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Fifty Shades of Grey Addition

Charlie Harber is Deputy Lead Adviser for Primary Mathematics at Herts for Learning.  This blog aims to take the seemingly simple operation of addition and demonstrate how we can vary presentation in order that pupils see connections and do not fall into shallow procedural thinking. 

grey1

Would you agree that these are all shades of grey? (Who hasn’t spent time deliberating between shades on a paint chart?)  They are all different. So how can they be all grey?

We are generalising what is meant by GREY. Oxford online dictionary define grey as ‘Of a colour intermediate between black and white’ – so under this definition all these colours, despite being different, are grey because they share this similar property. For children to understand what grey is, what would you do? Would you just present them with a single shade? More likely, one activity you would do is present them with a range of objects in lots of different colours including shades of grey and ask them to sort the objects based on their colours. In mathematics we could refer to this as part of generalisation – the ability to see what is common amongst a range of different situations. Continue reading “Fifty Shades of Grey Addition”

KS1 Mathematical Recording is not just for Ofsted.

Siobhan King is a Mathematics Adviser at HfL.  It’s probably fair to say teachers feel that have been hearing mixed messages about what pupils’ maths books should look like in Keystage 1. In this blog Siobhan gives teachers plenty to think about and argues that recording is a key mathematical skill at any age.

A question I often get asked, particularly by KS1 teachers, is…

“What should it look like in their books?”

I completely understand where this question comes from, as I know how hard teachers work to do the best for their pupils and over time, a misconception seems to have developed that books are all about providing evidence to external viewers.  With this, teachers have felt a pressure to supply evidence of every learning activity that pupils have undertaken.  In KS1, where fine motor skills and writing skills are being developed, this has sometimes translated into maths books full of photographs of children waving around plastic maths resources, which actually provide very little useful evidence of what has been learned.

Therefore, to start my answer, I might ask:

“What should it look like, for who?  What is the purpose of the recording?” 

 Let us consider first who may use our childrens’ books and unpick what they really want to see.  I will start with the easy one – Ofsted.  If you are recording in a particular way for Ofsted, you need look no further than the Ofsted Myths clarification for schools (link here: https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/463242/Ofsted_inspections_clarification_for_schools.pdf ) or Sean Harford’s (National Director, Education) twitter account:

Ofsted Myth – Ofsted need photographic evidence of children’s work.

Ofsted Fact – We don’t.  We’re happy to speak to children during an inspection about what they have learned.  We’re very aware of teachers’ workload.

If Ofsted do not need maths recording in a particular way as evidence – and they don’t – why should SLT, or you for that matter?  I guess what we all want is evidence that our pupils are learning and building an increasingly deeper understanding of what we teach them.  Yes, books may be one source of evidence to support this, but there are others, not least (as Sean Harford says) talking to children.

Does this mean it is not worth recording anything in maths?  Only if you consider the sole purpose of maths recording to be about providing evidence for an external body.   I would argue that there are many reasons to record in maths, that mathematical recording is integral to our children building a deep mathematical understanding and that it can be useful for teachers too.

So why is recording in maths important? 

Firstly, recording is a necessary part of building mathematical understanding.  We know that depth of understanding is strengthened through transferring between concrete, pictorial and abstract models so recording alongside other created models supports deeper learning.  It is through pupils representing their understanding that they explore and make sense of what they know.  This is borne out in research by Carruthers & Wothington (2010) in which they noted children’s own recording supported, “deepened thinking about the mathematics in which they are engaged, and significantly, about their use of symbols and other visual representations to signify meanings. They enable children to build on what they already know and understand”.

In addition, recording while working on a problem can be helpful for pupils to reduce cognitive load by using jottings or identifying key facts, which may be used later.  This type of recording may not be intended for anyone else to read, but can form a log of how pupils have worked a problem through and can be incredibly useful for teachers to identify misconceptions and the route of pupil mistakes.

Recording can be about developing a skill.  For example, making use of abstract symbols and numerals, requires learning their formation and practice in using and recording them as well as learning about their meaning.

Recording can also sometimes become a mathematical tool in itself, helping pupils to explore problems and develop reasoning skills.  Through recording, pupils can expose underlying patterns and structures, which lead to greater understanding or further questions to explore.

For pupils, recording can provide the opportunity to communicate with an audience.  Being asked to explain and prove understanding to an audience provides an opportunity to develop precision in reasoning and again deepen understanding.

What is selected for recording can also affect pupil perceptions of how things are valued and support them to focus on different aspects of the learning they are undertaking.  If pupils are asked to record how they tackled a problem rather than the answer to it, then they are much more likely to think, talk about and focus on these.  By doing this, the teacher can show pupils the range of different approaches to the same problem and draw out discussions around different choices, evaluate strategies and consider the range of possibilities.

Going back to the original question: “What should it look like in their books?”   It depends on the purpose of the mathematical recording.  Is it to make connections between models, practice a new skill, record the journey through a problem, develop precision in reasoning, focus on reflection and evaluate strategies…?  I can tell you one thing – it should not be simply to provide evidence for Ofsted!

In the Nrich article “Primary Children’s Mathematical Recording” (2013) there are some useful reflections as to how all teachers could think about making the most of mathematical recording:

Do we always make it clear to learners what the purpose of their recording might be and who it is for?

Do we value all types of recording and mathematical graphics? 

Do we discuss a range of recording strategies, for example by asking, “How else might we record this?”

On reflection, I think the question many KS1 practitioners are actually asking is,

“How is it achievable to develop manageable, meaningful recording in KS1?”

and perhaps this relates to what we are expecting, but also to the opportunities we provide and how we are supporting its development.  In my next blog, I will try to capture how current practitioners are developing pupil recording at KS1.

References:

Carruthers, E. & Worthington, M. (2010) “Children’s Mathematical Graphics: Understanding the Key Concept”, Published on the Nrich website. Nrich Primary Team (2013)

“Primary Children’s Mathematical Recording” Published on the Nrich website.


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Bar Modelling is a Leap of Faith

Charlie Harber is the Deputy Lead Adviser for Primary Mathematics at Herts for Learning.  A passionate advocate of bar modelling, her last blog on the subject RUCSAC pack your bags, dealt with a KS2 SATs question.  Here Charlie turns her attention to KS1 bar modelling.

In my last blog on bar modelling I used an example from the 2016 KS2 test. Subsequently, I had had a number of requests asking for a similarly worked example of a KS1 test question.

leap2 Continue reading “Bar Modelling is a Leap of Faith”

Changing Mindsets – Teachers as Action Researchers

The following blog is proof positive of how teachers as researchers in their classrooms are a force to be reckoned with.  Three of our advisers, Gill Shearsby-Fox, Nicola Randall and Louisa Ingram worked with just such a group and we’re thrilled that research from 17 schools is now ready to read as case studies on the Herts for Learning site.  Thanks to Jasleen Dhillon HfL researcher also for her support. The following blog by Gill shows how our advisers approached the project. 

The Great Maths Con Action Research Project

On Friday 18th September 2015 the Herts for Learning maths team hosted a national conference with Jo Boaler, Professor of Mathematics at Stanford University, as the key note speaker. Many Hertfordshire teachers attended the conference to find out more about developing mathematical mindsets and be inspired to continue improving opportunities in mathematics for their pupils. Continue reading “Changing Mindsets – Teachers as Action Researchers”

Greater Depth at KS1 is Elementary My Dear Teacher

Rachel Rayner is a Mathematics Adviser at HfL and is one of the KS1 Number Sense and Fluency project leads in Hertfordshire and Cambridgeshire.  The project aims to support teachers to develop pupils’ retention of facts and how we can help them use learned facts flexibly.  The project has been hugely successful and findings have been presented at conferences and journals. In this blog, Rachel turns her attention to what the greater depth judgement actually means and what kinds of opportunities can be used to foster it.

I’ve spent a lot of time in schools recently considering with teachers whether they have pupils working at greater depth in Year 2 but also what that might look like in Year 1.  Part of this work has, understandably been with schools who are fully expecting to be moderated this year and would like their books to reflect evidence for pupils they suspect could achieve the greater depth tag.  Why so nervous?  Well the landscape for maths has shifted in terms of expectation, whereas before L3 might be judged by acceleration into new coverage, speed and accuracy which seem easier to tick off, now ‘Greater Depth’ seems a little hazier, perhaps just out of reach. Indeed the language of judgement gives us the weightiest indicator with the change from high attainer to working at greater depth.  In terms of scaled score versus Interim Teacher Assessment Framework (ITAF), there seems a difference in expectation too. This has left schools feeling uncertain about their own judgements.  I have plenty of sympathy for schools and the greatest respect for the teachers who are questioning their judgements and recognising the shift. Continue reading “Greater Depth at KS1 is Elementary My Dear Teacher”

Tried and tested: Diminishing the difference at UKS2

Nicola Randall is a Primary Mathematics Adviser for Herts for Learning where she has been working on closing the gaps projects with disadvantaged learners, working with schools to champion their needs.  She has worked on behalf of the Virtual School.

Over the past few years, I have been working with schools across Hertfordshire to accelerate the mathematical progress of pupils in receipt of the pupil premium funding. The DfE research paper (November 2015) suggests that the 3 strategies which have the greatest impact on the attainment of disadvantaged pupils are: paired or small group additional teaching, improving feedback and one-to-one tuition. Whilst these are clearly helpful for closing gaps in understanding or knowledge, sometimes it’s the small tweaks to whole class practise that can make the biggest difference. Having worked alongside teachers, I have looked in great depth at many pupils’ work and together with teachers we have reflected on how the pupils conduct themselves during maths lessons.  What is clear, is that each individual pupil comes with their own experiences, strengths and areas of difficulty yet interestingly, through this approach, I found there were some common barriers holding this vulnerable group back in mathematics. Continue reading “Tried and tested: Diminishing the difference at UKS2”

Times Tables Tests: A little bit of history repeating

Our advisers at Herts for Learning are privileged to work in many schools in and beyond Hertfordshire and on our travels meet real gems. One of these chance encounters was with the writer of this hugely interesting guest blog.  Alan Parr contacted us after reading our blog ‘When Timestables get Political‘  where the following comment resonated strongly for him

Those halcyon days just did not exist; it has ever been true that some pupils find learning their tables harder than others…

Alan let us know that he had made some pretty intriguing finds about the learning of times tables from school log books…it seems some things really never change. Read on for the shocking truth.  Continue reading “Times Tables Tests: A little bit of history repeating”

When Mathematics Gets Political – Times Tables

Times tables have been a hot potato politically for a long time, there seems to be a slightly rose-tinted view that once upon a time every child in Britain learned them by heart and still remember every fact today. Yet clearly not, if the famous example of politicians such as Stephen Byers and George Osbourne are anything to go by then even the great and the mighty parliamentarians  have a wobble now and then.  Those halcyon days just did not exist; it has ever been true that some pupils find learning their tables harder than others – we can give you roomfuls of teachers, teaching assistants and parents from all walk of life who can bear testament to this, most of whom can name the fact that remained a blindspot. Last week Nick Gibb announced that there would after all be a times table test for Y6 pupils beginning in 2019.  Inevitably this is likely to increase the emphasis on the learning of these facts – no bad thing as recall of these facts reduces the cognitive load enabling the focus of learning to be attended to, but it pays to be mindful that we build in meaningful learning sequences before we introduce the low stakes tests that allow teachers and pupils to identify gaps in fact recall and teach for them.  

The following blog considers one such possible approach.

Kate Kellner-Dilks is a Primary Mathematics Adviser  at Herts for Learning

In many of the schools I have the opportunity to visit and work with, the age old questions of multiplication facts comes up. Lots of schools have a ‘times tables challenge’ or variation of, which usually equates to some level of weekly testing. In my last blog ‘Are the boys really better at mathematics?’ I questioned whether multiplication facts (times tables) needed to be tested before they are learned, particularly against the clock, because of the anxiety you see in those pupils who don’t (yet) know them. These children often end up using inefficient strategies based on counting, to figure them out, rather than the intended memory recall. The test does not necessarily help build their memory recall and often, in my experience, reinforces their counting strategies, (a child said to me her ‘mind goes blank’, so she panics and counts up rather than using memory). Continue reading “When Mathematics Gets Political – Times Tables”

SATs chat: Why did my children revert to written methods on Paper 1?

 

Rachel Rayner is a Primary Maths Adviser at Herts for Learning

It’s a good question. In my experience working with schools nationally, pupils default to the written method often unthinkingly.   See the lovely examples here of just that happening.

As teachers we value mental fluency and we want our pupils to have it.   But are we working in the right way to engage our pupils over a sustained period of time, out of unthinkingness and into causing pupils to think deeply enough about the facts and skills they are adept at and how they might use them to form a strategy?    Continue reading “SATs chat: Why did my children revert to written methods on Paper 1?”

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