Developing Mastery in Mathematics (2)

(Featured image: “Image Provided by Classroom Clipart“)

An Action Research project by Jodie Johnson

The aim of this project was to explore different ways in which we could embed the new ‘mastery in maths’ curriculum into our day to day teaching. The curriculum has changed dramatically for Key Stage 3 and 4 in terms of the way students will be assessed; while the content is largely the same the way in which we teach the new curriculum has to be adapted to this new style if our students are to be successful .

Our aims were:

  • To clarify exactly what ‘mastery’ means for our subject
  • What this means for us as a faculty as a whole and our teaching styles; we then wanted to work on how this should directly impact on our individual lessons and assessments
  • To begin to think about how we could allow our students the opportunity to be more resilient in our subject and therefore more ready to face the new style of questioning that they will be challenged with
  • Finally, we worked on how the mastery curriculum could be embedded more formally into our schemes of work.

Background

Looking formally at ‘Mastery in Mathematics’ is vital for our department at this time as our current year 10 are the first to face the new mastery curriculum at GCSE level. It was essential that we took the time as a department to focus on the shifting focuses of the new curriculum; it was important that we did this together and that we did it now. In our initial meeting we wanted to address the differing opinions we had in terms of what we thought mastery was and then whether this mirrored what the new curriculum required. Once we had clarified this for ourselves it was important to us that the students could articulate what we meant by mastery.  Finally and most importantly we needed to work on how this would impact on our day to day teaching methods so that our teaching style was adapted and in turn we were preparing our students as best we possibly could for the challenges they would be facing.

Context

We began our discussions at the beginning of the year by each focusing on a couple of specific classes that we would ensure had a ‘mastery plenary’ as often as possible and that we would use as a group to compare to the rest of our students. However we quickly realised that this would leave those that were not picked at a huge disadvantage in terms of preparing them for their assessments so we decided it was important that all of our students (in years 7-10) were experiencing ‘mastery’ style lessons.

While we felt as a department it was vital that we started to look at Mastery this year for our students, I have also been interested in this style of teaching for a while. I have become more and more conscious since I began teaching that the mathematics we were delivering to our students wasn’t necessarily preparing them for the real world but for an exam that we could pretty much second guess in terms of what it would look like. Like most other mathematics teachers I have worked with, I felt the problem solving skills and fluency that we should be teaching our students was being lost and replaced with teaching students how to answer a seemingly random set of questions in order to pass exams and this meant that they did not have a deep understanding of the concepts they were being taught. In my opinion, Mathematics should be an exercise in problem solving, it should stretch a person’s mind to work in a way that no other subject does and this was being lost as result of the pressure which falls heavily upon teachers shoulders to hit target grades. The new mastery curriculum while daunting for maths teachers in the short term, I saw and still see, as an exciting and hugely beneficial thing in the long term for our future generation of Mathematicians. How exactly this would look in my classroom, how I could ensure I was preparing them to problem solve and enjoy mathematics, while at the same time preparing them to pass their exams in maths is something I was grateful to have the time to do while preparing this Action Research Project.

Actions

As a department there are several ways in which we have modified our teaching since working together as a learning focus group1.

Research into Mastery and how this affected our work

All member of the department undertook their own individual research into what mastery was and we the brought it together in our learning focus meetings. We found that the most important factor when teaching the mastery curriculum was that of fluency between topics. We decided after our reading that for our students, especially those that would be facing the foundation curriculum this was something that we were not currently doing successfully, building their resilience in mathematics was paramount.   If they were to be successful mathematicians we needed to instil some confidence in them that it is completely fine to get things wrong in mathematics.

We also discovered various ways in which other countries have approached the teaching of Mathematics. We looked at the potential impact adopting Eastern Asian styles of teaching would have on our students and decided that some time would need to be dedicated to our students ‘learning’ facts and methods in maths so they had access to them at all times when completing more open ended tasks. Things like learning times tables for our younger students is something we often presume the students know from primary school but this is often not the case and we spent some time with our weaker students actually learning things like this as home works or in class.

We discovered after conversations between the team that articulating mathematics is something that is important for our students in order to ‘master’ a topic and that again our current methods weren’t necessarily allowing enough opportunity for this skill to be developed. We have therefore spent much more time on questions where students have to prove answers and in my lessons I questions students in a slightly different way, emphasising the importance of clarity in their working, asking questions like “Are you sure about that? Prove it to me as your current working doesn’t convince me”. This form of questioning also forces my students to think more precisely about what they are writing and the way in which they are presenting their work.

We researched different methods that we could use every lesson that wouldn’t necessarily link directly to individual topics. For example, asking questions like:

“Where does this fit into what we did last week?”

“Can you show me another way to do that?”

“Is that the only way to do that question?”

Adapting Assessments

At the beginning of the year we were working from a scheme of work called ‘Kangaroo’. We have worked on this for the least 4 years as a faculty but with the new curriculum changes Kangaroo have also updated their aims and lesson objectives. We continued to follow this scheme of work but adapted our assessments to include mastery style questions that we found on the Kangaroo website as well as the AQA website (which is the GCSE board we will be following) at the end of each unit of work. This meant that our students now needed much more fluency between subject areas and we were working at dispelling the myth that ‘a Pythagoras question looks like this’, ‘an expanding brackets question looks like this’ etc. We were starting to force our students to think of Mathematics as a puzzle and that each individual topic was just one piece and that they would need all the pieces to answer these new style questions.

Over the last 3 years we have been developing our schemes of work to incorporate more and more ‘Nrich’ challenges (Nrich is a website created by Cambridge University which has open ended questions and what we now recognise as ‘Mastery challenges’). While we have informally taught Nrich lessons once a fortnight for the last few years, one member of the department has now formally added appropriate Nrich lessons to our schemes of work where they naturally fit into the subject areas we are teaching. The rationale behind this is that the students will get used to being ‘stuck’ (no Nrich challenge is a 5 minute problem with a yes or no answer – each one takes at least an hour and the students will become more and more familiar with getting themselves unstuck as part of the experience). One adaptation I made during these lessons during the year was to only allow students to ask 3 questions of me the teacher per Nrich lesson. This forced them to have to really think about whether they needed to ask the questions or whether they were actually being too teacher reliant.

While this year was very much an experimental year in terms of the best way to adopt ‘mastery’ in the classroom, one thing that we were keen to get right was our assessments. We felt it was essential that the assessments the students were doing to inform our data on their learning resembled closely what their final assessments would look like in order to make our data as accurate as possible. In some cases (especially in year 10) this has meant students’ progress data has taken a hit, however we felt preparation for the new mastery curriculum was paramount. This also meant that we could build resilience, not just in the classroom when we are teaching and when they have the luxury of checking their answers and ideas with their peers and teacher but when the students needed to transfer this to the exam hall and feel as though they needed to at least attempt questions (especially the larger 6-8 mark questions which we have not seen before) without fear of getting them wrong.

Changing plenaries

In order to prepare our students for the new style curriculum we started to use plenary questions that paired more than one topic with that which was taught during the lesson. In the Appendix you will find two plenaries which show how mastery could be demonstrated once a topic has been covered.  There is also a full lesson which shows Levelled learning objectives and how we now must link subject content to other areas to secure ‘mastery’. Hopefully these will show how fluency between topics is now essential to completing most of my planned plenaries this year. While there was some resistance from students initially, the students do recognise the importance of doing this and have adapted accordingly.

Peer Observations

In order to help each other and compare our work, myself and another member of the faculty paired up to complete some peer observations. We used the time to discuss ideas and how the topics taught could be connected to other areas of maths.  This helped both of us to plan appropriate mastery style questions for the main bulk of the lessons and the plenaries. The joint planning that went into these lessons allowed us to think about the fluency between other areas and topics, as well as standardising the way we delivered our plenaries and most importantly, the different ways in which we were trying to build resilience in the subject.  As a faculty we plan to complete at least one peer observation per term to see how mastery is developing.

Impact and Conclusion

The impact our actions have had on the faculty will be felt in time. While there is no concrete evidence that can be shared in this document, I think that from my perspective, it has forced me to think about my practice and the fluency and links I make when teaching. My mathematics has certainly improved as a result of teaching the new curriculum, especially since I have a very able top set in year 10, who need to be challenged to reach their potential – the new assessments that we have even challenged me, which has been great!

While many students are still not comfortable with the new curriculum and style in which we now have to teach mathematics it is definitely improving, my students, especially the most able, are always very excited when they realise we are having an ‘Nrich lesson’ and now ask me at the start of lessons whether that is what we are doing today. This is an improvement on where we were at the start of the year since they didn’t tend to enjoy and therefore excel in these lessons because they were being pushed out of their comfort zone.

My key stage three classes have improved greatly in terms of their resilience and are now much more able to access mastery plenary questions that I give them to practice. At the beginning of the year many, especially my least able in year 7 and 8 would simply freeze when they were confronted with a questions that didn’t directly relate to the subject we had been focused on during the lesson. It is a gradual process but it is certainly a picture that is improving.

As I have mentioned above, the first mock our year 10 students took in June did not necessarily show strong progress, however in terms of my class, their reaction certainly showed maturity and resilience which is what this new curriculum requires our students to show. They worked solidly on their mock papers once they had been marked to understand as many questions as possible. Since they now understand the importance of keeping going – they are keen to do so.

Next Steps

Continuing our work on mastery is essential if we are to mould students to being successful not just in maths but in terms of their resilience to tackle problems and overcome their fear of getting things wrong. It is important that our work continues over the next few years and that any new team member understands why this is so important. Next year we will continue developing our lesson plans and assessments.  We will continue to work on Nrich challenges with our students and the peer observations that myself and another colleague completed will be rolled out to all members of the faculty. The standardisation of our lessons is important so that our students recognise that when they come to the maths corridor they will be challenged and need to have access to all areas of maths, not just those that they have been taught in the last 45 minutes.

This project is certainly an ongoing piece of work that we need to build on over the next few years. Our students will certainly become more comfortable with the mastery curriculum as we move forward, especially as this year Key Stage 1 & 2 have also been introduced to the new ‘mastery curriculum’ at their level, which should mean students are being moulded to move more freely between topics and solve problems independently. I look forward to seeing how our students develop as our teaching styles become more accustomed to the new curriculum.

Footnotes

  1. Learning Focus Groups – For professional development purposes staff work in small groups who share a common interest in developing an aspect of their teaching practice. These groups provide a forum for discussion, support, sharing and joint activities to help each teacher develop their own individual Action Research project.

Appendix

Plenary 1

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Plenary 2

 

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Full lesson

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Plenary

 

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References

Department for Education (DfE). (2013a). National Curriculum in England: Framework Document. London: Department for Education.

Kilpatrick, J. Swafford, J. & Findell, B.(eds.)(2001). Adding it up: Helping children learn mathematics. Mathematics Learning Study Committee: National Research Council.

NCETM (2014a). Developing Mastery in Mathematics. [Online] Available from: https://www.ncetm.org.uk/resources/45776 [Accessed: 28th September 2015]

NCETM (2014b). Video material to support the implementation of the National Curriculum. Available from: https://www.ncetm.org.uk/resources/40529 [Accessed 28th September 2015]

NCETM (2015). National Curriculum Assessment Materials. [Online] Available from: https://www.ncetm.org.uk/resources/46689 [Accessed 28th September 2015]

Ofsted  (2015) Better Mathematics Conference Keynote Spring 2015. Paper presented at the Better Mathematics Conference, Norwich, Norfolk.

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Engaging Disaffected Learners (1)

(Featured image: ‘English Dictionaries’ by John Keogh is licensed under CC BY 2.0)

An Action Research project by Anna Watkins

Project overview

Working with two other members of the department (Hannah Gale and Megan Dunsby), we established the following aims:

  • To develop our pedagogical understanding of the reasons behind disaffection;
  • To establish and broaden a range of strategies to re-engage disaffected learners;
  • To build resilience and inspire self-confidence.

These give an overview of our foci, and from here we endeavoured to use different strategies to engage disaffected learners. We decided to all concentrate on year 10 students and boys in particular. Having a shared year group meant that we were able to support each other and help each other to develop strategies.

There are a myriad of reasons why students become disaffected, but in the experiences that we had had in our year 10 classrooms we felt that these were the central reasons why:

  • Disaffection hides a literacy weakness;
  • Pressure of year 10 GCSE (especially with current changes);
  • English is compulsory and relevance isn’t obvious to all.

Background

“…by GCSE, for achievement at grades A* to C in English, the gap [between boys and girls] is 14 percentage points” (National Literacy Trust).

So, why is this gap so big and what can cause boys to become disaffected learners in English in particular? Firstly, the changes to the examination system at GCSE mean that students must sit examinations at the end of Year 11 in which they must recall and apply two years’ worth of learning. This is an overwhelming and stressful prospect for many students, who are immediately disengaged by their own assumption that they will fail at this challenge. This can be a huge cause of disaffection at the beginning of Year 10.

The English curriculum has also become more traditional, favouring more 19th century literature and classic British literature, which means that students are working with challenging texts and unfamiliar language. Some boys in particular find it difficult to understand the purpose of studying these texts, which can provoke disaffection, particularly given that English is a compulsory subject that students have not opted into. Indeed, Caroline Bentley-Davies suggests that a teacher must “signal exactly why you are doing something” (2010, p.165) when improving standards of boys.

In addition, GCSE assessment has become more rigorous; to achieve a Grade 5, students are expected to have a command of subject terminology and an ability to use a range of punctuation and sentence structures with accuracy and for specific effect. Those with weak literacy skills can therefore become disaffected to mask their difficulties.

My personal project overview

After a number of discussions with Megan and Hannah, I decided that I would look at ways in which it is possible to re-engage students through marking strategies. After reading David Didau’s comment that ‘“…apparently, 70% of all feedback given by teachers to pupils falls on stony soil,” I knew that I had to do something different and adapt my normal marking style. Therefore, my aims for this were:

  • To make DIRT more effective with my year 10 boys who initially rejected it;
  • To challenge them through my marking and feedback to ensure progress;
  • To use marking as a means of building relationships.

I had two year 10 classes and therefore used boys from both classes, who were clearly disengaged, to try different strategies with. Recent research points clearly to the importance of valuable feedback as shown in the diagram beneath:

According to Hattie and Timperley (2007) feedback is one of the most powerful influences on learning and achievement, but this impact can be either positive or negative.

In a recent paper on formative assessment, Dylan Williams (2014) suggests that:

“Indeed, assessment can be thought of as the bridge between teaching and learning—only through some kind of assessment process can we decide whether instruction has had its intended effect.”

The idea of assessment being a ‘bridge’ between teaching and learning struck a chord with me, and it was from this research that I based my project. In the same paper, Williams also advocates the idea that marking and assessment should be a process which ‘activates students as owners of their own learning’. I decided that I wanted my year 10 students to feel more in charge of their learning, and I planned that my marking would be the means by which I would try this.

Strategies tried and an evaluation of their efficacy

I decided that throughout the year I would try different strategies to re-engage some year 10 students. In fact, these evolved and developed as I got to know my students better, and as I learned what did and didn’t work. Here are the three most prominent techniques I tried:

  1. Highlighting WWW (a department initiative) and providing a code based on a clear success criteria.
  2. Providing regular feedback (in lesson and through marking).
  3. Giving clear tasks or questions of what I wanted them to do to improve, and using this as a form of differentiation.

I soon realised that the first strategy did not have the intended effect I had hoped for. My hope was that if the students could see precisely where they had got it right, they would be clear as to how they could improve. However, I made the common mistake of trying to provide both summative and formative feedback, resulting too often in a disengaged attitude towards marking. The boys who I was targeting also did not appreciate that they had to copy down the highlighted code from the Interactive White Board, and this actually caused a lot more hassle than it was worth. This is an example of a disengaged student’s response to my marking:

aw-ys1

And so I realised that the ‘little and often’ approach was necessary for these boys, and I endeavoured to provide them with feedback as much as I could. This was both throughout the lesson and after during marking time. I wanted them to recognise that I valued their written work, especially extended responses and I tried to mark their work as quickly as possible. This definitely helped to improve engagement, and a more positive relationship was created based on their work.

The final strategy that I developed (born out of the failure and success of the previous two) was to provide very specific feedback on what exactly needed to be done to improve. I realised that these boys needed to feel a sense of success, and it was only once I had really got to know them that I could do this accurately. I learned that my marking needed to be a balance of stretching these students enough, without making them feel like they couldn’t do the work. This really helped me to form relationships as they became much more engaged in both the lesson and their own progress. Here are two examples of particularly great work:

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Conclusion

The most valuable element of this research for me was to fully recognise the power of feedback in establishing high expectations and good progress. It was through developing good relationships with these students that I was able to understand their strengths and areas for development, and I then used this knowledge to inform my marking. By allowing these students to feel like they can succeed in English, I believe that their engagement in the subject has improved.

References

Bentley-Davies, C. (2010) How to be an Amazing Teacher. Carmarthen, Wales: Crown House Publishing.

Didau, D. (2015) The Learning Spy. Website: http://www.learningspy.co.uk/

Hattie, J and Timperley, H. (2007) The Power of Feedback. Review of Educational Research 2007 77: 81

National Literacy Trust (2012) Boys’ Reading Commission. All-Party Parliamentary Literacy Group Commission.

Williams, D. (2014) Formative assessment and contingency in the regulation of learning processes. Institute of Education, University of London.

Stretching and Challenging Pupils

As teachers, we know it is not enough for our pupils to coast through lessons. We want the very best for all of our pupils. We want them to love learning, to be stimulated by the lessons we teach and to develop intellectually.  So it is vital that we strive to stretch and challenge them.  This is true for all but of particular importance for more able pupils.

Ideally, we should be aiming to deliver lessons that take pupils just beyond the point they have already reached – something just at the edge of their capabilities. This idea comes from Lev Vygotsky’s notion of the zone of proximal development (Vygotsky’s Zone of Proximal Development: https://www.youtube.com/watch?v=0BX2ynEqLL41). This is the essence of education, where pupils are put in a position that enables them to move beyond their existing knowledge and understanding. This is the point at which challenge excites interest without the difficulty of a task overwhelming a willingness to engage.

Never underestimate the importance of having high expectations of all pupils. We know that every pupil can make progress, given the right set of circumstances (including a great teacher) but equally we must always seek to ensure that our most able pupils are being stretched.

There are a number of ways in which we can convey high expectations and raise the level of challenge in our lessons:

Reasoning

When pupils share their opinion during discussion, push them to explain what underpins that opinion. Do not let unsupported assertions escape without asking “Why?”, “What reasons do you have for thinking that?” You can even train your pupils to start asking these questions of each other.

Product

Plan your lessons so that pupils are building up to creating some kind of product – an essay, perhaps, an extended piece of drama or a presentation. This will imbue your lessons with purpose and show them that you believe they can create significant pieces of work.

Success criteria

Make sure that every pupil knows what the success criteria are for each major piece of work you ask them to do, and that it is possible for everybody to attain them. In a positive classroom, pupils will be more likely to push themselves to excel.

Socratic questioning

Socrates, an Athenian philosopher born in 469BC, appears in the dialogues of ­ Plato interrogating his fellow citizens to draw out the assumptions, errors and misconceptions in their thinking. We can use Socrates’ methods in the classroom to challenge the thinking of all pupils – particularly the most able.

There are four roles that Socrates takes on when asking questions: the gadfly, the stingray, the midwife and the ignoramus. Teachers can move between these in order to question the views, opinions and judgements held by more able pupils.

  • The gadfly: Mimic the practice of the gadfly, which nips away at larger animals. This involves asking lots of little questions intended to push thinking and avoid sloppiness: “What do you mean by that?”; “But, what if…?”; “What evidence do you have?”; “Does that always apply?”; “How can you be certain that is true?”
  • The stingray: Administer a shock to ­pupils’ traditional way of thinking in the same way a stingray unleashes its sting: “Imagine if X was not the case, what then?”; “What if everything you’ve said was turned on its head?”; “What if a great change happened?”
  • The midwife: Ask questions that help give birth to ideas: “That’s an interesting idea; could you explain it a bit more?”; “How might that affect things?”; “What made you think of that idea?”
  • The ignoramus: Emulate a character who has never encountered the topic you are discussing and play dumb to encourage explanation: “What does that mean?”; “I don’t understand – can you start from the beginning?”; “So, do you mean that…?”
  •  

Evaluation:

This is all about making judgements: “What do you think and why?”; “Is this better than that, or vice versa?”; “Which option should we go for and what reasons do you have to support your choice?”

Really good evaluation demonstrates a mastery of the topic. Pupils will be able to highlight the strengths and limitations of the issue before making a judgement about what ought to be done or what they believe is the best perspective on an issue.

Nearly every activity you undertake in the classroom can be supplemented by an evaluation task, directly or tangentially associated with the topic.

Use evaluation command words – appraise, argue, assess, critique, defend, evaluate, judge, justify and value – to frame questions and tasks for pupils who finish ahead of their peers.

2Mike Gershon, Elongate their learning: http://mikegershon.com/elongate-their-learning/

Working at the next level up:

Choosing a task from a higher level of study can be a productive way to both stretch and motivate more able pupils. Choosing a GCSE question for a Year 9 class to answer or giving an A level question to the most able Year 11 pupils can be an effective strategy.  It may be appropriate to provide support with critical research, resources and planning for the task but as a means of getting the best from a more able pupils and stimulating interest in higher level study in your subject it has a lot to commend it.

To achieve all of the above we need practical ideas that can be incorporated into our lessons, whether as planned activities or as strategies that we can adopt mid-lesson to provide challenge when and where it is needed.

The following are a range of such ideas:

Starters

  • Get an able pupil(s) to recap on the previous lesson’s learning
  • If your starter activity requires students to find examples set able pupils a higher target e.g. level 5 students find 5 examples, level 7 students find 10 examples
  • Learning logs can be used with all pupils as a plenary or a starter (to review the previous lesson)

Main tasks

  • If you are taking feedback during the lesson, enlist an able pupil to record the ideas on the board
  • Get more able pupils to TEACH their peers
  • Use an able pupil(s) to model their writing or thinking, by explaining their answer/ solution to a task to a neighbour
  • Ask the school librarian to produce a reading list of texts and electronic resources to encourage wider reading or research around a class topic
  • Set an independent task, such as a further investigation in maths or science, or a different class reader from a selected list and invite pupils to decide how they would like to demonstrate their learning to you or the rest of the class after an agreed length of time
  • Use GCSE questions with able Year 9 pupils
  • Provide opportunities for pupils to respond in ways other than writing: display work, role play, short video films etc.
  • Remember that ‘less is more’ is some cases. Prescribe the number of words to be used to make more able students think hard before they write, and make every word count
  • Ascribe the role of chair-person or lead-learner to able students who will then take on the mantle of responsibility and help maintain momentum and focus during tasks
  • Get able pupils to summarise instructions to the whole class
  • All pupils have a red and green card. If they do not understand the content of the lesson or want the teacher to slow down they show their red card, if they are following the lesson they show their green card. If you see a red card you stop the lesson to find out what the problem is. You then select a student who is showing a green card to respond to the issue that has been raised. Plan your groups carefully. Sometimes able students learn most productively together, sharing and extending their more developed thinking; sometimes it is helpful to allow them to advise a less able pupil and have to work harder to successfully articulate their ideas
  • Often, questions arise in the classroom that cannot be answered straight away. These can be valuable learning opportunities, yet are often not revisited. Children could be encouraged to write these on a sticky note and put them onto a laminated picture of a light bulb in your room. At a spare moment in your lesson, or at home, able students can be asked to research the question and report back to the rest of the class. This helps to create an atmosphere of self-motivation and self-challenge

Plenaries

  • Alert an able pupil(s) at the start of the lesson to be ready to summarise the content of the lesson at the end
  • Ask an able pupil(s) to come up with questions at the end of the lesson to test other pupil’s understanding of the lesson
  • At the end of the lesson pose a question which is based on next lesson’s work. All students have to write a response to the question on a piece of A6 card and hand it in on the way out. You use the pupils’ responses to inform your next lesson and as a result of this give able students a more demanding task

References and Links

1Vygotsky’s Zone of Proximal Development: https://www.youtube.com/watch?v=0BX2ynEqLL4

2Mike Gershon, Elongate their learning: http://mikegershon.com/elongate-their-learning/