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.

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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…professional 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. Previous posts were about displays, learning generally, maths and reading. Next up – professional learning.

What should leaders prioritise?

With likely a range of often conflicting priorities, deciding what to work on is tricky.  Subject leaders will strive to keep their subject’s nose in front of the rest but ultimately, leaders must be able to zero in on what it is that the children need.  Once that is known, leaders can think about what teachers might need to do differently in order for those outcomes for children to be realised.  The list of things that teachers (could) do day to day is endless so leaders must be able to judge, through experience or by leaning on research, which of those things are worth pursuing and which need to be jettisoned because they take up our time and mental effort for no significant impact.  Research such as that by Hattie is useful but are the interventions described in such research too broad?  For example it is obvious that feedback can have a significant impact on learning but only if it’s done well.  Consider the difference between these scenarios:

  • training on implementing a new feedback policy
  • training on providing feedback on persuasive writing

Or these:

  • training on clear teacher explanations
  • training on explaining how to add fractions clearly

There is a difference between being research led and research informed.  Research should be considered in combination with the needs of children and teachers so that leaders get teachers thinking about effective ways to teach.

This would go some way to ensuring that teachers’ subject and pedagogical knowledge is developed, in line with the Sutton Trust report into what makes great teaching. It’s relatively straight forward to ensure that the focus is on those things, however ensuring the impact is a lot trickier. It makes sense for leaders to have from the outset a very clear idea of what they want that impact to be. Phil Stock’s post on evaluating impact (based on  Guskey’s hierarchy of five levels of impact) is very useful here in terms of leaders planning what they want to happen as a result of professional learning and the rest of this post details how one might do that.

Intended impact on outcomes for children

The intended outcomes for children should be set out so that there is no misunderstanding of the standard to be achieved. Using resources like Rising Stars Assessment Bank for maths can help teachers to gather the types of questions that all children will be expected to answer.  The same can be done for a unit of work on reading – find or write the questions about a text or texts, including the quality of response that you’d expect in order to demonstrate age related expectations.  Something similar can be done for writing.  Find or write a piece that would exemplify the standard that you’d expect from children.  Whatever the subject, leaders working with teachers to clarify what exactly children will be able to do and what their work will look like is the goal.

Individual questions would serve as criterion based assessment but for reading and maths, these questions could be compiled into an overall unit assessment and a target could be set for all children to achieve in the first phase of a unit of work. Gentile and Lalley, in Standards and Mastery Learning  discuss the idea that forgetting is the inevitable consequence of initial learning even if it is to a high standard of say 80%+ .  The problem is that for the most vulnerable children, who don’t achieve that initial mastery of the content to anywhere near that standard, forgetting happens more quickly and more completely.  If children don’t initially understand to a certain level, their learning over time is far less likely to stick and will make subsequent planned revision not revision at all but a new beginning.  Therefore, the expectation of the impact on children of any professional learning simply must be that all children achieve a good standard of initial understanding, whether that is judged as absolute through criterion referenced assessment or by a percentage on a carefully designed test.

Now of course, meeting the standard set on an assessment means nothing unless it is retained or built upon. This initial assessment would not be at the end of the unit of work but part way through.   I’d expect, on an end of unit test, higher percentages compared to those that children will have achieved on the initial assessment.  This is because that initial assessment will have served to tailor teaching to support those that require further instruction or practice.  And I’d expect that intervention to have worked.

To summarise, teachers and leaders first set the assessment and the standard to be achieved.  The unit of work is taught until all children can attain the standard, then the unit continues, deepening the understanding of all which is then checked upon at the end of the unit and beyond. The DfE’s Standard for Teachers’ Professional Development (July 2016) identifies the importance of continually evaluating the impact on outcomes for children of changes to practice and so assessments of what children have retained weeks and months after the unit of work are crucial – they ‘ll inform at further tweaks to teaching and professional learning.  When there are clear milestones for children’s achievement, the professional learning needs of teachers comes sharply into view.

Intended impact on teachers’ behaviour

Once it has been decided what the intended impact on outcomes for children is, attention needs to be turned what teachers will do in order for children to achieve those outcomes. Such behaviour changes may be desired at the planning stages of a unit of work, for example in the logical sequencing of concepts related to addition and subtraction over a series of lessons. The behaviour changes may be desired during teaching, for example explaining and modelling how to create suspense in a piece of writing. Finally the behaviour changes could be desired after lessons, for example where teachers receive feedback on how children have done by looking at how they have solved addition and subtraction problems in order to amend the sequence of lessons.  Another example could be providing feedback on their writing to make it more persuasive either face to face or by writing comments in their books.  The key here is that behaviour change is specific to the unit of work.  Having said that, leaders must support teachers to think in increasingly principled ways so that over time, principles can be more independently applied to other units of work and subjects.  As such, intended changes to behaviour must be iterative and long term, with opportunities to make connections between topics and subjects through coaching and shared planning.

For any behaviour change, teachers must see the outcome.  They must see someone doing the things that are expected of them.  This live or videoed teaching needs to be deconstructed and then summed up concisely which acts as success criteria for teachers. For example, in a unit of work on place value, desired teachers’ behaviours could include (and this is far from exhaustive; simply to illustrate the point):

  • Plan for scaffolds (and their removal) so that all children can partition and recombine numbers fluently and accurately.
  • Intervene on the day if a child shows significant misunderstanding of that day’s learning.
  • Use concrete manipulatives and pictorial representations to model and explain the concept of place value.
  • Co-construct with children success criteria appropriate to the type of leaning objective (open or closed).

Having such success criteria ensures that both leaders and teachers are clear of what is expected in order for the desired impact on children to be realised. It can also be used to focus practices like lesson study and coaching conversations, which are crucial to keep momentum going and embed change.

Intended impact on teachers’ knowledge

If leaders require teachers to develop certain practices, for many there will be a knowledge gap that inhibits such development. The DfE’s Standard for Teachers’ Professional Development identifies the importance of developing theory as well as practice. Subject and pedagogical knowledge, as well as knowledge of curriculum or task design are all vital for teachers to be able to refine aspects of their practice.   This could be as straightforward as analysing the types of questions that could be asked to get children thinking deeply about place value before teachers write their own which are appropriate to the year group that they teach. Or it could be ensuring that teachers understand and can articulate the underlying patterns of addition and subtraction in the maths unit coming up. It could even be knowing the texts that children will be using for reading and writing in depth in order for them to dedicate future thinking capacity to pedagogical concerns. By setting out the intended theoretical knowledge to be learned and by providing opportunities to gain that knowledge in ways that do not overly strain workload, leaders can set teachers up for successful changes to practice.

Organisational evaluation

For children to improve based on teachers’ developing subject and pedagogical knowledge, there must be great systems in place that allow such development to happen.  Leaders need to be very clear about what it is that they will do to ensure that teachers are supported to act on the advice being given.  Some examples include:

  • Making senior leaders or subject specialists available for shared planning
  • Providing access to a coach (and training for coaches)
  • Arranging for staff to access external training
  • Ensuring that observations are developmental
  • Planning professional learning using Kotter’s change model

These items become success criteria for leaders implementing long term change.  They can be self evaluated, of course, but external validation of school culture is valuable here.

Reaction quality

The final strand of planning for impact concerns how teachers perceive the professional learning in which they’ll engage. It goes without saying that we’d like teachers to find professional learning not just useful but transformative – a vehicle for improving outcomes for children, personal career development and increasing the school’s stock all at the same time.  One can only create the conditions in which another may become motivated and by taking into account what drives people, we can go along way to ensuring a thriving staff culture. Lawrence and Nohria’s 4-Drive model of employee motivation is very useful here, describing four underlying drives:

The drive to acquire and achieve

If staff are confident that the professional learning will lead to them acquiring knowledge, expertise and success, then they are more likely to feel motivated.  Professional learning then must appeal to this drive – spelling out the knowledge and status that can be achieved through the planned work and never underestimate the power of distributed leadership, carefully supported, of course.

The drive to bond and belong

The school’s vision is key in keeping everyone focused and pulling in the same direction and this can certainly be reinforced with a common school improvement aim as the focus of professional learning.  Finding ways to ensure supportive relationships is crucial.  Culture is the result of what we continuously say and do so leading by example in developing good working relationships will go some to making it the social norm.  Leaders must also look for and iron out any pockets of resistance that could threaten the desired culture.

The drive to comprehend and challenge

This refers to providing opportunities for staff to overcome challenges and in doing so grow.  Setting out each individual’s importance in the school and how they contribute to its success is an example. This is often a long game, with external judgments being made in exam years or in external inspections, so leaders must find quick wins to acknowledge the impact of teachers’ work on the development of the school.

The drive to define and defend

By drawing attention to the good that the professional learning will do not just for the children but in turn for the reputation of the school, we can create a fierce loyalty.  If we get our principles right an articulate what we stand for, this momentum can be very beneficial for implementing professional learning.

This is the job of the leader, striving for improvement in outcomes for children whilst developing staff and building a culture of success. Any professional learning has to have clear outcomes and its only then that they can be reliably evaluated and tweaked to inform the next iteration.


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What I think about…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:
Stages of a unit of work

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.


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:

Maths structures

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:

Empty box

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:

Bar model

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’:

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:

Subtraction problems

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:

Sorting 1

Sorting 2

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.

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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:


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:



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.


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.


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.


Filed under Memory

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:


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.


Filed under Display

The teaching of fractions

There are certain prerequisites for children to develop a solid understanding of fractions.  First, they must understand, through work on additive reasoning, that a whole can be split into parts and that the sum of those parts is the whole.  There’s a short step into multiplicative reasoning from here – that a whole can be split into multiple, equal parts and that the whole is the product of the size of each part and the number of parts.  Once this is understood, children can begin to think about the whole being worth one and the parts being fractions of one.  The ideas that follow are broadly sequential in terms of conceptual development.

Prior concepts

Children will need to manipulate various representations of fractions, for example making them with fraction tiles (as both bars and circles); taking strips of paper and ripping them in to equal parts; and drawing bars and circles, dividing them into equal parts.  It is worthwhile to get children to do lots of judging by eye and marking equal parts of a whole as well as using squared paper to do so accurately.

Of course, there is a lot of language to work on whilst manipulating these models of fractions.  Children need to be shown clearly the link between the total number of parts and the language (but not yet necessarily the written form) of the denominator: two parts – halves; three parts – thirds; four parts – quarters etc.

With a secure start in the basics of splitting a whole into equal parts, children can work on the idea that fractions always refer to something.  A third, for example, doesn’t stand alone.  It might be a third of an apple or a third of twelve sweets or a third of one whole.  Modelling these full sentences and getting children to speak in this way should solidify their understanding of proportion.  Through the sharing out of objects, even very young children can work on the concept of fractions of numbers – sharing six sweets between three children means that each child has the same number of sweets and that two sweets is one third of six sweets.


Once children are comfortable with the idea that an object or a set of objects or a number can be split into equal parts, and that each of those equal parts can be described as a fraction of something, that object or that set of objects or that number, they can go on to work at greater depth.  By comparing strips of paper or bar models that are the same length yet are split into different fractions, children can look at the relationship between the size of each part and the number of parts.  That is, the greater the number of equal parts, the smaller the size of each part.  Children should be expected to think about how ¼ is smaller than ½ because ¼ of one whole is one of four equal parts whereas ½ of one whole is only one of two equal parts.  Then, questions like this should be relatively straightforward:

Covered strips

The understanding that unit fractions with larger denominators are smaller than unit fractions with smaller denominators will contribute significantly to work in comparing fractions later on.

Children could begin to look at improper fractions and mixed numbers next.  Using ¼ fraction tiles, they could make one whole and then see what happens if you add another ¼.

Mixed number

This lends itself to counting in unit fractions but we should exercise caution.  Children may be able to chant ‘Three quarters, four quarters, five quarters…’ but early conversion to mixed numbers as well should help to secure their understanding of the relationship between them.  Manipulatives like fraction tiles and multi-link cubes are great for representing improper fractions because they can trigger accurate mathematical talk to describe the improper fraction (the total number of cubes as the numerator and how many cubes in each whole as the denominator).  The same can be done to describe the mixed number (the number of wholes, then what is left over as a fraction of a whole).

Returning to additive reasoning, children could generate complements to 1 whole and record them as addition and subtraction statements.


A slight change to the representation used can support children to work with complements where denominators are different:

Complements 2

Placing two bar models of equal length one on top of the other is great scaffold for comparing fractions.  When the denominators of the fractions are the same, the bars should not even be necessary but when they are different, the image can help to structure thinking.


When dealing with fractions with different denominators, the practice that children had earlier of judging by eye to split a whole into equal parts and marking the divisions themselves becomes crucial, otherwise, things like this could happen:

Inaccurate bars

A standard fraction wall is all that is needed to begin work on equivalence and the first step is of course shading one fraction and looking up or down the fraction wall to find fractions of equal size.  When children are comfortable with that, they can begin to look at patterns in the abstract representations, particularly the link between times tables, numerators and denominators.

Fraction wall

Using the language of simplifying or cancelling fractions without first talking more generally about the concept is a mistake.  If children are well versed in using a fraction wall to find equivalents to a given fraction, it is only a slight tweak to talk about finding the equivalent fraction that has the fewest total parts.  It would be tempting to talk about finding the equivalent fraction that is ‘closest to the top’ of the fraction wall but this would be a mistake too.  The language of simplifying or cancelling can be used to attach to the concept of finding the equivalent fraction with the fewest total parts to get children thinking conceptually soundly.

One further aspect of thinking of fractions is to consider them as numbers.  To do this, plotting fractions on a number line directly beneath the bar model is a good way of linking the two representations.

Number line

Representing fractions as a proportion of one, as a part of a quantity and as a position on a number line significantly supports children’s development of proportional reasoning and ensures that future tricky concepts such as calculating with fractions can be built on a secure foundation.


Filed under Maths