As teachers, we work according to time constraints. For teachers at all Key Stages there is the need to stay in line with schemes of work in the drive towards consistency. For exam groups, there is the added pressure to cover the curriculum in time for the ever-earlier exams.
This creates a pressure to move on to the next topic or unit, even if we know some of our students have not done enough practice on the previous topic, have gaps in their understanding, or have not had time to review what they have learnt. Time pressure often results in us making a decision to move on to the next topic, even though we know it might not be in the best interests of our students.
There is a vast amount of research into the positive effects of practice and review on building understanding and memory, some of which I will outline in this post. And if we believe that review is crucial, we must find a way to create space for it in our daily lesson planning.
In his book Lean Lesson Planning, Peps Mccrea calls this the ‘economics of practice’ (p78):
“Spaced learning and practice take time. Often, our first response as teachers is: I’d love to do more of this, but I have so much to cover that I just don’t have time.
However, I’m not convinced this argument is economically sound. Not planning for memory will cost you more time in the long run. Your pupils will just end up revisiting topics again and again, struggling to develop longevity or fluency.”
Two types of daily review
Daily review means specifically allocating time in our daily lesson structure to review. In Principles of Instruction (2010), Rosenshine describes his research findings on daily review:
The most effective teachers in studies of classroom instruction understood the importance of practice and they would begin their lessons with a five- to eight-minute review of previously covered material. Some teachers would review vocabulary, or formulae, or events, or previously learned concepts. These teachers provided additional practice on facts and skills that were needed for recall to become automatic.
The plenary is often thought of as occurring at the end of a lesson, although mini-plenaries during a lesson are also common. However, it is not the timing that defines a plenary; it will be simpler to define a plenary as a review of previously learned material. A plenary can happen at any point during a lesson, or during a subsequent one.
Plenaries can be used to organize prior learning, address misconceptions, and make connections between what has been, and what is about to be learned. Plenaries might be in the form of exposition, worked examples or a set of questions for learners.
Although most teachers agree that plenaries are vitally important, they are often squeezed out if things don’t go according to plan - we will hold a plenary at the end of the lesson if there is time. I will argue that we must ensure that plenaries are given higher priority, and are not left to the end of lesson if there is time - plenaries are vital to building understanding and memory.
One role of the plenary is to organise, to help students make sense of what has been learned so far. In the book Mathematical Discovery, Volume 2 (1962), George Polya describes the importance of organising what has been learned (p85):
A well-stocked and well-organized body of knowledge is an asset to the problem solver. Good organization which renders the knowledge readily available may be even more important that the extent of the knowledge.
Learning – in the sense of gaining knowledge about the world - is frequently seen as a progression that starts with acquiring some basic facts and goes on through building ever more complex and advanced forms of knowledge out of, or on the grounds of, simpler forms. Although learning does work like this in some cases, and can be forced to work like it in others, we do not believe it to be the prototypical path to learning. In our view, learning proceeds, as a rule, from an undifferentiated and poorly integrated understanding of the whole to an increased differentiation and integration of the whole and its parts. Thus, learning does not proceed as much from parts to wholes as from wholes to parts, and from wholes to wholes. To put it very simply, in order to learn something you have to have some idea of what it is you are learning about.
The initial undifferentiated and unintegrated wholes that the learner grasps when embarking on learning something are likely to appear confused and erroneous when judged against the criteria of received wisdom. But on closer examination, these wholes, the learners initial ideas, turn out to be partial rather than wrong. They are the seeds from which valid knowledge can grow.
This view is corroborated in the book The Teaching Gap (1999), in which the authors Stigler and Hiebert analysed Maths lessons in Japan, Germany and the US following the 3rd TIMSS study. In the study, Japanese students outperformed their US and German counterparts. The authors observed that one of the main differences between Japanese and US/German lessons was what they call ‘content coherence’. They found that (p63):
96% of Japanese lessons contained explicit statements by the teacher connecting one part of the lesson with another, whereas only 40% of German and US lessons contained such statements.
Another important function of the plenary is for the teacher to address misconceptions, in reaction to assessment of pupil work. This can happen during a lesson, but I would argue that it is more effective to assess pupil work after a lesson (marking). The benefits of marking is that it allows the teacher to systematically check the understanding of all students, without the distractions of managing a full classroom, and then plan the plenary carefully to meet the needs of different students. See this article by David Didau @learningspy for more on marking as planning.
In summary: The plenary has a crucial function in helping learners to organize, to make connections, to address misconceptions, all of which help the student gain a deeper understanding – in short, to learn.
The second form of review we will aim to incorporate into our daily lesson structure is delayed practice. This is typically a short set of questions that require learners to recall previously learned material, so that learning is spaced out across weeks or months, designed to build memory. This is similar to a test in that it is a series of questions on a previously learned topic, but is different in that it may not be carried out under test conditions, must be ‘low-stakes’, and forms a part of the daily lesson structure.
The ‘spacing effect’ is not a new idea in education. It has been known for over 100 years that revisiting/recalling subject matter at a later time improves future recall. Educational Psychologist Robert Bjork describes it here as follows:
It is common sense that when we want to learn information, we study that information multiple times. The schedules by which we space repetitions can make a huge difference, however, in how well we learn and retain information we study. The spacing effect is the finding that information that is presented repeatedly over spaced intervals is learned much better than information that is repeated without intervals (i.e. massed presentation). This effect is one of the most robust results in all of cognitive psychology and has been shown to be effective over a large range of stimuli and retention intervals.
The theory of spacing would predict that an optimal learning schedule is one with expanding retrieval practice, rather than equally spaced practices. With successive practices, information is better learned and becomes inaccessible more slowly. As the greatest learning occurs when information accessibility is low (but not impossible), increasingly longer lags between retrieval practice lead to better long-term learning.
An NCER report Organizing Instruction and Study to Improve Student Learning (Pashler et al., 2007) suggests that teachers should employ spacing:
Make sure that important curriculum content is reviewed at least several weeks, and ideally several months, after the time that it was first encountered by students. Research shows that a delayed review typically has a large positive impact on the amount that is remembered much later. The benefit of delayed review seems to be much greater than the same amount of time spent reviewing shortly after learning.
Taking both of these sources into account, we at BSix aim to implement spaced learning in two ways. Firstly, we will set an assessment on each unit (around every 3 weeks). Secondly, and perhaps more importantly, we will carry out delayed practice as part of our daily lesson structure, revisiting topics from earlier in the year, months after first learning. Daily practice will happen before any other activities take place, giving it an important place in the lesson structure.
What, and how, should we practice?
In his book Why Don’t Students Like School? (2009), Cognitive Scientist Daniel Willingham states (p124):
Not everything can be practised extensively. There simply isn’t time, but fortunately not everything needs to be practised. If practice makes mental processes automatic, we can then ask: Which processes need to become automatic? In general, the processes that need to become automatic are the building blocks of skills that will provide the most benefit if they are automated. Building blocks are the things one does again and again in a subject area, and are the prerequisites for more advanced work.
Once we are agreed on the topics we need to practise, can we say what form this practice should take? Should we have lots of mixed questions or just focus on one topic?
In his book Practice Perfect (2012), Doug Lemov states that good coaches (teachers) differentiate between drill and scrimmage. Drill “deliberately distorts the setting in which participants will ultimately perform in order to focus on a specific skill under maximum concentration and to refine that skill intentionally”, where scrimmage “is designed not to distort the game but to replicate its complexity and uncertainty”. Perhaps we could think of drill as being a task that focuses on a single aspect of a topic, and scrimmage as a task covering many topics. He then asks:
When and how much should we drill? When is scrimmaging the best choice? [Award-winning Basketball coach John] Wooden provides some insight. Given the benefits of twenty players on the court with five balls in an engineered and predictable learning environment as compared to ten players on the court with one ball in an unpredictable series of interactions, Wooden chose to drill more – and scrimmage less – than most coaches. He was aware of this discrepancy and thought it was the key factor in his teams’ success.
After initial drilling, many people are eager to take their skills into the game or engage in scrimmage, but often this does not go smoothly. While scrimmage is often fun, it can lead to practice without a clear objective. It can be frustrating and chaotic. A graduated practice is what set the champion coaches apart from the merely good.
It must be better to practice specific, individual skills in order to diagnose accurately and intervene efficiently. There are also benefits to memory in concentrating on a low number of tasks/topics concurrently. A vast amount of research shows that working on a low number of tasks concurrently reduces the load on working memory, allowing us to become more fluent on the specific skill being learned.
The research into so-called variation theory compliments this view. The practical implications of variation theory are that we should design tasks that focus on, and then vary, the critical aspect of the object of learning, whilst keeping those less critical aspects invariant. The theory is then that leaners will focus on that which is being varied, thus making learning more predictable, and helping ensure that learners learn what you hope they will learn.
For more on this in a mathematical context, see the paper Seeing an exercise as a single mathematical object: using variation to structure sense-making (2006) by Anne Watson and John Mason. In this paper, Watson and Mason describe the process by which variation theory can be used to design exercises that improve students understanding. They conclude:
Our conclusions after three years of work … are that control of dimensions of variation … is a powerful strategy for producing exercises that encourage learners to engage with mathematical structure, to generalize and to conceptualize even when doing apparently mundane questions.
Note how the design of the questions employs the theory of variation. The variable is the value of the exponent; the invariant is the base of the exponent (4). Starting from (hopefully) securely known facts, the learner is encouraged to use these facts to answer questions involving fractional indices. Notice also how I have tried to avoid the temptation to include too many variables - I was very tempted to include negative indices here, but I can leave that for another example set; I wanted to try and ensure learners are only thinking about fractional indices.
The design of a set of questions like this raises fundamental questions about how we think students will learn. How would you have designed these questions? Would you have included the surds, or is this already too much variation? Is there too little variation? Would you have included an example that is not a simple power of 4, such as 6, in order to show students what ‘is not’ as well as what ‘is’, and to address the common misconception that 6 = 4^(1.5)? Would you have included the review question at the end? Would you have included more ‘problem-solving’ elements?
An interesting feature of the questions is the worked example for 32. I included this following the research on interleaving, which is described in Pashler et al. (2007):
When teaching mathematical problem solving, we recommend that teachers interleave worked example solutions and problem-solving exercises—literally alternating between worked examples demonstrating one possible solution path and problems that the student is asked to solve for himself or herself—because research has shown that this interleaving markedly enhances student learning.
Notice also that the learner is encouraged to generate two examples of his or her own. It has been shown that students learn mathematics effectively by testing what they have learnt through the creation their own examples. For more on this, see Mathematics as a Constructive Activity: Learner Generated Examples (2005), by Watson and Mason. The teacher could opt to use the more interesting student examples as a question for the rest of the class if he or she desires (a technique which matches what was witnessed in many Japanese lessons described in The Teaching Gap).
Finally there is a ‘review’ question at the end, where the learner is encouraged to reflect on what they have learnt. This is designed to make the user look at the exercise as a whole, in line with the previous discussion on building understanding. Here is what I have predicted learners might do when confronted with these problems (apologies to non-mathematicians):
Of course, we can never know how a student will approach a task. Different learners learn differently – see Marton and Booth (1997) for an excellent description of how different approaches to learning directly affect what is learnt - but that does not mean we should not carefully consider the various possible approaches of learners, try to predict how they might experience a given task, identify the misconceptions they might form, and try to build tasks that will reduce these misconceptions and guide them towards what needs to be learned. Variation theory is a means of attempting to guide students towards the critical aspects of a topic.
I have outlined a framework and rationale for incorporating daily review into our lesson structure. I propose two main types of review:
- Regular well-planned plenaries to build understanding.
- Delayed practice in the form of carefully structured example sets for the first 5 minutes of every lesson, to build memory.