What remains when the lesson ends

What the science of memory, mastery and transfer tells us about effective teaching

Recently, while tidying up some old filing, I came across a collection of academic papers that I have accumulated, read and returned to over many years. Each has influenced my thinking about teaching and learning, sometimes profoundly, and I thought it might be useful to bring a selection of them together and explore what they tell us about the practice of teaching.

Anyone who knows me will be familiar with my admiration for John B. Carroll, whose 1963 paper A Model of School Learning remains one of the most important influences on my educational thinking. I have returned to it countless times, and its central insight, that differences in learning are intimately connected to the time required and the conditions provided, continues to shape how I think about schools, curricula and the responsibilities of teachers. But Carroll is not alone in this collection. The work of Perkins and Salomon on transfer, Baddeley and Hitch on memory, and various researchers investigating mathematical understanding and problem solving have been similarly influential in helping me make sense of what happens when we teach.

Reading these papers together again, I was struck by how closely their ideas connect, despite their different origins and purposes. They illuminate different aspects of a question that has occupied much of my professional life: what actually happens when somebody learns something, and what must a teacher do to make that learning possible, enduring and useful? What follows is an attempt to draw some of those threads together. It is not intended as a comprehensive review of the literature, but rather a reflection on a handful of papers that have helped me think more clearly about the extraordinary business of teaching.

 

We devote enormous attention to what teachers do, yet comparatively little to the precise nature of the change we expect their teaching to produce. Lessons are planned, explanations delivered, activities completed and assessments administered. Curriculum content is covered, progress is recorded and pupils move onwards. Yet none of these things, individually or collectively, establishes that the intended learning has occurred. Or, indeed, that there ever was any learning intended.

A pupil may solve a mathematical problem successfully on Monday and be unable to solve an equivalent problem on Friday. Another may complete a page of fraction calculations accurately but fail to recognise the same mathematical relationships when they appear in a problem involving measurements. A third may reproduce a teacher's explanation almost verbatim without possessing the understanding necessary to use it independently. In each case, the outward appearance of successful teaching conceals a more complicated reality.

These are not peripheral difficulties. They concern the very purpose of education. Schools exist to bring about lasting changes in what pupils know, understand and can do. The question is not simply whether pupils can perform successfully under the conditions in which something has been taught, but whether the knowledge acquired becomes a durable and useful part of their intellectual equipment.

A substantial body of research, developed across several decades and drawing on educational psychology, cognitive science and mathematics education, provides a compelling framework for understanding this problem. The work of John B. Carroll on school learning, David Perkins and Gavriel Salomon on transfer, Alan Baddeley and Graham Hitch on memory, and researchers investigating mathematical schemas and representations points towards a conception of teaching that is both more ambitious and more precise than much contemporary pedagogical discussion allows.

At its centre is a simple proposition: effective teaching must change the internal organisation of knowledge in the pupil, and do so in ways that make that knowledge increasingly accessible, connected and applicable.

The problem of transfer

In 1989, Gavriel Salomon and David Perkins published a paper with an arresting title: Rocky Roads to Transfer: Rethinking Mechanisms of a Neglected Phenomenon. Their subject was one of the oldest and most troublesome questions in educational psychology. Why is it that people so frequently fail to apply something they have learned when they encounter a situation in which that learning would be useful?

The problem had been recognised long before. At the beginning of the twentieth century, Edward Thorndike challenged the assumption that studying intellectually demanding subjects, such as Latin, necessarily strengthened general powers of reasoning. The evidence for such broad mental improvement proved disappointing. Learning something in one context did not reliably produce improvements in apparently related activities.

The difficulty is profound because almost everything education hopes to accomplish depends upon some form of transfer. We teach mathematics partly so that pupils can use mathematical ideas beyond the particular exercises in which they first encountered them. We teach history in the expectation that pupils will eventually be able to interpret unfamiliar historical circumstances. We teach scientific principles because their value extends beyond the experiments through which they were introduced.

If learning remains confined to the circumstances of its acquisition, much of the wider ambition of education remains unfulfilled.

Perkins and Salomon (1989, 1992) proposed that transfer could occur through two distinguishable mechanisms. The first, which they called low-road transfer, involves the relatively automatic application of well-practised knowledge or behaviour when sufficiently similar circumstances are encountered. A learner who has become fluent in a mathematical procedure may recognise an opportunity to use it without substantial conscious deliberation. Repeated practice across varied but related situations makes the response increasingly accessible.

The second mechanism, high-road transfer, requires deliberate intellectual effort. The learner must abstract a principle from one situation, recognise its relevance to another and consciously establish the connection. Where low-road transfer depends substantially on well-established routines and familiar cues, high-road transfer depends upon the capacity to look beyond surface appearances.

Consider a pupil learning proportional reasoning. Repeatedly solving problems involving recipes may establish considerable fluency in adjusting quantities. Yet a problem involving map scales, exchange rates or similar triangles may present an entirely different appearance. Successful transfer requires the pupil to recognise that the same multiplicative relationship is operating beneath the different scenarios.

The distinction is important because it exposes the inadequacy of two competing simplifications of teaching. Practice alone does not guarantee that pupils will recognise the wider applicability of what they have learned. Equally, discussing general principles does not eliminate the need for extensive practice through which knowledge becomes sufficiently secure to be used reliably.

The two mechanisms are complementary. One develops fluency; the other supports deliberate abstraction. Effective teaching must attend to both.

Subsequent research has complicated the picture further. Susan Barnett and Stephen Ceci (2002), reviewing a century of work on transfer, argued that the familiar distinction between near and far transfer conceals considerable ambiguity. Transfer can involve changes in knowledge domain, physical setting, social circumstances, time, functional purpose and other dimensions. Two situations may be close in one respect and distant in another.

A pupil applying a mathematical technique to an unfamiliar question in the same classroom is not undertaking the same kind of transfer as someone applying that knowledge years later in a workplace. Both may involve knowledge acquired through schooling, but the conditions under which it must be retrieved and applied differ substantially.

Barnett and Ceci's contribution is therefore more than a refinement of terminology. It reminds us that transfer cannot be treated as a single accomplishment. The ability to apply knowledge in one unfamiliar situation does not establish that it will be available in every other situation. Nor is it reasonable to expect all forms of transfer to arise through identical teaching methods.

This makes transfer an essential consideration in curriculum design. Teachers must ask not only what knowledge pupils should acquire, but also where and under what circumstances that knowledge should eventually become useful.

Beneath the surface of a problem

The importance of recognising underlying structure becomes particularly apparent in mathematics.

Consider three questions. A school has 240 pupils, of whom three-fifths attend a concert. A farmer plants three-fifths of a 240-hectare field. A manufacturer completes three-fifths of an order for 240 components. The circumstances differ, but the mathematical relationship is identical.

For an experienced mathematician, this common structure is immediately apparent. For a novice, the surface details may dominate. The pupil may attend to the school, the field or the factory without recognising the shared multiplicative relationship.

This difference between surface appearance and underlying structure lies at the centre of research into schema-based instruction.

A schema is an organised structure of knowledge that enables a learner to interpret information and recognise meaningful relationships. In mathematical problem solving, schemas help pupils identify the relationships between quantities and connect those relationships to appropriate mathematical operations.

Rather than teaching word problems as a succession of disconnected exercises, schema-based instruction seeks to develop recognition of common mathematical structures. Pupils might learn to distinguish between situations involving changes in quantity, comparisons between quantities and the combination of parts into wholes. Diagrams can make these relationships visible, allowing pupils to connect the information presented in a problem with an equation expressing its mathematical structure.

Research by Fang and colleagues (2015) examined a simplified form of schema-based instruction with elementary pupils. Their intervention did not require pupils to identify formal problem categories, but instead concentrated on the relationships necessary to solve the problems. The participating pupils improved their performance and maintained the acquired skills.

The finding is interesting because it raises an important distinction between possessing a mathematical schema and merely knowing the name of a problem category. A pupil might correctly classify a question without understanding its mathematical structure. Conversely, a pupil might understand the relevant relationship without being able to supply a formal label.

The educational objective is not classification for its own sake. It is the development of organised knowledge that makes mathematical situations intelligible.

Julie Reneau's (2012) investigation of fraction word problems provides a particularly revealing qualification. Her study combined schema-based instruction with concrete and virtual manipulatives, visual representations and equations. All five participating pupils made some progress on problems resembling those encountered during instruction. However, the three pupils who successfully connected problem schemas with equations demonstrated especially substantial gains.

When pupils encountered problems with novel features, including difficult vocabulary, irrelevant information or unfamiliar conceptual demands, the improvements were much smaller.

The result illustrates both the promise and the limits of carefully structured teaching. Pupils can acquire useful methods and representations without necessarily becoming capable of applying them under substantially changed circumstances. The crucial achievement appears to involve more than familiarity with the teaching materials. It requires the development of connections between the representation, the mathematical relationship and its symbolic expression.

A meta-analysis by Rayner, Bernard and Osana (2013) provides broader, although qualified, support for this direction of teaching. Drawing on 53 studies and 116 effect sizes, the researchers examined interventions intended to promote transfer in mathematics. Approaches involving schema activation, particularly when combined with worked examples, produced comparatively large reported effects.

The findings were highly heterogeneous, and the different categories of intervention should not be treated as a definitive ranking of teaching methods. Nevertheless, they suggest that helping learners recognise and organise mathematical structure can be an important means of improving their capacity to apply knowledge.

The implication extends beyond mathematics. Knowledge becomes more powerful when it is organised around meaningful relationships rather than retained as an accumulation of disconnected examples.

The purpose of representations

The research on schemas also helps illuminate a longstanding question about the use of physical materials, diagrams and symbolic notation.

In mathematics education, the concrete-pictorial-abstract approach has become familiar. Pupils encounter mathematical ideas through physical objects, then through visual representations, and eventually through conventional mathematical symbols.

There is an intelligible rationale for this approach. A physical model may make a relationship accessible that would otherwise remain obscure. A diagram can preserve the important features of that relationship while removing distracting details. Symbolic notation can then express the same structure with greater economy and generality.

Yet the presence of a representation does not guarantee understanding.

A pupil may manipulate fraction tiles correctly without recognising the relationships they embody. Another may draw a bar model successfully because the teacher has demonstrated the procedure, while remaining unable to explain why the diagram represents the problem. A third may understand an equation perfectly well and gain nothing from being required to reconstruct it using physical objects.

Reneau's findings are instructive here. Concrete and virtual representations assisted some pupils but not others. The strongest improvements were associated with pupils making meaningful connections between schemas and equations.

The purpose of a representation is not to provide an alternative activity, nor to satisfy a prescribed sequence of instructional stages. It is to illuminate mathematical structure.

This suggests a more flexible conception of representational teaching. Sometimes pupils need to begin with physical objects. Sometimes a diagram provides the most direct route to understanding. Sometimes symbolic notation itself offers the clearest representation, and a physical or visual model becomes useful only when a misunderstanding needs to be exposed.

The movement between representations need not be exclusively in one direction. A pupil might move from an equation to a diagram, from a diagram to a verbal explanation, or from a physical arrangement to a general mathematical statement.

What matters is that the pupil recognises the invariant relationship expressed through these different forms.

Mathematical abstraction is not achieved merely by removing the physical materials. It is achieved when the learner understands what remains constant as the representation changes.

Time should be the key variable

If the research on transfer and schemas concerns the organisation and application of knowledge, John B. Carroll's A Model of School Learning (1963) addresses the conditions under which learning becomes possible.

Carroll's contribution was deceptively radical. Rather than treating differences in achievement as the straightforward expression of differences in fixed intellectual capacity, he proposed a model in which learning depends substantially upon the relationship between the time a pupil needs and the time that pupil actually spends engaged in learning.

He identified five variables: aptitude, perseverance, opportunity to learn, ability to understand instruction and quality of instruction.

The most consequential feature of the model is Carroll's conception of aptitude. Instead of defining aptitude simply as the amount a pupil is capable of learning, he considered it principally in terms of the time required to learn a particular task under favourable instructional conditions.

This reframes the familiar problem of differences between pupils.

In conventional schooling, the time allocated to learning is frequently held relatively constant. Pupils encounter the same curriculum content during the same period, complete an assessment at approximately the same point and then move onwards. The amount learned consequently varies, sometimes substantially.

Carroll's model asks: What if the intended learning were held relatively constant, while the time and support available to accomplish it were allowed to vary?

The implications are profound. A pupil who has not learned something within the allocated time has not necessarily demonstrated an inability to learn it. The pupil may require further explanation, additional practice, a different representation, better feedback or simply more time engaged in productive learning.

Equally, a pupil who masters something quickly should not be required to spend unnecessary time repeating material already understood. The model permits differences in learning rate without requiring those differences to become permanent differences in educational ambition.

There are important qualifications. Carroll did not suggest that time alone guarantees success. He distinguished elapsed time from time genuinely spent learning, and recognised that the quality of instruction and the learner's ability to understand it influence how much time is required. Perseverance also matters. Pupils may disengage when learning becomes frustrating, confusing or insufficiently challenging.

Nor did Carroll claim that every individual could necessarily master every conceivable task given unlimited time. His model is a theoretical account of the factors influencing school learning, not a proof of universal learning capacity.

Nevertheless, it provides a powerful challenge to an educational system organised primarily around the passage of time.

If the purpose of teaching is to bring about learning, then the failure to learn should prompt an investigation into the conditions of learning. It should not automatically result in a judgement about the limits of the learner.

I have long argued that all pupils must be part of the intellectual life of the school. Not merely present in classrooms, occupied with activities or accommodated within an educational system, but genuinely engaged in the acquisition of powerful knowledge. This is as true of the pupil who struggles to grasp an elementary concept as it is of the pupil whose understanding races ahead of the curriculum. Both are entitled to teaching that takes their intellectual development seriously. Carroll's work reminds us that differences in the rate at which pupils learn should not become an excuse for differences in the ambition we hold for them.

The model also exposes a tension at the heart of curriculum design. A curriculum that specifies an ever-expanding quantity of content without providing sufficient time for its acquisition risks confusing exposure with education.

Thomas Good (2008), writing about mathematics teaching in American elementary classrooms, raised precisely this concern. He argued that successive educational reforms had repeatedly added new expectations to the curriculum without adequately considering what should be removed. The result was an increasingly crowded curriculum, with insufficient time to teach everything well.

Good's criticism is especially significant when considered alongside Carroll. Curriculum breadth and learning depth cannot be treated as independent variables. Every additional demand competes for instructional time, and the consequences are particularly serious when new material depends upon knowledge that has not yet been securely acquired.

Ambition cannot be measured simply by the quantity of content included in a curriculum. A curriculum is ambitious when it enables pupils to acquire increasingly powerful knowledge. A lengthy catalogue of material that pupils encounter but do not adequately learn may be impressive administratively while remaining educationally impoverished.

The misleading comfort of immediate success

There is another difficulty in establishing whether teaching has succeeded. Learning is not directly observable. What teachers observe is performance, from which learning must be inferred.

The distinction is particularly important because performance is influenced by the immediate circumstances in which it occurs.

In 1993, Alan Baddeley and Graham Hitch revisited the recency effect, the well-established tendency for recently encountered information to be recalled particularly successfully. Earlier explanations had often associated the effect with information remaining temporarily available in a limited-capacity short-term memory store.

Baddeley and Hitch proposed a broader account, connecting recency with implicit learning and explicit retrieval. Their argument sought to explain recency effects across different timescales and cognitive circumstances, rather than treating the phenomenon simply as the persistence of the last few items in a temporary store.

The paper was not a study of classroom teaching, and it would be mistaken to derive a particular pedagogical programme directly from it. Nevertheless, it contributes to a wider understanding of why the accessibility of information at one moment should not be equated with its secure and enduring availability.

A pupil who successfully reproduces a mathematical procedure immediately after an explanation may be benefiting from the highly favourable circumstances of recent exposure. The relevant knowledge has just been activated, the teacher has supplied the context, and the sequence of activity may strongly suggest which method is required.

Remove those conditions and the task changes.

Several days later, the pupil must retrieve the knowledge without the same immediate cues. Several weeks later, the pupil may need to distinguish it from other procedures learned in the meantime. When the knowledge appears within an unfamiliar problem, the pupil must first recognise its relevance before attempting to use it.

These are increasingly demanding accomplishments, and success at the first does not guarantee success at the others.

This is why assessment must be understood as more than a means of checking whether pupils can reproduce recently taught material. Immediate checks are valuable because they reveal misunderstandings while corrective teaching remains possible. But they provide only part of the evidence needed to establish learning.

Delayed assessment helps establish whether knowledge remains accessible. Mixed and unfamiliar tasks help reveal whether pupils can identify when particular knowledge is relevant. Explanations and comparisons can provide evidence about the organisation of understanding.

None of these forms of assessment is individually sufficient. Together, however, they offer a more convincing picture than immediate performance alone.

The implication is not that teachers should distrust successful classroom performance. It is that they should interpret it carefully. What a pupil can do with extensive prompting, immediately after demonstration, is different from what the pupil can do independently, after a delay, in an unfamiliar situation.

Teaching should aim to close the distance between those accomplishments.

Beyond the search for the perfect method

These findings also cast doubt on the enduring tendency to organise educational debate around competing instructional formats.

Whole-class teaching is contrasted with tutoring, explicit explanation with investigation, procedural practice with conceptual understanding, and teacher direction with pupil curiosity. Such distinctions can be useful when describing particular approaches, but they become misleading when treated as fundamental divisions between good and bad teaching.

Thomas Good's analysis offers an important corrective. Drawing on research into mathematics instruction, he argued that the quality of an instructional format matters more than the mere fact of its adoption. Small-group teaching can be effective or ineffective. The same is true of whole-class instruction, individual practice and technology-supported learning.

Good also emphasised active teaching, while acknowledging that even this could be implemented well or poorly. His argument was not that one method should replace all others, but that teachers need to attend carefully to what pupils are learning and how instructional choices influence that learning.

This conclusion is consistent with the wider research.

Explicit explanation may be particularly valuable when pupils lack the knowledge necessary to infer important relationships independently. Worked examples can reveal the structure of unfamiliar problems. Practice can establish fluency. Comparison and discussion can support abstraction. Physical representations can make otherwise inaccessible relationships intelligible. Independent problem solving can provide opportunities to apply and consolidate what has been learned.

These activities serve different purposes. Their value depends upon the relationship between the learner's existing knowledge, the intended learning and the demands of the task.

A novice confronting an unfamiliar mathematical concept may require substantial guidance. A pupil who has already acquired the relevant knowledge may benefit from greater independence and more demanding applications. The same pupil may require different forms of teaching at different stages of learning.

This is an argument I have developed in my own model of learning, which describes seven stages through which a learner may progress, from initial awareness of something through to maturity of understanding. Each stage presents different intellectual demands and, consequently, calls for different pedagogical choices. The explanation, modelling and guidance appropriate to a novice are not necessarily those that will best serve a learner approaching expertise. Nor should we assume that a pupil who has developed expertise in one domain will be similarly advanced in another. The task of the teacher is to understand where the pupil stands in relation to the particular knowledge being taught, and to select the teaching that will most effectively move that pupil onwards.

It follows that no instructional technique can be judged adequately without considering the learning it is intended to produce.

This also restores an important place to the teacher's professional judgement. Teaching is not the mechanical implementation of a sequence of approved activities. It requires an understanding of the subject, the pupil and the relationship between them. The teacher must determine what is already known, what needs to be learned, which representations and explanations will be most helpful, how much practice is necessary, and what evidence will demonstrate that the intended change has occurred.

The expertise lies not in allegiance to a method, but in the intelligent selection and adaptation of methods in pursuit of learning.

The architecture of a curriculum

If knowledge must be organised, practised, retained and transferred, then curriculum design becomes a matter of constructing the conditions under which increasingly sophisticated understanding can develop.

A curriculum cannot be understood adequately as a list of topics arranged into a sequence of lessons. It must account for the relationships between what pupils already know and what they are expected to learn next.

Some knowledge provides essential prerequisites for later learning. Some ideas recur across different contexts, becoming progressively more general and powerful. Some procedures require extensive practice before they can be used fluently within more demanding intellectual activity. Some concepts require repeated encounters through different representations before their underlying structure becomes apparent.

These relationships are the architecture of learning.

Consider the development of proportional reasoning. Its foundations lie in earlier understanding of multiplication, division, fractions, equivalence and multiplicative relationships. Later, the same mathematical structures appear in percentages, similarity, rates, scale factors, algebraic relationships and numerous scientific applications.

A curriculum that treats these as isolated topics risks obscuring the continuity between them. A curriculum that makes their connections explicit gives pupils opportunities to recognise the increasing reach of ideas they have already encountered.

This is not simply a matter of revisiting material. Repetition without intellectual development may do little more than reproduce earlier performance. Productive revisiting should strengthen fluency, deepen understanding, extend connections or reveal new applications.

The transfer literature is particularly important here. It suggests that teachers cannot assume pupils will spontaneously identify relationships between different parts of the curriculum. Connections that are obvious to subject experts may be invisible to novices.

Curriculum design must therefore consider when those connections should be made explicit, when pupils should compare different applications of a common principle, and when they should be expected to recognise those relationships independently.

The same consideration applies to assessment. If a curriculum intends pupils to develop increasingly powerful and transferable knowledge, assessment must do more than measure success on the most recently taught material. It should establish whether earlier knowledge remains available and whether pupils can bring it to bear on increasingly complex problems.

This creates a more coherent relationship between curriculum, teaching and assessment. The curriculum identifies the intended development of knowledge. Teaching provides the experiences through which that development becomes possible. Assessment supplies evidence about whether it has occurred and what further teaching may be necessary.

The three cannot be separated without weakening the educational enterprise.

Teaching as the deliberate production of learning

What, then, does this body of research imply for everyday classroom practice?

It suggests that effective teaching begins with precision about what pupils are expected to learn. The teacher must identify not merely the topic to be covered, but the knowledge, relationships and capabilities pupils should possess as a consequence of instruction.

It requires careful attention to prior knowledge. New learning does not occur in isolation. Pupils interpret unfamiliar information through what they already understand, and gaps or misconceptions in that understanding can prevent subsequent teaching from succeeding.

It demands explanations and representations that reveal the structure of what is being learned. The teacher's task is not simply to demonstrate a procedure, but to make its meaning and conditions of application intelligible.

It requires sufficient practice for essential knowledge to become fluent and reliably accessible, together with opportunities to compare, generalise and apply that knowledge in different circumstances.

It calls for responsiveness to differences in learning rate. Pupils who require further teaching should receive it, while those ready for greater challenge should be able to progress into more demanding intellectual territory.

And it requires assessment that looks beyond immediate performance to consider retention, understanding and application.

These are not tenets of a single pedagogical movement. They are conclusions that emerge from different traditions of research concerned with different aspects of the same underlying problem.

Nor should their apparent coherence conceal the limitations of the evidence. Carroll's model is a theoretical account of school learning. Perkins and Salomon offer a framework for understanding transfer. Barnett and Ceci clarify the dimensions along which transfer can be investigated. Baddeley and Hitch examine mechanisms of memory. The mathematics intervention studies provide more direct evidence about particular teaching approaches, but their findings vary in scope and strength.

Together, these papers do not establish a universal formula for successful instruction. They do, however, identify a set of questions that any serious account of teaching must address.

What knowledge is being developed? How is it organised? What makes it accessible? How much time and practice are required? Under what conditions can it be retrieved? How will the pupil recognise when it is relevant? What evidence would establish that the intended learning has genuinely occurred?

These questions direct attention away from the visible choreography of lessons and towards their intellectual consequences.

That change in perspective is needed in England today. Educational improvement has too often been pursued through the promotion of preferred activities, the addition of curriculum content or the adoption of new organisational arrangements. Such changes may be useful, but none can substitute for an adequate account of learning itself.

A lesson can be orderly, engaging and apparently productive without producing durable understanding. Equally, demanding intellectual work may involve struggle, revision and temporary failure while contributing to substantial eventual learning.

The quality of teaching cannot therefore be inferred reliably from its immediate appearance. It must be considered in relation to the knowledge pupils acquire and the capabilities that knowledge subsequently makes possible.

The enduring purpose of education

There is something both modest and ambitious about this conception of teaching.

It is modest because it rejects the idea that any single technique, fashionable innovation or organisational arrangement can solve the problem of learning. It recognises that knowledge develops through complex interactions between the learner, the subject, the instruction and the time available.

Yet it is ambitious because it refuses to confuse participation with achievement, exposure with understanding or immediate performance with enduring learning.

The ultimate accomplishment of teaching is not that pupils can reproduce what they have just been shown. It is that something has changed within them. Knowledge that was once absent has become available. Relationships that were previously obscure have become intelligible. Procedures that once demanded conscious effort have become fluent. Ideas encountered in one setting can now illuminate problems arising in another.

The learner's intellectual world has expanded.

The research on transfer reminds us that this expansion cannot be taken for granted. The research on schemas shows why the organisation of knowledge matters. The work on representations illuminates the connections through which abstraction develops. Carroll demonstrates the importance of time and instructional conditions. The research on memory cautions against mistaking temporary accessibility for lasting possession.

Taken together, these contributions point towards a conception of teaching as the deliberate construction of increasingly powerful intellectual capabilities.

That should be the organising principle of classroom practice, curriculum design and the professional development of teachers.

The central question is not whether the teacher has covered the material, followed the prescribed method or completed the planned activities. It is whether the pupil has acquired knowledge that will endure, become connected to other knowledge and remain available for intelligent use.

The true measure of education is not simply what a pupil can do at the conclusion of a lesson. It is what becomes possible for that pupil thereafter.

 

References

Baddeley, A. D., & Hitch, G. J. (1993). The recency effect: Implicit learning with explicit retrieval? Memory & Cognition, 21(2), 146–155.

Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn? A taxonomy for far transfer. Psychological Bulletin, 128(4), 612–637.

Carroll, J. B. (1963). A model of school learning. Teachers College Record, 64(8), 723–733.

Fang, H., Herron, S., Zhou, Q., Hartsell, T., & Mohn, R. (2015). The effects of simplified schema-based instruction on elementary students' mathematical word problem solving performance. Journal of Mathematics Education, 8(1), 37–55.

Good, T. L. (2008). Teaching mathematics in grades 3–5 classrooms. Teachers College Record.

Perkins, D. N., & Salomon, G. (1992). Transfer of learning. In International Encyclopedia of Education (2nd ed.). Pergamon Press.

Rayner, V., Bernard, R. M., & Osana, H. P. (2013, April). A meta-analysis of transfer of learning in mathematics with a focus on teaching interventions. Paper presented at the American Educational Research Association annual meeting, San Francisco.

Reneau, J. L. (2012). Using the concrete-representational-abstract sequence to connect manipulatives, problem solving schemas, and equations in word problems with fractions. Doctoral dissertation, West Virginia University.

Salomon, G., & Perkins, D. N. (1989). Rocky roads to transfer: Rethinking mechanisms of a neglected phenomenon. Educational Psychologist, 24(2), 113–142.

Schema Based Instruction (SBI). Undated instructional pamphlet supplied with the research collection.