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A Lesson in KS1 Greater Depth – Simple Complexity

Rachel Rayner is a Teaching and Learning Adviser for Primary Mathematics at Herts for Learning.  She has previously blogged about greater depth at KS1 here,  and after pictures of a session she ran at one of the schools she supports became very popular on Twitter, we thought it might be useful to share her approaches. The lesson was taught to a mixed group of Year 1 and 2 pupils at Huntingdon Primary School, Cambridgeshire.  

I’m going to come completely clean here, it wasn’t my idea this problem.  I found this from the NCETM Teaching for Mastery – Questions, tasks and activities to support assessment document for Year 1. Like any good magpie not all my ideas are completely original – I’m always looking for simple little items that glitter.

KS1GD - 5

But with meeting the needs of all learners in mind – I began by thinking about access.

What would allow pupils to explore this deeply more quickly, to get to the heart of the problem without distraction?

Firstly I felt that the ‘squiggles’ were too abstract alone and thought that Numicon (or unicorn as some of the Year 1s called it on the day) would be useful for a number of reasons.

  • the shapes can be moved, allowing for adjustment without the commitment of putting pencil to paper
  • the holes in the shapes can be used for estimation – Do you think there are more holes in this line or this line?
  • the holes are countable and can be subitised – when young pupils get tired they can revert to counting but the holes might help them stay calculating for longer by subitising
  • they can be easily arranged to test equality
  • pupils at the school I knew were familiar with the resource

I also began with a pre-teach – I simply drew 5 boxes in a line horizontally on a whiteboard and arranged Numicon 1-5 shapes in them.  Then drew another 5 boxes underneath and arranged the Numicon shapes in a different order.  I wanted to check that pupils understood that the sum would remain constant whichever order we placed the shapes in.  Pupils seemed convinced and so I modelled the problem – incorrectly of course at first as we don’t want to give the crown jewels away completely.  And off they went, often in mixed age pairs which we (the teachers and I) all found fascinating to observe the dynamics of.

And off they went – some of the pairs literally did not raise their heads for another 40 minutes, so fascinated were they with trying to find magic numbers! They estimated, calculated (and then counted to check as they started to tire).  We asked the pupils to record, which they all did beautifully, very differently in each pair and sometimes with the lovely idiosyncrasy of children – joyous (one pair decided the lines of the cross were  sleaping lines and standing lines in their written recording).  Some pupils began to notice the balancing arms of the problem – a feature I wasn’t sure they would.  As they noticed this they began to work differently, purposefully considering the shape in the middle and how they would balance the remaining four shapes equally.

In the pictures you can see pupils solving the problem finding magic ten, nine and eight.

At this point I felt I could take the learning two different ways.  Either draw their attention to the fact that the shape in the middle was always odd and direct them to find out if they could make the problem work if an even number was in the middle – then consider why, or we could apply what they were thinking about in terms of balancing the arms with this simple case to a slightly larger case.

I decided on application as I felt this was a stronger focus for the pupils in this lesson.  Simply we used a new larger cross and Numicon shapes 1-9.

gdks1-4.jpg

Within ten minutes one pair had produced this example with the Y2 child in the pair explaining to me that they had made all of the arms equal nine – and he also knew the magic number was 27. Other pairs were working in the same way and soon after another  reached the same conclusion.

Sadly the hour ended too soon – that lovely simple complex activity that pushed beyond just adding single digit numbers.


References

https://cdn.oxfordowl.co.uk/2015/07/22/13/54/09/24/Year1_TeachingforMastery.pdf


Herts for Learning is a not for profit organisation that provides a wide range of training and CPD courses, events and conferences  to support teachers and school staff in their professional development and also offers an extensive range of resources to support their offering through the HfL e-Shop.  Please visit the website for more information.

Differentiation – How different does it have to look?

Nicola Adams is an adviser for Primary Mathematics at Herts for Learning.  In this, her first blog, she considers how differentiation or meeting the needs of all learners in the classroom is crucial but not always evident to those observing a lesson. She builds on Rachel Rayner’s blog FOMA – Fear of Maths Accountability to demonstrate how three boxes for differentiation is missing the point, and that observers must engage with the teacher before making judgements.

Picture this. Somebody is coming in to observe your maths lesson and what they see is all of the children doing the same thing. They all have access to the same manipulatives; they can all see the same working wall; they are sat in mixed-ability partners, they are playing a mathematical game… and there is conversation happening. The horror! Are they going to say that you are not challenging your more able? Are they going to ask why your lower ability children are not being supported by an adult? Are they going to say that your more able children simply don’t need the same manipulatives as the others? Just where is the differentiation? Continue reading “Differentiation – How different does it have to look?”

Help Parents’ Needed! Parental Engagement Has Big Benefits.

Kate Kellner-Dilks is a teaching and learning adviser for primary mathematics at Herts for Learning.  Kate has been successfully working with schools to increase the engagement of parents in supporting their children with mathematics.  Here she shares this advice. 

From conversations I have with primary school teachers and leaders, they often want to engage parents in the mathematics they are teaching. They have a feeling it will help the children, but are not sure the best way to go about it.

Research tells us that children are more successful at school, if their parents support their learning; ‘Family engagement in school has a bigger influence on a pupil’s achievement than socio-economic background, parents’ education level, family structure and ethnicity…’ (www.engagingwithfamilies.co.uk, 2017) Continue reading “Help Parents’ Needed! Parental Engagement Has Big Benefits.”

FOMA – Fear of Maths Accountability

Rachel Rayner is a Primary Mathematics Adviser at Herts for Learning.  Reflecting on her travels around schools, Rachel shares her observations about what can cause teachers to focus on aspects of planning that hinder teachers’ delivery of lessons. 

At the beginning of every year I work with new schools and teachers who are new to my existing schools.  In my last blog Teachers Reclaim Your Inner Artisan, I talked about how schools are changing their view of planning, how they are being unshackled from the ‘accepted planning routes’.  But I have also spent time with teachers over the last few weeks who are or feel they are, and that is an important distinction here, tied to a certain proforma, a definitive process for planning.   Continue reading “FOMA – Fear of Maths Accountability”

Teachers – Reclaim Your Inner Artisan

Rachel Rayner is a Primary Mathematics Adviser at Herts for Learning. In this blog Rachel explains why she thinks that teachers can save time by spend less of it looking for ideas online and more time crafting effective examples for themselves. 

So you are probably thinking, well all she has to think about is mathematics, and that would be a fair representation of my day every day.  But, to coin a phrase from the excellent Miranda Hart, please ‘bear with.’  As my schools will testify, if I can teach I will, I get a very real buzz from working with children of all ages on mathematics.  And like all of the advisers I work with, we do have a realistic view of life in schools and the barriers faced.  One of the areas I frequently see as patchy is the curriculum across the school.  Where we look in books across the year groups at one strand, say for example fractions, then it is typical that pupils are engaged in fairly similar content between Year 2 and Year 4.  And if I trawl through popular sites for worksheets I can see why – lots on colouring in shapes and finding fractions of amounts.  Continue reading “Teachers – Reclaim Your Inner Artisan”

Finding Maths in Storybooks – A Tale of Turning Training into Good Practice

In the Summer term 2016 Nicola Randall and Gillian Shearsby-Fox, Teaching and Learning Advisers for Mathematics at Herts for Learning, created and delivered a day of training on how to use books in maths. In this guest blog, Raj Khindey, an inspired maths subject Leader and Year 6 teacher at Chater Junior School,Watford; set about introducing the range of ideas she learned across her school.

In this blog she explains which ideas she trialled in her own class, as well as how she shared this good practise throughout KS2.

Following the training I was inspired to use a variety of fiction books that were recommended by Nicola and Gillian. I wanted to share this with the rest of the staff so the children as well as teachers could enjoy a different dimension to a traditional Maths lesson! So I held a staff meeting in Autumn Term and trialled some of the activities delivered in the course. Continue reading “Finding Maths in Storybooks – A Tale of Turning Training into Good Practice”

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.


Herts for Learning is a not for profit organisation that provides a wide range of training and CPD courses, events and conferences to support teachers and school staff in their professional and also offers an extensive range of resources to support their offering through the HfL e-Shop.  Please visit the website for more information.

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”

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