Tag Archives: interleaving

What I think about…learning

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.  First was displays.  Next up – learning.

Asking teachers what learning is surely throws up disagreements of varying degree from polite dispute to outright warfare.  What makes sense to me is that learning is a change in long term memory.  Too often, children don’t manage to transfer concepts from working memory to long term memory and without that internalisation, we cannot say that they have learned.  All we can say is that they have done some work.  Now that work might well have been good, but teachers and leaders need to be aware of the difference between short term performance and long term internalisation.

Performance vs learning and the importance of desirable difficulties

The key paradox is that to improve long term retention, learning has to be made more difficult in the short term even to the extent of being unsuccessful.  We remember what we think about and learning happens when we have to think hard about content.  If children are thinking about things other than what we have intended for them to learn (a distracting context, for example) then that’s what they’ll remember.  If they haven’t had to think too hard, they may well produce some decent work but the thinking behind it is less likely to be retained.  So what does this mean?  Units of work and individual lessons need to be planned around what it is that children will be thinking about.  Each decision about what the teacher will do and what the children will do needs to be justified with that question mind and amended accordingly.  We all get better at what we habitually do – we become more efficient – so if we require children to be able to remember knowledge, procedures and concepts, we must give them ample opportunities to practise remembering those things.  The efficacy of the testing effect has robust evidence and seems to work because testing (either yourself or a teacher posing questions) triggers memory retrieval and that retrieval strengthens memories.  Flash cards are a perfect example of this in action.

What’s important is that this testing is low stakes – no grade, no mark at the end of it, just practice in remembering and feedback on responses.  Feedback can take two forms.  Firstly the feedback can be from teacher to child and is as simple as telling the child what they were good at and what they misunderstood, then correcting those misconceptions.  Secondly, feedback can be from child to teacher and involves the teacher using the information to plan what to do next to develop understanding further.

Low stakes testing is a desirable difficulty – one way of making learning difficult (but not too difficult) so that children have to think hard.  Other desirable difficulties apply more to curriculum design:

  • Interleaving (switching between topics)
  • Spacing (leaving some time between sessions on a particular topic)
  • Variation (making things slightly unpredictable to capture attention)

By presenting content to children little and often, with increasingly longer spaces in between, teachers can instill the habit of continual revision rather than only revising when some sort of exam is approaching.  As such, concepts are internalised and retained rather than forgotten.  Robert Bjork’s research on desirable difficulties can be found here:

Knowledge

The idea of knowledge can be divisive.  Recalling knowledge is often described as lower order thinking and many are keen, quite rightly, to get children to do higher order thinking. This can be dangerous because knowledge is necessary but not sufficient.  Higher order thinking skills rely on a sound basis of knowledge and memory so teachers must ensure that these aspects are fully developed before expecting success in higher order thinking.  Knowledge needs to be internalised too.  It’s not enough to be able to Google it.  The more a child knows, the easier it is to assimilate new knowledge because more connections can be made:

Knowledge

Scaffolding

Children are more alike than different in how they learn.  Attempting to teach to a child’s perceived learning style is nonsense.  Everyone, no matter what we are learning, requires three things: knowledge, practice, and feedback on how we’re doing.  It is of course true that children come to a lesson with varying levels of prior knowledge and to a certain extent have different needs in order to be successful.  Teachers may have (and many, I’m sure, still do) differentiated tasks three, four or more ways – an unnecessary burden on time and a practice that reinforces inconsistency of expectations, particularly of the perceived ‘lower ability’ children.   For those children that are behind their peers, if they are not supported to keep up with age related expectations, they will be perennially behind and will never catch up:

Keeping up Differentiation

If we only cater for their next small step in development, we’re failing them.  Instead, all children should be expected to think and work at age related expectations.  Teachers should scaffold tasks appropriately so that all can work at that expectation and we do not have a situation where ‘that’ table are doing something completely different.

Scaffolding

For children that grasp concepts quickly (not our ‘most able’ children – heavy lies the crown…), teachers provide opportunities to deepen their understanding before acceleration into subsequent year groups’ content.  Undoubtedly, there are a small number of exceptions to this.  There are some children that have a lot of catching up to do before we can even think of getting them to keep up with age related expectations.  But if they are removed from lessons to carry out this catch up work, then everything will always be new to them – they’ll miss seeing and hearing how children are expected to think and work.  It is much better to precisely teach, and get them to practise, the basics that are not yet internalised in short bursts and often so that they remain with their peers as much as possible, experiencing what they experience but having the support needed to catch up.  This could be basics such as handwriting and number bonds, for example, and teachers should work closely with parents where there is a need to catch up to set short term, focused homework until the basics internalised.

Intervening

When children misunderstand something, when the work in their books is not to the standard expected, is a crucial time.  Paramedics talk of the golden hour – one hour after an accident – where if the right treatment is given, the chances of recovery are significantly higher.  With children’s learning, if we leave misconceptions to embed or even thrive, we’re failing them.  Even if we mark their books and write some wonderful advice for them to look at and act upon the next day or the day after, we leave holes, holes which children can slip through.  When there is a need, we should intervene on the day so that children are ready for the next day’s lesson and are keeping up.  This of course requires flexible and creative used of TAs and non-class based staff but from experience, it works. Interventions focus on the work done that day.  For some children, pre-teaching may be more beneficial.  Before the school day starts, they are shown the main content of the day’s lesson and carry out a couple of practice examples so that when it comes to the lesson later on, they have some prior knowledge which will improve their chances of success in that lesson.  This concept is in contrast to pre-planned, twelve week intervention programmes where children are removed from other lessons for significant periods of time.

Learning is complex and relies on many interrelating and often unpredictable conditions.  That said, there is much that we can control and doing so greatly increases the likelihood that what we intend to learn is learned – really learned.

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Filed under Memory

Mastering maths curriculum design

Recently, Bruno Reddy and Michael Tidd have written about their experiences in designing a mastery maths curriculum. It seems that there are lots of us designing overviews for year groups with the same goals in mind: long term retention of knowledge and concepts with problem solving at the heart.

It is fascinating to see what others have come up with working on the same project, and also a little reassuring. Bruno provides six tips for creating a mastery curriculum, some of which I’d agree with wholeheartedly and some I’d adapt slightly for a primary curriculum. He advocates lots of practice, separating minimally different concepts, teaching concepts in a sensible order, and spending more time teaching fewer things. All of which sound like good advice to me.

Whereas Bruno suggests going back to basics in the first part of Year 7, I’d say that for a primary curriculum, the spotlight on place value, number facts, mental arithmetic and written algorithms needs to be relentless; spaced out and returned to many times over each year. In lower key stage 2, I’d have these units of work at least termly and more likely half termly. Each time the topic is revised, the expectations can be upped, with more and more time devoted to solving problems. This would enable the teacher to spend the time on good modelling and practice, while gradually increasing the expectation in that topic over the year and introducing different styles of problem. I’d say that this is preferable to organising revision through starters, homework etc alone.

Like Michael, I started with the mathematics mastery example overview.

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Those arrows denoting the continued study of a concept over the year do not provide enough guidance for a teacher as to when and what to specifically revise. For the draft that I’m working on, I wanted to change that. While Michael describes spending a whole half term on fractions, I’d suggest splitting fractions into different units and linking back each time, like in the draft year 4 example below:

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I thought that Children would move from comparing pictorial representations of fractions, to looking at comparing fractions and decimals, to finding fractions of quantities throughout year 4. Each time there’s a unit of work related to fractions, the teacher would check up on prior knowledge (with some sort of test – see benefit number 3 on test potentiated learning) and spend an appropriate time remodelling and getting children to practise previous content before linking to the main work in the topic.

A maths overview like this would be fine for maths specialists but for some, and indeed NQTs and teachers new to a year group, a little more guidance might be necessary. To supplement the overview, I’d have some more detail on the topics themselves including the core knowledge that children must have and be able to recall in order to master the topic and a brief idea of what children should be expected to do:

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This pair of topics will be one unit of work within autumn term 1. These topics will be repeated over the year, so that practice will be distributed or spaced, but another layer of spacing can be introduced by switching between these two topics: a couple of days on place value, a couple of days on mental addition and subtraction. See Robert Bjork on desirable difficulties and a recent post of mine on the same ideas.

All of this, of course, is still a draft. The order of concepts could do with a review, as could the expectations for what children should be able to do in each unit. Also, the overview needs to emphasise flexibility. Although topics are arranged to be revised as shown, there will be occasions when some topics don’t need as much revision while others will need more.

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Filed under Curriculum, Maths

Spacing, Interleaving and Retrieval Practice in Primary Maths

In the last few weeks there has been a flurry of posts written on spacing, interleaving and retrieval practice. It seems that this flurry has in part been triggered by @miss_mcinerney’s Touchpaper problems. Two that stand out are Joe Kirby’s and Mark Miller’s. Both digest the research before summarising with great clarity what seems to be optimal conditions for learning. I first came across the ideas reading David Didau’s blog, and have been working on Year 6 maths planning to benefit from the effects of spacing, interleaving and retrieval practice. It’s split into 2 parts: longer term curriculum design and shorter term lesson planning.

Curriculum Design

OVerview

This screenshot is a section of the Year 6 Spring Term overview. The overview is split into units of work which consist of two topics. Sometimes, these topics compliment each other in order to show children links between areas of maths: working walls depict these links and they are referred to often. Other times, there is no link between them. This is a first draft of a curriculum overview and although there are probably more meaningful combinations of topics, it will take some time to reflect and switch things around. In this instance, I’m not sure how significant the benefit would be to deliberate too much over this.

The superiority of spaced rather than blocked practice is well known, and this overview plans for spacing in two ways. Some topics are repeated regularly as additional teaching blocks. The Pareto Principle, or the ‘law of the vital few’ describes the imbalance of effects of different causes. The theory applied to this situation would suggest that twenty percent of the content of the curriculum provides eighty percent of the value: there are certain topics that have much greater value than others. Knowing number facts such as times tables as well as being able to calculate quickly and reliably would certainly be within that twenty percent. As such, these vital few topics are repeated often.

Day to Day Planning

The other way that spacing is set up is through the switching between the two topics in each unit of work. Deciding when to switch is contextual – a natural break in one topic is the switching point.  For example, a few days on converting betweeen fractions and decimals before switching to working on calculating unknown angles would provide a few potentially fruitful opportunities.  It gives the teacher a bit of time to assign any extra practice (perhaps for homework) to help some children to be ready for ordering fractions and decimals.  It also gives the teacher a chance to delay feedback for a couple of days, which could be well worth experimenting with, as David Didau suggests here.

But what of the topics that are not in the vital few? These need to be spaced too if they are to be encoded into long term memory. A relic from the National Strategies is the oral / mental starter which could be tweaked to provide spacing and retrieval practice. Each lesson, an old topic is selected to work on where children use a model or image to practise recalling a concept, before working through a series of questions to practise recalling procedural knowledge. This not only spaces out learning but gives the teacher the opportunity to see what children can still do or what they have forgotten; to give feedback on known and likely misconceptions; and plan for revision sessions.  In the example below, children had, within the last few weeks, been working on calculating the area of compound shapes.  The success criteria that we developed at the time was shown on the screen and children used the images to recall the steps needed.  After that, they had the opportunity to practise.  The questions got progressively more difficult from left to right and children either chose to start from ‘column 1 or 2’ or were directed to the appropriate questions:

 MM PerimeterMM PErimeter 2

Factual recall is crucial in order to think with clarity about a concept. For example, if children are to be able to compare fractions, decimals and percentages, they have to be able to quickly recall conversion facts. For situations like these, the mental maths session would include individual use of flash cards, like these.

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Children look at the prompt then say the decimal and percentage conversion. They turn the card over to check and make two piles. One pile of facts that they can reliably recall accurately, and one pile of facts that they have not yet internalised. When putting the cards away, the ‘wrong’ pile gets put on top to practise first next time. Often, having practised an area of maths, a short problem solving task is presented for children to work through, like in the screen shot below.

FDP Q

What next?

My organisation of the spacing is still fairly arbitrary. Whether there are optimal spacing times is not yet clear and certainly, trying to engineer optimum times would be difficult and perhaps not worth the opportunity cost, especially if it turns out to be non linear.

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Filed under Maths