Mini-interventions

A ‘Sharing best practice’ post by Richard Noibi (Mathematics)

Reading time: 2 minutes

When we speak of interventions in education, particularly in the areas of literacy and numeracy, we typically think of large scale initiatives.  Schemes that might run across a whole school.  Interventions that have been meticulously planned with supporting documentation, layers of responsibility and financial accountability.  These are important but for many  pupils, it can often be the small-scale interventions teachers make, that can have the greatest impact in overcoming a barrier to learning in a particular lesson.

One example I use is the ‘mini-intervention’.  This is a way of supporting a pupil who has missed a lesson or not understood a key step in their learning.

Here’s how it works in my maths lessons.

Step 1

At the start of each lesson I give my class a bell-work/starter activity to get their mathematical brains warmed up.  This might reinforce the learning from recent lessons, give them a chance to demonstrate their mastery of an aspects of maths, or get them engaged with a new area of study.

Starter

Figure 1. An example of a starter activity

Step 2

While the majority of the class are working on the starter I will sit down with a pupil who was absent for the last lesson and go over the work we have covered in a 1:1 ‘mini- intervention’ and using a block of post-it notes to provide a brief explanation and  summary of the key learning points they have missed.

Fig 1

Figure 2. An example of some ‘mini-intervention’ post-it notes on trigonometric ratios given while the rest of the class work on their starter problem

Step 3

The pupil now has a greater chance of succeeding with the lesson ahead.  They can still seek support but they have enough information to make a start on the work set for them and often this is enough to let them catch up with the rest of the class.

Fig 2

Figure 3. Work on  multi-step trigonometric ratio problems completed by the pupil who received a ‘mini-intervention’ in figure 2. who has caught up with the rest of the class

By providing a pupil with a  post-it intervention they have a reference point to help them tackle the work, rather than having to repeatedly seek help once the main part of the lesson has started.  This helps them to be more independent in catching up with the rest of the class and allows me to focus my attention on the needs of other pupils in the class.

Fig 3

Figure 4. An example of a ‘mini-intervention’ post-it on the transformation of functions

Fig 4

Figure 5. The work completed independently by the pupil in figure 4 following their min-intervention

Why not try some mini-interventions yourself?

Featured image: ‘post its/ideas’ by B-G on Pixabay.  Licensed under CC0 Public Domain

 

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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

 

 

The Awkward Mole

A sharing best practice post by Jodie Johnson (Mathematics)

This activity sharpens up pupils’ ability to precisely follow a particular process to complete a specific task.  These examples come from Maths but they could apply equally well to any process in any subject.  For instance, ‘constructing a perpendicular bisector on a line’, ‘bisecting an angle’, ‘drawing an equilateral triangle’ etc., etc.

awkard-mole

Step 1: Pupils A and B sit back to back with Pupil A facing the teacher/board with an incomplete worksheet (see above)

Step 2: The teacher silently demonstrates the process to complete a task on the board.  Pupil A copies the teacher’s demonstration onto their worksheet.

Step 3:  Without changing position Pupil A now explains to Pupil B how to complete the process on their worksheet by giving clear verbal instructions (they are not allowed to look at what Pupil B is doing)

Step 4: Pupil A and B look at the results and discuss the instructions given (were they specific?, were they clear?, how could they be more precise? how could they be improved), in order to refine and perfect them.

Step 5: (Here is where the ‘awkward mole’ comes in!)  You now invite a ‘random’ pupil to come up to the front and follow the instructions they are given by another member of the class to demonstrate how to complete the process in front of the class.  Unknown to the rest of the class you have primed the ‘random pupil’ to be your ‘awkward mole’ and instructed them to be as awkward as possible when following the other pupil’s instructions – to take instructions literally, to deliberately ‘misunderstand’ ambiguous instructions and so on.  The onus is then on the pupil giving the instructions to refine their thinking and instructions until they succeed in getting the mole to ‘get it right’!

In one case a pupil instructed the mole to ‘draw an arc’, so that’s what he did with Noah and the animals too!

You can prime more than one pupil to be your mole in the lesson and don’t forget to reverse the roles for pupils A and B so they both get a turn.

Featured image: Mick E. Talbot, Mr Mole, CC BY-SA 3.0

 

Mastery in Mathematics (5)

An Action Research Project by Elizabeth Drewitt (Mathematics)

Focus

In this report I aim to share how our departmental research into Mastery in Mathematics has impacted on the students I teach.

Context

There is no argument to the value of mastery as a life skill:

Director Dr Helen Drury says, “In mathematics, you know you’ve mastered something when you can apply it to a totally new problem in an unfamiliar situation”¹.

What better way to prepare our students for life after school than to give them the confidence to approach new situations and problems with confidence.

Mastery enables students to:

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

I decided to focus on the techniques we can use as teachers to get our students ready for the road to mastery.

All teachers have experienced that:

‘Students learn better when they are curious, thoughtful, determined and collaborative.’ (Nrich)

We spend vast amounts of energy nurturing these traits within our classes. But for some students, the experience of failure or fear of failure shuts down any chance of curiosity. Expecting failure often means students cannot even consider an alternative outcome and therefore determination, thought and collaboration are pointless and avoided. It is a self-fulfilling prophecy. I feel this is particularly the case in mathematics, often voiced by parents at Parents’ Evening, ‘I can’t do maths’. Here I propose that maths is not something that ‘can be done’ or ‘cannot be done’. I would like to challenge the parents as to whether they know their times tables or not. It is highly likely the case that it is not maths these parents struggled with but their times tables. They did not have the basic tools to face the rest of the subject with and so encountered difficulty at every turn. I believe that for many, it was not the PROCESS of expanding brackets that caused a problem but actually the MULTIPLYING.

DEVELOPING BASIC MATHEMATICAL SKILLS

The importance of times tables within the mathematics curriculum cannot be underestimated, yet the importance of learning times tables is still under debate amongst professionals:

Jo Boaler argued that the UK Government position, that every child must memorise their times tables up to 12 x 12 by age nine, is ‘absolutely disastrous’. In contrast, Charlie Stripp stated knowing the times tables supports mathematical learning and understanding.

“Here at Mathematics Mastery, we believe children who have a strong grasp of their times tables are more confident when learning new mathematical concepts and, importantly, enjoy the subject more.” But note here I’ve said ‘strong grasp’ and not simply ‘memorised’.

Here I put forward the view that the road to mastery must start with each student being equipped with a tool box and in that tool box must be curiosity, resilience and……times tables!

Some students cannot/have not/will not memorise the times tables. Some think that if they do not know the answer then that’s the end of that. Full stop! If we do not give these students tools and tricks to work out the answer then we are closing the door on Mastery, opportunity and the growth of a learning identity.

SOLUTION

Never accept I don’t know my times tables. WORK it out. ‘Not knowing’ does not equate to ‘can’t find out’.

TRICKS

Explicitly TEACH how to work them out.

  • count up in two’s on your fingers,
  • count sticks/dots,
  • write out the times tables each time,
  • use your fingers for the 9 times table.
  • Do 10 x {?) then add 2x [?] for the 12 times tables.

ACTIONS

  • Time must be dedicated to times tables each week if we are to provide each student with a fully operational toolbox.
  • Bell work: fill in 5 x 5 times tables grids with random numbers. Students self-differentiate by choosing different coloured grids that represent basic times tables, reverse times tables, lots of mystery headings, larger numbers, decimal numbers.
  • For KS3 or lower ability classes, 10 minute multiplication and division challenges, results recorded and tracked.

RESULTS

  • Practice makes perfect, whether they are memorised or worked out.
  • Students get familiar with the method they choose to work it out.
  • Students see an increase in speed, ease at completing grids and see their own scores improve over time.
  • Take control of their own Bell Work, empowering, safe and challenging.
  • Pupil ‘A’ counting up in twos on his fingers. yr11!!!
  • Pupil ‘B’ in Year 8 showing a peer the 9x table trick using your fingers

Ultimately, we have removed a massive stumbling block that lurks on the road to mastery!

DEVELOPING STUDENT CONFIDENCE

So many students do not know their times tables and believe that is the end of it…but now we have challenged this idea. Just as some people say they can’t do maths…now we can move on and challenge the idea that ‘I can’t do maths’. Mastery teaches students to move away from these barriers and

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

BUT to articulate their reasoning they must first have an opinion. To discuss their ideas they must first have an idea. To solve a problem they must first want to find a solution. They must first form an identity as a learner. Self-worth and confidence play an enormous part. Teaching lower ability classes can often (but not always) mean the students are largely disaffected. Through perceived/experienced failure their confidence has been eroded. We must challenge the perception of mathematics being all about right and wrong answers to build up a self-esteem that is positive enough to support the mastery platform.

When I asked a new group of Year 10s to GUESS the size of ten angles they were shown, half the group did not commit to paper, stating that they did KNOW the answers. Therefore these learners denied themselves the chance to feel good – others who guessed were thrilled when their guess was close but interestingly were not crushed when their guess was way off. Their learning identity was positive and it grew in a very simple exercise. I too joined in to prove that I do not KNOW all the answers, but have the tools to either guess or work it out.

Year 8 Extension task: (LOWER ability) Having studied the rule for adding and subtracting directed numbers, I asked students to write down what THEY THINK the rule could be for multiplying and dividing directed numbers. Some students wrote, ‘I don’t know, we haven’t done this yet’. Again, they didn’t have an opinion and again these students reinforced the negative image they have of themselves as learners. They needed choices pointing out to them and then they were able to take ownership of their choices and make it their idea by giving an example. Imagine their delight when some had predicted the correct rule. Again, those who had predicted in error were not crushed – it was just an idea. The students who had developed their own idea were keen to tell everyone what their prediction was, irrespective of being right or wrong, purely because it was their own idea.

TRICKS

  • Give students opportunities to GET IT WRONG and show it doesn’t matter.
  • Insist (‘encourage’) students commit an idea to paper- to have an OPINIION. Having an opinion gave them a vested interest in outcome which in turn made them more likely to come up with an outcome AND remember it.
  • Admit that as a teacher/ human/ adult we don’t know everything. I am not expecting my students to KNOW everything, the joy is in the working it out.

RESULTS

  • Students are prepared to guess, think, form an opinion, take risks.
  • Students are more likely to see a method through to the end to see if they were right (a win-win situation)
  • Students are more likely to have confidence in the next unfamiliar learning episode.
  • One Year 10 pupil could not even say true or false to a probing question. She has no confidence in maths and so does not think about maths, has no ideas about maths, cannot possibly articulate maths………I sat down with her and asked her to guess (we’ve been working on this idea). She chose False. I encouraged her to use an amount of money to see if she was right or not. We worked through the calculation and proved it to be False. She was thrilled, smiled (!!!!!) and wrote in her book ‘so I was right!!’. Anna believes she has been very successful and her confidence and enjoyment of maths has changed enormously in just a few weeks.

REFERENCES

¹ Drury, H. (2014) Mastering Mathematics. Oxford University Press, pp8.

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: ‘Central City Times Tables’ by Derek Bridges (www.flickr.com) CC. BY 2.0

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)

Developing Mastery in Mathematics (4 )

(Feature image: ‘You can and will be successful here’ by Enokson.  Attribution: PhotosForClass.com licensed by CC BY 2.0)

An Action Research Project by Richard Noibi

Introduction

‘Mastery is the process by which Maths is taught incisively depicting route(s) to finding solution(s) to a problem. This process is complete when the pupil(s) discovers other routes leading to the Solution’, Richard Noibi, September 2016.

Mastery is also the ability to apply knowledge gained from various areas of Mathematics, i.e. Algebra, Geometry, Probability and other topics in solving Mathematical problems.

With the new mastery curriculum in Mathematics and its demands on application rather than  teaching a systematic approach to solving problems, it is imperative that teachers like me adopt new strategies and approaches to teaching Mathematics to students.  

Gone are the days when teachers can predict exam questions.   Now, we have to teach ‘the why’ more than just ‘how’ to solve questions. Our students must now be able to think outside the box rather than just follow a systematic route to answers.

Aims of the Project

This project seeks to answer the following questions:

  • How can I incorporate Mastery into my teaching to facilitate the development of my students to meet the demands of the new curriculum?
  • What can I learn from other Schools and colleagues to facilitate Mastery in my lessons?
  • What method(s) can I adopt so that my students can remember formulas they need for answering new GCSE questions?
  • How can I encourage Innovation in lessons?

Focus Group

My focus group was a Year 9 class, which had 31 students with various abilities ranging from level 5a to 7b. I chose this class because:

It has students who usually struggle with resilience – a quality needed for mastering Mathematics.

Some of the high achievers were still working below their challenge grades.

Most of the students also found applying Mathematical know-how to real life situations a struggle. This skill is needed to excel with the new strategy.

The Route to Mastery

 Good foundations

For my students to have Mastery skills in Mathematics up to the level required, they must have the solid background knowledge of the Subject. This is the foundation of the plan and the stage at which misconceptions are corrected.

Thinking outside the box

Students can no longer be one way learners but need to be dynamic in their approach to solving Mathematical challenges.

Creating a Culture of Mathematics

‘Consistency and Practice makes perfect’. This was developed through thorough Maths ‘skills drills’.

Mastery in other Schools

As a tutor to students from another school (School ‘A’), I noticed that the teaching styles have changed recently. The past focused on 3 part teaching (Starter, main, plenary) is now divided four parts (revision activity based on quick recap of the most challenging part of the last lesson; then a starter activity, which is usually a modelled question targeted as yardstick for success criteria; main teaching and class work; a plenary based on checking how many of the model questions could be answered at the end of lesson).

In another school I am familiar with (School ‘B’) they use 3 part teaching but there is a modelled test at the start of a new topic. Then after three lessons the students are given a test with the same questions but by this time they have had the opportunity to familiarise themselves with the test. The outcome of the test is used to gauge the understanding of the students. This method embraces consistent practice and developing a culture of Mathematics amongst their students.

Both schools use homework to consolidate learning. Sometimes homework is used for correcting errors students make in the lessons not undertaking new work.

Both strategies endorse the above listed route to Mastery. Thus I adopted aspects of both in my lessons with my Year 9 focus group.

Adopting the strategies

To create a culture of Mathematics with Year 9, I combined the method of teaching in School ‘A’ with the assessment programme of School ‘B’.   I made sure I informed parents of the outcome of the end of Topic tests I give pupils once a fortnight. This improved enthusiasm and effort from my students.

Also, I adopted a method I learned from my country (Nigeria) which helps promote independent thinking and application of Maths to real world situation. I started giving the class a Mastery question as a starter once a week.  I say to the class I am not interested in the final answer but how to start solving the question. What is the first step to take before solving this question? What if the question is changed to ………? How would you start? This helps many who struggle to know where to start a question during exams or tests. As we all know, marks are not only awarded for the final answer but the steps taken to reach it.

The results

Seventy per-cent of my Year 9 class achieved a grade higher than their challenge grades, with three making exceptional progress.

Developing Mastery in Mathematics (3)

Featured picture: http://www.freeimages.co.uk/

Maths mastery – exploration and implementation

An Action Research project by Julie Silk

Aims of the project

The aim of the investigation was to explore the changes to approaches in the teaching of Mathematics to incorporate the new style of questioning and understanding known as Mastery.

The Key Stage 3 and 4 curriculum has drastically changed, particularly with regard to the style of questioning in assessment.

Our aims

  • To clarify what “mastery” means
  • To identify changes needed to teaching styles and learning outcomes
  • To implement changes
  • To observe one another to assist with team planning and sharing good practice
  • Embed mastery in our Schemes of Work

Background

In 2015 the new Mathematics curriculum was launched. Numbers replaced grades and a new style of examination was introduced by the examination boards. Our current Year 11 will be the first to face the challenge of the new curriculum. It was, therefore, essential that as a department we gained full understanding of what the changes were and how this would impact on our teaching. There were two main changes: curriculum content and mastery. Exam boards, education experts and teachers across the country were all offering a variety of opinions as to how this would look. It was for this reason that the faculty as a whole decided to carry out action research that would assist with this process.

Context

Our initial discussions began with us selecting a couple of classes to work with in order to build resilience and mastery skills using plenaries that based on mastery style questions.   At the same time we set out to research more fully the definition of mastery.  It quickly became apparent that we would need to use our plenaries with all classes or some of our pupils would be disadvantaged.  In consequence we extended this practice to all classes in years 7-10.

The emphasis on moving from predictable questions where you can teach a few “tricks” to get enough marks to get a C, to a real understanding of how to problem solve with Maths is , I believe, an excellent step forward. I have always considered teaching maths to be like coaching a football team. You show them lots of skills which they can practice and master but it isn’t until they are put together in a match that the full beauty of the game can be appreciated; in our case the “match” is problem solving.

Actions

  • Research mastery
  • Change plenaries
  • Change assessments
  • Observe each other teach in peer observations
  • Share good practice within the department
  • Share good practice outside the department

Research was shared and stored in a central folder in the Maths faculty for the benefit of all.

The new style of questioning needs quite a lot of encouragement for pupils to get started and we have to build resilience as up until this year, pupils were reluctant to get things wrong in Maths.

With the new style of questions we felt that it was important for the pupils to get a realistic idea of their understanding of the work. Our new tests provided by the exam board are very challenging and pupils need much encouragement to correct their mistakes. I felt it was vital for them to persist and so for every end of unit test we do, one week later they have a retest, same style of questions but different numbers. Pupils are adapting much better to the tests as confidence grows. The younger the pupil the better they are dealing with the changes. In year 10 the tests and end of year exams have certainly spread the level of attainment, many who would normally be 4/5 borderline are struggling to achieve anywhere near their target grade while the top-end are almost on par with their counterparts from last year. We can now see that our next step is to get pupils to write down the steps taken in each question and to at least start a 6-8 mark question that they feel is at the limit of their ability.

Peer observation

At the start peer observations were used to have a look at what we were each trying out with our classes. We have a full programme of paired observation for the next academic year to further develop our skills and share best practice.

Impact

The full impact of our findings will be more evident as time goes on.

  • Test results for my year 10 groups have shown that the more able the pupil the better they have adapted to the new style questions.
  • Resilience is key to gaining marks.
  • Showing working out is now more important than ever.
  • Adoption of the Shanghai style of teaching (learning key facts, peer support, moving forward together) is important as pupils need all the mathematical skills taught readily available.

Conclusions

  • In the long term, changes to the curriculum will increase understanding of Mathematics by pupils
  • Resilience needs to be encouraged and perfected
  • We’ve been fortunate that Nrich has been good preparation for some of the skills needed
  • Results will rise as we develop mastery further
  • The skills we have gained can be shared with others in other departments, other schools and Primary colleagues

Next steps

  • Continue to adapt lessons to incorporate mastery plenaries
  • Increase pupil response to tests and exams
  • Use peer support to raise understanding in lessons
  • Contact Primary partners to set up a support hub
  • Focus mind set changes on the middle ability pupils who seem to have been the most affected by exam changes

Sources and references

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]

National Centre for Excellence in the Teaching of Mathematics. October 2014. Williams, H. (2014) Approach, Research. Mathematics Mastery Acting Director of Primary

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.

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.