# There or There Abouts – Using Estimation to Build Number Sense

‘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

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.

# Building Conceptual Understanding and Number Sense

I’ve just completed the excellent open access ‘How to Learn Math’ course led by Jo Boaler (author of ‘The Elephant in the Classroom’). There were lots of ideas to think about on the course and lots that I’ll want to revisit and mull over in the coming days and weeks. For a more detailed overview of the course, Pam O’Brien has written a helpful summary.

One of the big ideas is that children learn maths best by exploring real problems with a context rather than learning and practising routines. This, I’ll admit is a challenge. I really warm to the idea of exploring maths in context and feel that embedding maths in cross-curricular work is something we’re not at all good at as teachers. I also love the idea of children exploring problems and challenges for their own sake rather than as opportunities to use and apply their mathematical skills. However, there is a more ‘old school’ part of me that feels that it’s important for children to acquire fluency in calculation skills and fairly rapid recall of number bonds and tables facts. Boaler points out that these skills are not what real maths is about and I agree with her there, and I’m sure there are examples of real mathematicians who struggled with basic arithmetic. However, I do feel that having these facts and skills at their fingertips is for most mathematicians, part of their ‘toolkit’ of mental resources. That’s not to say that other attributes aren’t even more important – confidence, perseverance and curiosity all spring immediately to mind. In the real world of education too, both the current and the new curriculum require these skills and I have no doubt that children will continue to be tested on them for the foreseeable future, so as teachers we need to think about how we can help children acquire them.

What I’m not saying here, is that these facts and skills should be taught in isolation. Boaler is keen that children build up a conceptual understanding of number and mathematical skills and I would absolutely agree with her on this. There is little point in children acquiring rapid recall of number bonds and tables if they have little or no idea of what it means to add or to multiply. Before learning any number facts, children need to be building up a good ‘number sense’, a feel for numbers and how they can be manipulated. For a child without this understanding, ‘3 + 2 = 5’ is a fact to be learned in isolation. Whereas for a children with good number sense, they may visualise a group of 3 combining with a group of 2 to make a group of 5; they may use their knowledge of doubles facts to work out that the total will be one less than 6 or one more than 4; or they may be able to use this fact to derive lots of other related facts: 5 – 2 = 3 or 30 + 20 = 50.

Building up number sense needs to be an ongoing process throughout primary education, and probably well beyond. We can help children to do this in lots of ways but one of the key ones is by giving children lots of opportunity to explore the way numbers work using concrete apparatus and helpful representations before plunging into the more abstract world of numbers in isolation. This might be by using dedicated mathematical equipment: counters, Dienes, Numicon, multilink, ten frames etc. Or it might be by using the opportunities that occur every day: counting the steps as we walk upstairs, working out how many cakes will be left if we each eat one, combining our pennies to see what we can buy at the sweet shop etc. This brings us back to exploring real problems with a context. It would be great if this was happening naturally at home for all our children long before they ever got to school, and for many I’m sure it is. Creating opportunities in school for this to happen is something that I suspect EYFS teachers are already very aware of, but perhaps it’s something for the rest of us to be challenged by as we plan our classroom environments and cross-curricular work.