I first heard of mastery learning around 2014. Back then I thought it was something interesting to do with CPA approaches – it isn’t. It also seemed to be something which was exclusively to do with countries in South East Asia. I was surprised to learn that it is an Anglo-American construct, to do with formative assessment – without a bar model in sight! Mastery is still very much a buzzword in education just now, however, the majority of what I see out there, as was the case in 2014, is not related to mastery learning as defined by Benjamin Bloom.
It is very important to understand what mastery learning is, as we only know that Bloom’s mastery works. We cannot say the same for interpretations of that word which do not encompass Bloom’s key ideas. CPA approaches are not mastery learning, however, they can be utilised in effective mathematics teaching. There is a significant evidence base for their effectiveness. When schools move to mastery learning they sometimes also look at their wider pedagogy. As such CPA often comes along in tandem with mastery learning. The powerful force in this partnership is very much the mastery learning cycle.
Around 2015, I first met Mark McCourt who introduced me to some of the key reading in the field of mastery learning (and mathematics teaching generally). Mark is a passionate advocate of mastery learning, believing that it is truly equitable. He adamantly believes that every child can learn. I’ve had the privilege of working with him over the past year and a bit at La Salle and have continued to learn from him, every time we talk. In this role I’ve had time to step away from the classroom, read and reflect lots. I’ve been able to visit primary and secondary schools across the entirety of Scotland. In this time, I’ve come to realise that I used to believe that every child could learn. But near 15 years in schools had shown me that they frequently don’t. I’ve seen neither conventional classrooms or mixed attainment grouping address this. I know of no example of a school in Scotland which has had a seismic shift in mathematics attainment. This is why mastery learning is so intriguing, it is rooted in the idea, no – the expectation, that the vast majority of pupils can and will achieve well. It is a rebuttal of the idea that socio-economic factors need be the key determinant of attainment.
I spent a few years as head of maths at Hillhead High, implementing a curriculum which drew upon the principles of mastery learning. As is always the case, we got some things right while some other aspects of our implementation didn’t go quite as planned. Some of it, I now know was simply incorrect. I’ve written about that experience extensively in other blogs. What the experience showed me was that the ideas were effective and could impact upon pupil attainment, in some cases quite dramatically. I wasn’t quite sure how to maximise the effect of what we are doing. There were also some logistical issues which I didn’t have the understanding to properly address. Despite this, the pupil outcomes, and my belief in mastery learning never waned. The opportunity to work for La Salle since the start of 2019 has afforded me the time to read much deeper into mastery learning, support schools in implementing it and consider the practical implications of doing so. This blog summarises some of what I’ve learned and what I now think a successful implementation would look like in a typical comprehensive secondary.
The modern history of mastery learning spans at least the past 100 years and you can read about this in Mark’s book “Teaching for Mastery”. For now, I want to focus on one paper from this long history.
In 1984 Benjamin Bloom published his paper on “The 2 Sigma Problem: The Search for Methods of Group Instruction as Effective as One-to-One Tutoring”. (1)
In that paper Bloom contrasts 3 different approaches:
Conventional Classrooms: Pupils learn in classes of around 30, where assessments are use periodically to grade pupil attainment.
Mastery Learning: Pupils, again, learn in classes of around 30. The actual teaching is no different to the conventional classroom, however formative tests are given for feedback after each topic, with time specifically allocated for corrective work and then a second formative assessment.
Tutoring: Pupils work one to one with a good tutor and do the same formative tests and correctives after each topic, as per mastery learning.
In the experiment, pupils were allocated to one of three groups and taught the same material, but using either the conventional, mastery or tutoring approach. In four replications of the experiment the same outcome was clear at the point of summative assessment. Pupils who engaged in tutoring performed two standard deviations above the performance of those in the conventional classroom settings. Tutoring, it is clear, is very powerful. Unfortunately, it is extremely expensive and does not scale up. We cannot afford to have a 1-1 teacher to pupil ratio!
Pupils in the mastery learning classrooms, fascinatingly, performed around 1 standard deviation above the the performance of those in the conventional classroom. This result is even more startling when one considers the following:
“90% of the tutored students, and 70% of the mastery learning students achieved the level of summative assessment performance reached by only the highest 20% of the students under conventional instruction conditions”
This is one of a plethora of studies into the effectiveness of mastery learning. It can make a significant impact on pupil achievement. It is not a panacea. Education is not deterministic, as such, there is no “catch all”. Nevertheless, we should always be aspiring for a “catch more”. Mastery learning is an idea with a vast canon of supporting research, and significantly, the only thing I’ve experienced in my career to date which has had a significant impact on the attainment on the majority of pupils. In this blog, and with an impending return to school, I turn my attention to how can mastery learning be implemented more effectively and lay out some key ideas for consideration.
I do not intend to go over old ground from previous blogs, or replicate Mark’s book. Instead in this article I will be drawing heavily upon my recent reading of Thomas Guskey’s book “Implementing Mastery Learning” (2) and the consequences for implementation of mastery which I think arise from this.
Guskey described the above as being indicative of what happens in a conventional classroom. We might refer to this as the treadmill approach. We move forwards without regard to how pupils are performing – there might be some formative assessment techniques employed, but the application of these techniques is not systematic. If we define mastery as achieving a solid grade A, of around 80% or above, then it is clear that the majority of the pupils do not achieve this. One might argue that this is the natural order of attainment – only the elite get the highest grades. Instead, in mastery learning, we assess what pupils can do and and what they understand, rather than how they have performed in relation to others. In mastery learning, the purpose is for every pupil to achieve the highest grade. There should be no adjustment of grade boundaries- If 80% of the pupils have achieved an A then that is just great! It should be indicative of great learning and teaching rather than being taken to infer that exams have been eased. In mastery learning the criterion for assessment is clearly defined and the objective is that every pupil is expected to learn almost all of it.
Mastery learning is actually a very simple, robust formative assessment cycle. It can result in significant increases in attainment without any other change in instructional practice. The diagram above, from another Guskey paper (3) lays out the basic idea.
At the end of an instructional unit of work, approx 1.5 – 2.5 weeks long, a formative assessment is issued to pupils. This assessment is used as part of the learning process. It gives pupils specific feedback on their learning. Pupils are expected to achieve highly on this, over 80% typically. If they fail to do so on any sub-part of this then some corrective activities are to be undertaken, before a second formative assessment a couple of days later. The second formative assessment covers the same concepts and ideas, but is not identical to the first assessment – it shouldn’t be about remembering answers, but instead, understanding. The second formative assessment gives pupils a second chance of success and can be highly motivational. Those pupils who do not need to undertake any corrective activity get to experience some form of meaningful enrichment activity to further develop their learning.
I will explore these practicalities and typical teacher concerns in more detail below. First, though, it is important to reflect upon a quite startling fact. This simple instructional change can result in significant learning gains for the majority of pupils. Why is it not the common practice in schools already? Dylan Wiliam has written at length about formative assessment. Between Bloom, Guskey and Wiliam we have a great range of tools to support learning. There are technical issues to be dealt with, of course, but even in schools where formative assessment of some sort is common place, we haven’t seen a transformation in attainment. Wiliam argues that Feedback has little impact. It is the time to actually do something meaningful with that feedback that matters. This is frequently not afforded in schools. Unless corrective activity happens then formative assessment doesn’t impact.
In his own words: “Feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. Indeed, the whole purpose of feedback should be to increase the extent to which students are owners of their own learning” (4).
In successful mastery learning classrooms, significant numbers of pupils reach high levels of attainment in each learning unit before progressing to the next one, with the end point of high attainment in summative assessments. We can ensure that each building block of the tower is secure before building more on top.
Of course, learning isn’t entirely hierarchical – maths is a complex web of interrelated ideas, but the building blocks analogy captures a significant amount of why mastery learning work. The complexity of the relationships between the ideas can be teased out through intelligent curriculum design.
How (and what) to teach
The positive effects of mastery learning and neutral with regard to what is taught and how it is taught. Teachers do not need to change their pedagogy. Instead, they can use this mastery learning process to increase the effectiveness of their current practice. The key determinant is that the formative cycle is employed. Guskey explains that certain approaches to direct instruction, Rosenshine’s principles (5) for instance, are effective in learning basic skills. Cooperative learning (6) and constructivist approaches (7)(8) to learning appear to be more effective in helping students develop complex problem-solving skills.
While mastery learning is very successful for learning facts, definitions and procedures, it is not designed simply for this. Bloom, of course, wrote about higher order thinking skills in his Taxonomy of Learning. (Not the pyramid, which was nothing to do with him!). We can only assess what we have defined. So, if we want pupils to be able to, for instance, match multiple representations of quadratics then then we need to state this clearly in the course plan so that it is taught. That is, a learning experience is planned such that the desired outcome is arrived at. If we don’t teach it, then how is it fair to assess it? It is important that assessments do not just focus on what is easy to assess, but instead, what is most important for pupils to learn. This doesn’t mean that assessments only cover “seen questions”. The sort of task used in the assessment instrument need not be the same as those used in class. The key issue is that the understandings and behaviours are well defined in the curriculum. If this is the case then it is fair to assess them.
The Formative Assessments
Teachers should be well versed in responsive, in the moment, formative assessment. Use of mini-whiteboards, exit tickets and hinge questions, amongst other strategies should be part of the day to do day practice. These strategies give information to the teacher about the effectiveness of instruction as well as providing feedback to pupils on their learning.
Mastery learning builds upon these strategies. Pupils undertake formative assessments which give them precise feedback on what has been learned well in the recent instructional unit. These assessments also reinforce what the key learning points from the unit were. Perhaps most importantly, these assessments give pupils specific feedback on what they need to spend time studying.
The formative assessments issued at the end of the instruction unit are one of the key development priorities for teachers who begin to implement mastery learning. According to Guskey these assessments should:
- Last no more than 20 – 30 minutes to assess the instructional unit
- Cover all of the instructional unit’s important concepts
- Use a variety of assessment formats, for efficiency: multiple choice, matching tasks and extended response
- Be marked by pupils, in class, so that work on correctives/enrichments can begin immediately.
- Include “spiralling” items. That is items that refer back to key ideas from preceding instructional units. (This builds upon the idea of spaced practice and the testing effect). These sorts of questions should be no more than 20% of a formative assessment.
The mastery standard is the level of performance which indicate a unit has been well learned. If pupils do not achieve this then they do corrective work rather than enrichment. This level of performance is what we expect of all pupils. Typically 80% is used, however, we may chose to set this higher for particularly important ideas.
Block (9) argues that setting the mastery standard at 100% is not a good idea, as it can have negative consequences on pupil engagement. Also, it assumes that the assessment instrument is perfect, which is unrealistic. We need to allow for a margin of error when we assess.
One approach would be to split the formative assessment in sections – each related to a big idea and demand mastery of each section. In the main the mastery standard would be 80%, occasionally rising as high as 90% for essential basic skills. I favour this approach.
Typically, the first formative assessment would occur at the start of a lesson and pupils would then self mark their assessments. Cheating isn’t normally an issue as pupils realise it isn’t a competition and they only harm their own learning by doing so. The class would then split off into different groups working on correctives and enrichment activities.
Correctives and Enrichment
Managing the corrective and enrichment process is probably the most challenging aspect of the process of implementing mastery learning. It is also the most critical aspect of the process. In most cases around 10% more time is needed in the learning unit for the assessment, corrective and enrichment process. Mastery learning is not about going slowly. It is about teaching purposefully while allowing the extra time for the formative cycle. This small increase in required time per instructional unit is what results in the rise in attainment. I have argued, in other places, that if we have coherent curriculum design and make more explicit connections in learning, the amount of content to be learned is actually less than we can sometimes think. I believe the 10% extra time can be reclaimed in this manner.
The correctives that teachers use give specific guidance and advice on how they can correct their errors and remedy their learning. The correctives should use some alternative approach from that used in the original instruction. The way the idea was originally encountered is unlikely to be the successful model for pupils if they haven’t understood it in the original instruction unit. For instance, with addition of negative numbers, the correctives may involve the use of number lines rather than the first choice of algebra tiles. The correctives provide opportunities for pupils to practice and to think about the ideas which haven proven difficult, in preparation for the second formative assessment.
Planning correctives is hard! There are three types that we might consider.
- Activities led by the teacher
- Activities to be done with a friend
- Activities to be done alone
Although the correctives might straddle more than one category, the key thing is that they present the idea in a different way and involve different tasks. Pupils need to be “involved” with the ideas in a different way.
Reteaching is, of course, a very powerful corrective strategy. The teacher can work with the pupils requiring corrective intervention by explaining the ideas again, but in a new manner. It is important that the reteaching is not simply repeating the previous teaching in a way which is slower, but louder!
One common approach is to use some form of cooperative learning, pairing pupils who have achieved mastery, with those who have not. This re-teaching benefits the pupil who hasn’t yet succeeded, but also has positive learning benefits for the pupil doing the teaching. Having to rethink ones own ideas to explain to another has positive learning benefits. Other times flexible grouping may occur. If several classes are working at the same level, teachers may redistribute pupils temporarily, each teacher focusing with a group specifically on either enrichment or correctives.
Another corrective approach would involve pupils using self directed learning materials such as websites with videos and tasks, or materials from schemes such as the old SMP materials.
Not all of the pupils involved in correctives need to use the same approach. It might vary upon the severity of the correction required or the distinct parts of the unit which require improvement. The activities, mainly, involve some form of reteaching and then always involve some form of practice tasks.
Guskey argues that in conventional classes only the keenest pupils engage in corrective activities. Normally this happens outside of school time. In mastery learning, every pupil has the opportunity to engage in correctives, if required. Every pupil takes on the core behaviours of the hard working, high attaining pupils: reviewing learning, doing additional practice and aspiring for a top grade.
Correctives, can be sold to pupils as “the one step it takes to become successful”. Over time Guskey suggests that some of the corrective process can move from class time to become homework. I think this would be particularly true/necessary with senior pupils.
Enrichment activities on the hand come in two broad types:
- Activities to be done with a friend
- Activities to be done alone
Beyond issuing resources, the teacher should not have to spend significant amounts of time with those pupils who are doing enrichment. This does not mean that these pupils do busy work. The tasks they undertake should be engaging and meaningful. The enrichment is not necessarily about harder questions on the same material, although this might be one particular approach.
It may be empowering and rewarding for pupils to be given some choice over their enrichment activity. Some pupils may chose to work with friends doing correctives, as outlined above. In mathematics I can imagine using materials from places like NRICH, Don Steward or UKMT to provide more meaningful practice for pupils. Inquiry prompts can also provide great opportunities for pupils to develop their thinking. Pupils may be presented with self directed learning materials on something beyond the normal scope of the curriculum. For instance, while some pupils are doing correctives on long multiplication, the pupils doing enrichment might be teaching themselves how to do long multiplication in different bases.
After the Second Formative
Guskey describes something which I battled with when I last tried to implement mastery learning. After the second formative assessment, there will sometimes still be a pupil or pupils who have not yet reach the desired mastery standard on a section. Sometimes it might be possible to offer a third or fourth mastery assessment, but there is always some form of time pressure in schooling. Only 1-1 tuition doesn’t have this same pressure. In this instance, the teacher may have to move on as the majority of the class have achieved mastery on all of the sections of the instructional unit. If there are a significant number of the class who have not, then it may well be worthwhile investing one more lesson to try and remedy the situation. In the case of a small number of individuals, some of the burden must be picked up by the pupil doing work at home. Teachers can set more learning activities and correction tools before pupils come back and try again. Supported study classes may also be available for pupils to engage with after school. In my experience, in a related situation a letter or phone call home could be very effective in supporting pupils, as parents would ensure they engaged with the materials.
The fact that spiralling items are included on the formative assessments, means that the teacher can keep a note of topics where not all of the class have achieved mastery and continue to drop these into assessments. This is another chance for many pupils to demonstrate their learning.
A pupil may lose confidence having been unsuccessful for a second time. It is important the teacher highlights the progress made between the first and second assessments to say that the mastery learning process is actually working – even if the mastery standard has not yet been achieved.
Guskey states “the mastery learning process is not a perfect teaching and learning process”, he continues “ mastery learning does offer the vast majority of students the opportunity to make great progress in their learning and have many more successful learning experiences”. In my experience, it was rare for a pupil to continually fail to achieve the standard for mastery. If this was the case then there could only be one conclusion: the pupil was working on the wrong mathematics. It’s not that the pupil cannot achieve, it’s just that the point that this student has been studying in the curriculum is inappropriate for them.
This issue, once again, gives rise to the debate around setting and mixed attainment groupings. Attainment is the point in the curriculum up to which pupils have achieved mastery. Pupils start secondary school with a wide range of previous attainment levels. It is almost impossible to apply Bloom’s mastery learning to a mixed attainment grouping where the gap between pupils is as wide as 7 years of learning. I would suggest that pupils can be grouped in classrooms with other pupils working broadly at the same curriculum level. Some groups on Third Level (KS3), some on Second Level (KS2) etc. While the pupils working at Third Level may do enrichment into ideas beyond the scope of the curriculum, it might be possible that those working at Second Level may do their enrichment on ideas at Third Level. As such, those pupils who are working at a high standard consistently at Second Level can actually begin to close the gap with those working at the level ahead. Even those pupils who sometimes require correction, working at Second Level, or below, will be making progress in the curriculum, and achieving well.
Bloom suggests that no matter how pupils are grouped for learning, there are some who have significant barriers to learning. This may include up to the lowest attaining 10% of pupils, by arrival at secondary school. (10) Specific pedagogical approaches may be required for such pupils to make required progress. Englemann’s Direct Instruction might be one such approach.
I mentioned at the beginning of this article, I’ve never seen conventional classroom teaching either through setting or mixed attainment groupings have the sort of impact that is suggested can happen in mastery learning. In the imperfect implementation I led previously, there were significant gains, despite the fact there were various aspects which deviated from what I now understand the evidence to suggest is optimal. The aim has to be for every child to achieve.
Summative Assessment
Learning progress must eventually, be evaluated. While a formative assessment looks at every objective in the instructional unit, a summative assessment draws from a sample of objectives across four instructional units. These assessments are longer than formative assessments and broader in scope. The summative assessments can be used, in one sense, formatively. If a class perform particularly badly on a test item, the teacher has to decide whether the teaching was poor of if the test item was badly constructed. For instance, it may have been poorly worded. If the test item appears to be fine then the teacher should look to do some corrective work on the idea being assessed in the poorly done test item. Guskey suggests that pupils who have consistently achieved the mastery standard do not stress about the summative assessments and don’t require much revision. In my experience I found that no in-class time was required for revision prior to summatives, although pupils were issued with revision packs. In future I would take Guskey’s recommendation of issuing the formative assessment scripts and scores back to pupils so that they could plan which topics and ideas to target in their revision at home.
It is important to state that the idea of mastery standard still exists in the final exam. Pupils are expected to be able to perform to a high standard, drawing upon ideas from a range of instructional units. In my experience they normally do. Most pupils in a mastery summative assessment would see 65%, for instance, as a poor score.
I would suggest bands such as those in the table below. These may seem high compared to conventional grading boundaries, however, this is the point of mastery learning. It is the idea that all pupils can and should be achieving to a high standard. These scores are realistic in a mastery learning environment, where pupils have been taught at the right level. I have witnessed entire classes score in the A-B region regularly – not just the “top” sets. The ‘A’ grade is what is desirable. We already know this as, at SQA level the majority of pupils who achieve an A at National 5 go on to pass Higher. Around half who achieve a B go on to pass Higher and the majority of C candidates fail. Aspiring for higher levels of attainment in the short term supports progression at the next level.
|
A
|
> 80%
|
|
B
|
70 – 79%
|
|
C
|
60- 69%
|
|
D
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50 – 59%
|
|
F
|
< 50%
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Curriculum Planning
In the past I had advocated that we don’t have strict fixed timelines. However, to some extent this was naive. It did take the time pressure off teachers, and ensured that they had the freedom to teach for mastery, however, progress of some classes was too slow. Further, the date of final examinations in S4 or S5 is the target point of the curriculum – this is fairly static.
I do still believe that fixed day by day timelines are ineffective. Although I now think that having a notional number of days required for a class working at a specific level to study is probably sensible. This didn’t vary much year, on year, in my experience so there is no harm in having it written down – it is a suggestion of what is typically possible. The time that is usually required to take a group of pupils at this level to mastery in the instructional unit.
I think that a planner should lay out on a single page, the learning objectives for the unit, some teaching slides (not obligatory to use, but they will contain the key models and teaching points for reference) and a variety of tasks. Some of the task are for developing procedural fluency, some for conceptual understanding and others for problem solving. Of course, many will be dual purpose. Hinge questions and on-going assessment tools should also be clear. Links to the formative assessments and grids for recording scores, as well as links to relevant materials for correctives and enrichment activities should also be present.
The tasks which are available, in some sense, define the learning experience for pupils. The assessments should reflect the sort of experience pupils have had. There is much more to mathematics learning than becoming fluent with procedures. Mark McCourt has the wonderful framework of Teach, Do, Practice, Behave. (11) What I written below is a significant nod in that direction. Once again, his clarity of thought has influenced my understanding and ideas.
My time at La Salle is coming to and end, and I will be heading to a new school as head of department after Easter (or maybe August if the rumours about school closures are true). I continue to read, reflect and learn about mastery learning and will no doubt blog, in greater detail still about how this materialises, in my new post. I’m fascinated by the work La Salle are doing with Complete Maths. This is an incredibly powerful piece of software which literally does include “the whole of maths”. I can see it being a powerful tool in helping bring this process to life in my own classroom in the future, as it already does in schools across the country.
References:
(7) Brooks & Brooks (1993) In search of understanding, the case for constructivist classrooms
(8) G Forman (1987), The constructivist perspective
(10) B Bloom (1974) An introduction to mastery learning theory
Starting Points Maths 
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