By 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 tuition class 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’.
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 companionship- but 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
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’.
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.
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 mean-ness of Greenwich mean time.
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 creativity,
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 have been prevented from expressing.
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 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 elsewhere,
(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/contained. 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 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 self-sensations). 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 learning-situation
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 puzzle.
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 relationship- with the consequence that it is not just the
pupil/teacher relationship which is anchored in early life experience, but the
pupil/discipline relationship too.
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 .
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?
And one of 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?
