Mastery in Mathematics (6): Research and lesson adaption to fit the new GCSE curriculum

An Action Research project by Rory McMahon (Mathematics)

Aims of the Project

The aim of this project was to research ‘Mastery in Mathematics’ and the implications its’ introduction would have on our Faculty in terms of:

  • The new AQA Curriculum
  • Adjustments to the Scheme of Work
  • Alterations to lessons to promote ‘Mastery’

Background and context

This project started in response to the recent changes to the Maths curriculum which take effect from the 2017 GCSE’s. As a Faculty we looked to change our practice in light of the recent changes. The curriculum changes are as follows:

  • There is more content to teach with harder topics being introduced.
  • There is a greater emphasis on problem-solving and mathematical reasoning, with more marks in the GCSE exams being allocated to these higher-order skills.
  • The total examination time is increasing with all exams taken at the end of the course.
  • Students will also have to memorise formulae.
  • There is a new grade structure from 9 to 1, with fewer marks at the lower grades and more marks at the higher grades.

Actions taken

Peer observations to gauge the level of Mastery evident in lessons in September/October

As a Faculty all teachers took part in peer observations during Term 1 in an attempt to see good practice in action as well as gauge the level of ‘Mastery’ evident in existing lessons. Positive and constructive feedback was given and a discussion on how ‘Mastery’ could become more visible in lessons was held during Faculty meetings.

Scheme of Work changed from Kangaroo to AQA

The decision was made in January to make the switch from the Kangaroo scheme of work to the new AQA scheme to attempt to get pupils used to the new format in time for the start of the 2016-2017 academic year. Although it was thought to be a better move in the long run, there were some challenges to this approach. Firstly, a comparison of the schemes had to be made and topics which were covered already had to be crossed off.  However with the level of many topics increased, we needed to pick out sections of topics which students had not been previously been exposed to and teach those separately. Secondly, the increased difficulty of concepts and the change in focus to ‘Mastery’ proved to be difficult for students to adjust to. We were hoping they would adapt quickly to the problem solving nature of lessons as this was a style which they had not been previously used to.

Adaption of End of unit tests to support Mastery

End of Unit Tests now include Mastery style questions to build up resilience and retests are available and encouraged, so that students now have the key skills needed to succeed at this form of questioning. This is a work in progress which has been embraced by the pupils as they can see progression from the first sitting of the test to the second. It also gives them more opportunity to sample the type of examination questions they will be expected to answer in the coming years.

Further peer observations planned to see how Mastery is developing and lesson adjustments

Again in Term 3/4 the Maths Faculty undertook peer observations to observe the increase in focus towards ‘Mastery’ in lessons as standard practice. The Faculty was unanimous in the conclusion that Mastery questions were most easily integrated into the bell-work phase of the lesson or alternatively and possibly most effectively, during the Plenary phase. Personally, I found giving the students a ‘Mastery’ question as their plenary always challenged the pupils to think about the skills they had learnt in that lesson in a different way. Once the students spotted this they began to widen their horizons in terms of spotting links between different concepts learned. Some examples of Lesson alterations can be seen below.

Example 1

Our pupils in this case would have spent the majority of the lesson learning about the sum of the interior angles of polygons. In this question, they have to apply that knowledge but also represent their answers as fractions in their simplest form.

interior angles

Example 2

Factorising 1

A standard lesson on Factorising Expressions would concentrate on embedding the relevant skills needed as above. However, the Plenary to this lesson looks like the following slide below.

Factorising

The students are encouraged to use a skill learned in the lesson to solve a different style of problem, thus establishing links between different concepts.

 Adoption of Eastern Asian styles of teaching (learning information)

 It is widely recognised that the countries of Eastern Asia out-perform their UK counterparts in relation to attainment of Mathematics in primary and secondary schools. International tests show that in these countries the percentage of 15-year-olds who are functionally innumerate – unable to perform basic calculations – was more than 10 percentage points lower than in England. As recently as 12/07/2016, news broke of a £41m support for 8,000 primary schools in England to adopt the approach which is used by the leading performers in Shanghai, Singapore and Hong Kong.

The Eastern Asian method has the following features:

  • Emphasis on problem solving and comprehension, allowing students to relate what they learn and to connect knowledge
  • Careful scaffolding of core competencies of :
    • visualisation, as a platform for comprehension
    • mental strategies, to develop decision making abilities
    • pattern recognition, to support the ability to make connections and generalise
  • Emphasis on the foundations for learning and not on the content itself so students learn to think mathematically as opposed to merely reciting formulas or procedures.

As a Faculty we have tried to integrate the techniques of embedding skills in the minds of our students and then getting them to apply these skills to problems. Previous lessons would consist of teaching skills and then getting pupils to practice these skills for the remainder of the lesson. Now, our attention has changed to using and applying these skills to problem solving for real-life situations. 

On-going adaption of the Scheme of Work to include NRICH activities to further develop Mastery 

Before the focus on ‘Mastery’, the Maths Faculty always felt that problem solving was a crucial attribute for students to develop. This was enhanced by our used of ‘The Nrich Project’ from the University of Cambridge.

“NRICH is a team of qualified teachers who are also practitioners in RICH mathematical thinking. This unique blend means that NRICH is ideally placed to offer advice and support to both learners and teachers of mathematics.”

NRICH aims to:

  • Enrich the experience of the mathematics curriculum for all learners
  • Offer challenging and engaging activities
  • Develop mathematical thinking and problem-solving skills
  • Show rich mathematics in meaningful contexts
  • Work in partnership with teachers, schools and other educational settings

For teachers of mathematics, NRICH:

  • Offer free enrichment material (Problems, Articles and Games) for all ages that really can help to inspire and engage learners and embed RICH tasks into everyday practice.
  • Help to promote RICH thinking in classrooms by offering on-line and face-to-face support at Primary and Secondary level.
  • Deliver professional development courses and workshops in rich mathematics.
  • Help teachers to think strategically about ‘next steps’ and progression in problem solving.

In 2014-2015 ‘NRICH lessons’ were held once per term to help enhance the problem solving skills of students. In 2015-2016 it was felt that the Faculty should conduct NRICH lessons once per fortnight as the shift in focus was becoming apparent at that stage. Moving forward, the Maths Faculty has created a bank of NRICH lessons to be used in conjunction with the new Scheme of Work for the academic year 2016-2017. Some snapshots of how these were integrated can be seen below.

sow-1.png

sow-2.png

Impact

As a Faculty, we have discussed the possible impact of our endeavours to adjust our teaching and learning to the new and challenging ‘Mastery’ curriculum. As this style of teaching and type of examination questions have been rolled out, students have become more familiar with the concept. Therefore, we can say there has been definite progress in the students’ familiarity with the style of future exam questions.

Secondly, we can state that the confidence of our pupils has increased with regard to structuring an answer for these questions. At the beginning of the year, receiving answers from students for bellwork and plenary ‘Mastery’ questions was a difficult ordeal! Gradually through practice and knowing they should be able to use some of the content they had covered in lessons, many were then able to attempt a reasonable answer. This developed over time so now we not only have our highest attaining students putting answers together but our bottom sets are also successful.

Finally, the AQA practice papers were an invaluable resource. As with the previous strategies, students found the change in structure and expectations very difficult to deal with. Therefore, we gave students the practice paper to attempt and gave them a grade. Once the papers were handed back, students could then go through the mark scheme with green pens to see where they could have picked up more marks. Also, answers that had four, five or even six steps were often broken down by the teachers for the class. Students then had the opportunity to re-sit the examination as a confidence building exercise. Slowly but surely the results for the first sitting of the tests began to improve but as a Faculty we realise this is a work in progress.

 Conclusions

  • The new AQA Curriculum has been rolled out and used for six months this academic year (2015-16) allowing teachers the opportunity to familiarise themselves with the format and tests.
  • The new Scheme of Work has been adjusted to accommodate ‘NRICH’ lessons which we see as crucial to embedding a culture of problem solving across the department.
  • New lessons have been created and existing lessons have been amended to include ‘Mastery’ questions in the bellwork or plenary phases.
  • There is a confidence in the Faculty that we are ready to begin the 2016/2017 secure in our knowledge of the new requirements to ensure the continued progress of pupils in the Mathematics Faculty.

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.

Featured image: original image ‘Map of Mathematics Poster’ by Dominic Wallman, licensed under CC BY-NC-ND 4.0 https://www.flickr.com/photos/95869671@N08/32264483720

 

 

Advertisements

Silent Conversations

A ‘Sharing best practice’ post by Jodie Johnson (Mathematics)

“Shhh! We’re going to have a silent conversation…”

An unusual instruction to a class but one that can help to focus thinking and forge collaboration amongst pupils.  How?  Well listen in…

Working in pairs, the class are given a series of questions of varying levels of difficulty.  Their challenge is to answer the questions in silence.  Partners can ‘ask’ each other as many questions as they like, as long as they do so in writing.  At the end of the activity pairs can then demonstrate to their peers or to the class, how they would solve the problem…in silence just like they will have to do in an exam!

By taking it in turns to solve each step of the problem everybody is engaged and by being allowed to ‘ask’ questions they can help each other get ‘unstuck’ when necessary.  The focus on the written demonstration of the solution helps cement the process needed to reach the solution.

Here’s an example of some worked solutions shared (in silence) by pupils with the rest of the class:

silent-conversations

Featured image:  ‘Silence’ (original image) by Alberto Ortiz on http://www.Flickr.com (license CC-BY-NC-ND 2.0)

Building Resilience in Students

An Action Research project by Ursilla Brown (Science)

[Featured image: ‘Resilience by Ron Mader’- ShareAlike 2.0 Generic (CC BY-SA 2.0) ]

Focus

This action research focused on the concept of resilience and how it impacts on learning among our students.

Background

Throughout my teaching career, the link between work ethic and success in students has been obvious. What is less transparent are the factors that lead some students to relish diving into a problem and being prepared to take the risk of charting unknown territory while others desperately cling to the edge, afraid to take the plunge. This fear can manifest itself in a multitude of ways. While some students are absorbed in the challenge of cracking a code or finding connections, reasons for or ‘what if’s’, those on the periphery of learning can be sitting passively, getting distressed, engaging in off task behaviour or defiantly declaring that the content is boring or pointless. With a critical mass of students in the latter category the teacher invariably works much harder than these students as she guides, cajoles, pleads and, yes, even sometimes threatens detentions for lack of effort. So, while the issue has been of long term interest to me, the catalyst to embark on a journey of discovery was the coincidence of the launch of this Learning Focus cycle of research in school with my first experiences of my Year 10 GCSE Chemistry class. Since the beginning I feel I have had a good relationship with the students. They are a friendly bunch and came to me as a class seemingly happy to be in the room but mainly passive and pretty hard to strike up a dialogue with about anything to do with Chemistry. My lesson starters engaged around half the class while the others sat in a frozen position, not doing anything wrong, but not learning or seeming to engage with the activity. My mission was to shake them out of their lethargy and take charge of themselves as learners.

Objectives

My aim was to cultivate resilience amongst the students. The success criteria for this were to get the students:

  • To be able to concentrate for long/longer periods of time. (not give up)
  • To be able to control their thoughts and emotions
  • To enjoy challenge and problem solving
  • To see failures/mistakes as part of the learning process and be prepared to have a go
  • To show initiative when ‘stuck’
  • To recognise that learning is a process and takes time

Context

The class was a middle mixed ability class. I teach them the Chemistry component of the Science GCSE.

These were my thoughts about the class at the beginning of the year:

  • Lovely class – friendly, polite but quite passive
  • Majority of ‘resilient’ students quiet and self-contained so maybe not obviously modelling to others
  • Happy to listen to instructions but want to be ‘spoon fed’
  • Not really making the connection between effort and achievement
  • I was working harder than them – re-directing, re-assuring, checking, cajoling in some cases
  • Many students would give up if they did not already know the answer

Actions

  • De-mistifying ‘being clever’. At every opportunity reinforcing to the students how the brain works and how we learn. I have explained to them and continuously remind the students how we commit information to long term memory and used two examples to unpick ‘being clever’ :
  1. How amazing we all are at speaking our own language compared to how
  2. challenging we find it to learn a new language in school. The students can see the clear link between mastery and frequent repetition, often getting things wrong initially.
  3. Me as a teacher – I reminded them why I appear to be so effortlessly good at what I teach and discuss the fact that I am immersed in it, teaching it many, many times. The reason I am an ‘expert’ is that I teach the subject matter often so my neural networks are well developed FOR MY SUBJECT MATTER
  • Resilience poster – This has become a whole school tool and it reflects the effort that is put into becoming an effective learner. I continue to refer to the iceberg at every opportunity.
  • During Directed Improvement and Response Time (DIRT – time dedicated to allowing pupils to respond to teacher feedback/making to correct, develop or improve their work) taking the opportunity to Facilitate reflection on progress and relating it to effort
  • Linking to Science of the brain – unpicking the reasons for repetition and consolidation for mastery with reference to my above examples or other skills and aptitudes. I have a visual representation of the neurone connections in the brain that I refer to when reminding the students of why practice is important and why things seem hard at first.
  • ALWAYS praising effort not achievement and linking this to life skills
  • Seating resilient students with less resilient ones and encouraging a climate of mutual support where students can move around when appropriate and support one another in their learning.
  • Liberating students from the fear of committing mistakes to paper by allowing them to write on the desks. This seems to be very effective at getting some students to take the plunge and ‘have a go’.
  • Avoid re-assuring answers to questions – reflecting back to students.
  • Scaffolding resilience training by having selected differentiated resources available to enable students to help themselves to become unstuck (Links well with SOLO)

Impact

The last column shows the actual results achieved in the GCSE. Bearing in mind the target grades are actually for Year 11, the majority of students made expected progress. It is hard to say how much is attributable to the emphasis on resilience but, anecdotally, the vast majority of the students are focused and open to giving the challenging Additional Chemistry content their best shot and, importantly, bouncing back and returning to the drawing board when they get things wrong. The pupils highlighted in red were ones I was still concerned about the level of commitment from at the time of preparing to share my findings with colleagues in our learning focus group meetings but subsequently the majority of these have sought out advice from their peers or myself to help them progress.

ub-stats

Conclusions

To summarise the findings of the ’Developing resilience’ Learning Focus group of which my research was a part:

  • We believe our strategies have made a difference but……it would be more powerful if the language of resilience was consistent across the school
  • This approach supports stretch and challenge you have higher expectations and avoid ‘helicopter’ teaching
  • This work supports pupil independence and less teacher dependence
  • Rewarding attitude and effort is crucial – sending the right messages about what we value

Next steps

I will continue to employ these strategies with the students I teach. I will continue to focus on resilience development in the next round of Action research and explore ways of embedding the language of resilience across the school.

Sources/references

‘Mindset’ by Dr Carol S Dweck

Lesson Plans for teaching resilience to Children by Lynne Namka

Promoting resilience in the classroom by Carmel Cefai

The Iceberg Illusion poster by Sylvia Duckworth

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

jj21

 

Plenary 2

 

jj22

Full lesson

jj-1

jj-2

jj-3

jj-4

jj-5

jj-6

jj-7

jj-8

jj-9

Plenary

 

jj24

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.

Developing Mastery in Mathematics (1)

(Featured image: ‘Multiplication sentence written in multiples of three’ http://www.freeimages.co.uk/)

Mastery in Maths: Research and lesson adaptation to fit the new criteria in Maths

An Action Research project by Clare Mondair

Aims

The aim of this investigation was to explore aspects of Mastery in Maths to improve my own understanding of what Mastery actually means and what it would mean for the students in my lessons. In addition, my aim was to change my own teaching where necessary in order to best help the students in my classes achieve of their best. As a Faculty we aimed to work together to develop lessons that would contain a ‘Mastery’ element as well as developing resources to add to the new Scheme of Work which was quite thin on the kind of Mastery resources required.

Literature Review

Although there are many differences between the education systems in England and Eastern Asia, the ‘mastery’ approach to teaching commonly followed in these countries can teach us much.

According to the National Centre for Excellence in the Teaching of Mathematics. (October 2014), the main principles and features characterised by this approach are that:

  • Teachers reinforce an expectation that all pupils are capable of achieving high standards in mathematics.
  • The large majority of pupils progress through the curriculum content at the same pace.
  • Differentiation is achieved by emphasising deep knowledge and through individual support and intervention.
  • Teaching is underpinned by methodical curriculum design and supported by carefully crafted lessons and resources to foster deep conceptual and procedural knowledge.
  • Practice and consolidation play a central role. Carefully designed variation within this builds fluency and understanding of underlying mathematical concepts in tandem.
  • Teachers use precise questioning in class to test conceptual and procedural knowledge, and assess pupils regularly to identify those requiring intervention so that all pupils keep up.

The intention of these approaches is to provide all children with full access to the curriculum, enabling them to achieve confidence and competence – ‘mastery’ – in mathematics, rather than many failing to develop the maths skills they need for the future. In addition, it has been recognised that for many schools and teachers the shift to this ‘mastery curriculum’ will be a significant one requiring new approaches to lesson design, teaching, use of resources and support of students. It also focuses on giving students the skills they need to make sensible choices and to use their knowledge to tackle problems in all sorts of situations. It also aims to develop their resilience for when the road gets tough.

There is a real need for a balanced approach here.  Of course having key facts at your disposal is very helpful when it comes to solving problems, especially when in an unfamiliar context but the need to be flexible and adaptable is too. Also helpful is being able to use what you do know to get you facts you don’t know.  For some students when learning their tables it can take a while to get 11 x 12 to “stick”, but if they are confident with 10 x 12 = 120 then they know how to get to the answer pretty quickly by adding on another 12.  Therefore, it is their understanding of the structure and not just my knowledge of the facts that helps out in tables tests. This is what sets students out on the road to mastery of the times tables!

Mastery of the Mathematics curriculum encourages ‘intelligent practice’ to enable students to develop conceptual understanding alongside procedural fluency.  It is important to use multiple representations to support this understanding and to encourage students’ reasoning.  Students are also encouraged to solve problems from the very start of the curriculum journey, not seeing it as some ‘add on’ that can only be considered when all the facts are in place.  The challenge is developing these skills and understanding alongside mastering different aspects of the Mathematics curriculum.

Asking students to think of more than one way to answer a question not only forces them to think more laterally but it also allows discussion of methods of true ’mastery’. These types of tasks enable students to:

  • Develop mathematical language.
  • Articulate their reasoning.
  • Share ideas on approaches to problem solving.
  • Grow in confidence when discussing ideas.

The key to this success is strong peer support which must be built up over time. In addition a good pedagogic tool to use in mathematical problem solving is instead of finding one way to solve a problem find three ways. Working in pairs is key to problem solving tasks as students come up with different ways of starting and after establishing one solution they are able to share alternative ideas. A core value of ‘mastery’ is partnership, listening to each other and showing respect for different views and ideas. Allowing students to explain their thinking, asking for and giving support and encouraging feedback is very important to establish and maintain mastery of mathematics.

Research Methods

Having researched and established as a Faculty what ‘mastery’ actually meant in maths we had to think about the way in which it would be developed. An agreement was made that we had already been attempting to develop mastery in terms of problem solving over the last year but lessons were intermittent and often lost out on due to other factors of school life.

An agreement was made that we would continue to develop lessons as normal but that a majority of our lessons would now contain an aspect of mastery. Nrich lessons would continue but would be incorporated in to the new Scheme of Work where possible (Nrich is a website created by Cambridge University which has open ended questions and what we now recognise as ‘Mastery challenges’). In addition, teachers in the Faculty would make lessons that they had produced available for the whole team and we would observe mastery lessons being delivered so that we would have a good understanding of what it ‘looked’ like. This would enable consistency throughout the Faculty. Many lessons would contain the format below so the students would know that was Nrich.

nrich-cm

The lesson would then progress through a series of steps so that there was enough challenge for everyone. Students were encouraged to work at their own level, however they were also encouraged to go for the higher level challenge where possible. This would result in those students who were less able working alongside those who already had a good ‘mastery’ of mathematics. The advantage of this system meant that students were working with and supporting peers and learning from one another in their language and at their own level.

find-a-whole-cm

The above would not apply to Year 11, but all other year groups should benefit from this thinking. It would of course be especially important for the Year 10 classes as they would be sitting a new exam which would call for resilience as well as thinking outside of the box in order to tackle some of the new material.

Main Findings

Asking students to think of more than one way to answer a question not only forces them to think more laterally but it also allows discussion of methods of true ’mastery’. These types of tasks enable students to:

  • Develop mathematical language.
  • Articulate their reasoning.
  • Share ideas on approaches to problem solving.
  • Grow in confidence when discussing ideas.

The key to this success is strong peer support which must be built up over time. In addition a good pedagogic tool to use in mathematical problem solving is instead of finding one way to solve a problem find three ways. Working in pairs is key to problem solving tasks as students come up with different ways of starting and after establishing one solution they are able to share alternative ideas. A core value of ‘mastery’ is partnership, listening to each other and showing respect for different views and ideas. Allowing students to explain their thinking, asking for and giving support and encouraging feedback is very important to establish and maintain mastery of mathematics.

Using Nrich we were able to appreciate that the current mastery approach encompasses two key aspects of mathematical learning, conceptual understanding and procedural fluency, which we agree are essential for nurturing young mathematicians. In addition there are five aspects of being able to be a Master at Maths, conceptual understanding; procedural fluency; strategic competence; adaptive reasoning and productive disposition (Kilpatrick, Stafford & Findell, 2001).

  • Much of the curriculum has been moved from higher levels to lower levels resulting in students now being expected to suddenly achieve at a much higher level than previously expected.
  • Many of our students do not have the basic mathematical fluency or reasoning skills in order to access much of the new curriculum.
  • Resilience in students is key in helping to ensure that students stay on track and improve.
  • Peer support and discussion is vital if students are to succeed in mastering some of the problem solving activities and questions which will come with the new curriculum.
  • Nrich allows students to explore Mathematics in a safe environment where they don’t feel threatened by their lack of basic knowledge.

Discussion and Conclusion

Mastery can only be developed over time and is unlikely to have much impact for the first two years of the new curriculum changes. The current difficulty we face is the fact that our Ks4 students have not been brought up with this habit of mastering Mathematics and it is therefore difficult to develop these skills and follow a Scheme of Work designed for a new exam which is already challenging to our average ability and less able students.

The mastery of Mathematics is however, being thoroughly embedded in the curriculum where possible for the Ks3 students and the impact of this should be felt when the current Year 9 group begin the GCSE course.

Mastery in Mathematics will enable students to articulate their ideas, build resilience, build mathematical fluency and think about problems from a different angle which in turn should have an impact on many aspects of life as well as Mathematics.

References

Drury, H. (2014) Mastering Mathematics. Oxford University Press, pp8.

Mathematics Learning Study Committee. Adding it up: Helping children learn mathematics. National Academies Press, 2001.

National Centre for Excellence in the Teaching of Mathematics. October 2014.

Williams, H. (2014) Approach, Research. Mathematics Mastery Acting Director of Primary.