I’ve already blogged about some of my hopes and fears for the new curriculum here. In this blog, I want to think more particularly about introducing the curriculum to schools and suggest some useful resources.
It’s important to plan the change carefully and make sure teachers are well prepared. As the KS2 assessment arrangements don’t change until 2016, our current Years 3 and 4 will be the first to be assessed under the new curriculum at the end of KS2. The expectations for them in some areas will be higher and it may well be that schools choose to start teaching at least some of the new content this year so that there is less for these children to catch up in Years 5 and 6. This is particularly true in the areas of written calculations and fractions. Similarly in KS1, the tests and reporting arrangements will remain the same until 2016. This means that from September 2014, Years 2 and 6 should still continue to be taught using the current curriculum but all other year groups will need to move to the new curriculum.
Expectations for fluency with number facts and calculation methods will be raised and it may well be worth tackling this with some whole school initiatives. Some schools are choosing to give an extra 10-20 minutes each day to focus on this in particular, outside of the maths lesson, rather in the way that phonics is often taught discretely. For number bonds and tables, it would be well worth listing exactly which facts your school expects children to learn in each year group and sharing this with parents. It’s also worth tracking the facts that children know so that children who are falling behind in learning these can be given extra support. It would be good to discuss as a staff just what you all understand by ‘rapid recall’ of facts. You may find that some teachers feel children know their two times tables if they can chant the table, whereas others would expect them to be able to answer mixed 2 times tables questions, answering 20-30 or more in a minute. I have suggested some ideas and resources for teaching number facts and tables here.
The NCTM has a growing library of resources to support introducing the new curriculum. In particular, their Resource Tool could be a useful starting point. So far, only material for Years 5 and 6 have been added, but we are promised other year groups’ material before too long. For each year group, the content has been divided into several different strands. Selecting a particular strand and year group and choosing ‘Show Selection’ brings up the information below the tool. So for each of strand and year group, there is information on subject knowledge, connections (to content in other year groups, to other mathematical topics and to other subject areas), articles about good practice in teaching that strand, some suggested activities that could be used in teaching it, exemplification of the expectations and videos that support aspects of the strand.
The subject knowledge resources may be particularly useful for teachers in UKS2 where raised expectations may mean that they need a refresher in eg. calculating with fractions.
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. For subject leaders or senior leaders based in the Midlands, you may be interested in a course I am running next month on preparing for the new curriculum. I have lots of ideas and resources to share!
Over the holidays, many teachers will be sitting down to make medium term plans for their maths lessons. This might be for the entire time or it might only be for a block of learning, perhaps two or three weeks, but I’d hope that they will be giving some thought as to what the learning will look like over the course of the whole term. They’ll be looking at assessment records or notes they’ve written on their weekly plans or elsewhere to see what the priorities are. They’ll be looking at the objectives for the year or for the next steps in learning and making sure there’ll be opportunities to visit these during the term. For Year 6 teachers, with the SATs in May drawing ever closer, there’ll probably be a particular urgency and focus on the gaps that need to be plugged before Easter.
Medium term planning is important in Maths. It has to be flexible. Even for a very experienced teacher, each class is different and some topics will end up not needing as long as we planned, whilst others will need longer or will need revisiting. Just like our weekly plans, our medium term plans should be working documents which regularly get annotated and highlighted throughout the term. However flexible it is though, it needs to be well thought through to ensure coverage of all the important topics and that will probably mean building in some contingency time, just in case several topics take longer than we imagined.
In recent years I have often been involved in monitoring planning in different schools – this is usually weekly planning rather than medium term planning, but usually I’ll be looking at several weeks at a time. Plans vary enormously in the amount of detail they give, and there are teachers whose plans give very little detail who manage to teach great lessons, and others who dot every i and cross every t but whose lessons are not so sparkling. By and large they are thoughtful and thorough and it’s clear that teachers are on the whole familiar with what their children already know and what they need to learn next. One thing that does concern me, however, is that over the last few years, I have increasingly seen a rather ‘scatter gun’ approach to planning: a lesson or two on one topic, another lesson or two on another topic and then perhaps a quick problem solving lesson on Friday. I believe there are two main reasons for this. The first reason is the way the renewed framework which was introduced in 2006 divides into blocks of learning meant to last for 2-3 weeks. In each block there is a whole plethora of possible material, and I don’t think it was ever the intention that anyone would try to teach all of this in such a short period of time, but there was such a lot there, that it was common for teachers to try to teach far too much far too quickly. The second reason I believe is the increasing pressure from Ofsted that so many schools are feeling now. I know there’s lots of debate about progress in lessons currently, but whatever the rights and wrongs of this, it has made many teachers feel they need to teach something new every lesson. It’s relatively easy to show progress within the lesson when you introduce something new – at the beginning of the lesson, most children know nothing about it, at the end they do. Whether they will remember anything about it next week, let alone tomorrow, if we don’t then revisit it, is much more open to question.
I really believe we need to slow things down. I’m not suggesting individual lessons should lack pace (although there’s possibly a debate to be had about that too), but that we should plan sustained periods of time where we focus on one topic or a few very closely related topics. Within that period of time, our children can certainly make progress within that topic, but each new step will be connected to a very recent one, and there will be time for new skills to be practised and consolidated.
As well as taking longer over topics, our medium term planning should be exploiting the natural links between different mathematical ideas and topics. So, for instance, it makes sense to teach finding fractions of amounts soon after work on division, so that children naturally see how the two things connect, and can further practise their division procedures with fractions problems. If you haven’t seen it before, take a look at the picture at the top of this blog and try to work out what comes next. Unless you see the connection between the symbols, it’s pretty hard to do (if you’re still struggling, imagine a mirror line drawn vertically through the middle of each one). Once you see the connection, it’s easy. Similarly, we can make things much more difficult than they need to be for our children if we don’t exploit the natural connections between different mathematical ideas.
So if you’re sitting down to do some medium term planning over the holidays, think about the connections you can make. Think about giving children time to really get to grips with a topic. If it’s possible, also think about the connections that can be made with other curricular areas. Would your technology topic give opportunities for weighing or measuring using different scales? Could your geography topic give opportunities for using negative numbers in context when comparing temperatures or comparing large numbers when looking at populations? Could the children show some of the information from their history topic using their newly acquired data handling skills? Don’t force it, try to make the contexts as real as possible, but teachers tend to have great imaginations so I’m sure you’ll find some creative connections.
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.
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.
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.