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
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.
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.
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.
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.