are probably some children with specific learning difficulties who will never become very fluent with these facts.  For these children all is not lost.  Returning to the analogy of my daughter’s teaching toolkit, it’s worth remembering that teaching was perfectly possible before the invention of sticky notes and laminators!  These children need to learn how to be able to work out the facts fairly quickly and use aids like tables squares and calculators to support them when using and applying their mathematical skills.   If you have several children who have real difficulties with learning tables, Steve Chinn’s book may be worth reading.

# Know Your Place – the Importance of Place Value

As a new term begins, one of the topics we tend to cover early in term is Place Value.  An understanding of this is central to understanding our number system and underpins most written calculation methods, so it’s something that is well worth spending time on.  In my experience, there are two approaches that really help to build children’s understanding of place value: using concrete apparatus and representations; and regular use of counting in a structured way.

Concrete apparatus and Representation

Base ten equipment, such as Dienes, is commonly found in KS1 classrooms and I would love to see it more widely used in KS2 too.  Working with this helps children to visualise the tens and ones (and later the hundreds and thousands) they are working with and see how they relate to each other.  The Gordons ‘Dienes and Coins’ program has lots of ways of using Base ten equipment virtually too.  This program also gives similar activities with coins, and coins are another good way of exploring place value with children – giving it a context which they may already be familiar with.  Don’t forget too that fingers usually come in handy sets of ten and for whole class work, building numbers using several children holding up all their fingers as tens and one child holding up as many fingers as needed for ones can be a good way of building two-digit numbers together.  When children have had some experience of exploring place value with concrete apparatus, place value arrow cards can be very useful in relating this to written numbers.  The Gordons ‘Place Value Chart’ program links these to place value charts which again can be helpful in moving understanding on.

Counting

Regular use of counting in a structured way really helps to build children’s understanding of the way the number system works.  Again, it’s something that tends to happen a lot in KS1 classrooms but perhaps not so much in KS2.  Counting supports so many different areas of maths, but to build place value understanding, it’s particularly important to get children counting in steps of 1 and 10, and later in steps of 100, 1 000 etc or in decimal steps of 0.1, 0.01 etc.  Make sure that you count up as well as down and as confidence grows, you choose lots of different starting points, particularly focusing on counting which involves crossing the tens or hundreds boundary (or whatever boundary is appropriate to the stage you’re at).  With younger children, make sure you don’t stop at 100 when counting in ones.  It’s amazing how many children, even in early KS2, I’ve heard count … 107, 108, 109, 200!  When children are confident in counting in tens or ones, mixing the two can be an extra challenge.  One activity I’ve used with different age groups is to give children a starting number and get them to watch me crossing the front of the classroom.  When I take a small step forward, they count up in ones, when I take a stride, they count up in tens; and similarly for stepping backwards.  This can of course be adapted for different step sizes.  When children are working with decimals, the ‘Decimal Number Line’ ITP can be very useful in helping them see the way that decimal numbers fit together.  It gives a number line counting in ones, tens or hundreds and then allows the user to ‘magnify’ one small step to see what happens within this step.  This can then be repeated to make the steps even smaller. Using a counting stick can help children visualise the steps they’re counting.  Having number lines around the classroom counting in different steps can be helpful, as can ‘washing lines’ of numbers where children can order the numbers or spot numbers that are missing.

Moving Understanding On

Once children have an understanding of our number system, they are often fascinated by really large numbers and enjoy writing these in digits and then ‘translating’ this into words or vice versa.  Nrich have some place value related challenges for KS1 and some for KS2.

There are more ideas for teaching place value on my pinterest board.