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Conference Theme: High Quality Maths Teaching

Richard Dunne

Additional Notes

Contents

Preamble [J] Is a culture of principled pedagogy possible? Introduction [K] Principled pedagogy in mmms. [A] Why am I using ‘division’ as an [L] A note about ‘reading’. illustration? [B] How does ‘division’ relate to the [M] Effective teaching theme? [C] What are ‘higher order mental [N]Some examples processes’? [D] What is ‘experiential knowledge’? [O]Further discussion of the examples [E] What is ‘theoretical knowledge’? [P] ‘Denomination’ as theoretical knowledge [F] What is ‘thinking’? [Q] Glaser’s Propositions [G] Can we teach ‘higher order mental [R] Recognition of complexity processes’? [H] Am I advocating a new curriculum? [S] Notes [I] Am I serious about ‘principled [T] Courses in ‘principled pedagogy’? pedagogy’?

Preamble

Many years ago (in 1979) I advocated that a ‘prescribed curriculum’ should be introduced to specify the content of subjects taught in schools. This was an unpopular idea and my articles were not accepted for publication.[1] I wanted a curriculum with the major part prescribed nationally and the remainder determined locally (within Local Education Authorities). The idea was that teachers could focus their energy on pedagogy - on the science of teaching- if questions of content were largely resolved for them. But it was only content that I wanted prescribed (and I was aware that specifying content without at the same time pre-empting teaching methods would require unusual sensitivity; it would also be brief). I was hoping to release teachers from mind-numbing debate about whether this or that should be in the syllabus and focus instead on effective teaching. I anticipated that as teachers developed detailed approaches to effective teaching it would be necessary to support that work with new content. This was the reasoning behind having a ‘locally prescribed’ element of the curriculum.

Teaching as Assisted Performance 1 Richard Dunne You could easily assume that I welcomed the National Curriculum and all that has followed. Regrettably: no. Each national initiative (the various versions of the National Curriculum and the literacy and numeracy strategies) has both increased the amount of mind-numbing work demanded of teachers and specified content in ways that dictate teaching methods. ‘Teaching’ has become ‘putting the strategy in place’ rather than focusing unerringly on the development of a science of teaching.

I want to discuss what I have always seen as the most pressing question for teachers of maths: what is involved in effective teaching? Improving test results is not particularly difficult; nor is improvement in test results necessarily an indicator of effective teaching. There is no doubt that if children do maths each day rather than just occasionally, receive direct instruction rather than be given access to materials, practise the skills rather than neglect them and are specifically prepared for the test then results will improve. All of this is the bread and butter of teaching. The absence of any part of it is unforgivable. But we have to be careful about how they are instructed, what they practise and how they are prepared for the test. Just because the bread and butter of teaching is in place, with its inevitable improvement in results, it does not necessarily mean that teaching is as effective as it could be.

When I talk of effective teaching I do, of course, intend to improve results. But I want more than that. I want to improve students to engage with ‘higher order mental processes’. There are two immediate reasons for wanting to do this. Firstly, I think that the development of ‘higher order mental processes’ is the key to ‘a humanising curriculum’ and has to be a fundamental aim of any formal education. Secondly, I am concerned that when teaching is focused only on improving test results it often actually reduces the likelihood of schooling being humanising. In other words, early ‘success’ (the wrong kind of success) can be counter-productive. [2]

How can we be sure that any teaching can improve the capacity to use higher order mental processes? [3] In a sense, it is asking the impossible. It means that we have to evaluate what we teach today in relation to students’ future performance. How can we be sure that what we do now will inevitably have the right sort of impact some time later? We could undertake a lengthy experiment to track a group of children for several years but a study of this type is fraught with difficulty because we cannot control the multitude of variables that impinge on children’s learning over the years. In any case, nor could we be certain that we could reproduce the favourable conditions for any future group of children. This kind of research (I like to call it ‘watching and spotting’ – and any sense of disparagement is intentional) is just not appropriate for the highly complex task of understanding how to make teaching humanising.

Innovation in education carries enormous responsibilities, the more so in a centralised system (because a single decision affects so many people). Innovation carries moral obligations, implicating questions about what is worthwhile and how we know what is worthwhile. Current practice in curriculum development can be charged with taking these responsibilities too lightly because it is based on identification and dissemination of ‘good ideas’ (and nothing more). It has become the norm to perceive teaching as a pragmatic occupation in which ‘theory’ is thought to be a very suspicious character. We are currently faced with clear examples of this in, for instance, reduction of the scope of the maths syllabus at ‘A’ level because students find it difficult. These are very dangerous times. Perception of best practice in teaching being a complex, intellectual

Teaching as Assisted Performance 2 Richard Dunne activity has been replaced by efficient delivery of prefabricated packages. And these are most often of dubious origin. This reductionism is the root cause of pupils’ boredom, disillusionment, under-achievement and poor behaviour. What is needed more than anything else is a clear idea of what is meant by ‘knowledge’ and ‘learning’ and ‘thinking’ These need careful definition and detailed attention in curriculum development.

Introduction

My speech at your conference emphasises ‘assisted performance’ as a predicate of learning and places it very specifically at the centre of schooling. I make it clear that effective learning, focusing on higher order mental processes, is related to the acquisition and application of knowledge and is within the compass of all learners. My argument applies to all aspects of maths teaching (and, in fact, to all aspects of all teaching) but constraints of time mean that I have to use illustrative and indicative illustrations rather than tease out every detail. When I show how the mathematical operation ‘division’ should be taught it is relevant to all topics in all phases of teaching maths. My demonstration illustrates how complex ideas need to be represented to novices with an emphasis on speech, signs, symbols and visualisation in relation to carefully designed or selected pedagogical objects. In this case the pedagogical objects are, simply, paper cups; but there are detailed psychological and pedagogical reasons for their selection and the manner in which they are used. I will indicate in my talk how the paper cups represent an ‘auxiliary denomination’ and why philosophical problems in the conceptualisation of ‘number’ underpin this episode of teaching. This may sound like complex pedagogical reasoning. It is. And it is this level of reasoning that, absent from modern curriculum planning, can be the province of all teachers – as a normal part of their professional work.

[A] Why am I using ‘division’ as an illustration?

The early part of my talk today will put some emphasis on the teaching of ‘division’. I am using this one mathematical topic only by way of example to make some general points about effective teaching relevant to all aspects of maths.

The major point I will demonstrate is that our school curriculum, through all key stages, fails to engage pupils in higher order mental processes: it is too heavily weighted towards experiential learning that is largely memorization and somewhat trivial development of ‘skills’ (rather than complex skill development). The reason for this is that ‘theory’ has been given a bad name at all levels of society; but it is theory that expands minds. There is too much emphasis in our curriculum on making topics ‘easy’ rather than making them accessible. A rather recent example of this in making ‘A’ level maths easier in order to attract candidates: a policy that is as doomed to failure as it is silly. The fact that it has been ‘floated’ as an idea indicates the appalling lack of knowledge about pedagogy of those who dominate curriculum design in England. It is all this that I will demonstrate by reference to ‘division’.

[B] How does ‘division’ relate to the theme of ‘assisted performance’?

My contribution today is about the place of assisted performance in schooling. My use of ‘division’ illustrates the nature of a culturally developed (ie, invented) way of categorizing what can otherwise only be tolerated as a succession of unrelated phenomena or

Teaching as Assisted Performance 3 Richard Dunne algorithms. ‘Division’, like ‘gravity’ and ‘force’ and ‘energy and ‘structure’ and ‘function’ and ‘adaptation’ and ‘nation state’, is not something that has been discovered: it is a way of connecting disparate ideas that has been invented. It has survived as an important idea because it is useful. It is, however, arbitrary: it is a culturally developed artifact that, being arbitrary, must be specifically taught. It is not a ‘pattern’ that can be ‘spotted’ from a range of experiences that are connected only because they have been arbitrarily connected by the invented manner of categorizing. It cannot be gleaned from ‘experience’: it is literally beyond ‘experience’. It is theoretical; it has been constructed to capture the essence of a range of otherwise disparate experiences. It is ‘theoretical knowledge’, distinct from ‘experiential knowledge’. The problem for schooling (the endlessly fascinating problem that should be at the centre of professional work) is how best to represent theoretical knowledge to novices. Culturally developed knowledge is ‘carried’ by language and symbols; and cognition, crucially, implicates verbal and symbolic reasoning.

I must be careful. I need to be clear what is meant by ‘cognition’ or it could be confused with any of its modern impersonators. Compulsory schooling must at be aimed at least at ‘humanising’ our pupils. The defining feature of being human is the capacity to engage in higher order mental processes: it is the routine use of higher order mental processes that I call ‘cognition’. Any curriculum development should be clear about what comprises higher order mental processes and what is entailed in ensuring pupils develop them. Although I would like to talk in detail about higher order mental processes like logical memory, selective attention and verbal / symbolic reasoning and how they differ from basic processes like memory, perception and stimulus-response, I do not have the time. This is the stuff of persistent study. Instead, and by way of example, I will refer largely to the nature of ‘thinking’ as a higher order process.

My demonstration will illustrate how ‘division’ must be treated as theoretical knowledge that is progressively applied to a wide range of instances of experiential knowledge. The defining feature of theoretical knowledge, in this case ‘division’, is that it is detached from any specific everyday context. It ‘stands alone’; and it is consistently used year on year as new contexts (small numbers, large numbers, vulgar fractions, decimal fractions, negative numbers, algebra) receive treatment. ‘Division’ taught in the Reception class is the same theoretical construct utilised in the division of polynomials in ‘A’ level (but be quick because someone will remove it from the ‘A’ level syllabus soon to attract more customers! – they may already have done so … I really am too distressed to find out).

[C] What are ‘higher order mental processes’?

My discussion of ‘thinking’ makes reference to experiential knowledge and theoretical knowledge. I will define them and use the abbreviations

Kex (for experiential knowledge) … and … Kth (for theoretical knowledge).

I will also refer to the need to make connections between Kex and Kth and will abbreviate this as C so that the diagram …

Kth Kex C

Teaching as Assisted Performance 4 Richard Dunne … becomes familiar as a definition of thinking: it is the act of incorporating an image of Kex within Kth via the connection C that constitutes the act of ‘thinking’.

[D] What is ‘experiential knowledge’? (Kex) ?

The acquisition of everyday concepts involves identifying similar features in a group of concrete objects or observable phenomena and labelling this feature with a word or phrase. It is the simplest level of abstraction and can be developed by habit formation. It is evident in very young children as ‘pattern spotting’. The formation of everyday concepts is essential; but the process is not sufficient for effective learning. It results in a static, inflexible form of learning: it is not amenable to generalisation. The formation of everyday concepts wrongly dominates the school curriculum in England. This damaging domination of the curriculum is a result of wrong-headed commitment to ‘experiential learning’ and to making learning both easy and fun. Unfortunately, it does neither.

[E] What is ‘theoretical knowledge’ (Kth) ?

The acquisition of theoretical concepts (I prefer to call them theoretical constructs because they are deliberately ‘constructed’) is quite different in nature. The process enables a student to reproduce the essence of an object in the mental plane. To understand an object or a process theoretically means to construct an ‘idea’, detached from any concrete object, and to be able to experiment with it. It implicates verbal and symbolic reasoning. Although the formation of theoretical concepts is necessary it is not sufficient for effective learning. Both everyday concepts (ie, concepts derived from experiential learning; those associated with experiential knowledge) and theoretical concepts (ie concepts deliberately taught as theoretical knowledge) are essential for effective learning. But even taken together, they are not sufficient. However, the major problem with schooling in England is the almost total absence of attention to theoretical concepts.

[F] What is ‘thinking’ (C) ?

The whole point of theoretical knowledge is that it organises (it makes sense of) experiential knowledge. Everyday concepts group together a range of individual experiences sufficient for reasonable amounts of knowledge to be ‘carried forward’ by the learner. Theoretical knowledge makes sense of (‘explains’) experiential knowledge. But this is only possible when deliberate attention is given to making connections between the two distinct types of knowledge. It is necessary and sufficient for effective learning in any domain that pupils acquire relevant experiential knowledge, attain appropriate theoretical knowledge and connect the two. Each aspect requires deliberate attention by teachers and curriculum planners. Almost none of this is apparent in modern curriculum design. Making a connection between the two different types of knowledge, shown here as ‘C’,

Kth Kex C comprises ‘thinking’.

Teaching as Assisted Performance 5 Richard Dunne Notice that ‘thinking’ implicates culturally constructed theoretical knowledge. This means that ‘thinking’ is itself a construction: it must be taught. But there is a not a universally applicable activity called ‘thinking’ (you cannot develop ‘thinking skills’ in isolation from the Kth that defines it). You may get better at ‘thinking’ in new areas of activity from your practice in developing ‘thinking’ in a range of curricular areas, but this is not because it is ‘the same thing’. Each subject area has its own meaning for the act of ‘thinking’: there are different tests of validity in each curriculum area. Ultimately, you depend on your experience of ‘thinking’ in order to capture and utilize ‘thinking’. But the ‘experience’ is at a different level from that I have referred to earlier: it is the experience of using higher order mental processes.

[G] Can we teach ‘higher order mental processes’?

Let me illustrate that we can by focusing on the mathematical topic of ‘division’ to make the important points. In my talk, I illustrate that ‘division’ can be taught detached from any specific type of number: it is simply ‘look at it and wonder how many piles of …’. This detachment is the defining feature of ‘theory’. ‘Division’ is a theoretical construct: it is an example of Kth and, as such, is applicable to all types of number we will meet (small numbers, large numbers, fractions, negative numbers, algebra). With school pupils I refer to ‘division’ as a ‘Big Idea’ in maths. ‘Big Ideas’ (ie , theoretical knowledge) organise the range of experiences we engage in maths. But division is one of only twelve ‘Big Ideas’ in maths. Mastery of the twelve ‘Big Ideas’ makes maths accessible to all pupils: they are used in the same form in all phases of schooling. Eight of them are introduced in KS1; a further two in KS2; and the remaining two in Year 7. Be aware that the introduction of each of the Big Ideas puts relentless emphasis on visualisation introduced via speech, signs and symbols (and eventually used as ‘pure thought’). This is crucial for enabling all pupils to achieve mastery and practised use of theoretical knowledge. This is the gateway to excellence and enjoyment. All this is in my work in maths that I call ‘Making Maths Make Sense’ (mmms). ‘Division’, represented in the manner of my demonstration, is a ‘Big Idea’ in mmms. See Section [K], below.

In every subject we can identify appropriate ‘Big Ideas’ (being selected aspects of theoretical knowledge). In different subjects there are subtle differences underpinning the manner of introduction but in every case it is the emphasis on teaching and using theoretical knowledge to organise experience that remains the consistent key to engagement with higher order mental processes for a humanising and motivating curriculum. Students need continual requirement and practice in ‘standing outside of themselves’, in all aspects of formal schooling, to examine their experiences. This is the nature of being human; it is the defining feature of being human (distinguishing humans from other animals). It is possible for humans because they are able to use language and symbols, proffering the possibility of engaging verbal and symbolic reasoning. But verbal and symbolic reasoning, the connection of theoretical knowledge and experiential knowledge, is not inevitable. It is the central purpose of schooling. Its absence from schooling is not merely unfortunate. It is the greatest abuse we can heap upon the next generation. I must be clear. When I talk about ‘examining their experience’ I am not (repeat: not) referring to the current fashion for students to ‘talk about their problems’ or for the curriculum to be ‘relevant’. No. Their practice in standing outside the experience can be in the consideration of, say, ‘division’ or utilizing the circle theorems. The experience in these cases can be shopping problems or their experiments in physics or their careful skill development in drawing chords and the perpendicular bisectors. It is

Teaching as Assisted Performance 6 Richard Dunne the appearance of ‘theoretical knowledge’ across the curriculum, for it to become commonplace to do so, that contributes to ‘humanising’. [I would add, in relation to ‘discussing personal problems’, that I have a concern about anything approaching ‘counselling’ that mere concentration on personal experience is counter-productive. In ‘counselling’, clients do need to articulate their experience, but the crucial thing is connecting the experience to ideas that are theoretical. Eventually, clients in counselling need to ‘stand outside of themselves’. Theoretical knowledge is the keystone to ethics and morality. But perhaps we are straying too far from ‘division’ …].

[H] Am I advocating a new curriculum?

No. It would be preferable and very easy to re-write the National Curriculum specifically with ‘higher order mental processes’ as its central ambition; but there is no need. The current content can be taught in a pedagogically principled manner (the detail of day-by- day contact with pupils can be improved) provided teachers resist seduction by the banality of model lessons (so to speak) that are provided by the QCA and the various strategies. These are most often extremely faulty. The existing curriculum can be made enjoyable via excellence defined in relation to ‘humanising’, but it does require key personnel to be highly knowledgeable about pedagogy. This approach economises on time so that, seemingly paradoxically, there would be more time for engagement with worthwhile experiences. Yes! More Kth really does mean more time for excellence in Kex because seemingly different processes year by year are seen as the same process. Division is the same process when dealing with two polynomials, at ‘A’ level, as it is when dealing with two small numbers in Year 1 (and similar advantages apply across the curriculum). But we must leave behind the cult of the primacy of the ‘good idea’ and move to a culture of detailed engagement with principled pedagogy. Abandon the ‘focus group’ mentality in curriculum development and opt instead for analysis.

[I] Am I serious about principled ‘pedagogy’?

Oh yes! Yes!!!! There is a crisis in schooling that from time to time surfaces as a concern about standards; but any concern is rapidly buried by what, in other spheres of government, can be recognised as ‘spin’. It really is not good enough to cite putative improvements in examination results as evidence of satisfactory standards. We must be clear that our examination system has followed (perhaps hastened) the abandonment of any emphasis on knowledge (do not be misled by contemporary babble about ‘skills’ replacing knowledge: there is a sad absence of detailed definition). I want to be clear that our pupils who perform well in examinations are doing very well because no matter what we teach and examine it will always be a struggle for pupils to attain the standards (because the required standards define what is ‘hard’). Any examination is ‘hard’ by definition. Our responsibility is to ensure that what we present as ‘hard’ is in fact worthwhile. It is ‘hard’ to spend hours on the preparation of coursework; but when that coursework, at the end of KS4, makes no more cognitive demand than that required in KS2, then it can hardly be cited as worthwhile.

[J] Is a culture of principled pedagogy possible?

Is it morally acceptable not to make it so? But it can only be done through the concerted effort of Head Teachers and their supporting colleagues in the L.E.A. We know how powerful this combination is when we study the manner in which the faulty National Strategy has been embedded. But that irresistible force needs now to turn attention to

Teaching as Assisted Performance 7 Richard Dunne the design of professional development for teachers emphasising pedagogical analysis rather than ‘implementing the strategy’ and desperately seeking solace in the ‘good idea’. This requires study of curriculum materials in relation to psychology and philosophy. It is psychology and philosophy that provide the theoretical constructs to make sense of experiential knowledge from teaching. But the psychology must not be merely the ‘pop psychology’ of the ‘thinking skills’ lobby or the ‘brain gym’ group or the ‘multiple intelligences’ gang – because they are simply non-theoretical. They are mere translation of limited experience into attractive catch phrases. Similarly, there needs to be great care taken in selecting the curriculum materials to be subjected to study: virtually everything you see presented as ‘good ideas’ lacks any potential for analysis. They are simply too trivial. But the excellence of principled study and application of pedagogy is the progenitor of motivation in teaching. What is good for pupils is good for teachers!

[K] Principled pedagogy in Making Maths Make Sense (mmms)

‘ Making Maths Make Sense’ (mmms) is an approach to teaching maths that is rigorously informed by psychological and pedagogical theory and has been refined and tested since 1966. I call this approach ‘Making Maths Make Sense’ (mmms). Demonstrations of episodes of teaching mmms are sufficient to convince of its potential; and these episodes can immediately be employed in classrooms. But there is a danger that each of these episodes will be interpreted as an isolated ‘good method’. I would regret this because each demonstrated episode is in fact merely illustrative of pedagogy that is coherent and consistent: mmms does need some dedicated study.

A key feature of mmms is the manner in which students benefit from ‘assisted performance’. Aspects of this are clear in the demonstrations but persistent study is necessary to illuminate this. Assistance includes emphasis on visual images and rigorous use of symbols but incorporates induction into maths (rather than ‘discovery’ of maths) via twelve ‘Big Ideas’. Curiously, most of the twelve ‘Big Ideas’ look rather mundane: it is in the manner of their introduction and their consistent use in subsequent teaching that singles them out as being so significant. Consider, for instance, the big idea of ‘multiplication’. Its importance, as with all other Big Ideas’ is that the manner in which it should be represented in KS1 (actually, Year 1) remains consistent throughout all maths teaching. No matter what ‘multiplication’ is applied to, it remains consistent with its introduction. ‘Ah yes!’ I hear you say: that is obvious. ‘Not to most pupils!’ I retort. No matter what novelty you introduce into your teaching, no matter what advice you receive, no matter how hard you work, no matter how insistent you are in your ‘Plenary’, students focus on the actual algorithms and miss any inherent consistency. They think of multiplication initially as ‘repeated addition’ (incidentally, this is a seriously poor way of introducing ‘multiplication’); then ‘rows of things’; then counting of zeroes or something akin to this; then moving digits across decimal points; then something to do with numerators and denominators; then (with negative numbers) something to do with the number line and then something mysterious about two minuses; then something about leaving out the sign in algebra. But mmms clarifies how all this actually promotes the need for ‘catch-up’ and ‘booster’ sessions that should actually be completely unnecessary.

When a ‘Big Idea’ is introduced it is used thereafter to clarify, make sense of, all subsequent use of it. The ‘Big Ideas’ in maths, appropriately taught, provide a category

Teaching as Assisted Performance 8 Richard Dunne system for learning maths that means there is really very little to learn. They ensure that all students can succeed (a crucial aspect of ‘inclusion’!). The twelve ‘Big Ideas’, listed in Figure 1 (below), are the basis in maths for pupils’ theoretical learning Kth . The various items they apply it to comprise Kex and this application involves making a connection C. This ‘thinking’ (as defined above) involving a wide range of theoretical and experiential learning gives meaning to school learning (especially when the experiences are cross- curricular). But this learning is not static; it is not inert learning. The initial experiences from which meaning was derived are re-visited and seen in a new light. And this new light is dependent not merely on their own experience but, being generated by careful teaching of ‘Big Ideas’ that reflect carefully honed ways of representing maths, provide ‘meaning’ that goes beyond experience. This process has been alluded to (without any reference to maths) by T.S.Eliot (in a 1970 publication) Collected poems (1909 – 1962) London: Faber and Faber Ltd. In ‘The dry salvages’ (p. 208) he writes:

We had the experience but missed the meaning, And approach to the meaning restores the experience In a different form … … the past experience revived in the meaning Is not experience of one life only But of many generations …

This, of course, reflects the idea of teaching as induction into culture. It needs deliberate attention. All maths teaching involves (i) demonstration by the teacher (but this is what needs such detailed attention to the manner of representation and presentation and consistency); (ii) practice by pupils; and (iii) deliberate, detailed articulation of emergent understanding. I refer to these three aspects as:  The demonstration phase  The modelling phase  The summarising phase I regret the fact that this pedagogical model has been trivialised as ‘Starter; Main Part; Plenary’ because this is an over-simplification and a nuisance. I also regret the silly contention that each timetabled session should include the three aspects (I do not understand how a conceptual model of teaching became a recipe for each lesson; or perhaps I do …). The demonstration phase may well take up a complete session (pupils can concentrate that long – it is all to do with expectation and training). In subsequent sessions these same pupils would revise aspects of this long demonstration and then undertake individual or group work to model what they have been taught (they practise it). The following day may repeat the same pattern. The next day may be devoted entirely to pupils’ articulation of their understanding in relation to the examples they have been working with: this involves individual pupils talking to the whole class: an essential aspect of maths (it is the summarising phase, when students make connections between their experience and the organizing theoretical knowledge). This is, of course, consistent with the instructions given by OfSTED in ‘Update: Issue 40’ (page 13):

Inspectors should note that the use of lesson structures promoted by NLS and NNS is no guarantee that teaching will be good or that pupils will reach the objectives set in the lesson. There is also no place for pre-conceived ideas about teaching methods. Although the NLS and NNS recommend particular methods, they are not statutory. Whatever approaches to literacy and

Teaching as Assisted Performance 9 Richard Dunne numeracy are used in the school inspectors should assess the effectiveness of the teaching.

In order to complete this brief summary of mmms let me list the big ideas (there are only twelve of them).

Figure 1

Big idea Notes When introduced: 1. Addition Get ready to get some more

2. Subtraction Get ready to take away Key Stage 1

3. Multiplication Do the same thing lots of times (+ - = in N and YR; 4. Division Look at it and wonder how many … × ÷ in Y1; 5. Equals Same value: different appearance ‘symbols’ and ‘logic’ in YR) 6. “The symbols Notice how speech, signs and symbols speak to you” assist performance in my lessons

7. “The logic of Maths must be seen as a logical language the language tells you the answer” 8. Denomination Pervades everything, of course, but will be seen clearly in ‘fractions’ where ‘denomination’ is much more important than ‘denominator’. 9. Ratio Note the use of sticks and the emphasis on Key Stage 2 ‘compared to’. 10. Infinity Introduced as a direct consequence of the (Y3) representation of ‘division’. 11. Proof A formal development of ‘the logic of the Key Stage 3 language …’ 12. Sample / The underpinning (largely absent!) notion in (Y7) population statistics.

[L] A note about ‘reading’.

The basic skill in the drive towards ‘literacy’, supporting all others, is critical reading. Essentially, it involves seeing the connotations of sentences. The critical reader gets beyond the material literally referred to and perceives that the sentence is relevant to a larger domain. The literal reading is an aspect of experiential knowledge (but this needs careful teaching involving not only de-coding but also dramatic reconstruction of text). The ‘larger domain’ is the theoretical knowledge that will provoke thinking and eventually provide ‘meaning’. The parallel between critical reading in all subjects and formal

Teaching as Assisted Performance 10 Richard Dunne reasoning in logic (and maths and science) is very close. The literal contents of sentences are the premises; the connotations are the conclusions. Students need to be taught to treat sentences as premises on which to base conclusions, a problem which is complicated by the fact that propositions in sentences are rarely arranged in syllogistic form. Moreover, there is nothing in the sentence itself to trigger this realization of connotation (there are no explicit clues) so the difficulty of alerting students to connotation is formidable. But that is the purpose of giving deliberate attention to Kth and Kex across the curriculum: students are encultured into higher order mental processes. It provides a humanising curriculum. I am reminded of Lionel Trilling, the American critic, whose recent collection of writings is entitled ‘The moral obligation to be intelligent’. I like that!

[M] Effective teaching One way of studying the predicates of effective teaching is to focus on the behaviour of expert learners. There are many studies of expert learners and their findings can be illuminating. In a review of the literature one of America’s most distinguished educators, Robert Glaser [6], offers a number of propositions including:  Expert knowledge is structured better for use in performance than is novice knowledge.  Experts represent problems in qualitatively different ways than do novices. Their representations are deeper and richer.  Experts impose meaning on ambiguous stimuli. They are much more ‘top-down’ processors. Novices are misled by ambiguity and are more likely to be ‘bottom up’ processors.

All this of course just sounds like common sense. It is easy to read it and feel ‘OK. So what?’ In fact, research findings presented in this rather glib way are increasingly one of the problems facing the education system. [7] Let me summarise Glaser’s propositions in a different way to try to bring out their significance; then I can illustrate their implications for maths teaching. As I work through my summary, note the italicised sections that repeat some of Glaser’s points above.

Imagine I am lecturing an audience of sophisticated learners. During the course of my lecture I mention, show or teach a hundred points. When members of my audience are asked, later, ‘What did he do?’ their reply is of the form: ‘He did this, this and this’. Expert learners summarise my lecture in about three very broad and powerful categories. They do this because as I introduce a new idea (which, because it is new, is rather ‘hazy’), they link it to something they already know. They link it to existing broad and powerful categories. As they do this their concern ‘What’s he talking about?’ becomes ‘Ah! It’s a bit like that; I have met something like that before’. They link the new ‘hazy’ idea to one they already know really well, and in so doing they begin to make sense of the new idea. But the idea they link it to is rather special. It is a broad and powerful category. It is because their current knowledge is stored in categories rather than as individual items that we can say their expert knowledge is structured for use in performance. It is accessible to them when they need to use it for dealing with new knowledge. The broad categories ensure that new pieces of information are thought of as ‘belonging to that type’ or ‘similar to those things’. Their expert knowledge involves representations that are deep and rich because it is in broad categories; and, vice versa, they have stored knowledge in broad categories because the representations employed to teach them have been broad and rich. They can impose meaning on ambiguous stimuli because, when I mention something that is unclear, they examine it in relation to the broad ideas

Teaching as Assisted Performance 11 Richard Dunne that structure their knowledge. They are ‘top-down’ processors in the sense that they do not work from the point I have just made, but they instead examine that point from their well-structured knowledge. They more or less ask themselves: ‘How does this relate to the broad categories I have already developed?’

Let us contrast this with a novice audience similarly faced with a hundred points during my lecture. They start by trying to work as hard as the expert audience. They hear the first point and try to learn it (which, for them, means to memorise it). They do the same with the second point; and they try to memorise the third. At this point they have as much in cognitive space as the experts, but they are now faced with a further ninety- seven points each of which they are unable to learn. They begin to feel stupid and demoralised. They learn that they cannot learn.

Why do the novices not act like the experts? It seems to me that there is a very straightforward answer. The experts have been taught big powerful categories and how to connect new ideas with them; the novices have not been taught. Typically, maths teaching provides learners with a succession of disconnected points. It does not teach representations that are deep and rich; it does not teach how knowledge is structured for use in performance; it does not enable people to impose meaning on ambiguous stimuli by becoming ‘top-down’ processors. Some people (those who become experts) learn very quickly how to do these things, but others need a very carefully planned teaching programme that focuses on providing deep and rich representations.

[N] Some examples We need to make sense of what all this means. I will describe a few examples of maths teaching to show how something as straightforward as the ‘three times’ table can be used as the focus for a powerful category system. I will illustrate how in this case maths can be represented as a logical language. This enables children to see connections among topics as seemingly diverse as large numbers, fractions, decimals and negative numbers, supported by another category of ‘numeration and denomination’ that can be used to support further connections. Look at the following classroom scenario. [8]

The teacher has displayed a copy of the ‘three times’ table.

3 × 0 = 0 3 × 1 = 3 3 × 2 = 6 3 × 3 = 9 3 × 4 = 12 etc

The children chant the ‘table’ by reading it with the teacher. Good. You know this. I am going to show you that you can see things in your mathematical brain that other people cannot see. Watch me. Do not guess what I am going to say. Just watch. The teacher points to the 3; then to the space immediately after the three; then to the × sign; etc, saying: [Point to 3] Three [point to space after 3] thousand [point to ×] times [point to 0] nought [point to =] equals [point to 0] nought [point to space after 0] thousand.

In this way the teacher proceeds as follows:

Three thousand times nought equals nought thousand.

Teaching as Assisted Performance 12 Richard Dunne Three thousand times one equals three thousand. [Teacher beckons children to join in]. Three thousand times two equals six thousand. Three thousand times four equals twelve thousand. Good. You obviously know the three thousand times table. Let me test you. [Points to 3 × 8 = 24] Three thousand times eight equals [children join in] twenty four thousand. Good. Now let me give you the big test. [Points to 3 × 8 = 24] Three million times eight equals [children join in] twenty four million. [Points to 3 × 8 = 24] Three hundred times eight equals [children join in] twenty four hundred.

What we see here is how knowledge of the three times table is used to ‘know’ other, related tables. But notice it is not only knowing the three times table that makes this possible. The three times table has been used but the children are in fact learning something much more important. They are learning the ‘big powerful category’ that the logic of the language tells you the answers. The use of the logic of the language is crucial in maths. The vocabulary of maths, and the way it is used, provides indicators for how to find answers and which ideas are connected. The example of the three times table is just one example of how a powerful category (typical of expert learners) can be taught to all learners.

Notice how this idea with the three times table is rapidly developed into:

Three thousand times four hundred equals twelve thousand hundred

Notice two other aspects. Firstly, that it is written as:

3 000 × 4 00 = 12 000 00

Secondly, that in such a lesson, in which the focus is the logic of the language, it would be wrong to muddle the idea by insisting that we say ‘one million, two hundred thousand’ when we see 12 000 00. The conventional way of speaking the numbers must be the subject of a quite separate lesson after the logical way of generating answers has been mastered.

At first sight this is a rather straightforward episode of teaching, but there are aspects of it that need further discussion. What has not been conveyed in the above script is that the teacher’s use of language is very deliberate. The number 3 000 is not spoken as ‘three thousand’ but with a pause between the two words (I represent this in print as ‘three [pause] thousand’). Similarly with 3 00 (‘three [pause] hundred) and 3 000000 (three [pause] million). The reason for this can be best understood by thinking of ‘three francs’ and ‘three pounds’. In each of these cases it is clear that we are thinking of a particular denomination (firstly, francs; then, pounds) and we are also indicating the numeration of that denomination. We are saying ‘three’ of those things called ‘francs’. ‘Numeration’ and ‘denomination’ are implicitly understood by most people when dealing with, say, money or weight. They implicitly understand that they are dealing with ‘three’ of those things called ‘francs’. The expert learner also implicitly recognises the ‘sameness’ of dealing with ‘three’ of those things called ‘thousands’; the novice learner does not. When the teacher deliberately says ‘three [pause] thousand’ she is teaching

Teaching as Assisted Performance 13 Richard Dunne the ‘sameness’ that is not apparent to novices. She recognises the importance of ‘numeration’ and ‘denomination’ and insistently teaches it. [9] The logic is in the language – provided the language is emphasised and provided every opportunity is taken to capitalise on its organising power.

Let me clarify that last point. The idea (implicit as it is) of ‘numeration’ and ‘denomination’ is not powerful if it is just waved at children in a rather cavalier way, to be immediately abandoned for other tit-bits. It is selected for its potential power and must be persistently used. And that means persistently taught in similar topics. We need to see how the teacher does this.

If I write 9 [writes 9] you say ‘nine’. Now, do not say ‘nine’ again, just watch to see what I write here [next to the 9]. Do not guess … wait … [pause; writes 000 to complete 9 000] thousand. Yes. That is right. Now do this one. [Writes 9]. Nine [pause; writes 00 to complete 9 00] hundred. Now do this one. [Writes 9]. Nine [pause; writes 0 to complete 9 0] -ty. Yes. That is right. You are very clever. When I wrote 9 000 [points to board] you said ‘nine [pause] thousand because you knew it was nine lots of a thousand. When I wrote 9 00 [points to board] you said ‘nine [pause] hundred because you knew it was nine lots of a hundred. When I wrote 9 0 [points to board] you said ‘nine [pause] -ty because you knew it was nine lots of ten. When you say ‘-ty ’you mean ‘lots of ten’. Now read these. Eight [pause] –ty. Seven [pause] –ty. Six [pause] –ty. Five [pause] –ty. Yes! When you say five-ty you know it is five lots of ten. I know you usually say ‘fifty’ but when you say ‘fifty’ you mean ‘five-ty’. Now these: four [pause] –ty; three [pause] –ty; two [pause] –ty; one [pause] –ty; nought [pause] –ty!

The point of this exercise is to look again at the ‘three times’ table and speak the words as follows (remembering each time that ‘-ty’ is preceded by a pause):

Three-ty times nought equals nought-ty: 3 0 × 0 = 0 0 Three-ty times one equals three-ty: 3 0 × 1 = 3 0 Three-ty times two equals six-ty: 3 0 × 2 = 6 0 Three-ty times three equals nine-ty: 3 0 × 3 = 9 0 Three-ty times four equals twelve-ty: 3 0 × 4 = 12 0

On each occasion the figures are written it is the idea of ‘numeration ‘ and ‘denomination’ that guides the teacher’s actions. She writes, for instance, 6 [pause; gap] 0; and she writes 9 [pause; gap] 0; so when she writes 12 [pause; gap] 0, and says twelve [pause] –ty, she is making it quite clear that she is exercising the logic of the language. She knows that Three thousand times four equals twelve thousand is logical. She knows that Three hundred times four equals twelve hundred is equally logical. She also knows that ‘Thirty times four equals one hundred and twenty’ is logical to experts but, to novices, is an extra (quite different) thing to learn. She introduces ‘Three-ty times four equals twelve-ty’ so that novices are taught that it is equally logical (it is incorporated within the same big idea). In planning her teaching she has not just taught ‘what is there’ but has carefully constructed the teaching to recover the logic of ‘three-ty’ from the vernacular of ‘thirty’. [10]

[O] further discussion of the example This is an example of attention to detail in teaching that has the potential to enable novices to adopt the strategies typically learnt only by experts. It is very different from teaching a host of ‘mathematical strategies' in order to mimic the seeming mass of

Teaching as Assisted Performance 14 Richard Dunne knowledge acquired by experts. It recognises instead that experts economise by imposing large categories on experience in ‘top down’ processing. [11] Note that this is a discussion of a single example but I can provide a similarly detailed discussion for every aspect of maths teaching. However, the single example of using the ‘three times’ table is so provocative of detailed analysis that I want to persist with it. Let us look again at the teacher with her class.

I want to do some more work with the ‘three times’ table but I want to start with something a little different. This table here [indicates empty table raised high enough for all children to see easily] is the maths table. Over here [indicates a separate table] is my resources table where I keep my teaching equipment [the table has about fifteen inverted paper cups placed on it]. Today, I am the boss and I need a helper [selects a child]. Because I am the boss I tell you [speaks to ‘helper’] what to do and you obey my instructions. I want you to start by putting three cups on the maths table [teacher ensures that the ‘helper’ picks up three cups from the resources table, walks to the maths table and places them in a row]. Did you see what he did? He put three cups on the maths table. What did he do? [Class replies ‘He put three cups on the maths table’]. Now watch me because I am acting. [Teacher acts being very excited]. I loved what he did. He put three cups on the maths table. I loved it. I loved it. I loved it so much I want him to do the same thing [motions ‘helper’ to resources table and ensures he picks up three cups. Teacher nods at the class in affirmation] lots of times. I want him to do it four times. He has done it [helper places the second batch of three cups on the maths table] two times. He must do it again [motions to helper and ensures he picks up three cups]. That is three times. Has he finished? No. He must do the same thing again [helper delivers another three cups]. That is four times. Yes. He has finished. He put three cups on the maths table. I loved it so much he did the same thing lots of times. He did it four times. Now look at the maths table. Count how much on the maths table. Twelve.

The teacher now insists on doing the same process all over again, but this time she pretends to lose her voice. She insists that it is still her job to tell the helper what to do, but she now has to do it without talking (although she maintains a running commentary throughout the next piece of acting in which she claims to be unable to speak!).

If I had my voice I would tell my helper to start by putting three cups on the maths table. I must do it without talking. When I start by writing this (3) on the board, my helper knows that I want him to start by putting three cups on the maths table. [Helper walks to the maths table with three cups]. If I had a voice I would tell him I loved it so much [acts being excited just as before] that I want him to do the same thing lots of times. I loved it! [Writes × on board to show 3 × and motions helper to pick up three cups from the resources table] I want him to do it [writes 4 to show 3 × 4 and motions to the helper] four times. [As the helper completes the fourth journey:] Now I want him to look at the maths table [writes = to show 3 × 4 = and ensures the ‘helper’ ostentatiously looks at the maths table] and count how much on the maths table. Yes. Twelve. [Writes 12 to show 3 × 4 = 12].

The teacher is supplying a physical meaning to the symbols 3 × 4 = 12 so children can come to visualise what is hinted at by the symbols. Notice the insistence on ‘seeing’ the first number (in this example 3) as three cups. The multiplication sign ‘speaks’ to the reader as do the same thing lots of times. Notice what this insists on. It requires that you stick with the same number, 3, and then you continue reading to see how many

Teaching as Assisted Performance 15 Richard Dunne times you have to do it: 4. This is very important because the 4 indicates the number of journeys you make. It is not ‘another four cups’; it is the total number of journeys. This is emphasised and taught in the deliberate way outlined above. But it makes another vital contribution to making sense of maths. The ‘three cups’ that you start with is the denomination to be used; the ‘four’ tells you how many of the thing called ‘three cups’: it is the numeration. What you have when you see 3 × 4 is four lots of ‘three cups’. [12]

When the teacher characterises ‘multiplication’ as ‘do the same thing lots of times’ she is aware of how this same approach will provide economy of effort for learners so that they become more like ‘experts’. When she writes the ‘three times’ table as

3 × 0 = 0 rather than 0 × 3 = 0 3 × 1 = 3 1 × 3 = 3 3 × 2 = 6 2 × 3 = 6

She is doing so because she is aware of the need to create a consistent and meaningful language for maths, supported by visualisation, rather than just a list of vocabulary. She wants the ‘times table’ to evoke the image ‘start by putting three cups on the maths table’ so that the ‘three times table’ always involves thinking about dealing with ‘three cups’. In this manner, children do the natural thing of thinking about objects before they think about what to do with them – otherwise they have to think what to do with something that has not yet been identified. This latter manner of thinking is ‘a cognitive step too far’ that prevents many children from making sense of maths. It is fine for experts who can defer having an object to think about but we need to give all children access to modes of thinking that experts adopt rather easily.

[P] ‘Denomination’ as a ‘theoretical knowledge’

The importance of this clarification of the meaning of the multiplication symbol is that it can be used in all areas of maths. Suppose, for instance, I have introduced the idea of ‘a fifth’ [13] and you are confident that a piece of cardboard

1 represents a fifth. Start by putting three fifths on the maths table. I loved it. Do 5 the same thing lots of times. Do it four times. Look at the maths table. Twelve fifths. By the way: you knew this. You know 3 × 4 = 12 so you know Three fifths times four equals twelve fifths.

Notice that when I refer to ‘three fifths’ I am asking you to think of three of those things called ‘a fifth’. The denomination I am asking you to think of is ‘a fifth’. It is ‘a fifth’ that is the denomination. When you see a fraction and say ‘the denominator is five’ you are really only recognising ‘the five on the bottom’ as a hint of the denomination. The denominator hints at the denomination. You can see how the consistency of ‘categorisation’ resolves the difficulties in maths brought about by piecemeal stabbing at vocabulary and methods.

The same visualisation for multiplication is used with whole numbers, fractions, decimals, negative numbers, algebra and measures. It is a broad category that makes it possible for all children to think like experts. It is just one example of the several broad

Teaching as Assisted Performance 16 Richard Dunne categories that underpin effective teaching of maths. I will not provide further illustrations of this type because I want to use the space to make an equally important point about this style of teaching.

The methods that have been all too briefly illustrated in this article are based on a detailed model of the nature of maths. When the teacher insists on working with ‘powerful categories’ she selects them so that they simulate (in principled ways) what ‘mathematics’ is. When she plans with ‘numeration’ and ‘denomination’ in mind she is clear how this relates to subsequent topics in maths.

[Q] Glaser’s propositions

Let us return to Glaser’s propositions. Recall that he proposed:  Expert knowledge is structured better for use in performance than is novice knowledge.  Experts represent problems in qualitatively different ways than do novices. Their representations are deeper and richer.  Experts impose meaning on ambiguous stimuli. They are much more ‘top-down’ processors. Novices are misled by ambiguity and are more likely to be ‘bottom up’ processors.

The small episode of teaching that I have analysed here illustrates how all children can be taught in ways that enable them the more to resemble experts. Detailed attention is given to teaching to make sure that it represents maths as deep and rich structures or categories. Think, for instance, of the ‘logic of the language’ and the idea of ‘numeration and denomination’ that draws together ideas that for most children at present are disconnected. Recognise, further, how these ‘powerful ideas’ enable children to become ‘top-down processors’ because they are enabled to see how successive ideas in maths are examples of already known categories. Of course, the single episode of teaching I have used here has been sufficient only for illustration, but think of a curriculum that is based on this idea rather than conceived as a succession of barely connected notions. Think, in fact, of a curriculum that is centred on enhancing the capacity to learn by assisting performance.

[R] Recognition of complexity

It is now appropriate to look again at my continuing concern that the National Numeracy Strategy is having an undue and inappropriate influence on teaching methodology. There has been such an obsession with specifying detail that (although not statutory) it is institutionalising a piecemeal view of the curriculum and making it less likely that powerful pedagogic methods will be developed. Why has this happened? It is because the most important aim for any national strategy has been ignored. The most important aim should be for teachers to be given the tools for developing the science of teaching and to avoid at all costs any imposition that reduces the complexity of the task of teaching. Teaching must become less mind-numbing. Its complexity must be recognised, retained and placed at the heart of its development. It must focus on the assisted performance of higher order mental processes as the basis of a humanising curriculum.

[S] Notes

Teaching as Assisted Performance 17 Richard Dunne [1] I developed these ideas further in my MA dissertation: Dunne, R (1983) A core curriculum for middle schools. Lancaster: University of Lancaster.

[2] There is evidence of ‘the wrong kind of success’ even among the highest attaining students. Nardi, E. (1999) (The Novice Mathematician's Encounter with Formal Mathematical Reasoning: A sample of findings from a cross-topical study. This can be found at: (http://www.bham.ac.uk/ctimath/talum/newsletter/talum9.htm) She studied the performance of students in the Oxford University tutorial system and concluded: Even when the students accept the necessity to formalise, they still struggle with its implementation: so, for instance, they assume in their proofs what is to them intuitively obvious or what they are actually being asked to prove, that is their arguments are insufficient or circular. Sometimes, their initial, over-zealous perception of what the rigour of university mathematics is, leads them to doubt their school mathematical practices to the extent that they avoid, for instance, even the use of some basic arithmetical facts. What they seem to need urgently is to clarify the distinction between rigorous and intuitive arguments, legitimate and illegitimate use of previously established knowledge.

[3] I have discussed this elsewhere as both ‘informed and deliberate teaching’ and ‘assisted performance’. A booklet (Dunne, R. and Brodie, D. (1999) is available as notes to accompany the video package Target Getting available from Birmingham Advisory and Support Service (BASS), Martineau Centre, Balden Road, Harborne, Birmingham.

[4] ‘Watching and spotting’ is what people have done with international comparisons and surveys of good teachers. There really is little point in watching teachers at work, spotting what they do and then recommending this as good practice to be adopted by others. It is profoundly unscientific. The absence in such work of any theoretical background renders it useless. The findings reflect either the peripheral aspects of effective teaching or the preference of the observer. When whole class teaching is recommended because researchers (‘watchers and spotters’) have seen it done in countries with high standards of attainment, they miss what is the essence of that method. ‘Whole class teaching’ is a case in point. I have always asserted that whole class teaching is essential, but it is what actually happens in that whole class teaching is crucial. The method needs very careful attention in relation to a detailed model of teaching and learning. It is not only the fact that it is ‘whole class’; it is that whole class teaching provides an appropriate forum for ‘assisted performance’ (see note [3]).

[5] Myth abounds in teaching. Myths are created and sustained by ordinary language. There is a widespread belief, for instance, that some people possess identifiable `gifts' or `talents'. These are assumed to be inborn, restricted to relatively few individuals and reliable predictors of future expertise. I am convinced that we need to treat this idea with considerable scepticism. I am equally convinced its effects on teaching are palpable, tending to render teachers less optimistic than they should be about the possibility of developing expertise in all children. It is for this reason that I assert that expert knowledge can be taught; and it is for the same reason that I contend that expert learners should be studied for clues about how we can teach novices to become experts. Howe,M.J.A,, Davidson, J.W. and Sloboda, J.A.(1999) (Innate gifts and talents: reality or myth)? have studied and criticised the assertion that ‘gifts’ and ‘talents’ exist. In an

Teaching as Assisted Performance 18 Richard Dunne exhaustive study they report that they were ‘unable to detect any results consistent with the view that innate attributes operate in a manner that would indicate the operation of the particular qualities attributed to innate gifts or talents [and there was an] absence of the kinds of early signs of special ability that … is consistent with the suggestion that innate gifts do play a major role’. They conclude that ‘there is a striking lack of evidence of such early indications of potential, and where unusual very early precociousness is encountered it is also found that children have been given ample opportunities and encouragement to gain skills earlier than usual, with the special opportunities generally preceding any signs of unusual ability’. In addition, they report that there is ‘striking lack of evidence pointing to large differences between individuals in ease of learning’. When they find large differences, they conclude that ‘this can be accounted for as a consequence of prior differences in knowledge, skills, motivation, or other factors known to affect performance’.

[6] Glaser, R. (1987) ‘Thoughts on expertise’, in C.Schooler and W. Schaie (eds) Cognitive functioning and social structur eover the life course. Norwood, NJ: Ablex. Glaser, R. (1990) ‘Expertise’ in M.W. Eysenk, A.N. Ellis, E. Hunt and P. Johnson-Laird (eds) The Blackwell dictionary of cognitive psychology. Oxford: Blackwell.

[7] ‘The OfSTED Reviews in Education’ are a case in point. Askew, M. and William, D. (1995) Recent research in mathematics education 5-16. London: HMSO offers a series of plausible ‘sound bites’ supported by very selective interpretations. We see, for instance (p.27), that a consideration of ‘activity, concept and culture’ is assumed to imply that tasks should be ‘ill-defined … carried out in a social setting … have several possible solutions and solution methods’. In fact, each of these claims (sound-bites) is highly contentious and based on a partial understanding of the cited research. The same cavalier approach is evident in the DfEE Standards and Effectiveness Unit has published guidelines for homework (DfEE (1998) Homework: Guidelines for Primary and Secondary Schools. London: DfEE). The claim is made (p.3), for instance, that ‘Research over a number of years … has shown that homework can make an important contribution to pupils’ progress’. The authors do not list the research that has so impressed them, choosing instead to cite, firstly, an OfSTED report of 1995 that 'many pupils and their parents saw work done at home as a valuable and essential part of school work’; secondly, the 1996/97 OfSTED Annual Report claiming ‘homework is important in all stages in a child’s education’. The objective reader may well find this evidence rather on the weak side, wondering how justified are the ‘many’ pupils and parents in making their claim. The same objective reader may well remain unimpressed on reading that ‘pupils in the highest achieving schools spend more time on learning activities at home than pupils in other schools’. Really? And in which direction must we assume causation? Is it the case that high achieving children choose to spend time at home on ‘learning activities’? Or is it the case that spending more time on ‘learning activities’ at home causes high achievement?

Any tendency to scepticism might be increased by reading (p. 25) that ‘The research summarised here has been carried out by, and on behalf of, OfSTED, for the DfEE’ with evidence collected ‘through case studies in 19 primary schools … and a telephone survey of 227 primary schools’. We are not treated to anything resembling analysis. Similarly, the claims made for the effect of homework are exemplified in the ‘Guidelines’ by six ‘Case Studies’ that are, in fact, rather bare descriptions focusing more on organisational features than hard evidence.

Teaching as Assisted Performance 19 Richard Dunne [8] This episode of teaching can be understood better from the video set Target Getting available from Birmingham Advisory and Support Service (BASS), Martineau Centre, Balden Road, Harborne, Birmingham. The booklet accompanying the video expands on the ideas discussed in this article.

[9] Most people think in terms of ‘numerator’ and ‘denominator’ only when dealing with fractions. Children are taught that, for 3/5, ‘3 is the numerator and 5 is the denominator’. This is accurate use of vocabulary but its meaning is lost on novices. Effective teaching of fractions emphasises that that 3/5 means ‘three of those things called fifths’. The denominator is five, yes, but the denominator is merely a hint about the denomination under consideration (‘those things called fifths’). It is in details like this (so many of them!) that teaching is encouraged to be careless and confusing rather than detailed and clear. This problem is being exacerbated by a renewed emphasis on vocabulary (witness the vocabulary lists!) without an attendant emphasis on meaning. There needs to be a distinction between vocabulary (which is limited) and language and meaning (which underpins understanding of maths).

[10] When we say ‘fifty’ you mean ‘five-ty’. ‘Fifty’ is a contraction of ‘five-ty’ that has arisen from use. When designing teaching programmes to enhance the capacity to learn, we need to recover the logic of the language from the convenience of the vernacular.

[11] This is a huge topic that I have used to develop a model of teaching and learning. I have discussed it extensively in my MA dissertation (see Note [1]) and in articles about ‘Activity Theory’ as well as in ‘A model for planning teaching’, both of which I can supply on e-mail on receiving a note on [email protected]. See also Buchmann, M. and Floden, R.E. (1993) Detachment and concern. London: Cassell.

[12] There is another reason for thinking of 3 × 4 = 12 in the manner discussed. It allows consistency (and therefore categorisation) of the four operations ‘add’, ‘take away’, ‘times’ and ‘divided by’. In each case (say, 6+2; 6-2; 6×2; 6÷2) the symbols are thought of as: ‘What you start with; the type of job; the size of the job’. You will need to study the video pack Target Getting detailed in Note 3.

[13] See Note 3.

Richard Dunne taught in schools for twenty years and then in higher education at Exeter University and Portsmouth University. He now provides consultancy in Instructional Design and demonstrates the nature of his work by demonstrating teaching in schools. His interest ranges from nursery to ‘A’ level work and he has published extensively about both mathematics teaching and mentoring. He provides a web-site at

www.rdid.co.uk

Teaching as Assisted Performance 20 Richard Dunne [T] Courses in a principled approach to pedagogy.

Professional Development with Richard Dunne

Curriculum Managers in Primary, Middle and Secondary Schools.

Introduction to Teaching as Assisted Performance

A series of practical illustrations of what is meant by ‘Assisted Performance’ in several subjects serves to introduce a new dimension to the work of Curriculum Managers. Focusing on assistance of cognition, motivation, learning strategies, language and social skills, it implicates theory-driven, highly practical methods of monitoring, assessment and reporting. Delegates will gain insights into how to raise standards of attainment of all pupils through a programme of professional development for colleagues. Above all, there is an emphasis on clarifying the nature of teaching for effective learning summarised as: Do not reduce the demands of the task: Make it clearer. Date: Thursday 13 November 2003 Venue: Norwich Area (details to follow)

Delegate fee: £125 (including course materials, tea, coffee, lunch) Register at: www.rdid.co.uk

Teachers of Maths in Primary and Middle Schools.

Introduction to Making Maths Make Sense

Demonstrations of teaching show how all pupils can learn maths effectively via detailed attention to how best to represent maths to novices. The demonstrations are simply startling (yet they are only indicative of how the whole maths syllabus can be made accessible to all pupils). Delegates may well be angry that they have not previously been introduced to the beauty and sheer simplicity of maths but they will appreciate how the twelve ‘Big Ideas’ in maths organise learning from the Reception Class to University. Yes. Twelve organising categories that are persistently used throughout school life and beyond: eight of them introduced in KS1; two more in KS2; and a further two in Year 7. This is how maths can be made to make sense to all pupils (and it really is all pupils). But beware: it implies a genuine improvement in learning that goes beyond the demands of compulsory testing. Does anyone still care about that?

Date: Wednesday 11 February 2004 Venue: Norwich Area (details to follow)

Delegate fee: £125 (including course materials, tea, coffee, lunch) Register at: www.rdid.co.uk

Richard Dunne provides courses in Secondary Maths:

* Introduction to Making Maths Make Sense in Secondary Schools * Algebra * Geometry * Statistics + Staistics Coursework * Problem Solving * Calculus

Contact www.rdid.co.uk for details and for other topics.

Teaching as Assisted Performance 21 Richard Dunne Richard Dunne provides customised INSET for individual schools or clusters of schools, including work for pupils with special needs, the gifted and talented, assessment, and much more:

Contact www.rdid.co.uk for details.

Richard Dunne teaches an extended course for groups, usually focused on L.E.A.’s or clusters of schools, interested in detailed study of the philosophy and psychology of mmms. This course extends over a period of twelve months with five 1-day sessions, five twilight sessions and school-based support. It involves detailed study of at least one topic in maths including significant school-based development.

Contact www.rdid.co.uk for details.

Teaching as Assisted Performance 22 Richard Dunne

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