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Defence mechanisms operating in mathematics selection and assessment The psychoanalytic take: Maths operates either as an instrument of defence against anxiety or as something onto which anxiety is displaced and which we then defend ourselves against. How far can we get in understanding mathematical identities if we use this approach? Specifically, how do selection and assessment fit into this? Laura Black & Yvette Solomon, with Margaret Brown, Heather Mendick, Melissa Rodd

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Jaques Nimier (1993): One psychoanalytic account “Phantasy can be taken as the mental expression of instincts, but also as a means of escape - an escape from confronting external reality or the frustrated reality within” – so that either (1) the anxiety and its associated phantasy are displaced onto mathematics, and defences are directed against mathematics, so directly containing the anxiety; or (2) anxiety is contained in some other way and defences are mounted against this anxiety, mathematics serving as an instrument of this defence.

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Defence mechanisms against mathematics: phobic avoidance Q1 I feel there is something that keeps me from getting at the problem, a barrier I can't get across. Q3 At the start of a mathematical problem, I feel as if I'm in front of a black hole. Q11 Doing maths is doing something which, to me, seems impossible. It is as if there was a veil... which separates me from what is there, if... I look into space, I don't know, it's faraway... it has already given me this impression... evidently, I don't think immediately of maths, when I look at the stars; but there, I am aware that it represents something similar, I don't know. To look at something and then reach a time when you can no longer see, close your eyelids, there is the veil which prevents us from penetrating any further, from reaching it... I think it's that. “The student cannot approach the mathematical object, something separates him or her from it - barrier, hole, or impossibility. No blame is placed on mathematics itself…. The student is looking for a way of avoiding mathematics because of the anxiety it provokes. The student self-protectively prefers to classify mathematics as impossible rather than confront it. This is a matter of the defence mechanism known as phobic avoidance.” (p.30)

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Defence mechanisms against mathematics: repression Q 9 Doing maths, it represents nothing, it's absurd. Q10 It is something that you are told to do and which you repeat, a bit like a machine. Q13 It is doing something obligatory, something imposed. “…mathematics doesn't represent anything. If it is done at all, it is because it is compulsory; the student is so uninvolved that it feels mechanical. All drive towards mathematics is missing and is repressed. There is no longer anything positive for these students, but rather "indifference". If they are forced into mathematics, it becomes negative (or absurd) for them.” (p.31) The x's and the y's, they represent nothing, it's completely abstract. You have to copy out theorems that are completely idiotic. I don't know what is really concrete. You calculate figures, you do anything! In the end, anything... to me that seems nonsensical... I don't remember anything, it is precisely because it is absurd that I don't remember anything

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Defence mechanisms against mathematics: projection Q16 In mathematics there is no place for personality; all that you do has been done before, everything has been seen already. Q18 Mathematics sometimes risks bringing destruction, you only have to think of the atomic bomb. Q23 When one is deep in maths work, it's difficult to get out of it, that's why it doesn't do to get involved with it. Q24 Doing maths oversimplifies everything, it takes the poetry out of things. Q25 Those who do too much mathematics some times risk not having your feet on the ground. “The feeling of danger can only come from a projection of the self onto mathematics…. The danger is inside the individual; to protect yourself you project the danger outward onto an object…. There is also the danger of losing your own physical boundaries, … of losing one’s way” (p.31) I know a shepherd. …He no longer knows anything about life. … He studies… here he is now, enclosed in a laboratory doing calculations. He suffocates in figures... He thinks mathematically. Well, I don't know, ultimately... we are mad... we are mad. … Fortunately, there's little chance of our going mad, because not many of us become top mathematicians.

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Defence mechanisms against mathematics: reparation Q2 I have the impression I am creating something when I am solving a problem. Q8 It means doing something basic which is the key to everything else. Q12 It is constantly discovering something new. Q19 Mathematics brings you the pleasure of creating something. Q20 Mathematics means another world in which I feel at home. “Construction leads to the discovery of new and important things (‘the key to everything else’), even to the construction of ‘another world’. This reparation could be the ‘wish and the capacity for the restoration of the good object, internal and external, that is the basis of the ego's capacity to maintain love and relationships through conflicts and difficulties’…. But reparation can also be fanatical: ‘in that its aim is to repair the objects in such a way that guilt and loss are never experienced’. ” (p.32, quoting Segal).. by doing maths, you are making something for yourself. You are making something. Well, it's for that reason... that we enjoy it; everybody likes to make something, I think, because if one doesn't manage to make something, in the end to find... to find peace, to have precisely the joy of having done it, I think that makes you do more... The start of a problem doesn’t come from you, but after doing what has to be done, the most important things is that which has to come from you.

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Defence mechanisms against mathematics: introjection Q14 Doing maths is a way of training my mind. Q15 It is trying to find connections between different things. Q17 Mathematics is a way of getting a strong character. Q21 Working with mathematics allows you to acquire a well- balanced personality. Q22 Mathematics allows you to develop good reasoning. “…mathematics helps to set up a certain stability in the personality. The search for connections, the disciplining of the mind, the acquisition of sound reasoning help to establish a balanced character and a strong personality. … Failure to respect the rules of mathematics hinders any use of the subject …certain students … [make] use of it as a way of structuring their personality. …mathematics gives them some order,.. it lets them struggle against the instincts they consider bad..” (p.32) With maths, I think that when one has to solve something, you should do it progressively without jumping… jumping in that way from one question to another when they are totally different, you know... And then, to see how far you can go, although I don't know myself! And it's precisely that, I am searching for a way of life... a stability...It is perhaps because I am looking for something that will make me stable, because I know very well that in myself, I am not.

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Defence mechanisms against mathematics: reversal into the opposite Q4 If I can't find the solution, I feel defeated. Q5 When I work something out, I feel like a void is being filled. Q7 If I find the solution straightaway, I feel really at peace. “[There is] a disagreeable feeling which could possibly be neutralised whenever a solution to a mathematical problem is found…. [This is] a reminder of the ‘narcissistic wound’..and the search for the transformation of this feeling into its opposite: defeat into victory, lack into replenishment, disagreeable feeling into feeling at peace. Failure in mathematics often calls into question the image that you have of yourself. … For students like this there are two sorts of people, those who understand and those who don't. They also say ‘you're either good or bad at maths’".(p.33) When you can't do something, it's not that it isn't normal, it's not that you have a grudge against it... I enjoy it when we're taught something, when we've something to do. To be able to do it. Because you have to be able to do it: it has to be at your level. You're given it, you have to be able to do it! Otherwise, as soon as there's a snag, something you can't do or you say, What's going on? Why can't I do it? I'm stupid, it's no good!'... That's how it is. When you're given something, you have to know how to do it. That's how it is in the course, in class: some can do it, others can't.

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Why does mathematics take this role so powerfully and frequently? “So mathematics, through its rigour, that is to say through its constant refusal to entertain ambiguity, will more than any other discipline revive anxieties arising from noticing individual differences.” (Nimier, 1993: 33) But there’s also a powerful set of discourses surrounding mathematics…. Lucey et al., 2003: 286: “social and cultural analysis needs an understanding of emotional processes presented in a way which does not reduce the psychic to the social and cultural and vice versa, but recognises their interweavement” Psychosocial approaches to identity allow us to focus on how people work and are worked by these discourses.

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We looked for a fit with Nimier’s picture in our own relationships with maths Similarities in our accounts: a focus on ability and its markers discourses of maths and cleverness competition with others (both academic and personal) gender themes. We looked for a fit with the defence mechanisms – we can see them up to a point but the major mathematical discourses are also important gender and institutional practices of assessment and selection seem to be highly relevant in understanding our relationships with mathematics.

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Competition – projection?..my brother. He also did maths and got his PhD in the end. I always thought he was cleverer than me (I think he always thought that I was cleverer than him).... I had a massive discussion/row with my brother about how come he found it so easy to say he was good at maths and I didn’t. The happy ending was that me and my brother still get on,.. and I can now say I’m good at maths. There must be sibling projective identifications like mad going on around maths.

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Comfort mathematics: Reparation? Introjection? Reversal into the opposite? … the puzzle is definitely the motivating factor when doing Sudoku (and all those algebraic equations years ago) – solving the puzzle gives me a good feeling, like I’ve sorted something out in my head and that life and the world is much simpler now because there are only numbers. I guess that maths was more of a direct line to pure abstraction and also instant gratification really in terms of the achievement aspect of the thing.

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Mathematics as protective – introjection? Maybe I have used maths as a defence against other anxieties – I do recognise ‘introjection’ in the sense of using maths as a defence against teenage and maybe adult emotional involvements – somewhere you can feel safe in the stability of rules and structure and away from the emotional uncertainties of life.

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Selfhood and change – defences and assessment discourses Processes of introjection: I had a neo-Platonic view of maths, I thought it was discovered rather than created, cos that was how it felt to me. It went in the first year of my PhD…. my sense of who I was and my place in the world shifted and so I acted different and was different. I had to lose maths as special and to lose myself as special too. Reversal into the opposite: Had I stayed at A I’d have been in ‘honours’ maths so I enrolled in ‘honours’ courses at B. A different league – the thought of those first few analysis classes still brings out anxiety. I stayed on for graduate school. Passed my qualifying exams for PhD. But I was so out of my comfort zone, so stretched by daily difficulties of doing it that hard maths stuff. B as doctoral student were extremely anxious times and it is strange how rigid my thinking about maths became.

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Anxiety about how others see us – reversal into the opposite? I’m always scared I can’t do it and make some terrible error which makes me look really stupid. I also recognise ‘reversal into the opposite’ – the frustrations of failure and the relief at reaching resolution – and the fact that failure is made worse by the knowledge that it will demean you in the eyes of others as someone who is stupid, since mathematics is aligned so strongly with intellectual capacity! I got my first though it was a close one and so I didn’t feel that I’d succeeded at maths. … I felt that I’d crossed the first barrier by learning stuff rather than doing the hard questions. My relationship with maths was never fully secure after that. I still loved it though and it still functioned as a safe space.

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The presence of collective influences - gender issues I admit that choosing to do Mathematics, Further Mathematics and Physics for A-level was partly a matter of feeling that this was a bit exotic/esoteric for a female and separated me out from the crowd. It also had a cachet of intelligence referred to by Jacques N. I also think I was fully aware at the time that it would give me privileged access to a wide range of males at university; the female competition would be tougher in arts subjects.. On reflection, I guess my interest in maths education has been influenced by all these successful men in my family who do or have done maths…. I don’t see maths as impossible – it’s not the knowledge itself which has made maths seem impossible to me but other external or pragmatic circumstances (but maybe I’m in denial!) … It does seem that my interest in maths ed could be a defence mechanism against other events in my family life but I’m not really sure about this.

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Individual disposition versus collective influences There appears to be a distinction between the unconscious phantasies identified by Nimier and the collective discourses evident in our maths biographies (including discourses about mathematics). For example, gender relations seemed to feature in all our stories about our relationships with mathematics – whether we do it and how it is valued. Comparison with others via selection and assessment also featured, and institutions played a part. But how/why does this happen, and how does individual experience fit into large structural influences such as gender ?

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Mathematics Discourses in ‘self-authoring’ We use Discourses which are constructed and perpetuated using devices such as assessment and selection to position ourselves: I got my first though it was a close one and so I didn’t feel that I’d succeeded at maths. … I felt that I’d crossed the first barrier by learning stuff rather than doing the hard questions. Unlike any other thing I’ve ever done, maths is the thing I want to score hundred percent in. … I’m not actually good at maths. … I never feel as though I really truly understand what I’m doing. … And I really do think that a lot of my understanding is somewhat ritualistic in that I don’t know how to apply it to new situations, so to me that means that I’m not really very good at maths.

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Selection/assessment as a positioning device Institutional Policy – audit/performance Discourse: “I got really engaged in solving the equations – page after page in the textbook. I remember being so engaged that I didn’t eat my lunch….I quite enjoyed doing maths at school and was planning to carry on doing it for A level as I decided that I wanted to be an engineer. However, when my mum and I went to the open evening to discuss my potential A level subjects my maths teacher told us that I wasn’t good enough at the subject to carry it on after GCSE (despite the fact I got a B at GCSE).” Sometimes other things such as institutional policies position us in a way which prevents us from acting on our dispositions…..

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Conclusions & questions Do we need to invoke Nimier’s defense mechanisms in understanding how people relate to maths? If we accept Nimier’s basic position that it is useful to identify and understand the varied psychosocial dispositions towards/against mathematics, how might we best integrate these with collective discourses such as assessment and performance? What can Socio Cultural theory and Discourse theory offer? How do they differ in their contributions?

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