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Guidance, ideas and examples to support schools in developing their curriculum, pedagogy, enrichment and support for more able learners, within a whole-school context of cognitively challenging learning for all. Includes ideas to support curriculum development, and practical examples, resources and ideas to try in the classroom. Popular topics include: curriculum development, enrichment, independent learning, questioning, oracy, resilience, aspirations, assessment, feedback, metacognition, and critical thinking.

 

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Improving assessment and teacher workload: exit and entrance tickets

Posted By Rob Bick, 09 December 2022

Rob Bick, Curriculum Leader of Mathematics and Assistant Headteacher, explains how the use of exit tickets has improved assessment (and teacher workload) at Haybridge High School and Sixth Form.

The maths department at Haybridge High School introduced exit tickets almost 10 years ago, inspired by a suggestion in Doug Lemov’s book ‘Teach like a Champion 2.0’. Here’s how it works in our department…

In general, students would be given a coloured piece of A5/A6 paper towards the end of a lesson. On the whiteboard their teacher would write a hinge question (or questions) to assess whether or not students have a reasonable understanding of the key concept(s) covered in that lesson. A shared bank of exit ticket questions is available, often using exam-style questions, but teachers are encouraged to use a flexible approach and set their own question(s) in response to how the lesson has progressed. We wouldn’t use a pre-suggested exit ticket for a lesson if that was no longer appropriate.

Students copy the exit ticket question(s) down on to their piece of paper and then write their answers, showing full workings. As students leave the lesson they hand their completed exit ticket to their teacher. The teacher will then mark the exit tickets with either a tick or cross, no corrections, putting them into three piles: incorrect, correct, perfect. Those with perfect (and correct if applicable) exit tickets are awarded achievement points. Marking the exit tickets is very quick and easy and gives the teacher a quick insight into the success of the lesson, whether a concept needs to be retaught, whether the class is ready to build on the key concepts, any common misconceptions that need to be addressed, whether students are using correct mathematical language…

After the starter activity of the next lesson, the teacher will review the exit tickets using the visualisers in a variety of ways. This could be to model a perfect solution which students can then use to annotate their own returned exit ticket, or to explore a common misconception. The teacher may display an exit ticket and say “What’s wrong with this?”. Names can be redacted but hopefully the teacher has established a “no fear of mistakes” environment where students are comfortable with their exit ticket being displayed. Students always correct their own errors using coloured pens for corrections to make them stand out. Annotated exit tickets are then stuck into books. 

Exit tickets can also be set to aid recall of previous topics. This is particularly helpful when the scheme of work will soon be extending upon some form of previous knowledge. For example, exit tickets could be used to prompt students to recall how to solve linear equations in advance of a lesson on simultaneous linear equations, or to review basic trigonometry before moving on to 3D trigonometry.

Other than marking formal assessments, this is the only other marking expected of staff and the expectation is that an exit ticket will take place every other lesson. In sixth form we turn this on its head and do entrance tickets, so questions are asked at the start of the lesson using exact questions which were set for homework due that lesson. This gives teachers a quick method of assessing students’ understanding and identifying those who haven’t completed their homework successfully. It saves a great deal of teacher time and yet provides a much clearer understanding of how our students are progressing.

Obviously, exit and entrance tickets are just one approach to check for understanding. We also use learning laps with live formative assessment during every lesson. We make extensive use of mini-whiteboards and hinge questioning to quickly assess understanding. We also only use cold calling when asking for a response from the class – all students are asked to answer a problem and then one is asked to share their response – rather than choosing only from those with hands up.


Share your experience

We are seeking NACE member schools to share their experiences of effective assessment practices – including new initiatives and well-established practices. To share your experience, simply contact us, considering the following questions:

  • Which area of assessment is used most effectively?
  • What assessment practices are having the greatest impact on learning?
  • How do teachers and pupils use the assessment information?
  • How do you develop an understanding of pupils’ overall development?
  • How do you use assessment information to provide wider experience and developmental opportunities?
  • Is assessment developing metacognition and self-regulation?

Read more about our focus on assessment.

Tags:  assessment  feedback  maths  pedagogy  progression  retrieval 

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Metacognition for higher-attaining learners: “the debrief”

Posted By Kirstin Mulholland, 15 February 2022
Dr Kirstin Mulholland, Content Specialist for Mathematics at the Education Endowment Foundation (EEF), shares a metacognitive strategy she’s found particularly helpful in supporting – and challenging – the thinking of higher-attaining pupils: “the debrief”.

Why is metacognition important?

Research tells us that metacognition and self-regulated learning have the potential to significantly benefit pupils’ academic outcomes. The updated EEF Teaching and Learning Toolkit has compiled well over 200 school-based studies that reveal a positive average impact of around seven months progress. But it also recognises that "it can be difficult to realise this impact in practice as such methods require pupils to take greater responsibility for their learning and develop their understanding of what is required to succeed”  .

Approaches to metacognition are often designed to give pupils a repertoire of strategies to choose from, and the skills to select the most suitable strategy for a given learning task. For high prior attaining pupils, this offers constructive and creative opportunities to further develop their knowledge and skills.

How can we develop metacognition in the classroom?

In my own classroom, a metacognitive strategy which I’ve found particularly helpful in supporting – and, crucially, challenging – the thinking of higher-attaining pupils is “the debrief”. The debrief as an effective learning strategy links to Recommendation 1 of the EEF’s Metacognition and Self-regulated Learning Guidance Report (2018), which highlights the importance of encouraging pupils to plan, monitor and evaluate their learning. 

In a debrief, the role of the teacher is to support pupils to engage in “structured reflection”, using questioning to prompt learners to articulate their thinking, and to explicitly identify and evaluate the approaches used. These questions support and encourage pupils to reflect on the success of the strategies they used, consider how these could be used more effectively, and to identify other scenarios in which these could be useful. 

Why does this matter for higher-attaining pupils?

When working in my own primary classroom, I found that encouraging higher-attaining pupils to explicitly consider their learning strategies in this way provides an additional challenge. Initially, many of the pupils I’ve worked with have been reluctant to slow down to consider the strategies they’ve used or “how they know”. Some have been overly focused on speed or always “getting things right” as an indication of success in learning. 

When I first introduced the debrief into my own classroom, common responses from higher-attaining pupils were “I just knew” or “It was in my head”. However, what I also experienced was that, for some of these pupils, because they were used to quickly grasping new concepts as they were introduced, they didn’t always develop the strategies they needed for when learning was more challenging. This meant that, when faced with a task where they didn’t “just know”, some children lacked resilience or the strategies they needed to break into a problem and identify the steps needed to work through this. 

As I incorporated the debrief more and more frequently into my lessons, I saw a significant shift. Through my questioning, I prompted children to reflect on the rationale underpinning the strategies they used. They were also able to hear the explanations given by others, developing their understanding of the range of options available to them. This helped to broaden their repertoire of knowledge and skills about how to be an effective learner.  

How does the debrief work in practice?

Many of the questions we can use during the debrief prompt learners to reflect on the “what” and the “why” of the strategies they employed during a given task. For example, 

  • What exactly did you do? Why?
  • What worked well? Why?
  • What was challenging? Why?
  • Is there a better way to…?
  • What changes would you make to…? Why?

However, I also love asking pupils much more open questions such as “What have you learned about yourself and your learning?” The responses of the learners I work with have often astounded me! They have encompassed not just their understanding of the specific learning objectives identified for a given lesson, but also demonstrating pupils’ ability to make links across subjects and to prior learning. This has led to wider reflections about their metacognition – strengths or weaknesses specific to them, the tasks they encountered, or the strategies they had used – or their ability to effectively collaborate with others. 

For me, the debrief provides an opportunity for pupils’ learning to really take flight. This is where reflections about learning move beyond the boundaries and limitations of a single lesson, and instead empower learners to consider the implications of this for their future learning. 

For our higher-attaining pupils, this means enabling them to take increasing ownership over their learning, including how to do this ever more effectively. This independence and control is a vital step in becoming resilient, motivated and autonomous learners, which sets them up for even greater success in the future. 

References


 

Tags:  cognitive challenge  critical thinking  language  maths  metacognition  pedagogy  problem-solving  questioning 

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3 activities to improve the use of mathematical vocabulary in your classroom

Posted By Ems Lord, 11 February 2022

Dr Ems Lord, Director of the University of Cambridge’s NRICH initiative, shares three activities to try in your classroom, to help learners improve their use of mathematical vocabulary.

Like many academic subjects, mathematics has developed its own language. Sometimes this can lead to humorous clashes when mathematicians meet the real world. After all, when we’re calculating the “mean”, we’re not usually referring to a measurement of perceived nastiness (unless it’s the person who devised the problem we’re trying to solve!). 

Precision in our use of language within mathematics does matter, even among school-aged learners. In my experience, issues frequently arise in geometry sessions when working with pyramids and prisms, squares and rectangles, and cones and cylinders. You probably have your own examples too, both within geometry and the wider curriculum. 

In this blog post, I’ll explore three tried-and-tested ways to improve the use of mathematical vocabulary in the classroom.

1. Introduce your class to Whisper Maths

“Prisms are for naughty people, and pyramids are for dead people.” Even though I’ve heard that playground “definition” of prisms and pyramids many times before, it never fails to make me smile. It’s clear that the meanings of both terms cause considerable confusion in KS2 and KS3 classrooms. Don’t forget, learners often encounter both prisms and pyramids at around the same time in their schooling, and the two words do look very similar. 

One useful strategy I’ve found is using an approach I like to refer to as Whisper Maths; it’s an approach which allows individuals time to think about a problem before discussing it in pairs, and then with the wider group. For Whisper Maths sessions focusing on definitions, I tend to initially restrict learner access to resources, apart from a large sheet of shared paper on their desks; this allows them to sketch their ideas and their drawings can support their discussions with others. 

This approach helps me to better understand their current thinking about “prismness” and “pyramidness” before moving on to address any misconceptions. Often, I’ve found that learners tend to base their arguments on their knowledge of square-based pyramids which they’ve encountered elsewhere in history lessons and on TV. A visit to a well-stocked 3D shapes cupboard will enable them to explore a wider range of examples of pyramids and support them to refine their initial definition. 

I do enjoy it when they become more curious about pyramids, and begin to wonder how many sides a pyramid might have, because this conversation can then segue nicely into the wonderful world of cones! 

2. Explore some family trees 

Let’s move on to think about the “Is a square a rectangle?” debate. I’ve come across this question many times, and similarly worded ones too. 

As someone who comes from a family which talks about “oblongs”, I only came across the “Is a square a rectangle?” debate when I became a teacher trainer. For me, using the term oblong meant that my understanding of what it means to be a square or an oblong was clear; at primary school I thought about oblongs as “stretched” squares. This early understanding made it fairly easy for me to see both squares and oblongs (or non-squares!) as both falling within the wider family of rectangles. Clearly this is not the case for everyone, so having a strategy to handle the confusion can be helpful. 

Although getting out the 2D shape box can help here, I prefer to sketch the “family tree” of rectangles, squares and oblongs. As with all family trees, it can lead to some interesting questions when learners begin to populate it with other members of the family, such the relationship between rectangles and parallelograms.

3. Challenge the dictionary!

When my classes have arrived at a definition, it’s time to pull out the dictionaries and play “Class V dictionary”. To win points, class members need to match their key vocabulary to the wording in the dictionary. For the “squares and rectangles” debate, I might ask them to complete the sentence “A rectangle has...”. Suppose they write “four sides and four right angles”, we would remove any non-mathematical words, so it now reads “four sides, four right angles.” Then we compare their definition with the mathematics dictionary.

They win 10 points for each identical word or phrase, so “four right angles, four sides” would earn them 20 points. It’s great fun, and well worth trying out if you feel your classes might be using their mathematical language a little less imprecisely than you would like.

More free maths activities and resources from NRICH…

A collaborative initiative run by the Faculties of Mathematics and Education at the University of Cambridge, NRICH provides thousands of free online mathematics resources for ages 3 to 18, covering early years, primary, secondary and post-16 education – completely free and available to all. 

The NRICH team regularly challenges learners to submit solutions to “live” problems, choosing a selection of submissions for publication. Get started with the current live problems for primary students, live problems for secondary students, and live problems for post-16 students.

Tags:  free resources  language  maths  myths and misconceptions  pedagogy  problem-solving  vocabulary 

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The importance of language in mathematics

Posted By Jemma Sherwood, 11 February 2022

Based on a post originally published on Jemma Sherwood’s website, The World Is Maths.

Back in 2017 (where does time go?) I wrote this post on the importance of vocabulary, where I argued for including subject-specific (what we tend to refer to as ‘tier 3’) vocab more in our lessons.

Since then I’ve obviously thought more about this and, following on from conversations with David Didau, I wanted to get down another observation.
In my experience, maths teachers can have a tendency to underestimate two things:

  • The vocabulary our pupils can cope with.
  • The effect of bypassing the correct vocab.

Let me elaborate.

The vocabulary our pupils can cope with

Our pupils are capable of learning lots of words. They learnt to speak as youngsters and acquired thousands of them, but we know that many of them don’t move past the basic or intermediate literacy skills to those they need to access more advanced material (Shanahan and Shanahan, 2008). Something happens to many students at secondary age whereby their language acquisition falters. If that is the case, then it falls to us to accept that we’re not teaching them this as well as we could. We must maintain the highest of expectations of all our pupils and part of that is building language acquisition into our lessons such that it is both integral and normal.

What do integral and normal look like? Integral means you value language acquisition as an essential part of your teaching, that you understand its necessity in an education. You seize every opportunity to teach new words, you make pupils practice them – saying them out loud, using them in sentences in context – and you carefully build this into what you do. Normal means language acquisition teaching isn’t an add-on and it’s not over-complicated. We don’t need fancy worksheets and analysis of etymology (although etymology is fascinating and all students should meet it). If we make the teaching of language (or anything, for that matter) too onerous or time-consuming it won’t happen properly. It must be a simple, everyday occurrence, as normal as anything else we do.

When the teaching of language is integral and normal you see that pupils are able to learn really rather complex and specific vocabulary very well and this, in turn, allows them to think more precisely and to communicate more clearly.

Returning to the paper referenced earlier, the authors spent some time talking to mathematicians, scientists and historians to determine what reading looked like in each discipline. There were specific elements of reading that were valued to a different extent by each. The mathematicians valued close reading and re-reading, specifically because reading in mathematics is linked to precision, accuracy and proof. I particularly like this quote:

Students often attempt to read mathematics texts for the gist or general idea, but this kind of text cannot be appropriately understood without close reading. Math reading requires a precision of meaning and each word must be understood specifically in service to that particular meaning.

If we want to take our students on a pathway to being mathematical, thinking like a mathematician, we should build in language acquisition and precision reading as a principle of this.

The effect of bypassing the correct vocab

Something I see very regularly in classrooms is teachers avoiding using correct vocab, I think (from conversations I’ve had) because they are worried that particular vocab will make it harder to understand a concept. This is best explained with an example:

Teacher: A factor is a number that goes into another number.

How many times have you said this? I know I have! I think it happens because of a perception that a “simplified” definition makes this word accessible to more pupils. However, I would argue that we are making the word specifically less accessible in doing this.

What does ‘goes into’ really mean? As a novice without a strong mathematical background I could interpret this in a number of ways. However, if my teacher tells me, “A factor is a number that divides another number with no remainder”, or similar, and accompanies this with examples and non-examples, I can make more sense of the word from the start. Moreover, if my teacher regularly refers to the word ‘factor’ alongside this definition, and asks my peers and me this definition, and gets us hearing it and rehearsing it, then I start to associate the word ‘division’ with ‘factor’ and I am less likely to confuse it with ‘multiple’. Eventually, it will become part of my fluent vocabulary.

In ‘dumbing down’ a definition, we work against understanding rather than for it. That doesn’t mean we have to go all-out Wolfram Mathworld* on our pupils, but it does mean we have to consider the implications of our own use of language and how we can make small changes that have a positive impact. It’s worth taking some time with your team to discuss where else we have a tendency to bypass proper vocabulary or definitions, and think about the specific negative effects this will have on our pupils. How can you, as a team, work towards increasing your pupils’ language acquisition and precision? What ideas or concepts do you want them to automatically associate a certain word with? Design your instruction towards that aim.

*“A factor is a portion of a quantity, usually an integer or polynomial that, when multiplied by other factors, gives the entire quantity.” Mathworld

Jemma Sherwood is a Senior Lead Practitioner for Maths, and the author of How to Enhance Your Maths Subject Knowledge: Number and Algebra for Secondary Teachers. Find out more about Jemma on her website, or follow her on Twitter.

 

Tags:  language  maths  oracy  vocabulary 

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Phiddlywinks: have you tried it yet?

Posted By NRICH, University of Cambridge, 04 June 2021
In March this year, NACE members had the opportunity to preview and trial a new maths game being developed by the team at NRICH – a University of Cambridge initiative providing free online maths resources that promote challenging, enriching learning experiences.
 
The game in question has now been launched, and in this blog post the NRICH team explain how it works, and how you and your learners can get playing.
 
Question: What happens when you bring together Tiddlywinks and football?
 
Answer: You get Phiddlywinks!
 
In this blog we’ll learn more about Phiddlywinks, including the charismatic mathematician who inspired the game and role of NACE members in bringing it to our screens.

What is Phiddlywinks?

Phiddlywinks is a strategy game for two players. The winner is the first player to get the white counter into the coloured region at the opposite end of the board. Player 1 is aiming for the blue region and Player 2 for the red region.
Phiddlywinks 
The game begins with the white counter in the centre circle.
 
Players take it in turns to either:
  • Place a black counter on the board or
  • Move the white counter.
The white counter moves by jumping in a straight line over one or more black counters. A player may be able to make more than one jump when it is their turn.
 
To get started, consider this screenshot from a game which is underway. Both players have chosen to use their turns to add black counters to the board (you’ll notice that the white counter remains in its starting position). It is Player 1’s turn. Can you see how Player 1 might move the white counter to win the game?
 
Phiddlywinks 
 
Here’s one possible winning move:
  • Player 1 clicks on 7E (or 8F) and the white counter moves to 9G
  • Player 1 clicks on 9F (or 9E) and the white counter moves to 9D
  • Player 1 clicks on 9C and the white counter moves to 9B
  • Player 1 clicks on 10B and the white counter will move to 11B, winning the game!
Do take some time exploring the interactivity. To help you learn to play the game, we’ve uploaded more mid-game scenarios here. You can also print off black and white or colour versions of the board.

Who was the inspiration behind Phiddlywinks?

John Horton Conway was a prize-winning mathematician who loved creating new games for all ages. He is best known to many for creating the Game of Life. He also developed a game called Philosopher's Football (also known as Phutball) which challenged players to manoeuvre a ball across a large grid towards their opponent's goal-line. Not surprisingly, the game soon became popular with his university students.
 
We have taken Phutball as the inspiration for our Phiddlywinks. We piloted the developmental version of the game with NACE members at a specially organised online event attended by both primary and secondary colleagues. The feedback from teachers attending NACE event, and the follow-up response from the classes of NACE members who kindly trialled Phiddlywinks with their classes, enabled our team to prepare the game for its release.
 
Phiddlywinks is almost identical to Philosopher's Football except that the white ball has become a white counter and the players have become black counters. The rules are the same but Phiddlywinks is played on a much smaller board. The way the counters move reminded us more of Tiddlywinks than football, hence the alternative name.
 
The NRICH team would like to acknowledge the support of NACE and its members who kindly trialled our initial version of the game, giving us invaluable feedback which informed the development of Phiddlywinks.
 
What maths games and activities have you and your learners been enjoying this year? Share your ideas in the comments below or in the NACE community forums.

Tags:  enrichment  free resources  maths  problem-solving  remote learning 

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New free maths resource to develop self-reflection and motivation

Posted By Ems Lord, 13 January 2021

NACE is proud to NACE is proud to partner with Cambridge University’s NRICH initiative, which is dedicated to creating free maths resources and activities to promote enriching, challenging maths experiences for all. In this blog post, NRICH Director Dr Ems Lord shares details of the latest free maths resources from the team, and an exclusive opportunity for NACE members…

In this blog post, I’m delighted to introduce NRICH’s new child-friendly reflection tool for nurturing successful mathematicians – part of our suite of free maths resources and activities to promote enriching, challenging mathematical experiences for all. We very much hope that you enjoy exploring it with your classes to support them to realise their potential. Such innovations are developed in partnerships with schools and teachers, and we’ll also be inviting you to work directly with our team to help design a future classroom resource intended to challenge able mathematicians (see below or click here for details of our upcoming online event for NACE members).

At NRICH, we believe that learning mathematics is about much more than simply learning topics and routines. Successful mathematicians understand the curriculum content and are fluent in mathematical skills and procedures, but they can also solve unfamiliar problems, explain their thinking and have a positive attitude about themselves as learners of mathematics. Inspired by the 'rope model' proposed by Kilpatrick et al. (2001), which draws attention to the importance of a balanced curriculum developing all five strands of mathematical proficiency equally rather than promoting some strands at the expense of others, we have developed this new model and image which uses child-friendly language so that teachers and parents can share with learners five key ingredients that characterise successful mathematicians: 


  • Understanding: Maths is a network of linked ideas. I can connect new mathematical thinking to what I already know and understand.
  • Tools: I have a toolkit that I can choose tools from to help me solve problems. Practising using these tools helps me become a better mathematician.
  • Problem solving: Problem solving is an important part of maths. I can use my understanding, skills and reasoning to help me work towards solutions.
  • Reasoning: Maths is logical. I can convince myself that my thinking is correct and I can explain my reasoning to others.
  • Attitude: Maths makes sense and is worth spending time on. I can enjoy maths and become better at it by persevering.

Using the tool during remote learning and beyond

This reflection tool helps learners to recognise where their mathematical strengths and weaknesses lie. Each of the maths activities in our accompanying primary and secondary features is designed to offer learners opportunities to develop their mathematical capabilities in multiple strands. We hope learners will have a go at some of the activities and then take time to reflect on their own mathematical capabilities, so that when full-time schooling returns for all they are ready to share their excitement about what they have achieved, and are eager to continue on their mathematical journeys.

At NRICH, we believe that following the current period of remote learning, success in settling back into schools will be aided by recognising and acknowledging the mathematical learning that has been achieved at home, and encouraging learners to reflect on how they see themselves as mathematicians. It may be that some learners will not recognise the value of what they have achieved while they have been out of the classroom, because what they have been doing at home may be quite different from what they usually do in school. We want learners to appreciate that there are many ways to demonstrate their mathematical capabilities, and to recognise the ways in which they behave mathematically. By inviting children and students to assess their mathematical progress on a broad range of measures, we hope to change the narrative to recognise what learners have achieved, rather than focusing on what they have missed. 

Get involved….

As the current period of remote learning continues, we’re continuing to develop new free maths resources, and we always value input from teachers. On 3 February 2021, the NRICH team is hosting an online meetup for NACE members during which we’ll share an exciting new classroom resource, currently under development, intended to challenge able mathematicians. The session will involve an opportunity to explore this new resource and share your insights to help inform its future development. We look forward to working with you. Details and booking.

Very best wishes for 2021 – the NRICH team.

Ref: Kilpatrick, J., Swafford, J. and Findell, F. (eds) (2001) Adding It Up: Helping Children Learn Mathematics. Mathematics Learning Study Committee: National Research Council.

Tags:  confidence  free resources  lockdown  maths  metacognition  mindset  motivation  remote learning 

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6 top tips to develop collaborative problem-solving skills

Posted By King Edwin Primary School, 12 March 2019
Updated: 06 August 2019
Helping learners develop collaborative problem-solving skills requires careful planning to ensure all are engaged and challenged. In this blog post, Anthony Bandy, Assistant Head Teacher at King Edwin Primary School, shares six top tips drawn from his experience of participating in the NACE/NRICH ambassador scheme.

Inspired by research highlighting key skills and attributes for the next generation of citizens and employees, NRICH has created free resources to help learners develop mathematical “habits of mind” at primary and secondary levels – focusing on resilience, curiosity, thinking and collaboration. Each of these four key areas is broken down into different strands of maths, making it easy for activities to be delivered as part of regular maths sessions.

When using these resources to help learners develop collaborative problem-solving (CPS) skills, here are six top tips for effective implementation…

1. Explore perspectives on collaboration

When conducting research on effective approaches to developing collaborative problem-solving skills, the NRICH team discovered something they hadn’t even thought of. When asked about working with numbers, one in three surveyed learners said they felt working together was actually cheating! This is useful to bear in mind. Spend some time exploring existing perspectives on collaboration in your class and school – you may need to work on changing learners’ (and possibly teachers’) attitudes to collaborative learning.

2. Use “think, pair, share”

Before some collaborative activities, some learners will need a bit of time to get their head around the problem. “Think, pair, share” is a great way to facilitate this, allowing time for independent thinking as well as collaboration. Learners start by working independently, thinking about the problem for themselves and making notes if they wish. They then discuss the problem in pairs and/or as a group, working around a shared large sheet of paper to discuss their answers, reasoning and strategies as they go along – great for developing maths talk.

3. Consider group size

Some learners do not like working in large groups. In addition, the smaller the group, the higher the participation level of each child; larger groups could initiate passive learning. Consider group sizes before delivering the session – perhaps offer the option to work in twos, threes or fours.

4. Allocate roles and responsibilities

One strategy for developing collaboration is to give learners allocated roles and responsibilities. This can be used in all teaching and learning sessions, giving learners a chance to try out different roles, and increasing participation levels. For example, you could have a Chief Noticer, tasked with noting down ideas using a whiteboard. Your Chief Questioner could be asking questions, such as “What do we notice? How do you know?” You could also have Chief Explainers, Chief Justifyers and so on…

5. Choose activities with different learners in mind

A common concern when planning collaborative activities: how are you going to stop one learner taking over? To ensure all learners are motivated and empowered to participate, try to choose activities that will appeal to different interests and strengths. For instance, in NRICH’s Olympic measures activity, learners who are not usually highly engaged with maths, but who love and know about sports, can become the most important people in the room.

6. Encourage learners to reflect

At the end of each session, ask learners to rate themselves and their partner in terms of collaborative skills. If not a five out of five, what was missing? Why? Build in time to discuss collaboration and what skills are needed to be successful.

Tags:  collaboration  free resources  maths  problem-solving 

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5 steps to develop collaborative problem-solving in maths

Posted By Ems Lord, 05 March 2019
Updated: 08 April 2019
Ems Lord, Director of the University of Cambridge-based NRICH project, shares five key factors to consider when planning collaborative problem-solving (CPS) sessions using low-threshold, high-ceiling maths resources.

Have you ever attempted assembling flat-pack furniture with a friend or family member? How did it go? And are you still talking to one another?

Being able to work with others is a key life skill, but not always as straightforward as we might like. Whether we’re assembling furniture, putting up an extension or navigating our way to a holiday rental, we need to be able to work together towards a common goal and recognise our own responsibilities in achieving that goal. Moreover, developments such as driverless cars and drones signpost an increasingly automated environment in which those with strong group-working and problem-solving skills will thrive.

It is essential that we understand how to help learners develop collaborative problem-solving (CPS) skills alongside a sufficient level of challenge – planning lessons that will stretch more able learners while being accessible to all.

To this end, NRICH worked with 10 Cambridgeshire schools and the Cambridgeshire Maths Team in a project sponsored by Nesta. We shared existing low-threshold, high-ceiling NRICH resources with participants, who then adapted these to develop CPS in their own classes. After visiting each school, talking with teachers and running focus groups with learners, we identified five key aspects of CPS to consider when planning maths lessons:

1. Use low-threshold, high-ceiling activities

First and foremost is the importance of low-threshold, high-ceiling (LTHC) activities and resources. These enable all learners to get started on a problem while also offering sufficient challenge. One of NRICH’s most popular LTHC activities is the Factors and Multiples Game, which challenges learners to work together to build as long a chain as possible. Be warned: it’s hopelessly addictive for adults too!

When choosing LTHC tasks, explore our free curriculum mapping documents for primary, secondary and post-16 provision.

2. Get learners hooked

Engaging tasks are key for CPS sessions; learners must want to solve the problem. At NRICH, we aim to engage learners by designing activities which have a clear “hook” – such as the interactive challenge Got It! and the sports-themed activity Olympic Records.

Got It! requires learners to pit themselves against the NRICH computer to be the first to reach 23. This challenging activity draws learners in and they often make multiple attempts at the problem. Several of our focus group participants said they later taught the game to older siblings and family members because they thought they could outwit them.

The group activity Olympic Records is particularly appealing to learners with an interest in sports, who can draw on their knowledge to support others to match sports to their graphs. It demands effective group work and a willingness to adjust initial responses once learners realise that gender is also an important factor.

3. Model individual roles and responsibilities

A group is only as good as its individual members. Every member of the group must know what is expected of them during the task, and which roles belong to others. Individual learners should not dominate the session but should focus on filling their own roles while supporting others.

Card activities often work well in developing these skills; for example Shape Draw. Be clear about roles; which individual is responsible for recording the activity, suggesting the next shape or rolling the die? Make sure everyone knows their role and consider rotating different roles around the group. Teachers participating in our CPS project stressed the importance of modelling different roles for group members before embarking on the actual group work.

4. Develop skills for group communication

While knowing their own role is important, learners also need to be aware of the overall aims of the group. This changes the level of challenge for any task from merely cooperating to fully collaborating. In particular, all learners should be prepared to feedback to the wider class about their task.

Useful activities which offer a high level of challenge for older learners and the opportunity to feedback and explore different approaches include Steel Cables and Kite in a Square. Younger learners might enjoy the challenge of Jig Shapes and Quad Match.

5. Build in time for reflection

CPS skills need time to develop. Timetables should allow for regular CPS teaching sessions, including time allocated for reflection. Building in this reflection time can be a challenge, as time is also needed to focus on developing the required mathematics and group-working skills – but the teachers in our project stressed that it was highly worthwhile.

Ask learners about how well they worked in a group. If they awarded themselves a score from 1 to 5, what would it be and why? Which areas of their group work do they need to develop further? From a teaching perspective, when will they get their next opportunity to work on those areas?

And for your own reflection… If your class attempted one of our tasks, how do you think they might cope? Which aspects do you anticipate offering the most challenges? More importantly, when are you planning to lead the next CPS session with your class?

Further reading

Ems Lord has been Director of NRICH since 2015, following a previous role leading one of the country's largest Mathematics Specialist Teacher Programmes. Ems has taught mathematics across the key stages, from early years to A-level Further Mathematics, and has worked in a variety of settings, including a hospital school. She’s supported schools as a leading mathematics teacher, local authority consultant and Chartered Mathematics Teacher, and has taught mathematics education on both BEd and PGCE teacher programmes. She is currently working on her PhD thesis, which explores approaches to improve support for those learning calculation skills, and is President-Elect of the Mathematical Association for 2019-2020.

Tags:  collaboration  free resources  maths  problem-solving 

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What makes a challenging GCSE science exam question?

Posted By Edmund Walsh, 05 February 2019
Updated: 22 December 2020

In this excerpt from the NACE Essentials guide “Realising the potential of more able learners in GCSE science”, NACE Associate Ed Walsh explores the components of a challenging GCSE science exam question – and how teachers can best help learners prepare. 

There is sometimes an assumption that it is the complexity of the content that is the key determinant in how challenging an exam question is; this isn’t necessarily the case. In fact, there are a variety of ways in which questions can be made more challenging, and in order to support learners with high target grades this needs to be understood.

When preparing your learners for the most challenging GCSE science exam questions, here are six aspects to consider:

1. Reduced scaffolding and multiple steps

Whereas some questions continue to be structured and are specific about what understanding or application should be demonstrated, there will be other questions where learners need to work out the sequence of stages to be undertaken. This might, for example, involve using one equation to calculate a value which is then substituted into another. As well as being able to (in some cases) recall the equations and use them, learners also need to work out the overall strategy.

Encourage learners to get into this habit by asking: “What’s a good way of approaching this question?”

2. Extended response questions

Extended responses are frequently marked using a level of response mark scheme. If there are six marks allocated, the mark scheme will commonly have three levels. If more able learners are to score five or six marks, they need to be meeting the level 3 descriptor as often as possible.

Help learners prepare by modelling extended responses and providing opportunities to practise this – considering a structure, selecting key words, using connectives and checking against the exam specifications.

3. Use of higher-order maths skills

Learners need to be able to apply maths skills in a variety of ways. This could be a multistep response in which learners, for example, plot points on a graph, sketch the (curved) line of best fit, draw the tangent and calculate its gradient. This requires both the necessary command of these skills, and the understanding of which to use.

To ensure learners have access to the necessary maths skills, develop dialogue with your maths department. Invite colleagues to jointly consider the maths skills involved in sample science questions, and how best to prepare learners for these challenges. As well as nurturing specific skills, focus on developing learners’ ability to identify effective strategies and sequencing.

4. Linking ideas from different areas

As part of the changes to GCSE science specifications, learners are expected to show they can work and think flexibly, linking ideas from different areas of the subject. Help them prepare by providing regular opportunities to practise this. Check out the specification and the guidance it gives about key ideas and linkage.

5. Applying ideas to novel contexts

Telling learners “If it’s not on the spec you don’t need to learn it” is dangerous – and untrue! Challenging them to apply their understanding to other contexts is part of the function of the exams and will continue to be so. Again, help them prepare through regular practise so they become accustomed to applying concepts to new contexts.

6. Varied command words

Each awarding organisation uses a particular set of command words in GCSE science exams. Some of these will already be in common parlance in your science lessons, others less so. Familiarising learners with the full range of these terms will prepare them to answer a wider range of questions. 

For example, a trawl through a selection of stretch and challenge questions from one suite of exam papers indicated the following usage: explain (x7), suggest (x6), compare, calculate (x12), give (x6), estimate, justify (x2), describe (x5), write (x2), use (x9), work out, draw, predict, complete (x3), show (x2), state.

Note that while these numbers show the frequency of each stem in one random selection, they don’t reflect the numbers of marks associated. It is useful, however, to reflect on the extent to which these form part of the discourse in science lessons – not just featuring in practice exam questions, but in all written and oral activities.

Tags:  assessment  GCSE  KS3  KS4  maths  science 

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6 steps to help more able learners excel in GCSE science

Posted By Edmund Walsh, 22 January 2019
Updated: 22 December 2020

In this excerpt from the NACE Essentials guide “Realising the potential of more able learners in GCSE science”, NACE Associate Ed Walsh outlines six key steps to improve provision and outcomes for those capable of attaining the highest grades in this subject. 

1. Make effective use of assessment data

While many schools devote a significant amount of time to assembling, applying, marking and grading periodic tests, there’s often scope for these to be used more effectively to diagnose areas for improvement. Question-level analysis can help both teachers and learners identify areas of low subject knowledge and skills gaps (tagged against GCSE assessment objectives) – informing feedback, self-assessment and goal-setting, interventions, evaluation of teaching styles and planning for future lessons.

Similarly, analysis can indicate how learners perform in multiple choice questions, shorter written responses and longer responses. Be prepared: if aspirational students are looking to develop in one of these areas, they’ll expect guidance as to how to do so. Woe betide the teacher who can’t provide a learner chasing a good grade either with more examples or effective strategies in areas identified as weaknesses!

2. Challenge learners to use a range of command words

Each awarding organisation uses a particular set of command words in GCSE science exams. Some of these will already be in common parlance in your science lessons, while others may not be used as often. Familiarising learners with the full range of these terms will prepare them to answer a wider range of questions. 

When revising a topic, prompt learners to suggest the type of questions examiners might ask; this will help them revise more effectively. Elicit the nature of each question, encouraging learners to consider the influence of assessment objectives (AOs) and to use a full range of command words.

3. Develop dialogue with the maths department

The quality of dialogue with colleagues in maths and the development of a whole-school numeracy policy has never been so important. (It may also never have been so tricky, bearing in mind the pressure that both maths and science teams can be under.) It can be tempting for a hard-pressed science department to want the maths team to fit in with their running order of topics. The maths curriculum is also driven by a sense of progression, but not necessarily the same one. Skills demanded in KS3 science may in some cases not be taught in maths until KS4.

Rather than reach an impasse, focus on exploring common ground. Set up a joint meeting and look at maths skills involved in sample science questions. Invite colleagues to explore potential strategies, terminology, likely challenges for learners and how they would deal with these. As well as nurturing specific skills, focus on developing learners’ ability to identify effective strategies and sequencing. More able learners aiming for high grades need to develop problem-solving skills as well as a mastery of individual skills.

4. Review the role of practical work and skills

When carrying out required practicals, ensure learners have access to a range of question types, including questions based on AO2 (application of knowledge and understanding) and AO3 (interpretation and evaluation). It is also important to look at the lists of apparatus and techniques skills in the GCSE specification. Questions relating to practical work are often based on these, even if the context isn’t one learners have met in the required practicals. Assess how good learners are at these skills and whether you can give them more opportunities to develop these. These have a strong relationship with skills used at A-level, meaning those progressing to further study will also benefit.

5. Develop the role of extended writing

Candidates will be expected to develop extended responses, especially on higher tier papers. Look at learners’ performance on such questions to see how it compares with other items. It may be useful to encourage learners to consider what structure to use before commencing writing. Model the drafting of an extended response, demonstrating how you select key words, use connectives, structure a response and check against the answer. AQA, for example, is moving towards the use of generic descriptors for types of extended responses.

6. Link ideas from different parts of the specification

As part of the changes to GCSE science specifications, learners are expected to show that they can work and think flexibly, linking ideas from different areas. Use questions that require this, identifying good examples to use in advance. One of the sample questions uses the context of a current balance, including ideas about magnetic fields and levers. Check out the specification and the guidance it gives about key ideas and linkage. As well as scrutinising the detailed content, look at the preamble and follow-up.

Tags:  assessment  feedback  GCSE  KS3  KS4  maths  science 

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