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

• EEF (2021). Metacognition and Self-regulation, Teaching and Learning Toolkit.
• Education Endowment Foundation (2018). Metacognition and Self-regulation: Guidance Report.

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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.

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.

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

• Luckin, R., Baines, E., Cukurova, M., Holmes, W., & Mann, M. (2017). Solved! Making the case for collaborative problem-solving.
• Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. The Journal of Mathematical Behavior, 15(4), 375-402.
• McClure, L., Woodham, E., & Borthwick, A. (2017). Using Low Threshold High Ceiling Tasks.

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.

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.