On Thinking Through

I was spiked to write this post by this tweet from @HenkReuling, mathematics teacher at Liemers College in the Netherlands (visit his web page here), where he posed the lovely little problem shown on the left in orange below — inspired by this problem, possibly, shown on the right in white below, one of many such problems created by @sansu_seijin.  We are asked to find the area of rectangle ABCD, given a semi-circle and a quarter-circle.

These geometric problems — and their like — demand the kind of interrogative depth from secondary (11-16) students that is necessary for, and that encourages, their mathematical maturation. They invoke many of the pedagogical conditions I contend we need to concern ourselves with if, indeed, we are concerned about supporting the development of our students' mathematical maturity.  As, indeed, we should be.  But it is important to recognise that such problems will only ever engage this (mathematical) maturation process if we as teachers, as custodians of the knowledge underpinning them, approach and guide our students' interactions with them carefully, sensitively and with intellectual, emotional and pedagogical humility.  In any other such philosophical hands, these problems have the potential to be more than mere missed opportunities, they can undermine the very effects that their design aims — and has the potential — to produce, posing a real and equally as potent threat to a student's mathematical maturation.  

Yes, 'we need to design learning that involves students making meaning,' as David Geurin succinctly puts it, rather than 'just accepting information' (see his blog post here), but here I am also arguing that we need to give as much thought to the design of our teaching to such problems — our interactions with students, how we guide students' own interactions with the problem, what we will say to them and what we will and will not tell them, how we will say what and at what point in the process we will or won't say it — just as much if not more than the thought we give to the design of the problems themselves, or, indeed, the thought we put into what students will do, as if the doing alone equates to learning.

To elaborate, simply doling problems like these out to the dareIsayit more able young mathematicians under our care, as I have seen done, in the name of 'extension', in the mistaken yet often held belief that we are feeding our students' intellectual appetite and that this alone is enough, is, in effect, differentiation's equivalent of a sneaky bag of crisps; the mars bar from the back of the cupboard.  Whilst exposure to and engagement with such multi-layered and thus relatively complex activities is unarguably crucial for mathematical maturation, we cannot expect all of our students to become mathematically mature (or, to put it more accurately, mathematically matur-er, matur-ing) simply from exposure to such problems alone.  We may not be able — or maybe, moreover, it would be silly and more than unproductive of us to try — to teach students how to be mathematically mature, directly, but we must teach our students the tenets of mathematical maturity, or, rather, we must model for them what it is to be mathematically mature, to behave in a mathematically mature manner when we are befuddled by the beautiful sixes and sevens of uncertainty.  

Such problems may on the face of it sate a student's appetite for intellectual challenge, initially, but the burgeoning maturation of the budding young mathematician will soon and at best stall — intellectually in nubibus — if s/he is left lingering, frozen in the fathoms of unfamiliarity, dizzied by the bends of conceptual deprivation.  And it is not to boot OK to simply show them how it's done afterwards — we need to problematise with and alongside our students, to expose them to the habits of thought, emotional responses and ways of approaching an unfamiliar problem that in essence define what it is to be mathematically mature.

It is often through problems such as these that our younger, more able secondary students experience struggling with something, for the first time, gently, at the limits of their knowledge.  They are often some of the first developmental experiences young secondary students have of realising, and specifying moreover, that there is a gap in their knowledge and/or understanding that they need to fill — Vygotsky's 'Zone of Proximal development' for real, perhaps — an information-gap 'feeling' that fuels the developmentally important (to students' mathematical maturity) tenet of curiosity (cf. Lombrozo, 2017; Golman and Loewenstein, 2016; Noordewier and van Dijk, 2015; Loewenstein, 1994.)

   (See Seth Chaiklin's paper and David Didau's great post problematising Vygotsky's ZPD)

Such problems are also often the problems that spike some of the first emotional reactions young secondary students have to an appreciation that no matter how 'clever' they are, how 'clever' they feel they are, or are — or have been — told they are, they still need the support and guidance of someone who knows a bit more.  And in the hands of a skilled teacher, these problems can also be one of the first experiences young secondary students have where the surety that you just absolutely know something, that you just know it's right but can't explain it, is — to put it bluntly — no longer good enough.  (As Mortimer Adler put it, 'The person who says he knows what he thinks but cannot express it usually does not know what he thinks.')  In short, in the right hands these problems are, for the budding young mathematician, potently formative ones, but in the hands of the unthinking, such problems have the potential to do damage, more so perhaps than we may appreciate.

Our teaching in mathematics is thus not — or should not be — just about transmitting knowledge, developing skills and deepening understanding, it is also about modelling the fundamentally emotional responses to what Andrew Wiles describes as the 'state of being stuck' (in Ben Orlin's typically wonderful post) — and thus the approaches to problems that may faze us, that may knock us for six, that dare to tell us we're not as smart as we thought we were — problems, in short, that remind us who's boss.  (Listen to — and be inspired by — Richard Feynman talk about persistence and problem solving here.)

It is in this respect, as I hopefully more thoughtfully proselytised in a previous post, that I contend that as teachers of mathematics it is incontrovertibly our job to encourage, explicitly with thoughtfulness and planning, our students' mathematical maturation.  It is our job to help our students discover for themselves the joys and, indeed, frustrations of mathematical thinking.  And we do this by actively promoting, through our curriculum (our teaching) — or by exposing, in other words, our students to — a rich variety of problems and situations, such as the examples given, that demand:

• An intellectually honest and creative approach to problem solving.
• Perseverance and persistence in the face of difficulty.
• Resilience and retrospection in the face of error, frustration and disappointment.
• Fearlessness, insight and intuition in the face of complexity and unfamiliarity.

Such problems may (or may not) be 'easy' for us, the teachers posing them, but this is — or maybe should be — a pedagogical irrelevance.  We must remember that there is a lot going on in such problems for young minds and our job is to help students learn how to deal with this messiness.  Indeed, this is fundamental to the mathematical maturation process and should be our default position, or mindset, if you will.  If we are going to encourage depth in understanding, if we are bothered about supporting the development of our students' mathematical maturity, as the curators and custodians of their intellectual capital we have to have the mindset that students will find such problems messy, difficult.  When thinking about our teaching in the frame of our students mathematical maturation, therefore, when planning the use of such problems with our students we need, in a nutshell, to adopt the 'Feynman Technique', as described with typical clarity by the ever inspiring Farnam Street Blog, and as I have adapted here:

Firstly, we imagine we are teaching a young child, focusing on ideas (rather than algorithms or definitions), and using simple language*, because this young child will not have access to the complicated subject-specific vocabulary that we would otherwise — in all likelihood, and perhaps to the detriment of deep real learning — use.  We then review what we have imagined, identifying the gaps we encountered in our own knowledge (pedagogical as well as subject), where we get 'stuck', or where we can feel that our students will get stuck, and which therefore allows us to go back to the source of our knowledge — the first principles — where profound learning occurs and mathematical maturation develops.  Lastly, we organise our thoughts, simplifying things where necessary and formulating them into a naturally flowing narrative, a simple 'story that flows'.  (*Do read this also; yet another illuminating post from the Farnam Street Blog, exploring extracts from James Gleick's biography of Richard Feynman, focusing on Feynman's ideas for the teaching mathematics to children.)

Our planning, thus, is (or should be) about crafting how our students will interact with the problem, and how we will best be able to guide them towards creating their (full) solution, with mathematical — if you like, first principle — integrity.  We can do this, the Feynman Imagine stage, by simply visualising talking to a student about / through the problem, akin to visualising moves in a chess game: the moves we plan to make are a function of how we anticipate our opponent(s) will respond.  Or, perhaps more pertinently — as we do not expect a problem to respond to our line of attack, as it were, in anything other than a mathematically consistent way — it is perhaps more akin to visualising the moves we would make on a messed up Rubik's cube to get to the most efficient solution.

The diagram below, my scribbles on the commute home, where I am 'Thinking Through' — in Feynman style — the problem from the perspective of a Y8 class, is key to my personal process of planning.  It allows me to anticipate where my students will get 'stuck', appreciate why, and decide  how to go about what it is I can do about it.  Note that the 'key conceptual step' scribbled down identifies the point in the process where I believe my students will be most 'stuck', will be most tempted to ignore their uncertainty and keep going without evidence, or proof.  This is key* because again it is the point in the lesson where the deepest, most profound learning will take place, and where mathematical maturation will thus be most influenced.  (*Perhaps hinge is a better word.)

   Thinking Through the problem from my students' perspective (scribbles on the commute home)

Whilst I am not advocating the scripting of lessons, I am suggesting that a key part of the planning process when addressing and sharing such developmentally potent problems is this cognitive visualisation, scribbled down or not — playing through of the lesson in terms of how students will interact with the problem and how we should want them to.  At the same time, I imagine myself talking through the problem with my students (rarely noted down as I have below for the purposes of this post), walking the classroom whilst doing so, noting 'stuff' down, modelling ‘movement’ on the diagram, and anticipating students' responses (or lack of).  Questions with 'can', 'what', 'would', and 'is' stems are invariably followed up, where necessary (i.e. if students don't explain themselves fully), with 'how', 'why', 'why not', 'are you sure' and 'what if' stems.

Thinking Through the possible question stream.

We work through the problem ourselves, careful to cognitively empathise with our students' likely experiences and feelings, even if we devised the problem, and establish a pedagogical narrative.  We identify the point in the problem that learning hinges on, and where other segués in learning are most likely to most naturally occur.  We then interrogate / test these segués for possible gaps in students' knowledge and/or conceptual 'leaps' that may hinder or undermine learning, and think about what we may need to do to support children through them, whether breaking away from the problem is necessary to develop the knowledge and/or understanding that is as yet insecure or missing, but that is needed if students are to fully engage with the problem.  In short, we guide our students' interactions with the problem carefully, sensitively and with intellectual, emotional and pedagogical humility, modelling moreover that it is not just OK to be bedazzled by the shock of complexity and unfamiliarity, that it is not just OK to be stuck — it's actually quite exciting.

Notes & (Select) Links:

On Mathematical Maturity

I first paid attention to the phrase ‘mathematical maturity’ some six years ago.  And like so much that seems sensible in life, and certainly in education, the phrase itself, enticingly loaded with the comforting promise of familiarity, seemed to be something I was already aware of — although it was in all likelihood the first time I had heard of it, and without doubt the first time I had actually considered its meaning.  

It was the last — makeorbreak — discussion I was having with Ofsted in my school’s seventh inspection in seven years, after having been deemed, seven years before, to require ‘special measures’.  We were in my insatiably neglected office, with papers and books and Rubik cubes and journals and blu-tac strewn across the floor, on the two frayed chairs, and on my desk, alongside the ever present Jengastic tower of behaviour reports and lost Mathlink cubes and dice and doctored playing cards, and Twix wrappers, and where only days before in the locked bottom drawer I discovered a mouse, dead, curled up on top of some GCSE statistics controlled assessments like a question mark.  Students' names were scrawled across the cheap glassboards on the wall, graphs and data and Venn diagrams — lovingly posterized and laminated — were pinned up alongside a few fading thank you cards, post-it notes and photographs, and a newspaper clipping taken four years before after we had fought and won a gruelling nine month battle to fend off the closure of the school.  Now, almost unimaginably given the school's turbulent history and troubled context, we were on the cusp of being judged to be a school that was providing an ‘outstanding’ education for its students.  

I don't remember much about the ensuing discussion until my interlocutor — the inspector, a mathematician by trade — ended it with the words “OK, I think I have everything I need," before bringing up the lesson he had observed me teaching earlier in the day.  This we both used as a proxy to just talk, unfettered as we now were now by the formality of the inspection: he from the need to gather evidence, me from the need not to mess everything up.  So we subsequently proceeded to put the education world well and truly to rights and, unsurprisingly, the state of the UK's mathematics education in particular.  The inspector told me about a paper he and a colleague had written — building on the 2004 Smith report — to argue that mathematics in the UK should not be a compulsory subject post-14 (Collins and Quigley, 2010), and I was, to put it mildly, flabbergasted, enough indeed to remember the conversation now, six years on, more than the conversation we had that sealed my school's 'outstanding' judgement. 

The argument, in short, was that in ‘attempting to raise the standard [of mathematics] of the majority we are neglecting the capabilities of the minority whose potential contribution to our nation’s growth is huge’ (ibid.).  Or, to frame it another way, whilst the mathematics taught up to the age of 14 is enough to ‘meet the challenges of everyday mathematics in the workplace’ (ibid.), forcing the disinterested or incapable beyond this necessarily limits the curriculum for the mathematically ambitious and capable and thus actively reduces the nation’s ability to produce the mathematically advanced workers the knowledge economy needs.  The authors suggested that 'the issue... is one of expectations rather than actual content,' and contrasted the experience of British and American mathematics students with their Indian and Middle Eastern counterparts by way of illustration, describing 'a class of twelve Indian 17 year olds expected to be able to use applications of differentials in real examples, to use partial differentiation to demonstrate Euler’s theorem, [and] to use Lagrange Undetermined multipliers to find maxima and minima in three dimensions.'  They went on, in admiration, before emphasising that '[t]hese are topics not generally found before the first year of mathematics degree courses yet were handled with considerable mathematical maturity [my emphasis] and competence by the students.'

Whilst I adored the radicalness of the argument, and was sympathetic to the need to address the problem it was attempting to, I was not seduced by it, and disagreed on a number of levels — not least (and in short) because I felt that the proposal was a result of an underlying conflation of mathematical content with pedagogy.  Philosophically the notion is not new, and continues to engage discussion today, Eugenia Cheng, for example, mathematician and author of a number of popular mathematics books, recently suggested that 'we should change what type of math is compulsory' for students at school (listen to her interview here).  The difference now, however, is that even when advocating a non-compulsory post-14 mathematics education, for economic reasons or not, in doing so we cite the complicity of an unfit pedagogy: 'it would be better not to teach maths at all than teach it the way we currently do' (ibid.).

What struck me most in my conversation with the Inspector, and what has endured professionally with me ever since, was the notion of mathematical maturity, and the latent pedagogical promise that comes from the assumed distinction (as made in the paper) from mathematical competence.  Indeed, the fact that the 'twelve Indian 17 year olds' could handle the topics with mathematical maturity and competence not only illustrates this assumed difference, but more than hints, moreover, at the fact that they were able to do so precisely because of the relationship that existed between their mathematical maturity and competence.  But surely, I surmised, the more mathematically competent you get the more mathematically mature you get, and vice-versa?  It's a chicken and egg situation and the answer is yes.  It is mathematical competency we ultimately want of course, but isn't this competency defined in a not insignificant part by a student's mathematical maturity?  Isn't the relationship between the two self-referencing?  Mathematical maturity is not simply a function of expectation in the sense of the level of mathematics a student is exposed to.  It is a function to boot of the manner in which we expose students to the mathematics, the manner in which we get students to think about the mathematics, the manner through which we teach the mathematics.  It is as much a function, thus, of the how we teach as it is of the what we teach.

Becoming mathematically mature may indeed be something that only the mathematically ambitious and technically competent are able, ultimately, to attain, but far from there being anything wrong with trying to make all of our students mathematically mature, or more mathematically mature, the benefits to all of making the effort to do so is, I contend, a pedagogical imperative.

Mathematical maturity then, a phrase whose meaning appears to have been first considered in any depth by Lynn Arthur Steen in 1983, and which has had surprisingly little pedagogical exposure since, is not, in and of itself, the esoteric, abstruse or even semantic notion it may initially appear.  Moreover, it is a notion that I contend contains real regenerative potential for the teaching of mathematics — not least by virtue of the very effort we would need to make to get our collective heads around it — both pedagogically, and in terms of our curricula.

Whilst mathematical maturity may not have had the pedagogical attention I am suggesting it deserves, making it thus appear a somewhat ill-defined concept (as shown by this reddit discussion about what mathematical maturity actually is), it nevertheless endures grail-like to the budding mathematician (as shown by this reddit discussion about how best to go about developing it).  It is indeed telling that we find the notion loitering most — and most menacingly — amongst the prerequisite requirements of a multitude of undergraduate and postgraduate courses.  The University of Oxford's course on Computational Learning Theory, for example, just one of many, states bluntly, almost colloquially even, that '[t]his is an advanced course requiring a high level of mathematical maturity [my emphasis]'.     

Even before we consider what it actually means to be mathematically mature, then, we uncontroversially sense that the mathematician must be.  But we sense just as keenly that whilst it is possible for someone to be mathematically mature without being a mathematician, their level of mathematical competency must still be fairly advanced.  Wikipedia's entry on mathematical maturity alone is revealing insofar as it exposes the difficulty in reconciling what mathematical maturity is with how you get it, surmising it as 'a mixture of mathematical experience and insight that cannot be directly taught', before unsatisfyingly reducing it to coming simply from 'repeated exposure to mathematical concepts.'  Steen (op cit., p100) framed it more carefully, though not too differently, when he cited the two traditional marks of mathematical maturity as 'the ability to bring experience and reason to bear on the solution of new problems', firstly through an 'ability to abstract, to glean the essential structure from a complex situation,' and secondly through an 'ability to synthesize, to create new ideas by effective use of old ones.'  The mathematically mature, he goes on, have 'technical, formal competence', but they also have a 'sophisticated [my emphasis], meaning-filled understanding of the relations of mathematics to the world around it that emerges' (ibid., p101).

Being mathematically mature would thus seem to mean having reached a certain level of technical competency in mathematics, as well as having developed a certain behavioural, attitudinal, and cognitive disposition towards the discipline that is encouraging of continued growth.  Whilst we can reach a relatively advanced level of technical mathematical competency, without whatever behavioural, attitudinal, and cognitive disposition it is that mathematical maturity must demand, without whatever frame of mind I guess it is we are talking about, our learning and growth and capacity to keep learning will inevitably plateau.  So where then does this approach come from, how does it develop, and how, moreover, can we as teachers of mathematics to the mathematically adolescent, as it were, support, encourage and positively influence maturity?

Whilst we can assume that, unless we teach a child prodigy, none of our students will reach mathematical maturity during our time with them, we need to recognise that they will be mathematically matur-ing, or not, as the case may be.  And whether they are or are not, and whether they or we like it or not, is — to put it bluntly — down to us, to how we teach.  It is in these formative years under our guidance (or spell, perhaps) that our students will develop the intellectual, emotional and I would argue aesthetic dispositions that will make their mathematical maturation more or less probable.

Inasmuch, therefore, as we cannot directly teach our students to become mathematically mature, our teaching does, nonetheless, have the power to inhibit or promote their mathematical maturation.  It is down to us to foster an influencing environment in which students are encouraged to be curious, intrigued and interested; an environment where an inclination to doubt, question and wonder is celebrated; where the pat on the back you get is not for trying, but for trying harder; and where making mistakes is OK because they make us think again, they make us learn.  As Steen argued, 'maturity will emerge only if cultivated in an appropriate, supporting environment.  Like a seed that can grow to maturity only in the right kind of soil — dry for some, damp for others — mathematical talent may well require an appropriate curriculum in which to mature' (ibid., p102).

But if we take technical competency as a given, there is no single factor that we can isolate and exploit to make a student a mathematically maturing student.  Such maturation is patently a complicated, messy and multivariate process, that also includes factors external to the influence we can have.  Nevertheless, as teachers of mathematics we know the influence that we can have can be profound, in one way or the letsbehonest other — we experience the ups and downs of the promise and absence of learning in our students from our teaching in our classrooms every day.

It is clear then that the mathematical competency and, indeed, maturity, that will 'emerge' from a curriculum is not solely dependent on how strong the curriculum is.  A curriculum can only ever be conducive to — and cannot produce — mathematical maturation; its effectiveness moreover is a function of the pedagogy associated with it.  Any pedagogical deficit, so to speak, will invariably inhibit mathematical maturation — irrespective, to reiterate, of how conducive to maturation the curriculum is.  In such a situation, even 'students with special mathematical aptitude leave [a course] with less mathematical maturity than when they entered' (ibid. p103). In a nutshell, therefore, if how we teach the curriculum is not 'appropriate', how strong the 'curriculum' is just does not matter, and in fact, moreover, makes no sense. Pedagogy, thus, drives curriculum, or, in Dylan William's words, 'Pedagogy is curriculum'.

The mathematically maturing student is a student who is increasingly taking control of his/her own learning, who is increasingly self-correcting, and whose curiosity is becoming near habitual.  They are learning how to harness their growing creativity within the constraints of the laws of mathematics, a 'game', as Richard Feynman put it, of 'imagination in a tight straight jacket.'  Being mathematically mature is therefore, at its very core, about making meaning.  It is about independence and authenticity of thought, of choosing what direction to think in, and of deciding whether or not that direction is taking you to the right place.  But to expect our students to mathematically mature by virtue of them doing mathematics alone is at best a pedagogical pipe dream, and at worst inhibitively presumptuous.  To mathematically mature, our students need our guidance, our knowledge, our example.  

Guiding our students towards mathematical maturity, however, is about more than simply adopting the pedagogical stance of what Carey notes to be that of the 'top down "progressives"... who want children to think independently rather than practice procedures by rote,' as advocated by their so-called 'bottom down "conservative"' counterparts (2014, p165, cf. this illuminating 'open letter' to Carey from 'Math with bad drawings').  Recall, indeed, is important to the development of students' mathematical maturity, embedded as it by definition would be by a depth of 'first principle' understanding.  As the National Centre for Excellence in the Teaching of Mathematics (NCETM) put it, when outlining 'The Essence of Maths Teaching for Mastery', it is important that 'key facts [are] learnt to automaticity to avoid cognitive overload in the working memory and enable pupils to focus on new concepts' (cf. David Guerin's blog post about deeper learning, and the distinction between learning by delivery and learning by discovery).  

But teaching for mathematical maturity is not some kind of pedagogical pick 'n mix either. Whilst, for example, and on the face of it, there may be parallels to some with the Shanghainese Singaporean so-called ‘mastery’ approach to the teaching of mathematics, there are some important and distinct differences with an approach to teaching and learning that emerges from an underlying concern with students’ mathematical maturation.  Take the 'Mathematics Mastery' programme, for example, which aims — as outlined by the Education Endowment Foundation (EEF) — to 'deepen pupils’ conceptual understanding of key mathematical concepts[, where, ] compared to traditional curricula, fewer topics are covered in more depth and greater emphasis is placed on problem solving and on encouraging mathematical thinking.' (See here for NCETM information on everything 'Mastery.')  These aims are anything but pedagogically incompatible with mathematical maturation, but to help a student mathematically mature is more than helping them 'master' (or tame, perhaps) the mathematics, it is more perspectival than that.  And, moreover, by making mathematical maturity our business we have the potential to reconcile the issues associated by some with the ‘mastery’ approach, or rather, the issues that come from the many (mis)interpretations of the ‘mastery’ approach — such as the impact, for example, on curriculum time and design, and concerns about restricting more able young mathematicians (see, for example, Tim Dracup's post 'Maths Mastery: Evidence versus Spin' in his 'Gifted Phoenix' blog, and also the National Association of Mathematics Advisers' 'Five Myths of Mastery in Mathematics').

As teachers of mathematics I contend that it is incontrovertibly our job to encourage, explicitly with thoughtfulness and planning, our students' mathematical maturation.  It is our job to help our students discover for themselves the joys and, indeed, frustrations of mathematical thinking. And we do this by actively promoting, through our curriculum (our teaching) — or by exposing, in other words, our students to — a rich variety of problems and situations that demand:

• An intellectually honest and creative approach to problem solving.
• Perseverance and persistence in the face of difficulty.
• Resilience and retrospection in the face of error, frustration and disappointment.
• Fearlessness, insight and intuition in the face of complexity and unfamiliarity.

Curriculum is of course crucial, but particularly, to reiterate, in the sense of us being clear about what it is we mean by it in pedagogical terms (cf. Cambridge Mathematics 'What exactly is a curriculum', and consider the recently renewed accountability focus on the importance of curriculum, on what curriculum means: The Guardian June 2017, Amanda Spielman's 'Festival of Education' Speech June 2017, SchoolsWeek June 2017.).  Students' mathematical maturation is dependent on us realising, in the words of Dylan William (op cit.), that 'what matters is how things are taught, rather than what is taught...  The greatest impact on learning is the daily lived experiences of students in classrooms, and that is determined much more by how teachers teach than by what they teach.'  

This is brought into no sharper relief than when considering the lot of our mathematically more able students.  Whilst mathematical maturity is a default aspiration that I argue we should hold for all of our students, there is a danger that those monikored the 'more able', those most likely in other words to reach mathematical maturity, are also the very students whose mathematical maturation is most at risk.  Are they achieving highly in spite of having little mathematical maturity (because of how we teach the mathematics), or are they achieving highly because of their mathematically maturing approach?  We should not assume that the high-achieving students under our care are maturing mathematically by virtue of the very fact that they are achieving highly: achieving high grades at GCSE and maturing mathematically are not necessarily complementary.  Highly achieving students will seem to grasp concepts more rapidly — possibly from a 'fluency illusion' that they and/or their teacher holds (cf. Carey, op cit., pp82-83), or a 'curse of knowledge' (Birch, et al., 2017) — and without our guidance will have an eagerness to quickly 'move on', perhaps too quickly, away from depth and into mere mathematical skirmishes that may actually inhibit their maturation — with inevitable consequences further down the line.  I would indeed go as far as to lay a not insignificant portion of the blame for what is regarded by many to be the 'problematic' transition from GCSE mathematics to employment, A-level and undergraduate courses (cf. Rushton and Wilson 2014, Kyriacou and Goulding 2006), firmly at the feet of mathematical immaturity — or rather, at the feet of an unconcerned pedagogy that reinforces and perpetuates such — particularly for our more mathematically competent students.

To this end, I tentatively suggest — after Steer (op cit.) and others, and in no particular order of importance — that the mathematically maturing student:

• Learns more from a 'first principle' style understanding, rather than memorization.
• Volunteers good, valid — and sometimes unexpected — questions.
• Tunes out the noise to expose key ideas, or missing but required detail. 
• Analyses and reflects on mistakes, rather than being fazed by them.
• Recognises when gaps in knowledge restricts their own capacity to solve a problem.
• Estimates intuitively, and appropriately, perturbing data to solve problems.
• Perceives pattern and applies the principles of symmetry across the disciplines.
• Communicates mathematically, confidently, using notation correctly.
• Sees opportunities for other mathematics generated by a problem.
• Is inclined and has the growing capacity to mathematize situations; and generalise.
• Moves between the disciplines, particularly the geometrical and analytical.
• Recognises a proof as a proof, and when a solution masquerades as a proof.
• Detects and avoids sloppy, lazy reasoning.
• Has an aesthetic appreciation, appreciates elegance in solutions or proofs.
• Is increasingly comfortable handling abstract ideas.

Framing our own work as teachers of mathematics in the context of the development of mathematical maturity — aiming, in other words, to help our students become mathematically maturing students — whether we can (or need to) agree on a precise definition or set of characteristics or not, can be transformative, both in terms of our own teaching and thus in terms of our students' experience of mathematics.  Rather than aiming solely for the high exam grades we obviously and rightly crave for the students under our care, why don't we aim for something higher, with an unashamedly more purist outlook?  Why don't we aim for our students to achieve their high exam grades through a mathematically mature (or maturing) approach?  Indeed, irrespective of the overriding pedagogical imperative, given the recent, profound qualification and curriculum reforms, there is a probabilistic imperative — namely, that is, that the likelihood of students achieving high examination grades today will improve if we ground our teaching in a concern for mathematical maturity.  

Notes & (Select) Links: