Picture This – The Importance of Representation in Teaching Maths

Teaching mathematics visually and actively

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

New curriculum main aims

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?

For more information about the new curriculum and some resources which might prove helpful when introducing it, my New Curriculum Pinterest board may be helpful.

Measuring Up – Teaching Measures

Capacity measuring equipment Young children are often fascinated by comparing and ordering the sizes of things.  Perhaps it appeals to their innate sense of justice to determine whose apple is bigger and their equally well developed competitiveness to see who is taller.  Early Years teachers build on this by providing lots of opportunities to compare and order things and begin to use non-standard measures to quantify.  How many grapes balance an apple?  How many cubes high is the toy garage?  How many cups of tea can be poured from the teapot?  At this stage, it’s important too to give children lots of opportunity to experience and use the language associated with comparison: more, less, fewer, higher, lower, taller, shorter, heavier, lighter etc.  I’ve put together a few ideas for activities which support developing comparison language and you can download the document from the link at the bottom of this post.

As children move on in their understanding of measures, we move to using standard units of measure.  Children often struggle with estimating length, mass or capacity using standard units and they need lots of practical opportunities to measure familiar things using these units.  Wherever possible, opportunities should be found outside of the maths lesson for these activities, perhaps as part of topic work, for instance, to give them a meaningful context.  Children can weigh out ingredients for their chocolate snack in technology or find the capacity of a liquid before an evaporation experiment in science, or measure how far they can jump in P.E.  Another activity that can support children in becoming more familiar with units of measure is to give regular opportunities for estimating, and use these as opportunities to develop the skill of working out an unknown measure by comparing it with a known one.  Estimation 180 is a great source of visuals to support this (I blogged about this site here.)

Another common difficulty for children is remembering just how many grams in a kilogram, centimetres in a metre, millilitres in a litre etc.  One activity that can support this is by including counting in measures in daily counting activities, alongside counting in whole numbers, decimals and fractions etc.  So, for instance, when children are counting in hundreds, also count in steps of 100 grams.  I find a counting stick useful for this.  Develop skills progressively.  So for instance, you might count up first of all from 0 to 1 kg in steps of 100g, moving backwards and forwards along the counting stick.  As children become more familiar with this, use different starting points so that they become familiar with what happens after 1 kg.  At this point you have a choice of ways to count: 1100g, 1kg and 100g, 1.1 kg or 11/10 kg, and I’d suggest you use all of these ways alongside each other so that children start to also understand the equivalence of these.  Doing this will also help enormously when children begin to convert units of measure.

Children often find reading scales challenging too.  Again, there is no substitute for practical experience, and if you are able to have analogue scales, measuring jugs, tape measures etc. continually available in your classroom, this can be helpful in making it easier to pick up on opportunities for measurement that arise in other subject areas – a trip to hunt through the maths cupboard will probably make you less likely to do this!  The Measuring Scales ITP and Measuring Cylinder ITP can also both be helpful for focused opportunities to practise measuring scales skills.  Again, counting can also be useful in supporting reading scales.  Most scales are in intervals of 1, 2, 5, 10, 20, 50, 100, 200, 1000 etc.  so regular opportunities to practise counting in these steps will help children to use these skills when reading scales.

One of the main problems with children working with measures, I suspect, is that we move far too quickly to working with abstract measures or with diagrams rather than working practically.  I’ve been guilty of this myself – practical work involves finding equipment, it can be messy (particularly when working on capacity).  But practical work can also be lots of fun and really help children connect their learning to real life situations, so I’d encourage you to do as much as possible.

There are other ideas and resources for teaching measures on my Measures Pinterest board.

Resources for teaching language of comparison

Power Tools – Useful Online Tools for Teachers

Power tool

My blog this week is not specifically mathematical.  Instead, I’d like to share some of the tools I’ve found useful in the classroom.  We probably all have our favourites and these are mine.  I’d love to hear other people’s favourites too.

 

Countdown Timers

 

My favourite timer is the Classtools countdown timer.  This allows you to choose from a great selection of tunes (including the Dr Who theme and the Mission Impossible theme) and timers from 30 seconds up to over 7 minutes.  There’s also the option to upload your own tune in MP3 format.  Children are often very motivated to tidy up, move to groups, line up in order etc. before the music stops.  A large countdown can be displayed on your IWB which adds to the sense of urgency.

 

Class Dojo

 

My class last year loved this behaviour management system.  You can upload your class names fairly quickly and each child will be given an avatar.  Later, you can give them the option of choosing a different avatar.  Then you can use the system to award both positive and negative points to children or groups of children.  You can choose the categories you can award or deduct points for, so for instance you could award points for helping others or good team work, or deduct them for failing to do homework or shouting out.  The system keeps track of how many points each child has earned.  There is also the option to allow parents access to the site so they can track their child’s behaviour.

 

Display Ideas

 

Word cloud generators create attractive word clouds from text you input.  They can be useful for creating a visual of key words for different topics.  Two good ones are Wordle and Tagxedo.  Primary English wrote a great blog about different ways to use word clouds in the classroom.  For merging together photos or other pictures with a collage effect, Autocollage works very well.  Or for displaying pictures on a big scale Block Posters will create large size pictures which you can print off in pieces to put together.

 

Random Name Generators

 

Again I like the Classtools version.  In this version, you can copy in the names from your class and it acts like a fruit machine and picks one randomly.   Class Dojo also has a random name picker option.  Of course, there’s always the low-tech standby version of names on lolly sticks too.  I’ve also used these in starter activities by eg. listing ways to improve a sentence and then using it to choose one at a time.

 

Creating Slideshows

 

There are a number of tools that can be used to create slideshows from your own photos fairly easily, perhaps for a presentation in assembly or an end of topic celebration of learning.  I’ve quite often used the Microsoft Live Movie Maker, but Animoto and Photo Peach are also well worth exploring.  Some of these have their own bank of music that you can use to accompany the presentations and there is usually the option of importing your own too.  One of the best ones I made was a simple but very effective idea for a leaving assembly.  I took photos of classes and groups of children, teachers and other staff, governers etc. all waving and then made a slideshow accompanied by Andrea Bocelli’s ‘Time to Say Goodbye’.  Although suitable music can usually be bought from itunes fairly cheaply, Youtube have also recently launched a free library of tunes that can be used in this way.  You can search these by their mood, genre or duration which could be useful.

 

Virtual Pinboards

 

Padlet is probably the best known of these.  You can quickly create a virtual pinboard and share the link.  This can be used, for instance, at the start of a topic to find out what the children already know about it, or for creating a bank of questions to explore.  It could also be used for sharing word problems created by children or responses to a challenge.  One of my classes used it to share memories at the end of our year together.

 

There are so many online tools that can be used to enhance our lessons.  All of these are very easy to use (believe me, I’m no ICT expert!).  I come across new ideas all the time, so I’ve created a Pinterest board which I’m sure I’ll keep adding too.  All the ideas I’ve shared on here are on it, together with a few more.

 

There or There Abouts – Using Estimation to Build Number Sense

Estimation spreadsheet

‘Explore MTBoS’ is a series of challenges put together by a group of experienced maths education bloggers to help those of us with less experience to find our way around the world of maths blogging.  I’ve found it a useful way of finding other people who blog about maths teaching and have already encountered lots of new tools to explore and ideas to reflect on.  This week’s challenge was to engage with some collaborative sites and although I was already familiar with some of these, many were completely new to me and well worth exploring.  I’m sure I’ll be coming back to them.

One that really caught my attention was ‘Estimation 180’.  This is a site put together by Andrew Stadel who teaches middle school maths.  He has posted hundreds of estimation challenge pictures which could be used as starter activities to lessons.  There is a handout that can be used to keep track of estimations over a period of time.  Students are encouraged to give an estimate that is too high, one that is too low and then their best estimate.  Importantly they are also asked to explain their reasoning, based on contextual clues or pre-existing knowledge.  There are lots of ways of using the challenges.  Students can submit estimates online and explore the answers that others have submitted and their reasoning.  They could fill in the handout each day and keep a record of their estimates.  Or the challenges could just be posted up by the teacher at the start of each lesson.  The challenges are varied – estimating heights and weights, number of objects, ages etc. and often build from day to day so that the answer to the previous day’s challenge can inform today’s estimate.  Key to using this effectively would be giving students the opportunity to explain and share their reasoning.  Sharing strategies and approaches could make a valuable contribution to building number sense.  I like the fact that many of the challenges involve measures as I often find children find estimating these particularly difficult.

The site is a very useful resource because estimation can be a tricky skill to teach.  Give children a typical sheet with pictures of objects and ask them to estimate and then count, and all but the most compliant will probably sneakily count first then make their estimate very close to the actual count (and the reasons why they are so reluctant to risk a wrong answer will probably make a whole new blog some time soon).  I’ve found the Primary Strategies Estimation Spreadsheet (shown above) useful as it can be used on an IWB, and the stars can be shown and quickly hidden before the children have a chance to count them.  It can be downloaded here.  Another interesting looking site is the ‘Guess It’ game on the Problem Site.  This gives children a series of estimation challenges by showing dots of different sizes and colours.  There is a timer which can be used to adjust the number of seconds the dots are shown for.

I also like the idea of having an Estimation Station in the classroom, a transparent container that is regularly filled with small objects.  Children then estimate how many objects are in the container and strategies are taught and compared.  Looking at the price of the Amazon one though, I think I could probably come up with a cheaper alternative!

Some of the resources I have mentioned in this blog, can be found on my Number Pinterest board

Less is More – Supporting Less Mathematically Able Children

Ten frames

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.

Rich List – Using Rich Tasks in Maths Lessons

Growing triangles

The new primary maths curriculum has been criticised for its focus on fact fluency and traditional written methods.  However, of the three key aims at the beginning of the document, only one focuses on fluency.  The other two are that we should ensure that all pupils:

reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.”

and

“ can solve problems by applying their mathematic to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.”

It’s very important that we don’t lose sight of these very important aims in the new drive for increased fluency in recall and calculation.  Regular use of rich mathematical tasks in our maths classes can really contribute to both of these.

So what is a rich task?  In her very helpful Nrich article, Jennifer Piggott describes the characteristics in some detail and some of these are:

  • accessibility – they should offer different levels of challenge to learners of different ability, giving opportunities for early success but also scope to extend learning for the more able (low threshold, high ceiling)
  • encouraging growing confidence and independence, often by working collaboratively
  • potential to link with other areas of maths or to introduce entirely new areas of maths
  • encouraging different approaches and creative solutions to problems
  • allow learners to pose their own problems and ask questions

Jennifer Piggott also makes the important point that a mathematical task, although it may have the potential to do many of these things, does not become rich unless it is well led by the teacher, asking timely questions and supporting the children just enough to start to construct their own mathematical understanding whilst avoiding ‘spoon-feeding’ them.  In practice, this can be difficult to do.  In a busy classroom, it can be very tempting to wade in when a child is stuck and show them how to do it, but if we can restrain ourselves and instead offer a hint or a question that might open up a new avenue to explore, the experience will ultimately be much more satisfying and beneficial to our learners.

One good example of the sort of activity that could be used in this way is the ‘Sticky Triangles’ activity from Nrich.  Children are presented with a growing pattern of triangles as above.  These can be made from lolly sticks or pencils or similar or just sketched.  You might like to present just the first two steps to start with and see if the children can suggest how to extend the pattern.  Then get them to work on their own or in pairs or groups to explore the patterns.  It’s probably best not to ask too many questions to start with.  Children often naturally start to notice things about eg. how many triangles are in each row, how many lolly sticks are needed to make each pattern.  It can be very interesting to watch the children and see how they approach things.  Do they work systematically?  Do they record anything?  After a while, you might want to suggest some possible avenues for exploration.  Can you see any patterns in the way the number of lolly sticks increases with each new row?  Can you predict how many triangles will be in the next row?  How many triangles would be in the tenth row?  How many lolly sticks would be needed by this stage?  What about the 100th row?  Can you suggest any good ways of recording your findings?  Encourage children to explain the patterns they see to each other and to you, and encourage the use of accurate mathematical vocabulary as they do this.  The notes on the activity on the Nrich site also offers some other possible ways of extending the task even further.

The Nrich site offers lots of these sorts of activities at all sorts of different levels.  As a teacher, I’ve found their curriculum mapping documents for KS1 and KS2 very helpful in identifying activities which might be linked to our other current work.  Another source of helpful activities is the BEAM resources which can now be found in the elibrary of the National STEM Centre.   You do need to register to access these resources, but registration is free and well worth while as there are a great wealth of resources in the elibrary.

For other suggestions for mathematical investigations, puzzles and challenges, have a look at my Pinterest board.

Wonder Walls – Maths Working Walls

working wall resources smaller

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.

 

Take it Away – Subtraction

Number line

At about this stage of the term, many teachers will be teaching calculation.  Look at most schemes of work or medium term plans and you will probably find roughly equal amounts of time given to covering addition and subtraction.  Yet, almost any diagnostic assessment will tell you that most children are far more secure in understanding addition than they are with subtraction.  Perhaps we need to make sure we give a little (or maybe even a lot) more weighting to teaching subtraction.

So why does subtraction cause so many problems?  Well let’s think about some of the ideas we use when learning about subtraction in school.

  • The ‘taking away’ idea – probably the first that children come across.  One group of objects is taken away from a larger group and typically we count what’s left.
  • The ‘difference’ or ‘counting up’ idea – we count up from the smaller number to the greater and find the difference between them
  • The ‘counting back’ idea – we count back from the bigger number by the number of steps in the smaller number
  • The ‘inverse of addition’ idea – we work out what must be added to the smaller number to make the greater number

No wonder our children get confused!  Our teaching needs to help them make connections between all these ideas and will need to involve lots of practical work and the use of models and images, particularly number lines.

The new curriculum puts much more emphasis on using formal column methods for calculation and on building fluency with these.  If these methods are to serve our children well, it’s vital that well before we move onto them we have laid the foundations by building secure conceptual understanding.

This will start in Early Years and KS1 classroom with lots of practical work, wherever possible using real life situations which connect with the children’s experience of life.  This is also the stage where it’s important to start building up children’s mathematical vocabulary by lots of careful modelling and opportunity for discussion.  Particularly important at this stage is the language of comparison: greater than, less than, more than, fewer than etc.  It will also help enormously if children start to get a ‘feel’ for numbers and the way they can be split apart in different ways.  Later on, regrouping is going to be needed and children will find this much easier if they are already comfortable with splitting numbers up in multiple ways.  My blogs on building number sense and learning number bonds give some ideas which might be useful.  After lots of practical experience, children can be taught to record their work using number sentences, but only once they have clear mental pictures to accompany these.

Counting is another important skill that lays the foundations for subtraction, particularly counting backwards and counting over tens and hundreds boundaries.

Once children are becoming confident with manipulating numbers, number lines may be introduced.  It’s important however that children are able to connect the counting that they do along a number line with the practical work they have done.  It is not obvious initially to many children that, for instance, 12 – 8 can be represented by counting up from 8 to 12 along a number line.  One way of visualising this is to scribble out the portion of the number line up to 8 (representing the part taken away) as above.

At first, children may use ready-made number lines but as they grow more confident, they will be able to draw their own number lines to suit each new calculation.  As understanding progresses, counting on along the number line can be used for increasingly large numbers and children will count on using larger jumps, usually to the next tens or hundreds number and then counting on in tens or hundreds.  This program can be helpful in illustrating this.  As the use of the number line becomes increasingly sophisticated, it’s important to keep making connections with other representations of subtraction.  How could you represent the calculation with base ten equipment, for instance?

Eventually children will be ready to move onto column methods and at this stage, it’s vital that we don’t just teach them a procedure.  We need to show them how it works by using models and images alongside the formal calculation.  This program shows how the expanded method works alongside the more formal compact method.  I would suggest also using base ten equipment to make what is happening even more clear.

For more calculation ideas, my pinterest calculation board might be useful.

Stocking the Toolkit – learning number bonds and tables

Tool kit - small

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.