One or our MA Education students was presenting to her group the other day on Jerome Bruner and she quoted from his famous text, The Process of Education, as follows:
“I was struck by the fact that successful efforts to teach highly structured bodies of knowledge like mathematics, physical sciences, and even the field of history often took the form of a metamorphic spiral in which at some simple level a set of ideas or operations were introduced in a rather intuitive way and, once mastered in that spirit, were then revisited and reconstrued in a more formal or operational way, then being connected with other knowledge, the mastery at this stage then being carried one step higher to a new level of formal or operational rigour and to a broader level of abstraction and comprehensiveness. The end state of this process was eventual mastery of the connexity and structure of a large body of knowledge…” (Bruner, 1977, p.141)
It’s interesting to see Bruner use the word ‘mastery’ here, a term that is back in vogue in England’s schools. You may consider me deliberately provocative, but in my view learning is actually a process of not mastering things – avoiding the ‘eventual end state’ that Bruner invokes. Instead, it requires being open to further change. Indeed, the famous sociologist Stuart Hall advocates that “you must have the modesty of saying, this is a bloody good idea but I probably won’t believe it in five years’ time”. The same is surely true of learning at any level. Isn’t the claim that ideas are somehow ‘completed’ a nonsense? Don’t we nearly always learn simplified forms of ideas in the first instance, only to later learn that there are new ways of understanding that we haven’t yet realised? Indeed, isn’t this actually Bruner’s point?
“you must have the modesty of saying, this is a bloody good idea but I probably won’t believe it in five years’ time”
Stuart Hall
Take some examples from primary education:
In English, we master grammar … only to then learn that many of the great authors are great because of the way they break the rules. Most famously perhaps, James Joyce’s Ulysses has a long section at the end, more than 24000 words, with almost no punctuation, creating a stream-of-consciousness effect. By way of example, try this …
‘that was a relief wherever you be let your wind go free who knows if that pork chop I took with my cup of tea after was quite good with the heat I couldnt smell anything off it Im sure that queerlooking man in the porkbutchers is a great rogue I hope that lamp is not smoking fill my nose up with smuts better than having him leaving the gas on all night I couldnt rest easy in my bed in Gibraltar even getting up to see why am I so damned nervous about that though I like it in the winter its more company’
Hope yet for the Y2 child who can’t use full stops or, dare I say it, the undergraduate who finds semicolons baffling! Or if it’s syllables and meter that they haven’t yet mastered, then Spike Milligan shows how comic genius does it:
There once was a man from Japan
Who’s limericks never would scan
When asked why this was
He replied “It’s because
I always try to fit as many syllables into the last line as ever I possibly can.
Meanwhile, over in the science block all theories are simply models of what we can’t yet prove to be wrong – the basis of the scientific method. To master science is therefore to realise that one must assume you haven’t yet mastered it! Nicolaus Copernicus: what if the earth isn’t at the centre after all? Florence Nightingale: what if we look statistically at battlefield hospital deaths to show that they are more likely to be from poor sanitation than battle injury?
And, of course in maths … we are taught to master counting, learning about place value and how numbers sit on a number line. Initially this goes only one way, into the positive numbers; but as part of ‘mastering’ this they begin also to learn that there is an identical set of ‘negative’ numbers that head off in the other direction. Ah, now I’ve mastered them … well no, because after GCSE one comes across numbers which sit not on a line but on a plane. These are the ‘complex’ numbers, with the horizontal axis, as before, showing the ‘real’ numbers, but with the vertical line showing the ‘imaginary’ numbers, which are multiples of ‘i’, the square root of –1 (I know, I know!). Any complex number is described by a point on this plane, for example (–4 + 5i) … And you thought you’d mastered counting?
Of course, in the right hands mastery is a useful notion, encouraging the idea that learning should focus on making connections to create deep, interwoven knowledge of a subject. For example, the National Centre for Excellence in Teaching Mathematics suggests that ‘mastering maths means pupils acquiring a deep, long-term, secure and adaptable understanding of the subject’. The inclusion of ‘adaptable’ is important here – not a linear, fixed body of memorised stuff, but a flexible, connected understanding of the things they have been learning about.
With this in mind one might imagine a curriculum filled with investigative tasks, albeit built on some more direct instruction where this is appropriate, yet I’m not sure that this is what we are seeing in English schools. Take the latest incarnation of the mathematics curriculum guidance from the Department for Education (2020). This guidance,
‘identifies the most important conceptual knowledge and understanding that pupils need as they progress from year 1 to year 6 … referred to as ready-to-progress criteria and [which] provide a coherent, linked framework to support pupils’ mastery of the primary mathematics curriculum’.
Hmmm? Provide a framework of K&U that they need. That sounds rather strange given the definition from NCETM with its ‘adaptable understanding’. It sounds to me more like a method for teaching a specific set of knowledge as efficiently as possible to as many as possible, defining success as passing an exam of routine questions. Take, for example, the instruction later in the same DfE document that ‘pupils should spend sufficient time working with unit fractions to achieve mastery before moving on to non-unit fractions’. This is a non-sequester since to really understand unit (i.e. with 1 on the top) and non-unit fractions one has to understand the relation between them. You can’t ‘master’ unit fractions ‘before’ seeing how they fit into non-unit fractions. In fact, as Sfard (1991) has argued convincingly, to really understand any mathematical idea you have to first accept it as an object that you don’t yet understand in order to ‘play’ with it in other ideas. Through this playing you then come to understand it as a mathematical object, with certain properties. Or, put the other way round, before mastering it you have to see how it works in the next idea ‘up’ the chain of mathematics. (Sfard’s article is worth reading in full, here). For example, the real numbers only come into focus in the light of seeing how ‘they’ are connected to the imaginary ones; and you can only understand addition as a concept by seeing how ‘it’ works as part of multiplication. Mastery isn’t learning one thing at a time, in a chain, as if all ideas are independent of each other; it’s about struggling with ideas, playing with them as objects that are part of related ideas so that, over time, a whole emerges, albeit one that is always provisional.
mastery has its roots in the behaviourism of Skinner and Thorndike and the dream of designing a teaching technology that taught pupils precisely, step by step
To appreciate the confusion here between mastery as an interconnected, personal understanding of a subject and mastery as a procedure for ensuring specific content is learnt efficiently it pays to take a look back as its history. Whilst the notion of interconnection and personal adaptability may echo the constructivism of Piaget and Bruner, mastery has its roots in the behaviourism of Skinner and Thorndike and the dream of designing a teaching technology that taught pupils precisely, step by step, checking understanding along the way and administering corrective tasks if understanding was lacking. This led, in the 60s, to personalised systems of instruction (PSI), ‘an individually based, student-paced approach to mastery instruction wherein students typically learn independently of their class’ (Block & Burns, 1976, p.9) and which would later be computerised.
At about the same time though, another American psychologist John Carroll took up mastery in a slightly different way focusing on the idea that instead of thinking about how much could be learnt by different children in a fixed time, one might ask how much time is needed to teach different children a fixed amount. The aim was to teach as many pupils as possible the desired content in the least time possible. This led to a model of mastery as follows (Carroll, 1963):
Without going into all the numerical and psychological baggage, I will simply quote Carroll’s claim, to give you a sense of how this was conceived, namely that:
‘the model involves five elements—three residing in the individual and two stemming from external conditions. Factors in the individual are (1) aptitude— the amount of time needed to learn the task under optimal instructional conditions, (2) ability to understand instruction, and (3) perseverance—the amount of time the learner is willing to engage actively in learning. Factors in external conditions are (4) opportunity—time allowed for learning, and (5) the quality of instruction—a measure of the degree to which instruction is presented so that it will not require additional time for mastery beyond that required in view of aptitude.’
You can read the whole paper online if you want to find out more (here), but suffice to say that this technology of learning feels a long way from the development of an adaptable, conceptual network – though Carroll was clear that his model relied not on the robotic tuning of instruction (as in PSI) but on reflective and thoughtful teaching and assessment. It must also be understood in the context of a period of history dominated by the space race – which itself was really about developing the capability to launch ballistic nuclear missiles at Russia. How history goes round in circles! Moreover, as Carroll makes clear,
‘the model is not intended to apply, however, to those goals of the school which do not lend themselves to being considered as learning tasks. Such [as], for example … attitudes and dispositions. Educating a child so that he [sic] has tolerance for persons of other races or creeds, respect for parental or legal authority, or attitudes of fair play, is thought to be largely a matter of emotional conditioning or of the acquisition of values and drives.
This is mastery then, but only of the conceptual, not of the soul – ironic given the importance of values in a world where people were learning to hurl nuclear weapons at each other. In his paper Carroll actually avoids defining mastery at all, simply referring to ‘progressing towards the mastery of a task’ and stating that ‘the model may be thought of as a description of the “economics” of the school learning process; it takes the fact of learning for granted.’
However, in its contemporary (Emperor’s new?) clothing, the EEF points out that mastery learning ‘should be distinguished from a related approach sometimes known as “teaching for mastery” [which] is characterised by teacher-led, whole-class teaching; common lesson content for all pupils; and use of manipulatives and representations.’ Again, we see attention swinging between related, but apparently distinct ideas: from mastery learning where material is adapted to address individual differences (PSI); to learning for mastery where time is altered until individuals come to a full understanding; to (now) teaching for mastery and a shift of responsibility towards the teacher, to adapt teaching to address the differing needs of pupils in the class.
Perhaps this is why the modern version of mastery is so confused. Where previously there was a political urgency for well-educated rocket scientists, now this urgency is driven by a perceived economic need, in which ‘education is the best inoculation against unemployment’ (Cameron, 2014). At the same time though, other, more progressive, constructivist cultural roots still lie deeply in our school system, reflected in Plowden (‘the best preparation for being a happy and useful man or woman is to live fully as a child’ (1967, para 506)), Cockcroft (‘Nor should an interesting line of thought be curtailed because ‘there is no time’ or because ‘it is not in the syllabus’ (DfES, 1982, para 250)) and in Bruner’s original quote. Add into these intertwined roots a multifaceted notion of ‘mastery’ and you get a heady mix of philosophy and practice.
So next time you come across mastery in the education system you might like to reflect on which angle it takes – and hence on whether you are involved in an individualised learning machine, a time-centred programme of individual instruction, or a form of adaptive class teaching. Whichever it is, you might consider too whether telling learners that they are to ‘master’ something is really such a good idea in the first place.
For those who want to follow this up…
References:
Block, J. H. & Burns, R. B. (1976) ‘Mastery learning’. Review of research in education, 4 pp. 3-49.
Bruner, J. S. (1977) The Process of Education. London: Harvard University Press.
Cameron, D. (2014) Speech on the economy. Available at: https://www.ukpol.co.uk/david-cameron-2014-speech-on-the-economy/
Carroll, J. (1963) ‘A model of school learning’. Teachers college record, 64 (8),pp. 723-730.
Department for Education and Science (1982) Mathematics Counts. Report of the Committee of Inquiry into the teaching of mathematics in schools under the chairmanship of Dr WH Cockcroft. London: HMSO.
Department for Education (2020) Mathematics guidance: Key Stages 1 and 2: Non-statutory guidance for the national curriculum in England. Available at: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1017683/Maths_guidance_KS_1_and_2.pdf.
Plowden, B. B. H. (1967) Children and their primary schools: a report of the Central Advisory Council for Education (England). Research and surveys. London: HM Stationery Office.
Sfard, A. (1991) ‘On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin.’. Educational Studies in Mathematics, 22 (1),pp. 1.
If you liked the Spike Milligan limerick then try this one too … https://youtu.be/q3eN6dJn4Cg
Thinking Education 