Tag Archives: Memory

What I think about…reading

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, learning generally and maths. Next up – reading.

I’m proposing a model for teaching reading grounded in the various books that I’ve read. The examples will be for a fiction text but I think the principles apply to teaching non fiction too.

Reading model

Some principles

The first principle to be mindful of is that the teaching of reading is not the asking and answering of questions about a text: that’s testing comprehension.  Sure, asking and answering questions is an important part of developing comprehension – it’s one way we get children to think hard about what they have heard or read – but there is much more to it than that.  Any reader constructs a mental model of the content of what they have read – we don’t usually remember text verbatim without rereading many times and deliberately trying to remember it word for word. Poor comprehenders construct weaker, less detailed and perhaps outright inaccurate mental models whereas good comprehenders construct more accurate and elaborate ones.  One goal of teaching reading then is to ensure children construct good mental models of what they have read. I’m making the assumption here that children can decode fluently and focusing solely on the development of language comprehension.

Simple view of reading

Good readers combine word recognition with language comprehension to be able to decode the print and understand the language it yields. Once fluent in decoding, it is depth and breadth of vocabulary and general knowledge that contribute to comprehension and so the teaching of reading must develop vocabulary and background knowledge.

Developing reading comprehension

Poor comprehenders share many similar characteristics which we need to understand and use to drive the teaching of reading.  Poor comprehenders:

  • have limited general knowledge
  • have a limited knowledge of story structure or don’t relate events in a story to its general structure
  • have a narrow vocabulary and don’t know the meaning of important words
  • read too slowly, without fluency or enough prosody to understand the content
  • focus on word reading without focusing on content
  • make incorrect pronoun references
  • don’t make links between events in the text
  • don’t monitor their own understanding of what they’ve read
  • don’t see the wider context in which the text is set
  • don’t build up a secure understanding of the main events in a story
  • misunderstand figurative language

When it comes to vocabulary, we can’t teach every word or phrase that children might not know and neither should we. If we do, not only would it be incredibly time consuming but we’d also greatly reduce the experience that children have of deciphering meaning from contextual cues. Some words and phrases need to be taught explicitly before or during reading while others can be learned implicitly during reading.  Either way, if children are to master the language, they must think hard over time about its use.  Put the dictionaries away and don’t start off with ‘Who knows what x means?’  These are both particularly inefficient uses of time and are ineffective.  Instead:

  • Model the use of the word in its most common form
  • Use an image (this post from Phil Stock is excellent)
  • Act it out
  • Model other common uses
  • Explain word partners (for example, if teaching the word announce you often see make an announcement together)
  • Show various forms of words including prefixes and suffixes
  • Show words that are similar to and different from the focus word

Lemov (Reading Reconsidered)

That last bullet point is not the same as using the synonym model for teaching word meaning.  Telling  a child that melancholy means sad robs them of the beauty of shades of meaning because it is similar to, not the same as sad.

Memory is key. We remember what we think about, so part of teaching reading needs to be giving children plenty of spaced practice in remembering word meanings, general knowledge, events from the text and details of the characters that are crucial to developing a sufficient mental model of the text. It could well be the case that a child who has shown poor understanding of a text is not unable to comprehend it, they just can’t remember what’s necessary to comprehend. Regular low stakes testing of key knowledge from the text is a strategy to ensure this retention and readiness to mind.  Joe Kirby’s knowledge organisers are very useful for this and here’s one I made for Philip Pullman’s Northern Lights. 


Stage 1 – oral comprehension

Prepared reading, or providing a brief structural overview, ensures that no child hears the story without some prior knowledge.  In the first instance, read aloud or tell children the story. Capture their interest. Retell it, perhaps in different ways.   Lemov, in Reading Reconsideredidentifies different types of reading and here I’d go for what he calls contiguous reading – reading without interruption from start to finish, experiencing the text as a whole.  It may be sensible to teach the meaning of some words that are crucial for overall understanding of the text but not too many at this stage.  I’ve compiled some thoughts on introducing texts and teaching vocabulary here.

What have children understood?

Clearly it is tricky for teachers to know what children have understood and by asking questions all we really know is whether they are capable of comprehending, not whether they actually comprehend independent of us. Before any specific questioning, it would be useful to get an idea of what they have understood by asking them to tell you about what they’ve just read. The decisions they make about what they say (or write)  reveal what they think is important and you can also judge the accuracy of their literal and inferential comprehension. Aidan Chambers’ Tell me gives advice on developing this in a slightly more structured way whilst still retaining the importance of open questioning.

The key to this stage of reading is the focus on oral language comprehension.  Difficulty decoding should not be a barrier to children experiencing and understanding age appropriate texts.  Lemov puts this beautifully:

Low readers are often balkanised to reading only lower level texts, fed on a diet of only what’s accessbile to them – they’re consigned to lower standards from the outset by our very efforts to help them.

Lemov (Reading Reconsidered)

This is one of the reasons why I’m in favour of the whole class teaching of reading and not the carousel type ‘guided reading’.  Listening to texts and using open questions to prompt discussions ensures that the focus in on language development in a way that is not restricted by poor decoding.  Having said that, those children who are not decoding to the standard expected will still need some sort of intervention running concurrently to this so that they catch up.  The benefits of focusing on oral language comprehension have been shown in the results of the York Reading for Meaning Project, written about in Developing Reading Comprehension by Clarke, Truelove, Hulme and Snowling and here.


Stage 2 – modelling the reader’s thought processes and shared reading 

The information that teachers can gather from the open questioning in stage 1 then focuses modelled and shared reading on specific parts of the text. The teacher can model the reader’s thought processes, and get children thinking about the tricky bits. This isn’t simply reading the text from beginning to end; reading will be interspersed with commentary, explanation or making links to general knowledge.  Lemov calls this line by line reading, with frequent pauses for analysis and allowing the teacher to show children that good readers think while they read in order to achieve an acceptable standard of coherence.  As children get older and texts get longer, teachers can’t lead shared reading of the whole text, so by initially earmarking sections that children are likely to misunderstand and by using information gathered from stage 1, shared reading can be focused on addressing misconceptions.  Again, Lemov puts it succinctly:

Shared reading mitigates the risk of misreading.

Lemov (Reading Reconsidered)

I’d expect children to then read the text independently, drawing on what they’ve heard from the teacher’s modelling and all the oral language work. Children should have the opportunities for multiple readings of at least the tricky bits.  These bouts of reading become iterative: children build layers of understating with each reading.  For those children whose decoding is weak, they can be directed to smaller extracts, practising decoding and fluency with a text that they should have a decent understanding of following all of the language work.  It’s important to continue to get children thinking about new words that were taught in stage 1.  If that vocabulary is to be reliably internalised, they’ll need multiple interactions.

This is also an ideal point to make some links to non-fiction that can supplement understanding of the fiction. Questioning that involves deliberate comparison between the fiction and non fiction complements understanding of both.  For example, if reading Robert Louis Stevenson’s Treasure Island, spending some time on books or extracts such as below will significantly aid comprehension.

Non fiction links

Written responses

Writing is thinking, and to paraphrase Lemov in Reading Reconsidered, not being able to record their thoughts about what they’ve read on paper does not make them invalid, but children are at a significant disadvantage if they are unable to craft an articulate, effective sentence explaining what they have understood.  To this end, returning to those original open questions and working with children to refine their responses and write them effectively is a valuable use of time.  The teacher can model scanning the text for the part needed to refine an idea, or to check a detail, and then children should also be expected to behave in that way.  This post by Lemov makes very interesting reading on that topic.


Stage 3 – targeted questioning

It’s standard practice to ask questions of a text after it’s been read but a great deal of care needs to be taken in choosing or discarding already written questions, or in writing them ourselves. Questions need to be text dependent, otherwise what we’re really doing is getting children to activate general knowledge. An example of this, from Understanding and teaching reading comprehension by Oakhill, Cain and Elbro, is:

Where does Linda’s pet hamster live?

  1. In a bed
  2. In a cage
  3. In a bag
  4. In a hat

The possibility of guessing the right answer here would tell the teacher very little of the child’s ability to comprehend text and so asking questions where understanding is dependent on what’s written or what must be inferred from the text is a must. Doug Lemov espouses the importance of text dependent questions in Reading Reconsidered.

When designing questions, teachers must also use knowledge of the characteristics of poor comprehenders in order to model corrective thought processes and to ensure children think in a way that helps them to comprehend more reliably.  For example, we should give them plenty of practice in working out to what or whom pronouns refer.

The education system we work within requires examinations to be passed which then provides opportunities.  Preparing children for success is morally imperative. Write questions in the style of SATs questions about the text, model the thinking process behind successful responses and give children practice doing just that.


Stage 4 – fluency and prosody

Don’t misunderstand – children should be supported continually to read fluently with appropriate intonation and expression. It’s just that to do that well, a reader needs to understand the text. At this stage, that should be the case. Reading for fluency and intonation using a text that children know very well should yield great results and not only that, it provides another opportunity to glean previously missed understanding.

So there it is. A model for teaching a text that moves from oral to printed comprehension; general to specific questioning; and oral to written responses, all the while practising fluency and developing language.

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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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What I think about…displays

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 up – displays.

Displays can take up vast areas of wall space and many hours of adults’ time, therefore teachers and leaders must be sure of the impact that they are having on learning so that what is on display is justified and not simply a waste of time and space.  Put simply, before a display goes up, we must ask: What will this display do to improve outcomes for children?  For this to be answered with any sort of reliability, the question must be framed within a sound knowledge of how children learn and what learning is – a change in long term memory.

Recognition vs retrieval

Information displayed in a classroom can lead children to recognise rather than retrieve the knowledge and concepts that they have been learning. Recognising information that they have spent some time thinking about is much easier than recalling it from memory and can give the illusion of understanding both for the child (‘Oh I know this…’) and for the teacher (‘Hurrah – she knows this!’).  Classrooms with lots of information displayed can become a trap, a trap where both children and teachers come to believe that children have learned what we wanted them to learn.  Research by Robert Bjork into desirable difficulties differentiates between short term performance and long term retention.  Children can quite easily ‘perform’ if they know where to look in a classroom to find information that they can recognise and use to show their teacher that they know something.  However, it is the act of retrieving that strengthens memories – after all if we deliberately practise remembering things, we get better at remembering them.  If we practise looking for things when we need to know something, we get better at looking for those things.  Some would argue little difference between those two scenarios but the difference is subtle.  If children have knowledge and concepts to mind almost immediately, that means that finite working memory capacity is freed up to focus on other things such as paying attention to solving more complex problems.

Key principles

Displays should serve three functions.  Firstly, they should act as memory prompts for the knowledge, concepts and ways of communicating and thinking that children are currently learning or have been learning.  Images, symbols and words should be used to trigger memories and scaffold thinking and talking, with children being given regular opportunities to use displays in this way.  For example, rather than displaying definitions of sentence types, display something like this:

IMG_4095

Then, get children to regularly use it to think and talk about the concept.

Secondly, displays should set a standard for the extent of knowledge and the quality of work expected of children.  When displays are beautifully set out and are talked about with care by teachers and leaders, it shows that we value the quality with which work is produced.  This is why neat borders, carefully spaced work and pride in what’s on display are important – it’s one way of setting standards of children’s work in their books.  If we allow irrelevant content, or not enough depth of content, or display boards to become tatty, then we’re hypocritical when we expect the those same things in children’s books.

Thirdly, they should make the classroom an inviting place that stimulates interest in the subject content to be learned.  They should trigger enthusiasm for learning – one of many hooks so that the teacher can work with receptive minds.

Pitfalls to avoid

Displays should not be used in an attempt to prove that a particular initiative is embedded.  Posters about mindset or school rules, for example, if displayed on a wall, do not mean that those aspects are established as part of the school culture.  Displays like that mean nothing unless the ideas behind them can be articulated by children, teachers and leaders.  It is important here to return the first idea of recognition vs retrieval: displays about mindset and school rules (to name just two – there are, I’m sure, many other applicable projects) can be useful as long as they are thought about carefully.  Use images, symbols and words and give children regular opportunities to think about and express their meaning.

With the sheer amount of content that children are expected to learn, it can be tempting to plaster every inch of wall space with some sort of display.  This is a mistake.  Children can only attend to so much from the environment around them before working memory is overloaded.  A result of this is that some displays barely even get looked at and if that is the case, why are they there?

Displays, if done well, can have a significant impact on children’s learning or they can be a colourful yet ignored decoration.  If we take into account what is necessary for children to learn and use those principles when planning displays, we’re more likely to create an environment that has a greater chance of contributing to long term learning rather than short term performance.

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How can a child catch up to learn times tables in one term?

Children should know all times tables by the end of year 4, but there are children that slip through the net, taking much longer to learn them.  There are also children that may seem to have learned times tables by the end of year 4, but forget and have to work into upper key stage 2 to relearn.

This post describes a plan to get children who are in year 3 and 4 and who are not on track to understand times tables by the end of year 4.  The plan is also for children in year 5 and 6 who still do not know their times tables.

A fact a day for a term

The basic structure of the plan is to work on one fact per day.  Working with commutative facts such as 3 x 4 and 4 x 3 together, and taking into account that familiarity with tasks should accelerate the work the longer it goes, a term is a sensible time frame to work in.  This will be systematic, working from x10 to x5, then x2, x4 and x8, then x3, x6 and x9, finishing with x7, x11 and x12.  This is to enable links to be made between times tables.  Within each times tables, we’ll work in increasing order of times tables (i.e., 10 x 1, 10 x 2, 10 x 3 etc.).  Of course, different children will have different starting points, not all starting with 10 x 1.  As days pass, children will consolidate their understanding of a times tables through repetition, multiple representations, counting and low stakes testing.

Multiple representations

For times tables to stick and to be useful in other areas of maths, they need to be rooted in secure understanding.  To allow this to happen, each fact will be represented in different ways, in the first instance by the teacher but increasingly by the child.  The first representation is Numicon, using the example of 4 x 5:

TT numicon

Using this we can explain that 4 x 5 means 5 lots of 4 and that by counting in multiples, we can find out that 4 x 5 = 20.  Children will have done this for 4 x 1, 4 x 2, 4 x 3 and 4 x 4 in the preceding days so they should be able to count in 4s.  However, they may need to do some skip counting, where they whisper or say in their head each number except for the last on each Numicon piece (1, 2, 3, 4; 5, 6, 7, 8; 9, 10, 11, 12…).  The Numicon also helps to lead into other representations:

Repeated addition: 4 + 4 + 4 + 4 + 4 = 20

Bar model:

TT bars

Number line:

TT number line

All the while, the child is practising counting in 4s, and thinking about how 4 x 5 = 20.

Commutativity

One more representation can lead the child into working on the related commutative fact.  An array gives a little further practice seeing how 4 x 5 =20:

TT Array 1

Rotating the array shows how 5 x 4 has the same product:

TT Array 2

This can lead into counting in 5s to get to 20 and showing that 5 + 5 + 5 + 5 = 20.  Then, repeating the representations of Numicon, a bar model and a number line will help to internalise the commutative fact.

Low stakes testing

Having worked on this new fact (and its commutative relative), the child can then work on remembering facts that have been previously worked on in days gone by.  Practising recalling times tables is of course a great way of ensuring that they come to mind immediately when needed.  Quick, effortless recall means that little cognitive effort is required to summon the knowledge, thereby keeping as much working memory as possible freed up to solve a problem that needs the times table fact in the first place.

There are two ways of working on quick recall of times tables.  The first is if the child has a reliably secure understanding of multiplication.  In this case, simple testing such as asking ‘What is 3 x 5?’ or the use of individual flash cards will be fine.  However, if a child is still not quite there with conceptual understanding, testing by using objects or images can help to get them to think mathematically instead of guessing.  The teacher shows any of the pictorial representations already described to prompt thinking about the number of groups, the size of each group and ultimately quick recall of the whole.

 

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Multiplicative reasoning

If a child understands additive reasoning and the relationship between the whole and its parts, it is a fairly straightforward conceptual step to understand multiplicative reasoning.  Multiplicative reasoning should be modelled as repeated addition in the first instance.  Adding multiple equal parts (for example 5) might look like this:

Array 1

5 + 5 + 5+ 5 is equal to twenty.  Children need to understand that multiplication allows for efficient repeated addition.  You  have your thing to be multiplied (5) and the multiplier (4): 5 +5 + 5 + 5 = 5 x 4.  Creating arrays and deliberately connecting repeated addition with multiplication makes for sound understanding.

How children work out the whole should not be taken for granted.  At first, children might count each item in the array.  Counting in multiples can be achieved by first skip counting.  Children might whisper the numbers while counting except for the last in each row, which is said out loud.  Then replace the whispering with counting in their heads and then simply saying the multiples.  Over time, given sufficient practice, children will internalise these times tables.

Commutativity is important here – the array used above shows 5 x 4 but rotated it shows 4 x 5.  Times tables taught systematically and with such conceptual support should be straightforward for children to learn comfortably before the end of year 4, especially when we consider it like this:

Times tables facts

Of course, children need time to practise well and multiple representations help children to make connections.  Graham Fletcher’s blog post describes the use of pictorial representations on flash cards – an approach that is a great form of low stakes testing to support the learning of times tables.

Flash card

This image supports the understanding of having a ‘thing to be multiplied’, a multiplier and a whole.  With practice, children will be able to subitise from glancing at the flash card, becoming fluent and accurate with times tables recall.

Some children will grasp all this quickly and can work at a greater depth while children that need more practice with the basics get it.  Still using the array, children can easily begin to think about distributivity simply by splitting the array into parts:

array 2

The part above the line is 5 x 2 and the part below the line is 5 x 2:

5 x 2 + 5 x 2 = 5 x 4.

There is lots of scope for systematic thinking about equivalence with a task like this.

Arrays are perhaps not the most efficient of representation so a progression is to get children to be able to represent multiplication in bar models.  First though, Numicon to work on the language of size of each part, number of equal parts and the whole:

Multiplicative reasoning2

Numicon is a great manipulative to represent multiple parts because of its clarity of the ‘size of each part’.  Multi-link cubes could work too, but children would need to organise the parts into different colours to differentiate between them:

MR3

Building worded statements using a manipulative will ensure children practise the language needed to internalise the concept of multiplicative reasoning.  Dropping in some of the  inverse relationship between multiplication and division could be useful here too.  Doing it systematically can also help keep times tables knowledge conceptual and not shallow:

MR4

MR5

MR6

MR7

Commutativity could be brought in again – showing that 3 groups of 4 is the same as 4 groups of 3 using manipulatives arranged with intent.  Alongside this, comparing the similarities and differences with the worded statements will get children to think with clarity about equivalence between two multiplicative expressions.

Bar models are a versatile representation that can be used to solve a wide range of problems later on, so getting children to sketch out multiplication and division statements using bars enables them to practice a versatile skill.  We should expect great accuracy in their drawings – they should be representing equal parts.  If children also represent the same expression on a number line beneath the bar model, we can encourage links between representations and lay the foundations for trickier calculations and problem solving as they progress through school.

bar and no line

Update: The NCETM recently published this account of teaching the six times table, with some great ideas for depth.

 

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Mastering the common problem types in maths

I’ve been wanting to combine some of the thinking I’ve been doing into cognitive overload, worked and partially completed examples, and the bar method as a pictorial representation of mathematical problems.  The lesson described below is what I did, with a substantial eye on the expectations of the new national curriculum and the idea of mastery.

A worked example

A good explanation clearly takes children through the steps needed in order to solve a problem, but these steps should be rooted in the deeper structure of a problem and not the superficial.  As such, the success criteria that the teacher works from and that children refer to should support that expectation of analysis.  In the example, I showed my class how they could solve a comparison problem where both the total of the parts and the difference between them is known, but the value of each part is not.  To solve this type of problem, they need to be able to pick out totals and differences, as well as get an idea for which part is bigger/smaller.  The success criteria I used was this:

SC

Then, I took one question and walked them through each of these three steps:

WE1

WE2

WE3

WE4

WE5

A further few questions can be prepared and ready if further modelling is necessary.

Partially completed examples: scaffolding for mastery

Children will need to deliberately practise representing the information pictorially.  Work into cogntive overload suggests that when children are overloaded, a number of things can happen:

  • They’ll complete the first or last instruction only
  • They’ll lose their place in the sequence of instructions
  • They’ll abandon the task

In an effort to prevent this things from happening, the work I gave most of the children in my class included partially completed examples.  These aim to reduce the cognitive load while still providing the opportunity for deliberate practice.  The first few questions had some information already transeferred onto the pictorial representation.  Gradually, there was less and less of this until children were solving problems with just the basic structure of the problem given.

PCE1

Undoubtedly, some children will need more practice with heavy scaffolding before it is removed and some will need much less.  Working in this way makes adapting the scaffolding for different children easy to manage.

Trickier problems

There will be children who are already able to solve this kind of problem, but the process modelled and the success criteria will still be of use to them to solve trickier problems.  In the tasks that I chose to give to my class, I gave them three parts to compare:

PCE2

Although the basic structure of the problem is sketched out for children, in this example I did not give them any labelled sections, as the children I was intending this work for I felt would not need it.  Were I to do this again, I’d have another variation with some labels provided so that the questions are partially completed.  Hopefully, this would enable some children to work with questions this complex that otherwise might not have been able to.

Sure, this is but one lesson and it will take time for children to master the underlying patterns of problems so that they can solve them efficiently.  So it got me thinking about when different problem types should be introduced or when they should be mastered by.

Mapping out the problem types

With all this is mind, I set out to allocate problem types to phases for when they shoud be introduced.  The idea is that children in KS1 will master the basic additive and multiplictive probem types;  lower KS2 will master more complex additive and multiplicative reasoning problems as well as multi step problems based on the basic additive and multiplicative types; and upper KS2 will master still more complex additive and multiplicative reasoning problems as well as a wider variety of multi step problems.  Of course, children in each phase wlll need to revise older problem types, the goal being that eventually, they can see the undelrying pattern of a problem, thinking their way clearly to an accurate answer.  There are not many problem types for each phase, so they can be practised over time in a variety of contexts.  Each time they are revised, the scaffolding provided can be gradually removed so that towards the end of the phase, children are solving the problems with no scaffolding at all.

Introduction of bar models for additive and multiplicative reasoning Year 1-6

The expectations here are high, but achievable.  Mastering the common problem types by the end of KS2 will set children up very well for the next stage of their education.

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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.

20140221-172811.jpg

20140221-172826.jpg

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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