I have mixed feelings about homework in primary school. As a teacher, setting, collecting, chasing up, marking and returning homework took up a lot of valuable time, and I wasn’t always sure it was entirely worth the time expended. However, most schools have a homework policy that stipulates setting at least one maths homework a week, so over the years I developed some ideas to make it as painless and productive as possible for me and the children.
Using online games
I’m always very keen for children to learn their number bonds and tables. The underlying conceptual understanding of these is vital, but at some stage sustained practice is needed. There are lots of online games that can help children to do this, not to mention apps that can be downloaded onto tablets and smart phones. Some schools buy into subscription sites like mathletics, but there are some very good free alternatives too like tutpup and sumdog. It’s possible to set up accounts on these sites as a teacher and track children’s progress so that if you set a homework of spending time on one of the sites, you can check it’s been done, or it may be sufficient to ask parents to sign a homework diary to confirm their child has done this. Of course, there will still be some children who don’t have ready access to the internet or access to tablets, but these days that’s often very few and it may be possible to accommodate these with a weekly homework club where they can use school devices, or give them an alternative homework.
Adopt a Shape
This was an idea I used when we were just about to embark on looking at 2D and 3D shapes. I gave each child a shape to ‘adopt’ – this gave me a chance to differentiate fairly easily by giving more familiar shapes to some of my less able children and challenging my higher ability children with less familiar shapes like icosahedrons. They were given the task of finding out as much as they could about the shape and presenting the information in any way they chose. I gave some suggested starting points, like finding the number of sides and corners etc. or finding the number of diagonals. The children really engaged with this idea and came up with some fantastic presentations, including 3D models in some cases. It made a fantastic starting point to our unit of work (not to mention filling up the working wall nicely!)
Write a worksheet
This was a task I used quite often when we’d spent some time looking at particular calculation methods. I would ask each child to write a worksheet for another child in their group. They had to include at least one worked example with an explanation of the ‘steps to success’. They also had to include some word problems and if possible give the worksheet a theme, possibly linked to our current class topic. Again, this was something the children usually responded well to. I would often give small prizes or stickers for the best ones and display these on the working wall and this appealed to the competitive streak in many children. Some would hand in beautifully illustrated sheets. Writing word problems to go with particular calculations really tests children’s understanding of that operation. A variation on this theme at the end of less calculation based units was to ask the children to make a poster to display their learning and be a learning resource for others. Again, this often produced some beautifully presented responses.
When beginning a unit of work on handling data, I would often start by recapping all the different ways the children already knew of presenting data. Depending on the age of the children, this might include tally charts, pictograms, bar charts, bar line graphs, line graphs, venn diagrams, carroll diagrams or pie charts. I might then ask them to collect as many examples as they could from newspapers, magazines or the internet and make a poster which they would annotate with explanations of what type of representation it was. For older children, I might also ask them to comment on whether that was a good representation and why that particular representation of the data had been used. This encouraged the children to notice how often data was presented in different ways in real life and start engaging with this.
Another data handling homework I would sometimes set in Years 5 or 6 was to give the children an opportunity to investigate something themselves. They were given a free choice of what to investigate and how they would collect the information. We spent some time discussing possibilities in class beforehand. They then had to collect the data and decide which was the best way of presenting the information. They also had to draw some conclusion from their data and lastly to reflect on their project and decide whether it was a true picture or whether their were factors which might have affected their results. This was a homework I set over 2 or 3 weeks, often over a half term holiday to give them time to plan and carry out their projects. I gave some helpful hints and prompt questions at the outset. This led to some really good work and some good discussion afterwards when the children shared their projects. One particularly memorable one was the boy who patiently recorded each visit to the toilet by each member of his family over a few days. Some definite trends emerged and the conclusions he drew were very entertaining!
Particularly when using measures, it can be good to set a homework which gets children using their newly learned skills in a practical context. So, for instance, homework could be to follow a recipe using metric units and record the result in some way (ideally be bringing in a sample of any particularly delicious results for the teacher to critique!) Or it might be to measure up a bedroom and plan an ideal lay out using furniture of given dimensions.
I hope this has given some new ideas to try. The beauty of many of these ideas is that they often take very little marking or can be used to stimulate discussion or as a learning resource in future lessons. They also tend to engage children much more than a traditional worksheet, and often get parents involved as well. Some can take a little planning to set – a bit more than photocopying a worksheet maybe, but I always make a point of keeping the prompt sheets etc and they can often be quickly adapted for a different age group or mathematical area.
This blog was my very first venture into blogging on the fabulous Primary English blog. I’m very grateful to them for publishing it last May which led to me thinking seriously about starting my own blog. Their site is well worth a visit and they also have some amazing pinterest boards on all sorts of themes.
Here is what I blogged back in May:
As a maths leader, I quite often have the privilege of doing planning trawls and looking at weekly and medium term planning from other teachers. I’m often very impressed by the thought and detail that goes into these. But there’s one section that seems very rarely to be given much thought. If your weekly or medium term planning format is anything like mine, there’s a small section headed ‘cross-curricular links’, and I hardly ever see it filled in, except perhaps with the suggestions given on the format itself, and these are nearly always Science based.
On the whole, we are very good these days at making cross-curricular links, particularly at bringing writing opportunities into a whole range of curriculum areas. At the start of units, topic webs are drawn up and connections made – but maths is often very difficult to fit in to these and so we agree that it’s probably best to teach this discretely.
I’d be the first to admit that it is often difficult to bring maths into our topic themes – although I do think it’s worth making the effort. It’s so important that children see the relevance of maths to their lives and the way that the skills they learn can be applied. However, one great way of linking maths to other curricular areas is by using story and picture books.
As I write, my daughter – in her first year of teaching – is spending a few days with us. Yesterday, she was starting to plan the maths for her Year 1 class for the Summer term and looking for activities in particular for time and money. She’d already planned to use Eric Carle’s ‘The Very Hungry Caterpillar’ as a starting point for learning and ordering the days of the week, and ‘What’s the Time, Mr Wolf’ by Colin Hawkins and ‘The Bad-Tempered Ladybird’ (another book by Eric Carle) for telling the time and sequencing the day. I was able to introduce her to the wonderful Mick Inkpen book ‘The Great Pet Sale’ and she then happily spent most of the rest of the afternoon having lots of fantastic ideas about how she could use this – her role play area for the start of term will be a pet shop with lots of opportunities for the children to practise paying for items, finding the correct money and giving change, but also stimulating lots of writing opportunities too – descriptions of their pets, instructions for looking after a pet, recounts of visits to a pet shop – like most teachers, given an engaging starting point, the possibilities she’ll find will be almost endless. She also found some fantastic resources to use on T.E.S. and some good labels for her pet shop on Twinkl as well as a reading of the book on Youtube.
I suspect KS1 teachers have always been quite good at using story and picture books in some of their maths work, but as a KS2 teacher I wasn’t so aware of good books with mathematical links until I was introduced to some by the Coventry Primary Maths team at subject leader training and also during my MaST training. A particular favourite is Anna Milbourne’s ‘How Big is a Million’ which tells the story of a young penguin eager to find out just what a million looks like. Big numbers tend to fascinate children of all ages and although younger children would love this book with its very simple story line, I’ve also used it very successfully with children in UKS2. Another is ‘The Rabbit Problem’ by Emily Gravett, again a very simple story attractively presented, but with some quite challenging maths to explore for older primary children. I’ve used this with UKS2 when we’ve been looking at number sequences to lead into looking at the Fibonacci sequence and algebra.
There are so many books that can be successfully used in maths, from simple counting books like the beautiful Anthony Browne book ‘One Gorilla’, through books about measures like Pamela Allen’s ‘Mr Archimedes’ Bath’, to books about working with very large numbers like ‘Anno’s Mysterious Multiplying Jar’ by Anno Masaichiro. For fans of the ‘Horrible History’ books, there is even a whole series of ‘Murderous Maths’ books written by Kjartan Poskitt.
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.
leading later to expanded column addition, 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.
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.
Harder, Faster, Higher? – Supporting More Able Mathematical Learners
The new curriculum, we are told, is a mastery curriculum. This means there is an expectation that instead of pushing our more able learners on to ever higher level curriculum content, the focus is much more on making sure all our children are secure with the core content for each year group. This leaves us with a challenge for our more able learners, but also with a great opportunity. With less pressure (we assume – the assessment procedures are not yet clear) to push these children through the levels, we have to think of different ways to challenge them – not so much higher and faster as broader and richer content. These are a few ideas about how we can support these children.
Open ended questions
Challenging more able children should not be primarily about moving them on to ‘harder’ questions with higher numbers. We want to extend their thinking by asking more open ended questions which challenge them to apply their knowledge in new ways. We also want to develop their reasoning skills by asking them to explain their reasoning. Using Bloom’s question stems can be helpful in planning this. Ask children to explain how to find the answer to a problem and decide which approaches would be best to use. Ask them to explain the rule for a growing pattern or to explain what would happen if … Get them to think of other ways of doing things and compare approaches to decide which is best. Challenge them to explain their reasoning so that a younger child could understand it.
Mathematically rich activities
Children need to learn to think mathematically and to apply their skills. The Nrich website provides lots of games, challenges and activities to encourage this. They aim for activities to be ‘low threshold, high ceiling’ ie. accessible to as many children as possible but with enough to challenge more able children. Their curriculum mapping documents are very useful in identifying activities which link to different curriculum areas.
Investigations can help children to extend their mathematical thinking in a more open-ended way. Typically in an investigation, children are given a starting-point and some ideas of how to get started, but they won’t know what the answer will look like. They need to look for patterns and identify what is happening. The Maths Warriors site has a number of interesting investigations suitable for primary aged children.
More able children often respond well to challenges. The ‘Mathematical Challenges for more able pupils’ have a number of challenges divided by age group. Another good source of challenges for KS2 are the Challenge cards on the Maths Warriors site.
Whenever a new skills is taught and learned, make sure the children have the opportunity to practise their skills in a real context by applying them to solving problems. It can be particularly meaningful to give the problems a context from another curricular area. As well as regular opportunities to solve problems as part of their maths lessons, children often also respond well to the challenge of a ‘Problem of the Week’ which can be displayed in class for a set time. The Nrich site is a good source of suitable problems, some of which are available as posters. The Numeracy Strategy Logic Problems also have problems at a range of levels.
Missing Number questions
For calculation in particular, once a calculation process is learned (eg. column addition), presenting questions with missing digits can extend children’s thinking about the process they have been using.
Other useful websites
Mathpickle has some interesting videos and other activities.
Mathsticks has lots of useful resources. There are lots of great activities which are free to download, and some premium resources if you can stretch to a membership.
7puzzle posts a new puzzle every day. The site also categorises the archived puzzles into Easy, Medium and Hard etc.
My Pinterest board has lots of other ideas for investigations, puzzles and challenges.
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-May. Teaching 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.
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.
‘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
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.
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.”
“ 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.