Mastery is a commonplace word now in mathematics education, and social media is awash with 'mastery lessons', 'mastery resources', and 'mastery curricula' - is this mastery? What do we mean by mastery? Is it a teaching style? Is it a curriculum design method? Is it an intervention strategy? The answer is, always, firmly, no!

When Benjamin Bloom and John B Carroll were squirrelled away codifying what Carleton Washburne (and others) had mapped out, they certainly did not intend for mastery to be distilled down into lessons, pedagogy or even how a curriculum should be written. Mastery in its purest sense is a way of schooling, a way of ensuring every child can succeed given the right conditions.

Using Carrol’s model of school learning we can formulate the degree of learning into 5 areas: perseverance, opportunity to learn, learning rate, quality of instruction and ability to understand. However, we strive to ensure that the function is equal to one; where an equal amount of attention is dedicated to each of these key principles. In this blog, we shall focus on ‘quality of instruction’ and all that is encompassed by that. It must be noted at this point that learning rate is often what we define as ‘ability’ and it must be clear that ability is only a measure of learning rate. For example, we can have pupils who are ‘low ability’ but in the same regard, ‘high attainment’, i.e. their pace of learning is fairly deliberate but can, and will achieve well - I very much put myself in this category!

So when I refer to ‘mastery’, I mean Bloom’s mastery. Mark McCourt describes teaching for mastery in his latest book, ‘Teaching for mastery’ and Chris McGrane has outlined how utilising the Complete Maths platform to teach for mastery and I plan to complement this with my experience of implementing mastery in a comprehensive secondary school.

Mastery hinges on responsive teaching and not only after summative assessments, but in the moment responsive teaching. Interventions must be made as soon as they are needed and the goal of mastery is to scale the one-to-one tutoring model of teaching to one-to-many. To explain my take on implementing mastery, I’ll exemplify what a learning episode might look like and how we make the mastery cycle work

In this episode the phasing model of learning is used, i.e. teach, do, practise and behave. Mark McCourt provides a sensible proportioning of content as follows:

Teaching and doing are blended with the opportunity to do purposeful practice also. The final phase of ‘behaving’ is somewhat the most challenging, but most important - this is what develops the mathematician and proves the understanding is deep and connected. I plan to explore this phasing model in more detail in upcoming conference presentations or at our public CPD events. Furthermore, to keep this blog somewhat succinct, I have omitted some of the detail but hopefully left enough for it to be comprehensive.

The learning episode is on directed number arithmetic. The basis of learning directed number arithmetic is best modelled using algebra tiles (usually with a visualiser) and allowing pupils to use the concrete materials to help conceptualise the ideas. I believe that at the early acquisition stage of learning (when knowledge is inflexible), example-problem pairs are a powerful tool and a plethora of research on this support their utility.

Above is one example-problem pair using algebra tiles. It must be noted at this point, that much of this is usually done with pen, paper and algebra tiles under the visualiser. Prior to this, a lengthy introduction to the idea of ‘zero pairs’ has taken place and we have explored this profound idea with pupils. Building on this example, the class and I would explore lots of different addition calculations and encouraging the pupils at every opportunity to create their own questions. There are two parts to self-generation: one, it alerts me immediately to lack of understanding for those who cannot do it and two, provides an insight to the depth of understanding at this point - this is far superior to just providing or asking questions in my opinion. You will see in the example above I like to include ‘Make another with the same answer’ box pop up when pupils are working on the problem question; this allows pupils to maximise their ‘up-time’ in lessons and creates space in the lesson to allow me to get between the desks. Once happy with addition then next comes the dreaded subtraction! Not anymore! We strive to teach children proper mathematics and in particular, proper arithmetic. No silly rules or sayings or tables to spot what signs they have in comparison to the magnitude of the number - just simple arithmetic.

In the example above we use the additive inverse property as a basis for what we usually refer to as subtraction; pupils love the conversation I have at this point to hook them in about ‘take-away’ and subtraction not being a ‘real thing’. We are proper mathematicians now and we only work with two operations: addition and multiplication. Again, like with addition, example-problem pairs are utilised and ‘make me another’ prompts but before I move to more demanding questions I like to check for understanding or more replication at this point and offer something like this:

A fairly standard set of questions, but maybe not as many as you would have hoped for? Before we delve into why that is, take note of the ‘think’ and ‘do’ prompts: here I offer some undoing style questions and a type of non-example to provide opportunities for pupils to turn inflexible knowledge into more flexible, usable knowledge. Furthermore, maximising ‘up-time’ in lessons is crucial and this ‘buys’ me time as a teacher to meet the needs of everyone in the room. Returning to my first point, the need for endless exercises on a specific skill is simply not required. Multiple studies by Rohrer and Taylor have shown that for over-learning to occur, pupils only need to answer two questions correctly and the skill is over-learned. However, it is noted that educators should probably err on the side of caution and offer more than two. In a simple experiment between two groups, where one group answered 3 questions and the other 9, the differences in far transfer are negligible and even more worrying, the drop in accuracy after only one week (both groups were observed to have a mean of 94% accuracy at the initial teaching stage) is mind-blowing:

Do we waste too much time doing repetitive questions? How can we properly practice mathematics?

Would a task like this help pupils practise directed number arithmetic and also draw upon reasoning skills and ultimately aid the transition from inflexible knowledge to flexible knowledge? Based on my experience, this does! With the practice phase, we also want to interleave previously taught ideas to take advantage of retrieval and method selection. Again, Rohrer and Taylor provide us with another study indicating the benefits to far transfer in ‘shuffling questions’ when practising mathematics.

We can see those who block practice (i.e. practise what has been taught explicitly) perform well initially but over time do not learn as well. Those who work on a mix of questions from previously taught content, perform far better over time. Once again in this study, the questions were very procedural and we can see accuracy drops significantly over time - this begs the question: what else can we do? Although I must note at this point, interleaving in its truest sense, is something as Maths teachers we already know and have always done but maybe we need to ensure opportunities to do so are embedded in our curriculum scheduling.

In my school, we utilised the custom diagnostic quizzes you can construct in Complete Maths for this very purpose. We creating a quiz, worksheet or task that is interleaved we need to consider what has been taught before and look for opportunities to call upon method selection, i.e. questions that might appear to look the same but have very different structures underneath. Also, we can include questions where on the surface they look nothing alike but when you drill down they are very similar. Interleaving is not using something like perimeter as a vehicle and changing the sides to decimals, fractions or algebraic terms - interweaving is a better description for this type of intention.

Moving to the behave phase of learning we want to draw upon previously learnt ideas and build connections to the current idea. We need to take care that when the problem-solving demand is high, the level of the mathematics needs to be fairly trivial (dependant on the pupils whom you have in front of you of course) to provide easy access but a high ceiling of mathematical opportunity. We need to consider maturation and it is suggested that 2 years is around the typical period, however in this case of directed number arithmetic a suitable behaving task is the classic always, sometimes and never true type of task. Pupils are handed statements about negative numbers, e.g.

“Two negatives make a positive”.
“A positive number is more than a negative number”
“Adding a positive number to a negative number will make a positive answer”

They must put each statement into the always, sometimes, and never true category, but also provide evidence of their decision. Using only prose statements pulls the maths out of the pupils' heads and getting between the desks to challenge them on their decisions (or sometimes settle debates) is a fruitful way in which to gather intelligence on your pupils’ learning.

However, we always need more when it comes to information on learning and in my opinion, building in as many different opportunities as possible to do so helps teachers make well-informed decisions and become more responsive. Back the mastery cycle! Our curriculum was designed in the following way:

Each block has three parts: number, algebra and another strand. Number and algebra are hugely important strands of mathematics and having opportunities to generalise was important to me when sequencing topics. Each block required mastery to be met or future topics would provide problematic. Teachers have complete autonomy to work across the block left to right, right to left or to be honest whatever is best for the pupils in front of them. Professional autonomy is the foundation upon the re-design of our curriculum and I was keen to foster the ethos of doing what is best for the pupils. We grouped the pupils by prior attainment, although not solely based on primary information and we ran diagnostic quizzes to ensure we had accurate data to determine where each child should start on their journey to learning mathematics at secondary school. In the interests of brevity, I go into much more detail on this in my conference presentations, but I must stress that the groupings were very flexible and I would regularly move pupils from class to class based on the judgement of the teacher. I used a model whereby once grouped, the teacher would keep this class for a minimum of four years. Relationships are critical in schools and even more so in secondary; short 50 minute periods are a significant change for pupils and in some cases, we don't see them each day.

Circling back to assessment: we offered diagnostic quizzes after each strand and a summative assessment after the entire block where we would look at quantitative data to check for mastery. Strand assessments were invariably multiple choice and the block assessment would more of an extended response. I had to consider workload of marking and the workload of pupils, continually doing assessments, however, teachers run experiments and gather data every period of maths, hence their judgement was of paramount importance to making the mastery cycle work.

Correctives are integral to making mastery work and often it is the barrier most schools face when attempting a global implementation. Something I considered, which is often never done in education was scenario-based practice - for the teachers! I created a scenario and we discussed how logistically we could make it work:

Teachers were faced with an example breakdown of quiz scores and then I indicated something that might be controversial but a very real problem: the concentration levels (and at times behaviour) during the five periods this particular year group attended maths:

I offered the following paragraph:

Think carefully about how you can correct learning before progressing. The topic of integers is an integral foundation upon which we build, hence careful thought and planning must be prioritised. We aim for pupils to sit the block assessment at a state of readiness, i.e. we are fairly certain they should all demonstrate mastery in the essential skills section. The earlier we can make interventions (or correctives) is of paramount importance to a successful implementation of the mastery cycle. What is your course of action?

Scenario-based practice is a powerful tool and used frequently in other walks of like, for example, special operators in the military often ‘rehearse’ the scenario they face to ‘iron out’ the kinks in their procedures and delivery - why not in maths teaching?

So how can we manage correctives and what did we, as a department, collectively decide as to the best course of action? Let’s take the following example of quiz data:

Some pupils have not grasped some of the key ideas in each of the three strands: number, algebra and integers and in my conference presentations, I go into more detail on what these assessments look like. Based on the data, it is clear that we need to run correctives to ensure every pupil is ready for the next block of work; remember, we do not want pupils beginning collecting like terms if they cannot work with integers. I am holding myself accountable and building this culture in the room. Often I would explain to pupils that clearly my lessons had not been impactful and I need to work with specific groups to ensure I correct this - I want you all to be successful and let’s fix this together. How can we organise this?

Above exemplifies two models of how correctives can operate in a ‘real’ classroom. Model one offers those who are secure with integers are offered enrichment of the topics contained within the numbers strand and allows the teacher to interview and work closely with those pupils who have shown remediation is required. Model two offers those who are secure on all areas, enrichment of all three strands and the remaining pupils who need remediation across the block are grouped to allow the teacher to design a plan for them. It must be noted that interventions in the moment, throughout the teaching of this block of work, are critical to making this model of correctives manageable - if you wait until the end of the block before diagnosing problems I would argue that corrective teaching will be very much impossible and probably too late.

I hope that this blog makes some sense but as you can gather, just like teaching and learning, mastery is a complex process and distilling it into a blog is not easy! If you would like to know the finer details of how all of this worked for us at St Andrew’s then look out for my presentations at the PT Conference on Friday 13 March and also the next MathsConf (22) in Manchester on Saturday 14 March.

What Next?

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