H. L. Mencken 1880- 1956
Raising attainment in mathematics is a complex problem. Developing independent and resilient learners is also a complex problem. There are no quick fixes. Offering extra supported study sessions (or Higher revision weekends like the one I am on just now) can help a pupil approaching a final exam. However, I compare this to treating the symptom while avoiding the root cause. Of course, supported study and exam revision sessions are important and add value for learners. However, I offer the opinion that a bigger impact upon senior phase attainment will be given by, not only the initiatives put in place for S4/5/6, but also what goes on in the broad general education.
The BGE should not be the after-thought that it so often is. It is the fundamental underpinning. It is the basis upon which all future attainment is built. For a learner to reach a Higher or National 5 class with a chance of success in maths, the BGE experience has to be very effective. It must equip the learner with all of the fundamental explicit and implicit skills that are required to study the senior qualifications.
This complex problem does, indeed, require a complex solution. In this post I will discuss our curriculum design principles, discuss some work which exemplifies the quality of resources we will require to support our curriculum and then, finally, set out our rationale on ability setting.
Lessons – Pedagogy and Tasks
In my previous post I stated that the Mastery cycle was the defining characteristic of a Mastery curriculum. Implementing this alone, according to some research can have an effect size of 0.5. This is unsurprising, given the amount of formative assessment and feedback that is built in to the cycle.
Successful implementations of Mastery curricula not only involve the cycle, but also demonstrate similar pedagogical approaches, best encapsulated by two words: conceptual understanding.
With this in mind I propose the following as the guiding principles of our curriculum. These principles are a refection of various reading on effective mathematics teaching approaches and curricula.
The curriculum will:
1. Build on the knowledge learners bring to lessons.
The Mastery cycle, automatically considers pupil prior learning through the use of pre-teaching diagnostic assessments and corrective teaching. These diagnostics should also seek to expose any of the common misconceptions pupils have on topics.
2. Build procedural fluency from conceptual understanding.
There is a significant body of evidence which suggests that effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems. The development of conceptual understanding will promoted through teaching advice in our schemes of work and supported by high quality resources. A time investment will be required for professional dialogue and learning, to increase our own understanding of how best to aid the development of conceptual understanding.
3. Encourage effective questioning and formative assessment
To support teachers, our schemes of work will offer suggested question stems for each topic. A source of inspiration for this is the Kangaroo Maths schemes of work (freely available online)which support a Mastery approach to the English curriculum. Much of the questioning support material from these schemes can be incorporated into our own. Many of the question stems in Kangaroo are from the work of John Mason and Anne Watson – another couple of names unfamiliar in Scotland, who are outstanding academics in the world of mathematics education.
Dylan Williams mentions that formative assessment is an activity of the short term, medium term and long term. In my experience, many teachers are good at short-term formative assessment – it is almost instinctive. In our course planes, this will be supported by the incorporation of hinge questions and the question stems from Kangaroo in addition to other formative assessment tasks. Our medium term formative assessment will be supported by the diagnostic tasks that are built into the Mastery cycle. Longer term formative assessment will be supported by both learning logs and post assessment feedback sheets.
4. Promote cooperative small group work/rich collaborative tasks
While a Mathematics exam may be an individual pursuit the classroom is an environment where productive collaboration between learners can be very effective for learning and teaching. The use of open ended and or challenging questions are often very suited to such situations. These sort of tasks tackled in pairs, particularly, can have a significant impact on learner understanding. I tend to agree with literature that takes a constructivist viewpoint. A significant part of learning is based upon learners constructing their own meanings and interpretations. Learners will do this whether they have learned through a rich collaborative task, direct instruction or any other approach. A quality task, with proper scaffolding, used in a situation where on-task discussion is allowed can allow learners to develop their own meanings and interpretations. The role of the teacher is to ensure that the meanings do not include miscomprehensions. My personal preference is to have learners tackle problems individually initially and then ‘pair and share’ – this can ensure that both learners have engaged properly with the material.
5. Place an emphasis on methods rather than answers.
This is a fundamental in mathematics education. We already do it in class. “You must show all of your working” is the most common used phrase in maths classrooms. In secondary mathematics some errors are unsatisfactory e.g. basic arithmetic, however, as we advance further then the contrasting of approaches becomes the most interesting aspect. A simple example from my own teaching, inspired by the typical Shanghai lesson, is that when we are engaging with new material I will set a short initial task and will invite two pupils to do the task on the boards at the front while the rest of the class work. I will then write up my own solutions. This simple recent addition to my teaching armoury has had an incredible impact. This can result in the most terrific learning conversations. We have real pupil misconceptions on display. We sometimes have really unusual and alternative pupil approaches on the board. Very often teaching points I never even considered making are considered. Particularly with solving equations I find it useful to discuss two strategies, which have resulted in a correct answer – but open it up for discussion with pupils to decide which they think is more elegant and desirable. Of course there is the issue of classroom culture to consider. This approach might not be able to launched with an S4 class midway through the term. However, I have S1 and S3 classes of very different levels of ability who have really enjoyed and benefited from this approach after recent adoption.
6. Promote the Use of mathematical representations, the connections between representations and create relational knowledge between mathematical topics.
There is a range of evidence, which suggests effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving. High quality tasks, such as those mentioned below, can facilitate this. Our curriculum will be built with the idea of making connections between representations built into schemes of work. For instance, in addition to the allocated separate fractions, decimals and percentages teaching times there will be an additional part of the course plan where several lessons are spent looking at how they are different representations of the same thing. See below.
7. Support productive struggle in learning mathematics
Effective teaching of mathematics consistently provides learners with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships. Learners must also develop the ability to persevere and accept “stuckness” as a natural part of doing mathematics. Appropriate problem solving activities for all learners from the first moment of first year have to be encouraged. Learners need to be able to think for themselves and have the courage to attempt problems. I am sure you have heard “we don’t have an example like that in our notes” on many occasions. The reality is – we cannot, through exemplification alone, prepare learners for every possible situation that will arise in an examination. We can however, equip learners with experiences where they have to apply their knowledge to unusual situations. This can only serve to develop their confidence, resilience and enjoyment of questions where reasoning is required.
8. Support memory and recall
We need to take account of the research on how learning and memory operate (more of in a later blog post). The interleaving of topics will be a fundamental aspect of our curriculum. Simply put, when teaching new content the application of previously learned material will be a key component of the learning. Where possible pupils will encounter problems that develop and consolidate their algebra, negative numbers, fractions, decimals and percentages within the context of the current topic. For example: algebraic expressions in angle problems or area problems with fractional lengths. The key skills must be encountered in as many contexts as possible, not just when being taught explicitly. This will ensure that the key skills are kept fresh. Memory and recall will also be addressed through our homework policy and scheduled revision within the course plans.
9. Promote a culture were teachers allow learners to confront difficulties rather than seek to avoid or pre-empt them
The report Mathematics Matters from the NCETM states that ‘effective teaching challenged learners and has high expectations of them. It does no seek to ‘smooth the path’ but creates realistic obstacles to be overcome. Confidence, persistence and learning are not attained through repeating success by struggling with difficulties.’ This resonates with my view that we shoudn’t be looking to make the maths easier, but raising the ability of the learners to engage with it. There should be no magic tricks in maths. Ideas such as cross-multiplying to solve equations with fractions simply avoid confronting the real issue: lack of comprehension of fractions and equation solving. Instead of viewing the problem as an opportunity to develop skill in both of these vital areas the teacher is simply making maths easier. However, the learning is of limited, if any, sustainable value as this trick – like many others is applicable in only certain situations. Similarly, in my view it is shocking that a Higher candidate would be taught to do all trig equations in degrees and then convert to radians at the end – this is a convoluted practice which allows the learner to continue to hide from fractions.
Malcolm Swan as an inspiration
Malcolm Swan is relatively unknown in Scotland – at least judging by the small show of hands when I asked delegates at the Stirling conference if they had heard of him. This is unfortunate as, without exaggerating, his work is world class. This section will summarise a few of his ideas about mathematics teaching. His work has provided many points for consideration and much inspiration for effective learning and teaching. His work will provide a point of reference in the development of our Mastery curriculum.
Swan discusses two broad views of mathematics teaching: A transmission culture or a collaborative, challenging culture.
In a transmission culture:
Mathematics is seen as
- a body of knowledge and procedures to be ‘covered’
- an individual activity based on listening and imitating
- structuring a linear curriculum for the learner
- giving explanations and checking these have been understood through practice questions
- ‘correcting’ misunderstandings when students fail to
‘grasp’ what is taught
In a collaborative, challenging culture:
Mathematics is seen as
- a network of ideas which teacher and students construct together (the teacher is seen as a facilitator of pupil learning)
- a social activity in which students are challenged and arrive at understanding through discussion
- non-linear dialogue in which meanings and connections are explored
- recognising misunderstandings, making them explicit and learning from them
I think, in reality, these perspectives can be placed on a spectrum, with many teachers somewhere between the two. It is my opinion, after a lot of reading and a decade of teaching that the collaborative, challenging culture is the one that should be the aspiration for the learning and teaching within my own department. This fits comfortably our principles for curriculum design.
Much academic evidence supports my opinion. Boaler(1997) states that Learners who had engaged in collaborative work develop relational forms of knowledge that are more useful in a range of different situations (including traditional examination questions). Boaler also states:
- Textbook exercises or discussion of mathematical ideas are not merely vehicles for developing knowledge, they shape the forms of knowledge produced.
- Learners, who work through textbook exercises, find it difficult to use mathematics in applied or discussion-based situations.
In reality learners need a period of practice to hone and perfect routine skills. Fluency only comes through practice. I doubt there are many maths teachers who would disagree with that. Textbook exercises themselves are not intrinsically a bad thing. They can be extremely useful for learning and teaching. The danger, however, is that the textbook can become the course. In a number of our prominent Scottish texts problem solving, tasks that aid conceptual development, and high quality non-repetitive tasks are a rarity. The second bullet point, in particular, should be a point of reflection for many of us in Scottish mathematics education. No book alone should be the basis of our curriculum. The same can be said for any set of worksheets. A wide range of resources is required to support the fulfillment of the ambitions we have.
Swan proposes various styles of task:
- Evaluating mathematical statements
Swan suggests asking learners to decide if a given statement is Always, Sometimes or Never true? Having used his examples and similar of my own making I am convinced that these are of a high value. Swan states that learners are encouraged to develop ‘rigorous mathematical arguments and justifications and examples and counterexamples to defend their reasoning.’
- Classifying mathematical objects
- Interpreting multiple representations
- Creating and solving problems
The examples above a small snapshot of a wide range of Malcolm Swan's work. John Mason and others have produced similar standard of materials. These will in some cases be directly useful for us, and in other cases will serve as a basis for the development of our own tasks.
The examples above a small snapshot of a wide range of Malcolm Swans work. John Mason, Ann Watson and others have produced similar standard of work – be it tasks, questioning advice or pedagogical approaches etc. These will in some cases be directly useful for us, and in other cases will serve as a basis for the development of our own tasks.
I must also mention Don Steward and his outstanding blog ‘Median’. This is a source of excellent materials that will be of major use in the development of our schemes of work.
The example below will work nicely in our topic on the multiple representations of fractions, decimals and percentages.
Many people have asked, since my first blog post on Mastery, how we will be able to slow down but meet time pressures etc. As I stated previously, a lot of the pressure is put on ourselves. We have 5 blocks/phases of material in Third Level. Within that there are 4/5 topics in each. The key-differentiating factor is pupil pace of progress not expectation. I have included a PDF with our pathways. I am happy to take questions on these – but can assure you that these are a slowed down version of what has been happening previously. It should be noted that we do not present anyone for an S4 exam in our school, except S4 leavers. Unlike the west coast school you may have heard about in the news that tried this approach, the impact has been very positive to date. This model being in place in the school offers the maths department greater flexibility in panning long-term pupil pathways.
Setting – A confession of being pragmatic
I have a confession to make. A large number of the Mastery implementations I have read about are firmly run with mixed-ability classes. We will, against that notion, continue to set the classes. A number of considerations have led to this decision.
In the existing curricular model, the department has a “common course” which runs for the first few months of S1. That is all of the pupils, unset, do the same work – with everyone sitting the same test. I must admit that I didn’t enjoy my experience of it. I didn’t know how to resource it, manage the lessons effectively or give all of the pupils what they required. After ten years of teaching rigidly set classes, my own skillset was inadequate. Looking at pupil attainment it was clear that I likely wasn’t the only one of my colleagues who struggled to make it work. It was also a horribly negative starting point for a number of our learners.
All of the above is not to say mixed ability couldn’t work here, however, having pupils on First, Second and Third level in one room is something which most maths teachers in Scotland are unaccustomed to. In order to attempt that level of mixed ability teaching we would need to spend a lot of time developing new strategies and skills. Given that we will be investing so much time in developing our skills in Mastery and conceptual development, I feel this is simply too much to ask of colleagues at this point. Further, if implemented badly, any positive effect, which can come from mixed ability, would undoubtedly be wiped out. As I said in my previous post, there is no Mastery Curriculum. The principal teacher building a Mastery curriculum needs to be comfortable that the conditions for a successful implementation are in place and make any relevant adaptations as needed. Ultimately moving to an unfamiliar curriculum model – Mastery, AND moving to mixed ability in one year would be too much, too soon for all of us.
In future as we develop our skillset at operating multi-activity lessons, which will be required by our Mastery cycle (some pupils on rich tasks, some on re-teaching) then moving to true mixed ability might seem a smaller step to take.
I must emphasise: the decision not to do mixed ability classes is due to the fact that in the short term, I do not think it will make a positive impact in our context. Put simply: is it better to have a skilled teacher delivering lessons to a rigidly set class or a teacher who is unskilled working with a mixed ability class? I suggest that in this case it is best for the learners that the classes stay rigidly set. I am very open to how we arrange the classes but feel that, in our context, this is for the medium term.
A final point on setting is that, even though classes will be set, the differentiation is through pace of progress. Not through content or expectation of Mastery. Time is the variable, not the standard expected. Mark McCourt uses the excellent analogy of the driving test to describe this. The standard required to pass a driving test remains static. Some people achieve a pass after 10 lessons while others take years. However, each individual has passed the test, the passes are considered of equal value with both being awarded driving licenses.