Yesterday I attended the Scottish Maths Council annual conference at Stirling. This was, in my opinion, a pivotal day for mathematics education in Scotland as Professor Ruth Merttens made the first significant mention of Mastery learning in a Scottish context, in her keynote address. Professor Merttens’ speech was entertaining and passionately delivered. She is clearly an expert in the teaching of early years mathematics. However, having read her SMC journal article prior to attendance I knew that her reaction to Mastery, Shanghai and Singapore would be largely skeptical. I felt that a lot of this skepticism was, correctly, placed at the way these ideas have been politically hijacked in England. However, in the address there was no detailed explanation of what Mastery learning really is, what benefits it could have or any reference to its’ successes in a UK context. This was unfortunate given that Mastery is, at best, a peripheral idea in Scotland, and the majority of the audience most likely hadn’t read much into it or heard of it at all.
I am passionate about Mastery learning. My own presentation yesterday wasn’t about Mastery, per se, but more of a collection of thoughts on what makes effective mathematics teaching. (Slides available on request). The sort of approaches I discussed, as part of a well-implemented Mastery curriculum could have an incredibly positive impact upon pupil experience and final outcomes. I am currently working with some colleagues in the planning and development of a Mastery curriculum for CfE Third Level. I believe this to be only the second attempt at this in Scotland. I have spent the past seven months attending conferences in England, reading countless books, blogs and online articles, corresponding with experts in the field and reflecting upon Mastery learning and other aspects of evidence based approaches to mathematics teaching. Over the course of this and subsequent posts I will attempt to lay out the rationale behind my decision to proceed with a Mastery curriculum and discuss exactly how we are adapting this, and other notions, to fit with our own context.
What is Mastery?
Let us not confuse Shanghai and Singapore with Mastery. Mastery, as Professor Merttens correctly identified, is an Anglo-American concept, first prominently discussed by Bloom (of taxonomy fame). Shanghai and Singapore are cities that have mathematics education curricula, perform well in PISA studies. We can learn from the practice in Shanghai and Singapore but it would be incorrect to attempt to take their curricula transplant it to the UK as Professor Merttens correctly stated. We can consider Mastery without even thinking about East Asia. It isn’t all that relevant to my arguments for Mastery.
Mastery learning is the belief that students should master a skill before moving on to learn a new one. In contrast to the classic spiral curriculum, where students can race between topics without properly learning any of them, a mastery curriculum gives students the space to learn a skill, understand it conceptually, and practise until it’s automatic. Mastery is ambitious and inclusive for ALL learners at its heart. It is based upon the belief of Bloom that if teachers can provide the necessary time and appropriate learning conditions, nearly all students could reach a high level of achievement.
We can say a concept has been mastered if:
- The student can demonstrate or explain the concept orally, concretely, visually and abstractly.
- The student can apply the concept automatically, so that it is not dominating their working memory.
This idea of Mastery is very important because of its effect on working memory. Students who have mastered previous skills have their working memory freed to learn new ones, while students who haven’t get bogged down in the basics and don’t have the working memory space to learn something new. A student cannot afford to expend a high cognitive load on, for instance, multiplication facts, during a more complicated problem. The problem being studied should be the challenge. Not the underlying pre-requisite.
Bruno Reddy (@mrreddymaths) who is well known south of the border for his work in implementing a very successful Mastery curriculum at King Solomon Academy, London raises some flawed assumptions that teachers often have.
- ‘There is so much to learn that we have to keep moving on’
- ‘They might not get it now, but we’ll recap it again next year and *then* they’ll get it’
- Because pupils can follow the algorithm they have learned the concept.
I agree entirely with Bruno that this sort of thinking is a mistake.
In terms of the volume of work there isn’t actually as much as we would sometimes think. In my own department if a pupil embarks upon Third level at arrival in S1 they will have to learn 12 blocks of material in order to reach National 5 standard. For many pupils, in 5 or 6 years, they do not master 12 blocks. The progress required is only 2 to 2.4 blocks per year, for the median student aiming to leave school with National 5. In simple terms, 8 topics per year. How many Scottish school maths curricula have as little content as this for a year? We often rush on through a topic, but then end up having to repeat it the following year!
The idea of repeating the topic a year later is based upon a false assumption. Quite simply the chances are the student won’t improve the following year as there will be recap of the previous year and then a whole new set of concepts delivered.
If a student doesn’t master a topic first time around there must be a deficit in underlying skill, which is stopping the student accessing/succeeding with the topic. This is the start of a vicious cycle. The student fails to master the Third level content, due to Second level miscomprehension. A year or two later the same student finds him/herself studying the topic at Fourth level, with a shortfall in understanding which goes back two levels. This is then further exacerbated when said student is coached through National 4 and ends up in a National 5 class. By this point, all hope of achieving proficiency in the topic is gone as the work in question is now, often, three levels above the level the pupil is actually working at in that topic.
If you think this sounds far fetched then try a National 4 Mathematics or even National 3 Lifeksills paper, without preparation and coaching, with some of your senior N5 pupils who are on track to fail. Many of them will struggle to pass the levels below. Simply, because they haven’t mastered it.
Bruno’s final point is the crux of the matter in mathematics education for me. Just because a student can copy a series of steps, you have shown them, and can do this with practice questions with the only difference being the numbers doesn’t mean that they understand! What is required is the development of real understanding and conceptual development. Coupled with this, there is a requirement for teachers to utilise tasks which allow pupils to develop and demonstrate that real understanding. The work of Malcolm Swan showcases a variety of tasks, which encapsulate these principles.
How we teach really matters.
The tasks we assign really matter – maybe even more so!
The mastery cycle
The Mastery cycle is what distinguishes a Mastery curriculum. Professor Merttens overly simplified diagram and explanation wasn’t a fair representation of what a well-implemented Mastery cycle looks like. Mark McCourt @EmathsUK (check out his excellent La Salle education website) has collated 100 years of research into the diagram above.
Check out https://docs.google.com/drawings/d/1jxN60kmTjluFnKY7VTfLkadXt3U0OprCuKdwru0CT-w/edit?pref=2&pli=1 for the original version with comments (I have used some of these comments as stems for the following).
Some key elements of the Mastery learning cycle:
- Very precise learning objectives are part of curriculum design and shared with colleagues and pupils alike. In Scotland most of us have recently been through the idea of unpacking outcomes from CfE. In order to develop a Mastery curriculum those outcomes need to be decomposed even further.
- Diagnostic Pre-Assessment with pre-teaching. This part of the cycle is one of the key elements of a successful Mastery curriculum. The pre-teaching diagnostic is vital. We must ensure the initial conditions for success are set, before teaching of new material begins! Leyton, 1983 found that students who have reviewed missing prerequisite concepts and skills were far more likely to achieve mastery on new content. This is one of the big differences from conventional teaching. In a Mastery curriculum you do not start teaching new content unless students can demonstrate Mastery with the pre-requisites.
- High quality – whole class, initial instruction. The teacher is the single biggest resource in any classroom. A quality Mastery curriculum will encourage a variety of effective pedagogical approaches. The curriculum will support teachers in its abundance of excellent resources. The aim in our department is to develop this vision into reality. Every department meeting next year will have Mastery teaching on the agenda. This is new to all of us and we will need to share ideas, reflect upon and discuss curriculum content, resources and theory and how we are implementing all of this in our classes. Nobody wants an overly prescriptive curriculum. However, a helping hand with a substantial bank of excellent materials, which build upon our vision, surely cannot be a bad thing? Teacher autonomy is not in question. A starting point and teaching advice in a course plan does not eradicate this.
- Progress monitoring through regular formative assessment: Diagnostic assessments. Students must demonstrate a high level of performance AND understanding before proceeding. We have settled on a figure of 90% of the class demonstrating 80% Mastery before moving on. Mark makes the point that the threshold of 80% is only as useful as the design of the questions / tasks / problems. Designed badly, 80% (or any other score) can be utterly meaningless. This is such a key statement; it is easy to fool oneself into believing that the students have mastered a topic by setting an inadequate test and preparing students for it!
- High quality corrective instruction/interventions - this is not the same as re-teaching! It is easy to repeat the same words louder and call this corrective instruction, but it is not. A variety of approaches are required for pupils who have failed to master an outcome. Interventions include: additional teaching via alternative approaches/models and representations, peer support, small group discussions, homework (encourage parents), online resources – online lessons etc. Should be attainable in class time.
- Enrichment/extension activities. This poses a challenge as we design our curriculum. We must ensure that students are engaged in valuable learning experiences. However we do not proceed to the next topic until we have hit the 90% at 80% figure. Having students simply bide their time, doing more, harder problems or completing busy work while others are engaged in corrective instructions would be highly inappropriate. The enrichment tasks must provide opportunities for these students to extend their understanding and broaden their learning experiences.
- Retention – needs to be continually monitored and addressed. Adequate planning is required in curriculum design on how learned knowledge will be retained. (more of in a later post)
The notion of pace is something, which, in my opinion, is dangerous in the wrong hands. Depth and understanding matters most of all. As I said earlier we really should look at slowing down. 8 topics per year for the middle of the road kids is all we need to achieve. Yet the course outline will have as many as 15-20. A “teach once – deeply” approach means we have to alter how we look at our approaches to scheduling.
Mastery aims to avoid unnecessary repetition across years by regularly assessing knowledge and skills. Extra time is obviously required for pre-teaching diagnostics, development of deep pupil understanding, ccorrective instruction and remediation, problem solving and enrichment and formative assessment diagnostics. Academic and anecdotal evidence indicates that this shouldn’t concern us. Because pupils embark upon subsequent learning with mastery of pre-requisites, initial instruction in later topics can proceed more rapidly. This means that remediation time spent by students and teachers significantly decreases as the student reaches higher instructional units. That is, in essence, start slow and then accelerate later. Subsequent learning will be “easier” as pupils will have fewer gaps in pre-requisite knowledge. This then leads us to be able to be really ambitious with our final outcomes.
I will post our predicted curriculum time pathways in a later post, demonstrating how by slowing down to begin with, we can still arrive at the same end point. The arrival at the end point, contrary to now, should have been after a journey, which has fully equipped learners with the required skills for success.
I’d like to address two points raised by Professor Merttens yesterday. The claim was that meta-skills – ie those things which are hard to capture in a learning intention, don’t sit well within Mastery. Our model will address this by a variety of teaching approaches, interleaving of topics (which is a key part of the curriculum design) and a lot of rich tasks and problem solving. For me, the meta-skills argument is something which shouldn’t be an issue in a well-implemented Mastery curriculum. Professor Merttens talked about “instruction”. While it is true that much of the literature on Mastery uses this word, it can be readily changed to teaching. It is only instruction if you say that is all it is. What I have laid out above, and will do so in greater depth in future posts, is about outstanding teaching. Not lecturing! However, let us not pretend that direct instruction won’t be part of the mathematics classroom even within a Mastery curriculum. (John Hattie does show that is has a positive effect size after all.) It is just one of the many approaches that an effective teacher would utilise.
We must be clear that there is no Mastery Curriculum. Each and every implementation will be unique to the context. I would never expect a PT Maths to blindly implement some approach without making appropriate adjustments and enhancements.
In subsequent posts I intend to look more at the sort of lessons we want to include in our curriculum and the types of resources, which will be required to support this. I will also discuss our planned approaches to assessment (both formative and summative) as well as homework policy. Further posts will focus on academic supporting evidence, the development of a mastery culture within the department and the specific vision for what we are trying to achieve in our department. I will also look at how we will build in retention from topic to topic.
Finally I must thank the following, each of whom have, unknowingly in some cases, helped to develop my understanding of effective mathematics teaching, effective tasks, textbook-less curricula, curriculum design and culture. @MrReddyMaths @Maths_Master @mrbartonmaths @EmathsUK, @KrisBoulton, @BodilUK, Malcolm swan, @JohnMOxford, @MichaelOllerton and many others who I forget at this time.
I would love to hear from anyone who reads this. I am still at an early stage of my development of this curriculum and would welcome any comments and criticisms. I know I’ve a lot more to learn. However, I felt it imperative that Mastery had somebody arguing the case for it in Scotland.