ESRC Mathematical Relationships: identities and participation
Edinburgh seminar: The Curriculum
Jenny Shaw
I know very little about maths education per se, but have some indirect experience of it
having spent the best part of a weekend sitting at my kitchen table beside my son
struggling with the maths component of his electronic engineering degree. His girlfriend
was there too, and both us were feeling both his and our anxiety. When, eventually, he
finished the task it felt like a tremendous achievement, though he said he was sure he had
got it all wrong and would fail. Whether or not our sitting there had helped or not, it was
clear that feelings and unconscious states of mind were flowing around the room; and on
reflection I have wondered how far this might be compared to what psychoanalyst
Wilfrid Bion called ‘reverie’ or more specifically ‘maternal reverie’1. There was some
discussion, I gather, at the last seminar about Bion’s theory of thinking, and the terms K
and -K referring to types if learning and knowledge, and the ‘K link’ referring to the
relationship of mutual dependency between infant and mother were raised. This last
flows into the concept of reverie which was the term Bion used for the way mothers
metabolise, process and transform the painful experiences their infants are busy
expelling. Many of us know that just having someone there while you struggle can be
comforting, and some children, or adults, seem especially to need that companionshipbut it is not only comfort.
What I want to talk about today is obviously speculative, but internal states and dilemmas
probably have something to do with liking or not liking a subject, and whether someone
feels they are good at it or not. And both, together, account for a good part of the
disaffection or turning away from maths which is the theme of this series. Because the
psychodynamic approach has been chosen as one of the theoretical perspectives to be
carried through the series I take it that there is some acceptance that unconscious
1
See Wilfrid Bion (1967) Second Thoughts. Selected Papers on Psychoanalysis, London, William
Heinemann Medical Books
processes and associations have an impact, that early life experience is especially
formative, that feelings come before cognition, and that relationships are the fundamental
source or driver of many of life’s patterns. Generally speaking, I am taking an ‘object
relations’ approach, the development of which marked a departure from Freud’s stress on
instincts, but nevertheless want to mention Melanie Klein’s notion of the epistemophilic
instinct, the desire to learn about the world, but also the need to understand what inhibits
it. The child psychotherapist Hamish Canham notes that at the heart of all learning is a
tension between this desire to learn and a different desire not to know the truth and in
writing of this he quoted Ronald Britton on how ‘new knowledge arouses our hostility,
threatens our security, challenges our claims to ominisicence, reveals our ignorance and
sense of helplessness and releases our latent hatred of all things new or foreign’2. It is
easy to find examples of this, at the level of both the individual and society, we need only
think of Thomas Kuhn’s discussion of paradigms and paradigm shifts. But let me stay
with Canham who cites Roger Money-Kyrle’s claim that there are three key facts of life
which, if not accepted, impede learning, one of which is recognizing the centrality of the
relationship between parents to mental life. Evading knowing about this fact (and again
there is plenty of evidence that such evasion occurs as, for example, in the ‘family
romance’), can make for difficulties in learning and especially, Canham suggests,
biology, but also maths. And he then gives an example: of a 6 year patient whose parents
had separated, but were thinking of getting back together again. Based on some drawings
done in the session, of faces and lines connecting them (but not mother and father) and of
the girl then deciding to do some maths – he interprets the difficulty she has at doing a
sum she had set herself (1+46 = 46) as being due to her difficulty in seeing her parents as
a couple and that, as a result, the girl could only see half the sum. This is only one
example, but many of us have our own ‘funny’ ways of doing certain calculations which
are perhaps signs of an inner shaping of mental operations and of resistance to doing
them in the way we are taught/told.
See Canham (2002) ‘Where do babies come from?’ reprinted as ch 1 ‘What makes children want to
learn?’ in B.Youell (2005) The Learning Relationship
2
Numbers, tables, geometric shapes such as cones are all, as are words, capable of eliciting
associations- not all them negative. But some are, and negativity runs through the
language of mathematics which is full of destruction - cancelling, killing, subtracting,
taking away, dividing and, of course, negative numbers. And though there is adding and
multiplication too, the death-dealing words outnumber the more benign or creative ones
and, I am suggesting, resonate at the unconscious level. The power of maths is often
described and thought of as ‘awesome’, and something with an ‘awesome power’ is
potentially very threatening and easily associated with punishment. Words like ‘mean’
are ambiguous or have several meanings, and though they have a specific meaning used
in maths, they carry the other meanings too. Gianna Williams, for example, writes of an
adolescent patient who was very particular about time, and never let her forget the meanness of Greenwich mean time3. The binariness of some maths, of one and nought, does
not allow for any in-betweeness, and may encourage a sense of ‘sudden death’. For
paediatrician and psychoanalyst Donald Winnicott indeterminacy and imprecision was
especially important in fostering growth and creativity4, but maths at the levels studied in
school is very prescriptive, rule bound, and breeds the feeling that there is nothing
between being very good at maths or a complete failure. However creative maths can be
at the higher level it is not at the lower ones and our concern today is about how so few
get to enjoy maths in the most creative way. And one possibility, which I am suggesting
today, is that there is a binary quality about the way maths is learned in school which
inhibits the form of understanding which is based on identification and that unlike some
other subjects, say history, it is harder or even impossible to understand what it takes to
be good at maths. And just as someone who isn’t good in this way can’t see how those
who are can be so, reciprocally, those who are good at maths cannot imagine how those
who aren’t fail to see the light. The precision of maths may, at times, be a comfort, and in
the context of gender differences in subject choice it has often been mooted that because
of the ‘male wound’ (the social demand’ for boys to dis-identify with their mothers) boys
lean towards subjects which give some relief from the messiness of feelings which they
3
Gianna Williams (1997) Internal Landscapes and Foreign Bodies. Eating Disorders and Other
Pathologies, London, Duckworth. Tavistock Clinic Series.
4
Donald Winnicott (1971) Playing and Reality¸London, Tavistock.
have been prevented from expressing5. But precision can also represent heartlessness. It
is a terrible tension and Wilfrid Bion, a psychoanalyst mentioned at the Manchester
seminar, drew heavily on mathematical imagery in his attempt to explain the nature of
psychoanalysis was and, indeed, concludes his book Attention and Interpretation by
prefacing the index as follows: ‘This indeed, like the rest of the book, is the outcome of
an attempt at precision. The failure of the attempt will be clear; what may not be clear is
the following dilemma; ‘precision’ is too often a distortion of the reality, ‘imprecision’
too often indistinguishable from confusion’. A no-win situation.
My main point so far is that maths has connotations with death, and many of the anxieties
which come, for many, from doing maths, hark back to the primitive anxieties of early
life when all was chaos, unpredictability and often full of fear. The existential fears of
infancy are grounded in whether mother is there, in which case all is well, or whether she
is not, when everything falls apart. This is the ur experience, life or death, or ‘one or
none’. With the growth of a capacity for symbolisation, things change, and things,
material things, can come to stand for the mother, and she can be ‘a little bit there’. This
is a the point of the ‘transitional object’ as described by Donald Winnicott and the series
of tricks or illusions which help an infant build the internal apparatus for dealing with
separation and deprivation. It is indeterminacy, not precision, which Winnicott sees as the
critical feature of the ‘transitional object’, and in his account of what goes on between a
mother and her infant Winnicott stresses that what is given is experienced as found or
created. And this has led me to wonder, from a position of ignorance, whether there is
something of this underpinning the terms ‘constructivist’ or ‘fuzzy maths’? Both
Winnicott and Bion attempted to grasp and explain the flow or passage of feelings
between individuals and the psychological techniques, mainly projective identification
and introjection, which powered those flows. Which brings me back to my kitchen table.
I don’t know what happened next, what marks my son got, only that he has does not
appreciate the ‘digital marking’ which gives a tick or a cross, but not feedback which
See Jan Harding and Michael Sutoris (1987) ‘An Object-Relations Account of the Differential
Involvement of Boys and Girls in Science and Technology’ in Alison Kelly (ed) Science for Girls Milton
Keynes, Open University Press and Liam Hudson and Bernadine Jacot (1991) The Way Men Think:
Intellect, Intimacy and the Erotic Imagination New Haven, Yale University Press
5
would help him learn from the wrong answers. Earlier I suggested that the situation of the
shared feelings around my table might be comparable to Bion’s notion of maternal
reverie. Neither my son’s girlfriend nor I could think for my son, we weren’t making
sense of the problem he was struggling with, but we were both intently thinking about
him, we had him in mind and he was not left quite alone with the struggle. As with a
small child a mother may not explain something, but because she is emotionally engaged
with and tolerates it, this may enable the child to make the shift. The reverie concept
involves an exchange of experience between mother and infant where a painful feeling,
like an infant feeling it is dying, is projected into the mother and after some time there is
re-introjected in a way which make the feeling tolerable because it has been ‘shared’ which means the infant does not feel not alone with the terror. But if the ‘mother’ does
not accept the projection, and bats it away, the infant’s feeling is not returned in a more
tolerable understood form and loses all meaning – it becomes a ‘nameless dread’. In
Bion’s description the infant still projectively identifies with ‘the mother’, but instead of
having a sense of her as a receptive and understanding person the baby is presented with
a ‘wilfully misunderstanding object’; but, nonetheless, it is the one with which the infant
identifies. I have put ‘mother’ in inverted commas because I think a subject or discipline,
like maths, can be experienced in a similar way as a person and have argued this
elsewhere6, (and one reason the term ‘object’ is routinely used in the psychoanalytic
literature, is because things or parts of people can be our ‘object’ too). And even if this
example is a bit close to home, the obligation to use academic jargon may capture some
of the awkwardness of identification with a wilfully complex and misunderstanding
object.
This leads to another of Bion’s contributions, the distinction between, and pairing of, the
container/contained7. The point of the container (mother usually) is to contain the child’s
experiences and fears, and Bion writes that it permits ‘an emotional realization of a
learning experience which becomes progressively more complex as it constantly recurs in
different forms throughout mental development, finally encompassing the possibility of
6
Jenny Shaw (1995) Education, Gender and Anxiety, London, Taylor and Francis.
Wilfrid Bion (1970) ‘The Container and Contained’ in Attention and Interpretation: a scientific approach
to insight in psychoanalysis and groups,London, Tavistock Publications
7
whole hierarchies of hypotheses, and scientific deductive systems’. However it seems
that for Bion maths was, perhaps, the final test, and that if psychoanalysis could be
represented in terms of mathematics it would become the container. What may be
interesting for this audience is that Bion used mathematics as the idiom for expressing his
ideas about the nature of thinking. It is abtuse stuff and there is a schema or grid for
plotting how thought develops from a ‘preconception’ which meets/mates with a
‘realization’ to make a ‘conception’ and produces satisfaction, (a harvest of selfsensations). Bion emphasises that the ‘crux’ in the development of thought lies in the
‘decision between modification or evasion of frustration’ and that the development of
mathematical elements is ‘analogous to the development of conceptions’. But he also
writes that ‘Mathematical elements, namely straight lines, points, circles and something
corresponding to what later becomes known by the names of numbers, derive from
realisations of two-ness as in breast and infant, two eyes, two feet and so on….’
I fear I have laboured this point enough, and that what we need to focus on is what breaks
the link and the circuit of mutual understanding/thinking and leads to attacks. The link in
the first place is that between infant and the breast, or patient and analyst, and the
mechanism of projective identification-and the attacks on mathematics or the teacher of
it, is a result of non-meaning dominating and unmet needs being responded to with envy
and hostility – which were beautifully illustrated by some of Steven Blake and Tamara
Bibby’s transcripts from the last seminar. The relationship with the person, the teacher,
becomes the relationship with the discipline and the ‘object’, now maths, goes bad, and
‘pleasure’ comes only from reviling/attacking it and the person (teacher) associated with
it. When communication between the one ‘who can do maths’, (teacher) and ‘the one
who can’t’ (child or student), is ruptured, a different process is set up, misunderstanding
arises. Negative and toxic feelings are projected, and introjected, the ‘container’ doesn’t
understand what is going on, and with non-meaning dominant the infant has hostile,
envious feelings (thinks the mother/teacher does know, and has what is wanted, but won’t
give it- the good things in the ‘breast’ are withheld).
I am not sure what practical lessons we can draw from all this, but I want to end by citing
Margot Waddell noting that the kind of thinking which will be going on in any learningsituation is based in processes for which the mother/infant relationship is the prototype’,
and that there is inevitably a very ‘complex relationship between emotional and cognitive
learning and their underlying mental states’. Waddell then outlines three broad ways of
relating (‘primary identifying with’) another person: adhesive, projective and introjective
and gives an example of three ways a mother might help her child solve a jigsaw puzzle8.
Teachers, as I have argued elsewhere, are emotionally ‘in loco parentis’, and much of
what goes on schools and colleges is piggy-backed on to the parent/child relationshipwith the consequence that it is not just the pupil/teacher relationship which is anchored in
early life experience, but the pupil/discipline relationship too9. These traces and links,
though they become more attenuated and abstract as children move from primary to
secondary school, and have more and different teachers, as well as different rooms to
frame their learning, remain the template or paradigm for all later learning. The challenge
for maths educators is how to be aware of unconscious and primitive understanding of
relationships in ways which foster the manipulation of their more abstract
representations. I know policy makers like solutions from the moment the problem is
identified rather than understood, and that they like ‘trigger points’ too, as these give a
clue to points of intervention. I am loathe to either give suggestions or simply evade the
need for them so I would end on suggesting that there may be scope for re-thinking the
order in which maths is taught – the latency period for example with its characteristic joy
in amassing facts, and joy in being tested stands out but also further experiments such as
those teaching topology to primary school children such as reported by the Sauvys 10.
P.S. I had planned to start this talk with Howard Becker’s reverse engineering point – of
how, if we wanted to turn off students from maths, might we go about it?11 And one of
8
Margot Waddell (1998) Inside Lives.Psychoanalysis and The Growth of the Personality. London,
Tavistock Clinic Series. The adhesive mode is a form of mimicking or parroting where a child seeks to be
someone else, a teacher or parent maybe: the projective is where a child acts ‘as if’ they are another,
towards, say, a sibling; and the introjective is where by seeking understanding and engaging in the task the
child is enabled to build a secure sense of self derived from introjecting the thoughtful qualities of mind
nurtured by the mother/teacher or, in Bion’s world, the ‘thinking breast’.
9
See Shaw op cit.
10
Jean and Simone Sauvy (1974) The Child’s Discovery of Space, Harmondsworth, Penguin Education.
11
Howard Becker (1998) Tricks of the Trade. How to think about your research while you are doing it
Chicago, Chicago University Press
the most obvious answers might be by giving lots of tests and impersonal marking
(digital) systems. And though not all testing is counterproductive, it generally evokes
anxiety and it may be no accident that other subjects suffering ‘flight’, like languages, are
also ones which rely heavily on testing as part of their pedagogy. I was very shocked to
learn at the first seminar in this series, on testing and assessment, that any item which
most students could do was removed- a sign that ranking or stretching was prioritised
over learning, and that almost all mathematicians felt that they were failures. Is it any
wonder?