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A Mastery Curriculum in Scotland - A review of year 2


A Mastery Curriculum in Scotland – A review of year 2

This is the latest article in a series of blog posts on our implementation of a mastery curriculum from Third Level to Higher mathematics.  Previous installments can be found at:







What is going well?
Pupil attainment has improved dramatically.  All of the research about mastery learning suggested that there would be a shift in the distribution of attainment in the positive direction for a significant number of learners.  We believe that this shift is beginning to happen.  The graph below, is what much of the mastery literature suggests will happen over time, in a well implemented mastery curriculum.  Our pupils now finishing their second year are progressing incredibly well.  The level of sophistication in the mathematics our fastest moving classes are working on is far in advance of what was previously the case.  One of our key aims was to see significant improvements in our middle two classes.  This has now increased to the middle three classes – through ambitious learning and teaching we have increased significantly the number of pupils in the middle of the year group who are performing well.   Our pupils finishing first year have similarly performed well.  However, there are a number of concerns regarding pace versus attainment, which I will address below.




Our homework policy of “every pupil, every night” has continued with excellent engagement.  Whether or not this is having a significant impact on attainment or learning, I do not know.  However, culturally, this is very important.  I’ve written before about how it is easier to keep a pupil doing homework from S1 to S5 than it is to ask him/her to start doing it in S5.  It is excellent to see that almost the entire cohort of S2 pupils moving into S3 are used to doing 5/10 mins of maths every day at home.   We have no more than a couple of pupils who have opted out.  For an inner city school we are delighted with this level of engagement.  We are currently looking at improving the quality of the homework booklets we use to support this policy.  For our pupils who have moved into fourth level there are also formal homework exercises, in addition to the nightly, which are issued around every 2 weeks (they are written into the curriculum plans for when to issue).  These formals typically include 6 to 8 essential skills questions from prior learning and 2 challenging problem-solving questions.

We introduced some changes to our Summative Assessments Policy.   We record component marks for each of the four topics being assessed in each summative.  If a pupil scores below the expected standard in a particular topic they have an opportunity to resit the section a few weeks later.  Letters are issued to parents informing of this, with associated revision materials.  In almost all cases learners have shown improvements between first attempts and resits.  We always allowed resits of our mastery diagnostics.  Doing this for summatives seemed like the obvious next step.  The focus should be on helping the pupil arrive at mastery.  Rather than seeing a score of, say, 64% on a topic and saying “room for improvement”, we are now permitting an opportunity for the learner to engage in making that improvement.   This remediation is supported by the starter tasks in lessons and by teachers offering supported study after school.  Another change for summatives is that prior to assessments we will now be texting all parents a couple of weeks in advance and alerting parents to the fact their child has extensive revision materials to use. 

Before becoming a PT I was regularly frustrated by the fact that most meetings had very little impact on learning and teaching.  During my first 3 years as PT, Professional Learning in the department has been the key feature of our department meetings (DM).  Normally there is reading issued in advance, which we then discuss.  Clearly, reading alone is not enough to impact any change in practice, so, over the past year I’ve kept the focus on one idea over a series of meetings and used a series of follow up tasks to cement the learning.  For instance, when exploring variation theory I issued some examples and delivered a presentation on some basic ideas.  For the next DM, colleagues had to do some professional reading and then take turns to talk about their interpretation of the ideas.  For the next DM I tasked colleagues with creating a task based upon variation theory for use during a forthcoming lesson.  I compiled these tasks into a booklet and we spent a DM discussing the relative merits and areas for improvement in each of the tasks.  We then had another meeting where a second attempt at writing variation tasks was shared.  I believe this approach helps everyone to engage meaningfully with the professional learning.  A Principal Teacher cannot own the professional learning of his/her team – however, having some collective focus can help to develop a shared understanding, language and pedagogy which we, and ultimately our learners, are richer for.  We used a similar programme, which included paired observations, when improving the quality of our direct instruction through example problem pairs. 

Department colleagues have been off attending the conference circuit, taking in many interesting talks.  All of the department have a copy of Craig Barton’s book, which we’ve dipped in and out of where he has written about something we are focusing on.  I’ve regularly used blog posts, research papers and articles from the likes of the ATM as discussion points.

At times I’ve worried that the focus in our curriculum is too procedural.  However, there has been a massive improvement in the number of UKMT prizes.  A couple of years ago we seldom had more than 5/6 pupils receive a certificate – whereas with the most recent batch of JMC the numbers achieving at least a bronze have multiplied six-fold.  We now have pupils going forward for the Kangaroo and Olympiad rounds.  This might not be stellar compared to many schools – however, for us; this is one indicator of progress.

What could be better?
For the pupils who have the lowest prior attainment upon arrival from primary school our curriculum does not adequately meet their needs.  For next year we have decided to use the Maths No Problem series.  The year 5 textbook and workbooks provide a good basis for study before moving towards engagement with our Third Level mastery curriculum.  Interestingly, Block (1971) states that…

“Mastery methods have been more effective for first grade arithmetic and ninth grade algebra than eighth grade arithmetic.  The learning of first grade arithmetic requires little, if any, previous arithmetical training, and the learning of algebra requires only simple arithmetical skills which most students have acquired by the ninth grade.  The learning of eighth grade arithmetic, however, requires the arithmetic skills of grades one through seven, which many students may not possess.”

I am of the opinion that for the lowest performing learners, this year of fixing the previous learning, identifying and repairing gaps is important before working on newer material.  The materials from Maths No Problem are well designed and promote the sort of pedagogy we've been utilising (concrete and pictorial representations).

While our fastest classes are making good progress through the curriculum there is a concern that the pace is not fast enough with some groups.  Some classes have been given a lot of time to achieve near perfect mastery.  For instance in this year’s S1 there are classes which have almost every pupil over 90% in our, not trivial, summative assessment for phase one.  This demonstrates the power of mastery learning.  However, we need to be careful not to be spending more time than is required on topics. For the S1 classes mentioned, this excellent performance may have been at the expense of ensuring learners have adequate exposure to phase two in S1.   This is obviously a problem as we know pupils need to have learned enough to sit the final exams in S5 (Hillhead has no S4 presentation for any subject in any exam).    Covering 8% of the curriculum in a year is too slow, even if it is covered very well.  For next year, each teacher will have targets of coverage for the year.  This will be based upon our experience of the past two years; the understanding of how fast classes can go without reducing the attainment below a very good standard.  Mastery principles will still be adhered to, with the rigorous formative assessment applied.  It is likely that some extension tasks and learning may need to be skipped with some learners/classes, if mastering fundamentals takes up too much time.

To help to aid the increase in pace, we have been doing some primary transition work (more details below) and have made some curriculum content alterations.  The initial topic in the curriculum is whole number skills.  In coming years we will be skipping the first third of this topic with many classes, as the learners have already acquired this knowledge and mastered the key skills. 

Some content in the curriculum has been colour coded blue to show that the topic is not core knowledge expected of all.  It will be assessed in our summatives, but will not be weighted heavily.  It should only be covered if there is adequate time.  Similarly we’ve, reluctantly, had to remove certain learning intentions from the curriculum altogether.  This has happened where we felt the topic added little or no value to key understanding for later attainment in senior exams.  For example drawing and measuring angles has been removed.  Expressing a number as product of prime factors is no longer a process we formally expect learners to master.  While this is lovely maths that I’ve always enjoyed teaching, and still will, – it is time we need to use more wisely as it doesn’t add a lot of value to labouring this procedure for 3 lessons with our borderline pupils.  I spent a week rewriting our properties of shapes unit recently, but have subsequently decided that we are removing the topic in its entirety – any really important learning in it has been slotted in to appropriate places in other units.  Knowing the properties of the diagonals of quadrilaterals is just not core learning in the Scottish curriculum. 

I have been impressed by the improvement in procedural fluency in all of our learners – so too their conceptual understanding.  However, the development of mathematical thinking and problem solving skills is something I am looking to focus on more explicitly across the department next year.  John Mason’s Thinking Mathematically provides the framework for this.  In my own classroom I need to provide more opportunities, not just for learners to solve problems but also for them to have opportunity to reflect upon their experience of solving the problem.  I am keen to have learners discuss the internal dialogue involved in solving problems and also, inspired by the approach in Japanese classrooms, have learners spend time looking at alternative approaches.  I have read that sometimes, when teaching mathematical thinking, doing 1 problem 3 different ways may be more effective than doing 3 different problems 1 way.  Working together as a department next year, this is somewhere we can learn and improve the quality of our practice. 

The scale of the task in our comprehensive curriculum re-write seems to regularly increase.  Looking at some of the topics, which we put together in the initial months– there is a clear difference in quality with what we have written recently.  As more classes go through each topic in the curriculum and teachers feedback how pupils of different abilities are served by the resources we have included, there is a need to constantly update and refine.  This refinement takes many forms:  improved teaching slides, extra practice and drill sheets to ensure adequate practice, better rich tasks to ensure appropriate extension opportunities etc.  Further, we are constantly monitoring the effectiveness of our summative assessments.  These have to give a comprehensive gauge of pupil learning – but the variety of questions asked, and the associated marking schemes are regularly adjusted as to ensure that all the learning of all classes can be fairly evaluated. 

Our increasing knowledge and understanding around effective tasks, and variation theory, in particular, have led to us writing new resources to include in the course outline.   Less and less are we using materials found on the internet or textbooks – the vast majority of tasks and resources available are not good enough.  Writing our own seems to be the way to ensure that we meet our learners’ needs.  Of course, effective task design is hard and requires critical analysis from colleagues, while the most effective way of evaluating a task is to use it with a class.

We have an over-abundance of resources written in to the system.  In purely practical terms this can be a little frustrating for teachers as, for some lessons, as many as 6 different pieces of paper may be required for each pupil once the diagnostic assessments etc are accounted for.  In our most recent development work we have produced comprehensive workbooks to support the learning and teaching of topics.  For example we have recently developed department booklets for topics such as surds, quadratics, trig graphs etc.   These are not intended as the sole teaching resource, but should support teachers in delivery of the core material.  To further alleviate the paper overload, we have begun to experiment with diagnostics booklets – where all of the diagnostic tasks for an entire phase (4 topics) are included.  Some teachers find this useful, and the children have enjoyed recording their performance in diagnostics in the performance log inside.


What other changes?
We have recently began to work more closely with our cluster primaries.  We’ve shared the learning we would like learners to have acquired upon arrival and shared the baseline test we use for whole number skills.  Further, we’ve given the primary 7 teachers all of our Third Level learning intentions and asked them to highlight what has been covered.  We know we have likely spent time re-teaching things that pupils could already do.  We want to avoid not making best use of time in future.   

We’ve had visitors from over 30 other schools in the department this year, as well as a full inspection of the school.  We’ve used every single one of these occasions as an opportunity for learning and have been keen to tap in to the knowledge and opinions of those who have visited.   In addition to this I was involved in really useful triangular visits with @offpistemaths and @mrgraymath.  Going in to each other’s department, talking to staff and pupils and observing lessons moved all of each department along in our paths to improvement.  The most pivotal day in the year was the visit of @Emathsuk.  Mark is a colleague I hold in incredibly high regard and has had a massive influence on my career in recent years – he’s set me off reading in all sorts of different directions and introduced me to some of the top thinkers in mathematics education.  The day of his visit was challenging.  He acknowledged the progress we’ve made, but, like I asked him to do, he cast a critical and constructive eye over all aspects of where we currently are.  While we’ve had a lot of congratulatory visits and praise for our work, even from the inspectorate, we knew that to really improve we needed somebody who is a subject specialists but has a wealth of knowledge and experience far superior to our own.  Mark influenced some of the ideas things I’ve written about above and was the catalyst for me taking stock of the whole thing.  I’d “sort of” identified issues we had.  Mark came in, shone torchlight upon them and challenged me to lead the department in improving them.  

Mike Ollerton is somebody I have respected for a long time, and he continues to challenge my perception and understanding of what maths learning is.  Similarly @dannytbrown has become a colleague who’s Twitter feed and blog have become places I have found incredibly though provoking.  He continuously challenges my thinking and opens my mind.  He has influenced much of my own professional learning for the coming year.  I also have to thank Danny for encouraging me to launch the first ATM branch in Scotland and for sensationally getting the legends of mathematics education, John Mason and Anne Watson to be the speakers at our first branch meeting.  I hope that the presence of the ATM branch in Glasgow will be a catalyst for a different type of dialogue about what maths teaching is and how we go about it.  A beacon of a different way of thinking.   Personally I use a fair bit of direct instruction and subscribe to cognitive load theory etc., but I am all too aware that maths teaching is not as simplistic as this.  I would love to see us, as a nation, moving away form the TeeJay-ifiication inherent across the country, where too much focus is on doing sheets of drill without any sort of mathematical thinking or developing any conceptual understanding. 

Finally, I am currently developing a website called starting points.  This is intended to be a collection of tasks and starting points to help teachers plan for richer and more effective learning experiences. All of the tasks include suggested teaching points and questions for discussion with learners. The tasks are a collection of ideas for learner-generated examples, some rich tasks, intelligent practice, some examples of variation theory and various other curiosities.  If you have a task you would like to contribute or would like to be involved more by submitting tasks on a more regular basis please get in touch.  If you would like to see where how this is developing, drop me a message on twitter and I’ll send you a link. 

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