Inset on Monday was a session on developing subject knowledge and expertise in the teaching of fractions. In the session, we worked on planning some work to include:

a broad knowledge of expectations across the primary age range

a sensible sequence through the mini topics within ‘fractions’

appropriate models and images

finding realistic as opposed to contrived contexts to apply learning

a variety of work to include fluency of factual recall; necessary repetitive practise; reasoning; investigations etc

fine tuning of what success criteria will look like

deliberate practise of explaining concepts and generating success criteria

consideration of links to other mathematics

The reason that INSET time was invested in what is the ‘bread and butter’ of teachers’ planning was to free up day to day time in order to carry out action research into the principles of interleaving and spacing. Below are links to Robert Bjork talking about both interleaving and spacing and, if you prefer, an excerpt from his book:

We must be cautious with how we interpret research into education. With so many variables in our classrooms, it is notoriously difficult to pinpoint cause when seeing an effect. All we can do is strive to apply what has been claimed in our own classrooms in order to secure the best opportunities for the children in our care. Here’s Dylan William’s rallying cry on developing expertise:

An important thing to remember with work on interleaving and spacing is that it can produce benefits to long term memory, as opposed to children looking like they understand something. We’ve all furiously revised before an exam, which worked out OK, then forgotten the content completely soon after. We may not see quick results, but we must be brave enough to strive for expertise and excellence.

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10 thoughts on “Subject Knowledge and Expertise – Mathematics”

So what are the implications here? I like the idea of the interleaving – in terms of teaching writing that might mean that we need to do more where we ask children to compare different text types and think about differences and similarities? I’m sure Dylan Williams is right when he talks about how we can all develop our teaching. I’ve seen that happen time and time again.

Some implications revolve around organisation – although it would not necessarily mean any extra work to interleave different text types (or maths topics), it would mean more planning ahead. In my view the basis for this has to be secure subject knowledge in order to be flexible yet well prepared. It has to be manageable too.

As you say it opens the door for quality comparisons between text types whereas at the moment most of us probably only compare within a genre. If it works it should lead to greater internalisation of toolkits. I wonder how spacing could work to fit in with Talk for Writing? Perhaps a little spacing between innovating and inventing could be beneficial? We’ll give it a go…

In year 4, we have been interleaving fractions and multiplication. We have been spending 2/3 days on each topic and children have adapted well to the changes in routine. We have noticed that children have become less ‘robotic’ and it clearly reveals when children have misconceptions.

We have ‘chunked’ fractions into mini topics that build upon previous learning.

Using the counting stick, we have counted in fractions, including halves, quarters and tenths. For above expectation children, we have worked on conversions between fraction and decimals.

In mental power, we have focused on quick mental recall of previous learning within ‘number’. We have tried to include a whole class reasoning question followed by a set of questions that need to be completed within a set time (to support quick recall).

When we have moved children from fractions to multiplication, for example, some children have been unable to solve a confirming learning question that has been previously left by the CT (fractions). Although the continuity has been lost, it clearly shows who has the ‘deeper’ knowledge and who is still relying on robotic, predictable learning.

Variation:
Counting and mental power activities are varied on a daily basis.

Spacing and interleaving:
One lesson a week is now dedicated to revisiting a previously taught unit. This takes the form of either using and applying work related to a topic or progress in maths as a record of achievement. During the input for these lessons, children who are still having difficulty in grasping a skill are identified and work as part of a guided group with the class teacher or LSA. Fridays are dedicated to the children completing a mental maths test. During marking, appropriate strategies for solving problems are discussed. Class teachers make notes about which questions have caused particular problems and these are incorporated into the following week’s sessions. Children regularly scoring below 10 in mental maths tests are targeted for support during counting and mental maths using the conkers maths cards.

Good question about when to move on. I think that by splitting up a whole unit of work into smaller chunks, there are many opportunities for suitable end points. Try out different periods of time for spaces. If they have a fragile grasp of something perhaps shorten or extend the time before looking at it again. See what works!

Ok that makes sense, we will try that next term. Thanks Nick

Sakina, Pam, Jen and Sarah de R

In year one, we have from September allocated 2 lessons a week focussed on number and 3 lessons on our current unit. Our number lessons vary from week to week so that we vary the topics and review previous concepts. This allows us to review the children’s learning and if they have fully understood the topic and we then address any misconceptions.

In year 6, we have split the vast unit of work on fractions, decimals and percentages into smaller, shorter mini topics. For example: Fractions of shapes; equivalent fractions; ordering fractions; fractions of numbers. We interspersed these with similarly short units of work on area and perimeter. These seemed a good match, as there are some good links between them (What’s the area of 25% of the shape?). We still did not cover all of the fractions requirements in one go and we’re still returning to some bits that children need further work on.

We have used a variety of counting tasks: times tables; missing numbers; crossing boundaries; counting in mixed numbers; working out mid points between numbers.

Interleaving – we are teaching two topics each week, one for 3 days and one for 2 days. The lessons have smaller learning intentions, based on concrete learning before moving to abstract concepts. Topics are reviewed and revisited regularly during mental maths and starters.

Others in KS1 should try – Basing mental maths on testbase questions to help prepare the children for ‘test’ questions – group discussions and sometimes independent working using whiteboards.

We have also ensured that we are always differentiating work by each child’s previous day’s outcomes rather than the work that is planned for that “group” as the children within the same level grasp concept at different paces so they are therefore given work that they need, rather than what their “group” should be able to do.

Problems – Children that don’t internalise knowledge as quick as others could do with more time on each topic. Difficult to always fit in working with those that struggle.

Pie CorbettSo what are the implications here? I like the idea of the interleaving – in terms of teaching writing that might mean that we need to do more where we ask children to compare different text types and think about differences and similarities? I’m sure Dylan Williams is right when he talks about how we can all develop our teaching. I’ve seen that happen time and time again.

Nick HartPost authorSome implications revolve around organisation – although it would not necessarily mean any extra work to interleave different text types (or maths topics), it would mean more planning ahead. In my view the basis for this has to be secure subject knowledge in order to be flexible yet well prepared. It has to be manageable too.

As you say it opens the door for quality comparisons between text types whereas at the moment most of us probably only compare within a genre. If it works it should lead to greater internalisation of toolkits. I wonder how spacing could work to fit in with Talk for Writing? Perhaps a little spacing between innovating and inventing could be beneficial? We’ll give it a go…

Emma and Tracy O'SIn year 4, we have been interleaving fractions and multiplication. We have been spending 2/3 days on each topic and children have adapted well to the changes in routine. We have noticed that children have become less ‘robotic’ and it clearly reveals when children have misconceptions.

We have ‘chunked’ fractions into mini topics that build upon previous learning.

Using the counting stick, we have counted in fractions, including halves, quarters and tenths. For above expectation children, we have worked on conversions between fraction and decimals.

In mental power, we have focused on quick mental recall of previous learning within ‘number’. We have tried to include a whole class reasoning question followed by a set of questions that need to be completed within a set time (to support quick recall).

When we have moved children from fractions to multiplication, for example, some children have been unable to solve a confirming learning question that has been previously left by the CT (fractions). Although the continuity has been lost, it clearly shows who has the ‘deeper’ knowledge and who is still relying on robotic, predictable learning.

Katherine and Anna BVariation:

Counting and mental power activities are varied on a daily basis.

Spacing and interleaving:

One lesson a week is now dedicated to revisiting a previously taught unit. This takes the form of either using and applying work related to a topic or progress in maths as a record of achievement. During the input for these lessons, children who are still having difficulty in grasping a skill are identified and work as part of a guided group with the class teacher or LSA. Fridays are dedicated to the children completing a mental maths test. During marking, appropriate strategies for solving problems are discussed. Class teachers make notes about which questions have caused particular problems and these are incorporated into the following week’s sessions. Children regularly scoring below 10 in mental maths tests are targeted for support during counting and mental maths using the conkers maths cards.

Hanna, Juliette and MichelleInterleaving – money (addition and subtraction) and fractions

Variation – counting – different ways of counting /counting in fractions– table tennis, number square, using a ball

Variations Mental power – quick calculations – number bonds (response to not knowing) – 3 minutes to do activity

What others should try –/video problem/more real life – shops

What problems were encountered?- How much should we do before moving on? Just when you think they are getting it and then need to move on.

Nick HGood question about when to move on. I think that by splitting up a whole unit of work into smaller chunks, there are many opportunities for suitable end points. Try out different periods of time for spaces. If they have a fragile grasp of something perhaps shorten or extend the time before looking at it again. See what works!

HannaOk that makes sense, we will try that next term. Thanks Nick

Sakina, Pam, Jen and Sarah de RIn year one, we have from September allocated 2 lessons a week focussed on number and 3 lessons on our current unit. Our number lessons vary from week to week so that we vary the topics and review previous concepts. This allows us to review the children’s learning and if they have fully understood the topic and we then address any misconceptions.

Nick H and JoIn year 6, we have split the vast unit of work on fractions, decimals and percentages into smaller, shorter mini topics. For example: Fractions of shapes; equivalent fractions; ordering fractions; fractions of numbers. We interspersed these with similarly short units of work on area and perimeter. These seemed a good match, as there are some good links between them (What’s the area of 25% of the shape?). We still did not cover all of the fractions requirements in one go and we’re still returning to some bits that children need further work on.

We have used a variety of counting tasks: times tables; missing numbers; crossing boundaries; counting in mixed numbers; working out mid points between numbers.

Sarah C and AmyInterleaving – we are teaching two topics each week, one for 3 days and one for 2 days. The lessons have smaller learning intentions, based on concrete learning before moving to abstract concepts. Topics are reviewed and revisited regularly during mental maths and starters.

Others in KS1 should try – Basing mental maths on testbase questions to help prepare the children for ‘test’ questions – group discussions and sometimes independent working using whiteboards.

We have also ensured that we are always differentiating work by each child’s previous day’s outcomes rather than the work that is planned for that “group” as the children within the same level grasp concept at different paces so they are therefore given work that they need, rather than what their “group” should be able to do.

Problems – Children that don’t internalise knowledge as quick as others could do with more time on each topic. Difficult to always fit in working with those that struggle.