# Tag Archives: big ideas

## Maths — the big ideas

The beauty of maths lies in the interconnectedness of ideas and concepts yet this concept of relationships is often lacking in children who struggle with maths.  Mike Askew, Professor of Education at Monash University, Melbourne, has written about what he calls the big ideas of maths. These help children to connect different areas of mathematical understanding, yet are small enough to understand in their own right.

Read the rest of the article here.

Filed under Maths

## A place for everything and everything in its place

Place value is very often one of the first units of work for maths in most year groups and is absolutely fundamental to a good understanding of number.  By getting this right and giving children the opportunity for deep conceptual understanding, we can lay solid foundations for the year.

For the purpose of this blog I’m going to assume that children can count reliably and read and write numbers without error. If these things are not yet developed to the appropriate standard then targeted intervention needs to happen without the child missing out on good modelling and explanations of place value.

Children need plenty of practice constructing and deconstructing numbers, first using concrete manipulatives like base ten blocks or Numicon.  This is to show that 10 ones is equivalent to 1 ten etc.  While they’re making these numbers they should be supported to talk articulately about what they are doing, perhaps with speaking frames: ‘This number is 45.  It has 4 tens and 5 ones.  45 is equal to 40 add 5.’

Read the rest of the article on the Rising Stars Blog.

Filed under Maths

Moving schools and with more than an eye on headship is sure to get you reflecting.  The following posts are what I think about various things, in no particular order.  Previous posts were about displays and learning.  Next up – maths.

Any unit of work should be planned with end points in mind.  Teachers should start with the relevant National Curriculum statements but more importantly, the types of questions that children will be expected to be able to solve.  Teachers in Y2 and Y6, with their experience of end of key stage testing, may well have internalised the type of questions that would be appropriate for children to solve but resources like the Rising Stars Assessment Bank are invaluable for teachers.  Our expectations of what children will learn are vital. All children are capable of achieving age related expectations given the right support and sufficient time. If we begin a unit with lowered expectations for some children based on perceived ability then we are failing them.

Once the expectations of what children will be able to do at the end of the unit are clarified, teachers should then plan backwards, thinking carefully about what children will need to know and be able to do in order to solve those problems as well as figuring out a sensible conceptual sequence of those things.  For me, these include fluency with number, learning the underlying structure of the problems to be solved, the deliberate teaching of mathematical language and opportunities to reason.  The more I think about this, the more I’m settling on a sequence of development that units of work should be structured around, key parts of the sequence are developed and consolidated over time:

This model may not work for all topics but I’d suggest that units of work start with pure number.  Contexts can be stripped away to reduce load on working memory and children can get on with learning and practising fluency so that as soon as possible, they are able to recall necessary factual knowledge and manipulate numbers in calculations with little mental effort.  This is by no means rote learning – it should be carefully thought out so that children develop sound conceptual understanding, starting with concrete representations, progressing to more efficient pictorial representations and then on to even more efficient abstract representations.  It is important that teachers remember that the abstract representations are not the maths itself, merely the most efficient way of recording or communicating the thinking.

Fluency

If children can recall number facts and other basic mathematical knowledge within a few seconds, if children can calculate reliably without expending too much mental effort, and if children can recall varied mathematical knowledge, switching between topics, then they will be considerably more able to commit precious working memory capacity to problems that require deeper thinking.  It is for this reason that fluency must come first and continue to be practised in order for that recall to become increasingly accurate and efficient.  Flash cards are very useful here – they provide the opportunity for self-testing and, with a little training, can help the child to become more aware of the what they know and do not know, enabling them to focus their own study.  Teachers figure out what basics are required and deliberately teach those basics if they are not sufficiently internalised already.  If some children are already fluent, they can work on speed and efficiency, for children may well have fluent but inflexible strategies.

Underlying structure

Stories can be said to be a variation or combination of just seven basic plots and expert writers have a sound knowledge of these, enabling them to see stories at a deeper level.  I’d argue that there is a significant similarity in maths; that there are five basic problem structures.  These five structures are aspects of either additive or multiplicative reasoning and are classified based on what is unknown in a problem:

Knowing the structures is not sufficient.  Children must be able to identify the underlying structure from a given problem.  This is no mean feat so sticking to just number problems and avoiding distracting contexts for the time being is important.  Take a problem like this:

Hattie and Yates, in Visible Learning and the Science of How We Learn, said that experts see and represent problems at a deeper level.  A novice will only see the surface features in this problem: two numbers and an addition sign.  Consequently, they’ll solve this by adding the two numbers together.  Of course, this is a mistake.  An expert, on the other hand, will know that the whole is made up of parts and that you add the parts together to make the whole.  They may even ignore the numbers at first and read it as part + part = whole, realising that it is a problem where one of the parts is unknown.  They may see it or draw it like this:

They’ll use their knowledge of the relationship between the whole and its parts, plus the idea that to find a missing part, you subtract the known part from the whole, therefore calculating 564 – 327 to find the unknown.

It is this kind of thinking that we must get children to do.  Maths lessons should be planned with a sole priority – what will the children be thinking about?  In this first stage of a unit of work, children develop their fluency and then begin to reason about what is unknown in a numerical problem and how to figure out that unknown.

Deliberate teaching of language

The second stage of my model involves building on the number work by adding layers of mathematical language that enable children to talk like mathematicians and understand problems involving ambiguous language. We need to embrace ambiguity because it is in that grey area of language that we can really get children to think hard.  The image below is one I’ve seen many times in many schools and even had up in my classroom in a previous life:

What this kind of display tells children is that a word equates to an operation.  This is misleading at best and more likely disastrous for understanding.  Words like this only carry meaning in context, for example look at the phrase ‘more than’:

It does not mean that you have to add the numbers together.  It can mean that, but it could also mean finding the difference.  And what of the word difference?

The ambiguity of language must be deliberately taught and linked back to the underlying structures that children will have been working on.  Teachers model the thinking and ask: In the first  question, is the whole unknown or a part unknown?  What about the second sentence? Draw it…

Wider problem solving

Remembering Hattie and Yates’ assertion that novices see surface features of problems and experts see the same problems at a deeper level, consider this:

Anum and Jay have saved up their pocket money.  Altogether they have £35 and Anum has saved £18.50.  How much has Jay saved?

A novice would read that question and would say it’s a problem about pocket money – the surface feature.  An expert would look at the same problem and say that it’s one where the whole is known and so is one of the parts, but that the other part is unknown.

Once teachers know how experts think, it is perhaps a mistake to simply try and get novices to think like that.  Experts have a vast store of knowledge from which they draw on when analysing problems and so to get novices to eventually think like experts, we must first teach them the underlying patterns.

With this in mind, take a look at this common practice:

If we teach children how to subtract, then give them problems that only require subtraction, what are children really thinking about?  Not structure.  Not language.  And yet we can still mislead ourselves when children ‘get the right answers’ that they truly understand what they’re doing.  They may well do but we can’t be sure with tasks like this.  The tasks that we set show what we value.  Perhaps a better task is to ensure that children are thinking like mathematicians, sorting then solving problems based on their underlying structure:

Throughout the entire unit

Other considerations during a unit of work include the big ideas in maths.  Coined by Mike Askew, these are concepts that children develop throughout their time at school and are built on year on year:

• position on a number line
• estimation
• equivalence
• place value
• numerical reasoning
• the meaning of symbols
• classifying
• sequences

Opportunities should be created throughout a unit for children to think about content in these ways so that they can make connections between ways of thinking and different representations.

Finally, but by no means of least value, teachers must pay careful attention to success criteria. Before this though, objectives need to be sound. Shirley Clarke’s work on formative assessment is important here and there should be clarity between closed and open objectives. Closed objectives are absolute. They have either been achieved or they have not. Procedural success criteria are most appropriate here – steps to follow in order to be successful. Open objectives on the other hand are subjective in that they can be achieved to varying degrees of quality. A selection of tools is most appropriate for success criteria in this instance – strategies to choose from with the goal of efficiency, for example. Year 5 children could be given the calculation 5023 – 3786 and they should be able to, following great teaching, choose a subtraction strategy to solve that calculation in the most efficient way possible. All the procedural work, such as deliberately learning to count up, count back, round and adjust, or carry out column subtraction comes together and all those strategies form a toolkit from which children choose he best tool for the  job.

This model for teaching maths certainly covers National Curriculum aims but more importantly, it strives to get children to think and communicate like mathematicians.

Filed under Maths

## Multiplying fractions and integers

This is an account of a particularly successful topic from Y5 on multiplying fractions and integers. It’s fast becoming one of my favourite concepts to teach.
I showed multiple representations of a calculation, showing how to make it with cubes:

In this example, it was important to make four lots of two thirds first before rearranging the coloured cubes: later when they draw it on a number line, they’ll need to be thinking of four jumps of two thirds. The language of number of parts, size of each part and whole was supported by the bar model and came in handy for the empty box problems later.
Children needed to make other examples of calculations, talking about it with a partner:

Most children were fine with this and ready to move on to pictorial representation on a number line. Some needed a bit more consolidation with how the cubes represent a calculation. This is of a set of scaffolded questions where children made the calculation with cubes, then drew it:

For those ready for a number line, I explained a worked example before modelling how to do one from scratch:

The explanation had to be clear on how to construct the number line accurately, which involved making decisions about how many squares to leave between each whole number (using the denominator of the ‘size of each part’), and how many squares each jump needed to be (using the numerator of the ‘size of each part’). The final decision to make was how many jumps to draw, using the number of equal parts. The whole, and the answer to this particular calculation, was where the last jump ended up in the number line.

When children practised doing this, they chose from using a scaffold of minimally different questions or tackling varied questions:

Some children needed a few lessons to keep practising this and each time we returned to it, I started by showing multiple representations of the same calculation:

Some children grasped this concept quickly and needed to work on deeper tasks. I made some subtle changes to the language I used to describe what was happening (bottom right corner of the above image) and used the same bar model to introduce inverse operations with aim of learning to solve empty box multiplication of fractions:

Bar models helped to explain what was known and what was unknown, using the language of whole, number of parts and size of each part. I showed how it looked on a number line too:

Some managed to get on to dividing fractions and whole numbers, using bar models (which they were familiar with for integer multiplication and division) and number lines to show what was happening:

All the while, more children began to move from basic multiplication of fractions to some deeper thinking about multiplicative reasoning and fractions.

A few weeks later, children worked out the area of rectangles involving fractions and had internalised much of the language to be able to represent questions like this: