text:
Gareth Lewis
email: gl116@le.ac.uk

Emotions and problem solving by prospective primary school teachers

Keywords: Motivation, emotion, problem solving

This paper reports on a study into the motivations and emotions of prospective primary school teachers and how they change before, during and after a problem solving task. The results highlight the need to build the emotional intelligence or cognitive-affective competences, resources and strategies to overcome negative emotion and to scaffold learning.

Background

The link between emotion and the experience of learning mathematics has long been recognised [START_REF] Buxton | Do you panic about maths: Coping with maths anxiety[END_REF]. In particular, negative experiences, leading to negative emotions, are seen to inhibit or disable the learning of mathematics. Buxton's study provided evidence of the damaging effect of lack of confidence and competence that motivation and emotion play in the learning of mathematics. [START_REF] Skemp | The psychology of learning mathematics[END_REF] also described the part played by a range of emotions in successful and unsuccessful strategies for learning mathematics. Yet until recently, much research on emotion and mathematics has focussed on anxiety, and particularly test anxiety [START_REF] Hembree | The nature, effects, and relief of mathematics anxiety[END_REF]. Evans (2000, p. 108) claimed "there is still little or no explicit acknowledgement of the importance of the affectivefeelings of anxiety, frustration, pleasure and/or satisfaction which attend the learning of mathematics." More recently, there have been attempts to classify emotions related to learning mathematics [START_REF] Pekrun | The control-value theory of achievement emotions: An integrative approach to emotions in education[END_REF]. This characterisation does not account fully for the wide range of emotions reported in other research [START_REF] Schorr | Student's expression of affect in an inner-city simcalc classroom[END_REF]. [START_REF] Schutz | Inquiry on emotions in education[END_REF] point out the tendency of research to characterise emotion as trait, and that this in turn tends to promote more reductionist interpretations. They note the methodological problems in the attempt to study emotions, since they happen 'in vivo', and in the moment. Although not specific to mathematics, [START_REF] Meyer | Discovering emotion in classroom motivation research[END_REF] review of emotions in classroom motivation research argues for the key role of emotions in learning, using terms such as 'essential' and 'pivotal'.

In relation to emotion and disaffection with school mathematics, there have been few studies. One such is Skinner, Furrer, Marchand and Kindermann. They view disaffection as negative engagement, or, more specifically (2008, p. 767): "the occurrence of behaviours and emotions that reflect maladaptive emotional states." Amongst the emotions accounted for are boredom, anxiety, anger and shame. The framework takes account of these emotions, but offers no theoretical account of their genesis. [START_REF] Lewis | Disaffection with school mathematics[END_REF], using a Reversal Theory structure, has attempted to widen the range of emotions studied, and has reported on the existence of a range of negative emotions that inhibit or disable learning.

Emotions, then, have received less attention than other affective constructs. In particular, emotionas-state has been under-researched. The importance of state, as opposed to trait, or to more stable, cognitively-mediated constructs such as attitudes and beliefs, has been pointed out by Hannula. He says (2012, p. 155): "There is a clear imbalance in favour of studies that focus on traits over studies that focus on states.". He goes on to say (ibidem, p. 155): "In particular, studies that focus on the dynamics of emotional or motivational states in classroom or other learning community are still rare."

More recently, researchers have begun to pay attention to this deficit. The Cerme community, at Cerme 9, 2015, saw a number of papers address issues of emotion, and focussed on emotion-as-state, in rather innovative ways. Liljedhal (2015) collected data from 38 prospective mathematics teachers after an intensely negative experience. The results contribute to work in mathematics education that anchors emotions in a theoretical framework and links them to other constructs in the affective domain, particularly motives. Di Martino and colleagues [START_REF] Antognazza | The flow of emotions in primary school problem solving[END_REF] look at if, and how, young students' emotions change during problem solving, the factors behind the change, and the potential impact of certain emotional changes on mathematical activity. They investigate young students' problem-solving difficulties, and the links between affective and cognitive factors in context. They note that intrinsic causes seem to be attributed by students to positive emotions, and extrinsic causes to negative emotions.

Despite there being a number of significant differences between the studies, they share a number of features that help to address the difficulties of studying emotion 'in-situ'. They both involve polling participants before, during and after the performance of mathematical tasks, and both make creative use of open-ended responses. These studies can be taken together to form a conversation within the research community, where researchers respond to identified gaps in the field, and cooperate to move the research on at each stage. A number of points seem to me to be particularly worth exploring further. The first is the interaction between affect and cognition, which research is only now beginning to explore. Secondly, as pointed out above, we need to understand more fully how affective and emotional states help to facilitate or inhibit learning. Further evidence is needed of the dynamic progression of motivational and emotional states through the problem solving process, how these interact with cognition and cognitively-mediated constructs such as attitudes and beliefs.

The study

The aim of the study, then, is to investigate the motivations and emotions associated with the performance of a problem solving task. More specifically:

What are the motivational and emotional pre-dispositions of prospective primary teachers to performing a problem solving task? How do these change during the process of the task, and how are they interpreted after completion of the task? What is the role of self-regulatory skills in mediating negative emotions? How do motivations and emotions interact with cognition in the undertaking of the task? Prospective primary school teachers on initial teacher training represent a category of whom many members are lacking in confidence and a facility in mathematics (See [START_REF] Liljedahl | Emotions as an orienting experience[END_REF]. There is a lot of interest in this group for this reason. Looking to take and adapt the methods and protocols from the studies outlined above, I ran a session with a group of primary PGCE students in which these ideas can be explored. The protocol involves presenting them with mathematics problems, and polling them both prior to working on them and after the task, about their affective dispositions. The task was actually a set of graduated questions involving working out terms in series in which they were given the first few terms. By using a graduated set of tasks, I was hoping that there were tasks that were simple enough for everyone to get some right, but also some that would stretch the most accomplished.

Seventeen students on the Postgraduate Certification of Education (PGCE) programme for prospective primary school teachers at a UK University volunteered to attend a lunchtime session. They were informed only that I was interested in researching affect in mathematics education. All but two of the volunteers were women, and they were split equally between those who considered being mathematics specialists, and those who did not.

At the beginning of the session, students were briefly shown the task, and then given a questionnaire. Prior to the task, they were asked to rate the difficulty of the task on a 5-point scale from easy to difficult. They were also asked how they felt about the task, what they were thinking, and why. They then undertook the written task. After completion of the task, they were asked how well they did on the task, to describe their emotions and thoughts and feelings as they undertook the task, and about their most negative emotion, and how they dealt with it. The data was content analysed according to the categories of responses, as reported below.

Results

Initial thoughts and feelings

Ten respondents assessed the task as quite easy, and no one assessed it as difficult. Consistent with the Antognazza et al. study, students who rated the task easy or quite easy felt positively about the prospect of doing the task. When asked how they felt about the prospect of the task, prior to undertaking it, a range of positive feelings were expressed: I like a challenge; Excited; Confident; Relaxed; Curious; Anticipation These all reinforced the apparent perceived simplicity of the task. However, there were a few other responses (and these all came from volunteers who rated the task as of medium difficulty):

It makes me feel excited because I want to get it correct, but scared because I might get it wrong (sc) I think I will be able to do the first ones and then they will get harder and I probably won't be able to do them so anticipation (ja)

When asked what they were thinking, and why, most volunteers reflected confidence and excitement. The two exceptions were those again, who judged the task to be of medium difficulty.

How can I get it right and not look silly. (sc) I'm thinking that although maths isn't my strongest subject, I'm not being judged and marked so I feel more relaxed (an)

In terms of the task itself, volunteer scores were evenly distributed between scores of 4, 5 and 6 out of 6, reflecting their evaluation of the task as fairly easy. After the task, volunteers were asked how well they thought they did, and most seemed to judge that they did quite well. What is clear from the narratives is that the motivation to succeed at the tasks was strong. There are multiple mentions of determination and perseverance.

I really wanted to get the answer …… I didn't want to be defeated (ce)

My main emotion was one of determination (rp) I was just determined to get the right answer (cp) Added to this, not only is getting right answers important, but speed in doing so is also seen as a requirement. Thus we hear: How quickly I could work it out.. (cp) I wanted to get through it quickly (sj) I was quite upset when I took a little bit longer to do the last one (id) Fine until I felt rushed due to time (cw)

A number of responses suggests that getting ANY answer wrong is unacceptable, and causes negative emotion.

I didn't do well because I was unable to answer the last two questions (an)

Post task reflection

Although the questionnaire prompts participants to distinguish between emotions on the one hand, and thoughts and feelings on the other, including how they dealt with their most negative emotion, the responses seem to represent a 'package deal' in which the emotions and cognitions are conflated, thus demonstrating how intimately connected they are. In terms of emotions, a range of words and terms are used. They include: confidence; stress; panic; confused; happy; confident; uncertainty; worried; feeling worse; frustration; anger; annoyed; irritated. This list seems to indicate quite a narrow range of primary emotions, comprising variations of anxiety and anger, on the negative side, and happiness and confidence on the positive side. Confusion and uncertainty appear to be cognitive conditions with negative valence, that result in negative emotions such as anxiety.

Again, as with Antognazza et al., making progress and getting right answers are seen as a vital condition of satisfaction.

I was just determined to get the right answer, each time I solved one I was happy (cp) Mainly joy at being able to do the task relatively easy (gh)

The sense of satisfaction and positive emotion continues until the prospect of getting the answer right is perceived to be at risk, when the emotion turns negative. Getting answers right, and then not being able to get an answer, is expressed in emotional dualities: Happiness, success, proud I could do it. Annoyed when I had to take a few looks at the last one (id)

The negative feelings seem to easily initiate more deep seated negative and disabling thoughts: I felt quite happy and relaxed at the beginning when I was able to complete the sequences but later on I felt inadequate as everyone else seemed to know the answers (an) I began to doubt myself (nd) Frustration stimulating a negative thought process that I am not that great at maths (ao) I was happy and confident until I reached 'E'. At this point my uncertainty about maths re-surfaced. I have never been confident in maths and so the fact that I struggled on the last 2 questions made this emotion re-surface (nd) Other categories of narrative expression also emerge from the data. One category relates to the mathematical or heuristic strategies employed by the volunteers either in the search for pattern or answers, or in response to negative emotional conditions. Examples include:

My prior knowledge of sequences helped (cs) My confidence went up and down as I used trial and error, once I'd figured out the pattern that was fine (cp)

Try out a variety of methods until I found the one that worked (id) My most negative emotion was before the last question when I worried 'whoa' not sure I can do this, but I dealt with it by trying to think about the problem from a different perspective and take a different approach as I had with other questions (ce) I approached it very methodically, wrote things down to help remember what I was processing mentally. As I cracked each pattern I felt more willing to try the next (rp) I was thinking about the possibilities of how to work each one out. What different methods may I need to use? (hr) Since the tone of emotion changed from pre-task to post-task, and from mainly positive to mainly negative, it is interesting to examine how these students deal with the negative emotion. There is evidence here of significant self regulation, which is often mediated by self-talk: Can I do it? Can't I do it? (sj) I had to tell myself that I had tried my best (nd) Come on you can do better (cp) It's not a test so it is ok if and when I get it wrong (ja) Self talk also plays a part in the negative case: I couldn't find the sequence and therefore must be rubbish at maths (sc)

The following example illustrates such a negative pathway or series of responses: I was confused as soon as I couldn't find an obvious pattern and consequently panicked and guessed. That made me feel inadequate (sc) Note the sequence (with comments in parentheses):

1. I couldn't find an obvious pattern… [searching for pattern is a cognitive strategyin this case unsuccessful]

2. This made me confused…[ lack of success leads to lack of solution and the cognitive condition leads to high arousal]

3. Therefore I panic…[this induces negative emotion] 4. When I panic I guess…[ this leads to a poor behavioural response] Many of the accounts talk about confusion leading to panic, and the panic relates to the very strong (but unsatisfied) need for progress, often leading to inappropriate strategies for quick solutions:

I felt panicked to try and find a solution as quickly as possible (sc)

Motivational and emotional pathways in mathematical problem solving

By assembling the evidence, it is now possible to propose a model for the possibility space of the pathways for this motivational-emotional-behavioural nexus. To do this, I will draw on the reversal theory framework, but instantiated by data from this study, consistent with data from other, similar studies. More details about the eight motivational states, and their associated emotions can be found in [START_REF] Lewis | Disaffection with school mathematics[END_REF].

The motivational state combination determines how the experience or engagement in the task will be evaluated against the needs of the active states. If the need is satisfied, or if satisfaction is anticipated, it will lead to positive emotion. If not, it will lead to a feeling of frustration, and negative emotion determined by the specific state combinations active at the time. In this case, there is a behavioural dilemma which can be resolved in two ways. Illustrated visually, the space comprises the following pathways: In the serious self-mastery state combination, which evidence suggests is the dominant motivational disposition of students in problem solving contexts, achieving progress, getting right answers (quickly) and the associated feeling of power or competence, is necessary for a positive affective outcome. The likely behavioural response is attraction, leading to the desire to continue (A).

If progress or positive outcome is not achieved in the serious self-mastery motivational disposition, this will result in anxiety, anger (serious), or humiliation or helplessness (self-mastery, losing), whereas in paratelic (playful) state combinations, boredom or sullenness will result if arousal is low and excitement is unavailable.

From this situation, the student has choices available. One choice is to use strategies to override or mitigate the negative emotions. One such approach is the learnt behavioural response to 'call down' mathematical strategies or heuristics, as evidenced above. Another available resource is the application of metacognitive skills such as determination or perseverance. This allows the student to continue, even if it is painful or uncomfortable to do so.

There is also another process that appears to take place. Negative emotions such as anxiety, anger, (possibly in conjunction with helplessness or humiliation) seem to induce a strong need for meaning, significance or explanation for the failure. Since this is unavailable in the situation, it appears to initiate a search of cognitively-mediated constructs (such as attitudes or beliefs) in order to satisfy this need. This search for meaning seems to be strongly mediated by self-talk, and may result, as the evidence shows, in evaluations of self and capability ('You can do it' or, 'I am dumb'), or evaluations or attributions related to the situation ('these questions are too hard', 'mathematics is useless'). But as Antognazza et al. point out, negative emotion is more often associated with explanations extrinsic to the problem at hand.

If such evaluations about self or the situation are positive and enabling, they provide a kind of behavioural override to the negative emotion, and encourage further qualified or reluctant attraction and engagement, as expressed, for instance, in 'come on, you can do better', as illustrated in path (B). If, on the other hand, such evaluations are negative, they result in repulsion and avoidance or withdrawal from the task, as in path (C). We can see this last option in operation in statements of the form 'I am no good at maths', 'I feel less intelligent than the others.' Since all students will, at times, encounter negative emotions, it is important to understand in more depth what influences the choice between pathways B) and C). It seems clear that having a range of cognitive-affective resources are the key to the likelihood of students choosing pathway C). The evidence here suggests that these resources and strategies fall into three categories:

 A repertoire of mathematical or heuristic processes to enact in seeking progress in tasks and problems.

 Meta-cognitive and self-regulatory resources such as determination and perseverance in order to continue with a task when it is affectively uncomfortable or painful to do so.

 An architecture of positive or enabling cognitively-mediated structures or representations such as attitudes or beliefs, that provide a frame of confidence in which otherwise psychologically risky situations can be tackled.

Discussion

This study, then, has attempted to contribute to the understanding of affect, and particularly negative emotions and their effect on learning or not learning, mathematics. A number of interesting points arise. Firstly, further evidence to the Liljedahl study of just how prevalent negative emotion is among prospective primary teachers. Secondly, the evidence here shows just how intimately connected affect and cognition are in undertaking mathematical tasks, and the influence of affect on learning, or not learning mathematics, and indeed, the reciprocal influence of learning on affect. I have proposed a model of the mechanisms by which emotion and cognition interact when students are engaged in mathematical tasks, and in particular, ways that aspects of cognition and behaviour can be used to mitigate negative emotion, such that it doesn't disable learning. This is a modest study which has a number of limitations, especially related to the small sample, and the fact that they were volunteers. Because of this, no attempt has been made to make quantitative generalisations from the data.

Proceedings of CERME10

One of the key points to emerge that should inform teaching practice and pedagogy is that explicit focus and attention is needed to help students to build the emotional intelligence or cognitive-affective competences, resources and strategies to overcome negative emotion and to scaffold learning.


Renaud Leplaideur
email: renaud.leplaideur@univ-brest.fr

Benoît Saussol
email: benoit.saussol@univ-brest.fr

LARGE DEVIATIONS FOR RETURN TIMES IN NON-RECTANGLE SETS FOR AXIOM A DIFFEOMORPHISMS

Keywords: Mathematics Subject Classification. Primary: 37B20, 37C45, 37D20, 37D35, 60F10 large deviations, return times, thermodynamic formalism

For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.

Introduction

Recall that for any given measurable and ergodic dynamical system (X, T, µ), and for any set A with positive µ-measure, Kač 's lemma together with birkhoff ergodic theorem implies that the sequence r n A of nth return-times into A by iterations of the map T satisfies

lim n→∞ r n A (x) n = 1 µ(A)
for µ-a.e. x.

We are interested in fluctuations of order n of r n A around n µ(A) , so we want to prove a Large Deviations Principle (LDP for short), that is, we want to show the existence of the rate function Φ A such that for every u ≥ 1 µ(A) ,

lim n→∞ 1 n log µ r n A n ≥ u = Φ A (u)
and for every 0

≤ u ≤ 1 µ(A) , lim n→∞ 1 n log µ r n A n ≤ u = Φ A (u).
If this holds, we will say that the sequence of return-times into the set A satisfies the LDP for the measure µ. If the above LDP only holds for u ⊂]u, u[ with u < 1 µ(A) < u, we will say that the sequence of return-times into A satisfies a LDP for the measure µ near the average.

A standard method to get a LDP is to prove the existence of the scaled-cumulant generating function Ψ A , defined by the following limit

Ψ A (α) = lim n→∞ 1 n log e αr n A dµ,
and prove that it is differentiable. In this case it is well known that the rate function exists and these two functions form a Legendre transform pair, namely

Φ A (u) = inf α {Ψ A (α) -αu} . (1)
This is the approach that we are adopting in this paper, with the difference that we do not prove the differentiability of the function Ψ A . It is noteworthy that with our monotone approximation method this assumption is not required (see Proposition 5 for details).

Our result applies to Axiom A diffeomorphism and equilibrium state of Hölder potential, and for the sequence of successive return times into a Borel set A that satisfies a condition about the smallness of its (non-Markovian) boundary. Taking a Markov partition and the corresponding semi-conjugacy, we will state the result for subshifts of finite type, keeping in mind that this really corresponds to a result for Borel sets on the manifold.

Statements

Throughout, (Σ, σ) will denote a topologically mixing subshift of finite type. The set of vertices of the defining graph of (Σ, σ) is {1, . . . , N } with N ≥ 2. We denote by A = (a ij ) the N ×N -transition (irreducible and aperiodic) matrix associated to Σ; namely points in Σ are sequences x = {x n } n∈Z such that for every n, x n belongs to {1, . . . , N } and a xnxn+1 = 1.

Recall that if f is an Axiom A diffeomorphism of a compact manifold M then there always exists a subshift of finite type Σ and a coding map π : Σ → M such that π • σ = f • π.

Let φ : Σ → R be α-Hölder continuous. For a given σ-invariant measure λ, the φ-pressure is the quantity P λ (φ) := h λ (σ) + φ dλ; P λ (φ) will also be called the λ-pressure of φ. The unique equilibrium state for φ, i.e. the unique σ-invariant probability measure with maximal φ-pressure, will be denoted by µ φ . Its pressure is the topological φ-pressure.

For a set A ⊂ Σ and an integer n, we denote by ∂A its topological boundary A ∩ Σ \ A.

Note that ∂A can be empty; this holds for example when A is a finite union of cylinders. We let P φ (∂A) be the φ pressure of ∂A ; since ∂A may not be invariant we define it according to the variational principle: P φ (∂A) = sup h ν (σ) + φdν : ν ergodic and ν(∂A) > 0 Note that this does not correspond to the dimension-like definition of the pressure introduced by Pesin and Pitskel [START_REF] Ya | Topological pressure and the variational principle for non-compact sets[END_REF].

If D is any subset in Σ, and for x ∈ Σ, we denote by r D (x) the first hitting in D by iterations of σ (if it exists). Namely r D (x) is the smallest integer n > 1 such that σ n (x) belongs to D, and r D (x) = +∞ if no such integer exists. We also set r 1 D (x) = r D (x), and denote the nth return time r n D (x) the cocycle defined by

r n+1 D (x) = r n D (x) + r D (σ r n D (x) (x)).
Let max(A) := sup µ µ(A) and min(A) = inf µ µ(A), where the extrema are taken among all invariant measures µ. Then our result is:

Theorem. Let φ : Σ → R be any Hölder continuous function. Let A ⊂ Σ be a Borel set. We have:

(1) if for any σ-invariant measure µ, µ(∂A) = 0, then the sequence (r n A ) n≥1 satisfies the Large Deviations Principle for µ φ , except possibly for discontinuity points. More precisely, for any u ∈ 1 max(A) , 1 min(A) the rate function Φ A (u) exists and is finite, and for any u outside the interval [ 1 max(A) , 1 min(A) ], the rate function Φ A (u) exists is equal to -∞.

(2) if the φ-pressure P φ (∂A) of the boundary is strictly smaller than the φtopological pressure then the sequence (r n A ) n≥1 satisfies a Large Deviations Principle for µ φ near the average. We recall that our method is based on the existence of the cumulant generating function Ψ A (α) for every α in some open interval (α, α). For the statement 1, we will prove that the function Ψ A is defined on an interval (-∞, α). For the statement 2, we will only get the existence of Ψ A on some open neighborhood (α, α) of 0.

We emphasize that if f is an axiom A diffeomorphism of a manifold M and µ φ is an equilibrium state of a Hölder potential φ, and V is a Borel set then the theorem applies to A = π -1 V since ∂π -1 V ⊂ π -1 ∂V under the same hypotheses on ∂V . The hypothesis in statement 1 could seem very restrictive; however it should be satisfied quite often in some general situations, as the following example suggests.

Example 1. Let (M, f ) be an hyperbolic automorphism of the 2-torus, and consider the family of balls B(x, r) about a given point x ∈ M . Then, for all but countably many radii r > 0, the condition µ(∂B(x, r)) = 0 for every invariant measure µ is satisfied.

Proof. Let S be the boundary of a ball. Using hyperbolicity one can show that the intersection of S with its images f n (S) consists, at most, of countably many points. Hence the set of recurrent points R(S) in S is at most countable. If an invariant measure gives weight to S, by Poincaré Recurrence Theorem it implies that it gives weight to R(S) which is a countable set. Thus the measure must have an atomic part consisting of a periodic orbit. Hence S must contain a periodic point. Since the set of periodic points of such a map is countable there can be at most countably many boundaries ∂B(x, r) carrying a periodic point as r varies, which proves the proposition.

The hypothesis in statement 2 about the pressure of the boundary appears quite naturally in the thermodynamic formalism of dynamical systems with singularities. In the case of φ = 0 it simply says that the boundary ∂A does not carry full measure theoretical entropy. A more explicit condition can be given on the manifold itself. Proposition 1. Let f be an axiom A diffeomorphism of a manifold M and let µ φ be an equilibrium state of a Hölder potential φ. Let V be a Borel set and denote by U ε (∂V ) the ε-neighborhood of the boundary ∂V . Assume that there exist some constants c > 0 and a > 0 such that

µ φ (U ε (∂V )) ≤ cε a ∀ε > 0.
Then, P φ (∂V ) < P φ (M ). In particular the sequence of return times into V satisfies a Large Deviations Principle near the average. See Section 5 for further details. In particular we have: Example 2. Let (M, f ) be a C 2 volume preserving Anosov diffeomorphism and let V ⊂ M be an Borel set with piecewise C 1 boundary. Then the sequence of return times into V satisfies a Large Deviations Principle near the average.

Proof. Set φ = -log |D u x f |.
The equilibrium measure µ φ is the SBR measure which is here the volume measure, and the assumption in Proposition 1 is clearly satisfied.

Outline of the proof of the theorem: in Section 3 we recall how the LDP was obtained for the return-times in cylinders. In Section 4 we compare the cumulant generating functions of inner and outer approximation of our set A by union of cylinders. In section 5, under the assumption of the statement 2 of the theorem, we prove the existence of the cumulant generating function Ψ A on some interval. In section 6 we give a dynamical proof of the first statement of the theorem.

Large deviations for return time in cylinders

We first recall the local thermodynamic formalism introduced in [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF]. Then we recall how the large deviations principle for union of cylinders was obtained in [START_REF] Chazottes | Fluctuations of the nth return time for Axiom A diffeomorphisms[END_REF]. Finally, we derive a uniform mass concentration principle.

3.1. Induced systems and local thermodynamic formalism. For a given point x = (x n ) n∈Z ∈ Σ, the past (resp. future) of the point denotes the backward (resp. forward) sequence (x n ) n≤0 (resp. (x n ) n≥0 ). For x and y in Σ, when x 0 = y 0 , the point z := [x, y] is the point obtained when we take the past of y and the future of x.

In Σ, the cylinder [i k , . . . , i k+n ] will denote the set of points x ∈ Σ such that x j = i j (for every k ≤ j ≤ k + n). Such a cylinder will also be called a word (of length n + 1) or equivalently a (k, k + n)-cylinder. If x is in Σ, C k,k+n (x) will denote the cylinder [i k , . . . , i k+n ] such that x j = i j (for every k ≤ j ≤ k + n). By extension, C -∞,n (x) will denotes the set of points (y k ) such that y k = x k for every k ≤ n; similarly C n,+∞ (x) will denotes the set of points (y k ) such that y k = x k for every k ≥ n. By definition, the local unstable leaf W u loc (x) is C -∞,0 (x), and the local stable leaf W s loc (x) is C 0,+∞ (x). For n ≥ 0, a n-cylinder will denote a (-n, n)cylinder. The letter R = ∪R i denotes some finite union of (-n, n)-cylinders; in each of these cylinders we fix some local unstable leaf F i . There is a natural projection from each R i onto each F i defined by π Fi (z) = [z, x], where x is any point in F i . For convenience we denote by π F the map defined on R by

π F (z) = π Fi (z) ⇐⇒ z ∈ R i .
We denote by g the first return map in R, and by g F the map π F • g. We thus have g(x) = σ rR(x) (x). Note that if the maps r R , g and g F are not defined everywhere in R, the inverse branches of g F are well defined in the whole F .

We can thus define the Ruelle-Perron-Frobenius operator for g F : for x in F , we set

L S (T )(x) = y, gF (y)=x e S r R (y) (φ)(y)-rR(y)S T (y),
where T : F → R is a continuous function, and S is a real parameter. As usual, S n (φ)(x) denotes the Birkhoff sum φ(x)

+ • • • + φ • σ n-1 (x).
There exists some critical S c , such that for every S > S c all the following holds: L S admits a unique and single dominating eigenvalue λ S in the set of α-Hölder continuous functions. The adjoint operator L * S has also λ S for unique and single dominating eigenvalue; we denote by ν S the unique probability measure on F such that L * S (ν S ) = λ S ν S . We denote by H S , the unique α-Hölder continuous and positive function on F satisfying L S (H S ) = λ S H S and H S dν S = 1. We also denote by µ S the measure H S ν S , and by µ S the natural extension of µ S . We recall that µ S is a g F -invariant probability measure, and µ S is a g-invariant probability measure. At last, we denote by m S the opened-out measure: namely m S is the σ-invariant measure satisfying, m S (R) > 0, and µ S is the conditional measure m S (•|R).

The spectral properties of L S yield the existence of positive real constants C φ and ε S , such that for every Hölder continuous T : F → R, for every integer n ≥ 1 and for every

x in F (2) L n S (T )(x) = e n log λS T dν S H S (x) + O(e n(log λS -εS ) )||T || ∞ .
Note that H S is a positive function on the compact set F . In [START_REF] Chazottes | Fluctuations of the nth return time for Axiom A diffeomorphisms[END_REF], it is proved that the critical value S c is the pressure of the dotted system, with hole R, associated to the potential φ. Namely we consider in Σ the system Σ R := n∈Z σ -n (Σ \ R). Although not explicitly mentioned, the case Σ R = ∅ appears, when min(R) = 0. In this case one simply has S c = -∞ and the identity remains valid with the convention that the pressure of the emptyset is -∞. The proof in [START_REF] Chazottes | Fluctuations of the nth return time for Axiom A diffeomorphisms[END_REF] was done under the assumption that Σ R is mixing. We claim that the mixing hypothesis can be omitted. Indeed, any subshift of finite type can be decomposed in irreducible components, which satisfy the mixing property, but for some iteration of the map σ (see e.g. [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]). As we are considering first returns in R, note that the word defined by the cylinder C 0,rR(x) (x) contains no R i but at the first position. Now, two different irreducible components can be joined in Σ only by a path which contains R. Therefore, the word defined by the cylinder C 0,rR(x) (x) is an admissible word for a unique irreducible component.

Unicity of the equilibrium state in any mixing subshift (for φ) implies that the topological φ-pressure P φ (Σ R ) for (Σ R , σ) is strictly lower than the topological φpressure for Σ, P φ (Σ).

We now finish this subsection with some important characterization for the measure m S .

Lemma 3.1. The measure m S is the unique equilibrium state in (Σ, σ) associated to φlog λ S 1 R . Moreover, its φlog λ S 1 R -pressure is S.

Proof. For simplicity we set β := log λ S . The measure m S satisfies, (3)

h mS (σ) + φ dm S = S + m S (R)β.
We refer the reader to [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF], Proposition 6.8 for a proof. Moreover, the measure µ S is the unique equilibrium state for (R, g) associated to the potential S r(•) (φ)(•)-Sr(•), with pressure β. Let us pick some σ-invariant probability measure ν.

Let us first assume that ν(R) > 0. We have

h ν (σ) + φ dν -S = ν(R) h ν |R (g) + S r(•) (φ) dν |R -S r(•) dν |R , ≤ ν(R)β,
where ν |R is the conditional measure ν(•|R). This gives

h ν (σ) + φ dν -β 1 R dν ≤ S,
with equality if and only if ν |R = µ S (i.e. m S = ν).

If we assume that ν(R) = 0, then ν is a σ-invariant probability measure with support in Σ R . Therefore it must satisfy

h ν (σ) + φ dν -β 1 R dν = h ν (σ) + φ dν ≤ S c < S.
This finishes the proof of the lemma.

3.2.

Large deviations for return times in cylinders. In [START_REF] Chazottes | Fluctuations of the nth return time for Axiom A diffeomorphisms[END_REF], it is also proved that λ S → +∞ as S goes to S c . Moreover, the map S → log λ S is a decreasing convex map on ]S c , +∞[. There also exists some complex neighborhood of ]S c , +∞[ such that the map S → log λ S admits an analytic continuation on it. In particular the map S → log λ S is real-analytic on ]S c , +∞[. Finally, it is proved in [START_REF] Chazottes | Fluctuations of the nth return time for Axiom A diffeomorphisms[END_REF] that for every α < α(R)

:= P φ (Σ) -P φ (Σ R ), (4) lim
n→∞ 1 n log R e αr n R (x) dµ φ = log λ P φ (Σ)-α .
We shall show now that the large deviations for successive return time and entrance time is the same question; namely, the fact that we are starting from the set R or from the whole space to compute the integral does not make any difference. Remark 1. As mentioned in the introduction, this readily implies the large deviations principle for return times in the form given by (1) since the function Ψ R is differentiable.

The proposition is a weak consequence of the ψ-mixing property of the measure µ φ . Indeed, there exists M > 0 and κ > 1 such that if f and g are two integrable functions such that f (x) only depends on (x n ) n≤p and g(x) only depends on

(x n ) n≥p+M then (5) κ -1 f dµ φ gdµ φ ≤ f gdµ φ ≤ κ f dµ φ gdµ φ .
Lemma 3.2. If R and S are finite union of (-m, m) cylinder then for any n ≥ M + 2m and for

κ -1 µ(S) Σ e αr n-(M +2m) R dµ φ ≤ S e αr n R dµ φ ≤ κµ(S)e α(M+2m) Σ e αr n R dµ φ (α ≥ 0) κ -1 µ(S)e α(M+2m) Σ e αr n R dµ φ ≤ S e αr n R dµ φ ≤ κµ(S) Σ e αr n-(M +2m) R dµ φ (α ≤ 0).
Proof. For any n ≥ M + 2m we have

r n-(M+2m) R • f M+2m ≤ r n R ≤ M + 2m + r n R • f M+2m from
which the result follows by inequality [START_REF] Dembo | Large deviations techniques & applications[END_REF].

The proof of the proposition consists in applying twice the lemma : from the integration over S to Σ and then to R.

3.3.

Concentration of the mass. Let R be a finite union of cylinders. The large deviations principle holds for the return times r n R . It is well-known that this implies a kind of concentration of the mass. Proposition 3. Let R be a finite union of cylinders. Let α and δ > 0 such that

α + δ < α(R). Then for every τ > Ψ R (α + δ) -Ψ R (δ) δ we have Ψ R (α) = lim n→∞ 1 n log e αr n R dµ φ = lim n→∞ 1 n log {r n R ≤nτ } e αr n R dµ φ .
Proof. Take ε > 0 so small that -δτ

+ Ψ R (α + δ) + ε ≤ Ψ R (α) -ε.
Using Markov inequality we get

r n R >nτ e αr n R dµ φ = r n R >nτ e (α+δ)r n R e -δr n R dµ φ ≤ e -δnτ r n R >nτ e (α+δ)r n R dµ φ ≤ e -δnτ e (α+δ)r n R dµ φ ≤ e -δnτ e n(ΨR(α+δ)+ε)
= o(e nΨR(α) )

Existence of inner and outer approximations, their properties and consequences of their equality

In the first subsection we prove a monotonicity result about cumulant generating functions. The idea is to approximate the set A from the inside and from the outside by finite unions of cylinders, and show that the inner cumulant generating function Ψ in and the outer cumulant generating function Ψ out exist. Finally, we study the consequence of their equality on the cumulant generating function and the rate function for the set A.

Monotonicity of the cumulant generating function on rectangles.

For m an integer, let B m be the biggest union of m-cylinders contained in A and C m be the smallest union of m-cylinders which contains A. Then, we denote by D m the set C m \ B m (See Figure 1). As any (-m, m)-cylinder is a union of (-m -1, m + 1)cylinders, we have

B m ⊂ B m+1 ⊂ A ⊂ C m+1 ⊂ C m ; Therefore (D m ) is a decreasing sequence of compact set which converges to ∂A.
Following what is done above, there exists two analytic functions Ψ Bm and Ψ Cm respectively defined on ] -∞, α(B m )[ and ] -∞, α(C m )[. As it is said above, α(B m ) is the difference between P φ (Σ) and the topological φ-pressure of the dotted system Σ Bm . In the same way, α(C m ) is the difference between P φ (Σ) and the topological φpressure of the dotted system Σ Cm . Now, we clearly have Σ Cm ⊂ Σ Bm Proposition 4. For any 0 ≤ α < α in and for any sufficiently large m, we have

Ψ Cm (α) ≤ Ψ A (α) ≤ Ψ A (α) ≤ Ψ Bm (α).
For any α < 0 we have

Ψ Bm (α) ≤ Ψ A (α) ≤ Ψ A (α) ≤ Ψ Cm (α).

Proof. The double inclusion B

m ⊂ A ⊂ C m implies that r n Bm ≥ r n A ≥ r n Cm .
The result follows then immediately by integration and taking the appropriate limits.

Remark 2. Forgetting the set A in the previous proof, we have in fact proved that if B and C are finite unions of cylinders satisfying B ⊂ C, then for any α < 0,

(6) Ψ B (α) ≤ Ψ C (α),
and for any α ≥ 0 (but sufficiently small such that the functions are well-defined)

(7) Ψ B (α) ≥ Ψ C (α).
4.2. Existence of inner and outer cumulant generating functions. Let us pick some 0 < α < α in . By [START_REF] Ya | Topological pressure and the variational principle for non-compact sets[END_REF], the sequence of functions (Ψ Bm ) m is a non-increasing sequence of non-decreasing convex functions on [0, α[ (for sufficiently large m !). It thus (simply) converges to some limit function Ψ in . This function Ψ in has to be convex, thus continuous on ]0, α[. It also has to be non-decreasing, thus it must be continuous on [0, α[. Moreover the Dini Theorem yields that the convergence is uniform on every compact set included in [0, α[. This occurs for any 0 < α < α in , thus the limit function Ψ in is non-decreasing and continuous on [0, α in [ and the convergence is uniform on every compact set included in [0, α in [.

and Ψ

C ≤ Ψ A ≤ Ψ C + ε on the interval [0, β]. This implies that Ψ C (β) -βu > Ψ C (α * ) -α * u, therefore by convexity of Ψ C (α) -αu we have (9) inf α Ψ C (α) -αu = inf α∈[0,β] Ψ C (α) -αu ≥ inf α Ψ A (α) -αu -ε.
On the other hand r n C ≥ nu implies r n A ≥ nu, hence Φ A (u) ≥ Φ C (u). Since the large deviations principle holds for the set C (see Remark 1), we have Φ C (u) = inf α Ψ C (α)α and the conclusion follows from inequality (9). The case α * < 0 can be treated in the same way by considering the inner approximation Ψ in .

(iii) By (ii) it is enough to consider u < lim α→-∞ Ψ ′ A (α), but in this case inf α Ψ A (u)αu = -∞ which implies the result by [START_REF] Ruelle | Thermodynamic formalism[END_REF].

We remark that the exceptional value Ψ ′ (-∞) in statement (iii) is the slope of Ψ A at -∞ which is 1 max(A) . At this point the value of Φ A jumps from -∞ to finite values.

Note that if lim α→α Ψ ′ A (α) = +∞ in Proposition 5 then one gets the Large Deviations Principle and the formula (1) holds for every u > Ψ ′ A (α+). In Sections 6 we will prove under the assumption in statement 1 of the theorem the existence of Ψ A on the interval (-∞, α in ). However, we are not able to prove that the limit of the derivative at α in is infinite. We doubt that it could be finite when α in < +∞. Still, this is clearly the case whenever α in = +∞, which is equivalent to min(A) > 0.

Nevertheless, using next proposition which exploits the symmetry of our assumption (since A and A c share the same boundary), this will be sufficient to get the existence of the rate function Φ A on the whole interval (except at the discontinuity).

Φ A (u) = (u -1) Φ A c u u -1 .
In particular, if Φ A c is the Legendre transform of the cumulant generating function

Ψ A c then Φ A (u) = inf α<0 {-αu + (u -1)Ψ A c (α)} .
Proof. Observe that if u > 1 µ(A) ≥ 1 then r n A ≥ nu if and only if the orbit entered at most n times in A before the time ⌊nu⌋, which means that the orbit entered at least ⌊n(u -1)⌋ times into A c before the time ⌊nu⌋. Therefore

1 n log µ φ (r n A ≥ nu) = 1 n log µ φ r ⌊n(u-1)⌋ A c
< ⌊n(u -1)⌋ ⌊nu⌋ ⌊n(u -1)⌋ and the result follows by taking the limit as n → ∞.

Coincidence of inner and outer approximation in the case of a small pressure boundary

The goal of this section is to prove the statement 2 of the theorem. By the previous analysis (See Proposition 5) it is sufficient to prove the existence of some interval (α, α) ∋ 0 such that for any α ∈ (α, α) we have Ψ in (α) = Ψ out (α).

5.1.

A more explicit condition to get a small pressure boundary. Let K ⊂ Σ be a Borel set. We recall our definition of its φ-pressure:

P φ (K) = sup h ν + φdν : ν ergodic and ν(K) > 0 .
Note that it is not the same as the one defined as a dimension like characteristic with forward cylinders. That would satisfy us in the case of expanding maps, but for diffeomorphisms it would lead to a condition much too strong. Proposition 7. Let K ⊂ Σ be a Borel set and let V n (K) be the smallest union of (-n, n)-cylinders which contains K. If there exist some constants c > 0 and θ > 0 such that µ φ (V n (K)) ≤ ce -θn for all integers n, then P φ (K) ≤ P φ (Σ) -1 2 θ.

Proof. Let S n φ(x) = n-1 k=-n φ(x) denote the two sided Birkhoff sum and C -n,n (x) the (-n, n)-cylinder containing the point x ∈ Σ. Recall that since µ φ is a Gibbs measure, for some constant b > 0 and for every x ∈ Σ we have

e -b ≤ µ φ (C -n,n (x)) exp S n φ(x) -2nP φ (Σ) ≤ e b .
Let ν be an ergodic measure such that ν(K) > 0. We have

c ≥ e θn µ φ (V n (K)) ≥ K exp θn + log µ φ (C -n,n (x)) ν(C -n,n (x)) dν(x) = K exp 2n θ 2 + P φ (Σ) + 1 2n S n φ(x) - 1 2n log ν(C -n,n (x)) dν(x).
The Shannon-McMillan-Breiman theorem and the ergodic theorem implies the convergence ν-a.e. of the term into square bracket to the value

θ 2 + P φ (Σ) + h ν + φdν,
which cannot be positive according to Fatou's lemma.

Proof of Proposition 1. Take a Markov partition of sufficiently small diameter and denote by π : Σ → M the semi-conjugacy. We know that the diameter of the image by π of a (-n, n)-cylinder goes uniformly to zero at an exponential rate. Thus, setting A = π -1 V , since ∂A ⊂ π -1 ∂V , we get that the (-n, n)-cylindrical neighborhood of ∂A has a measure exponentially small, therefore Proposition 7 applies.

5.2.

Coincidence for positive values of α. Lemma 3.1 characterizes Ψ Bm and Ψ Cm : as soon as Ψ Bm (α) is defined, Ψ Bm (α) is the unique real number t = t(α, m) such that the topological pressure associated to the potential φ -t1 Bm , P φ-t1B m , equals P φ (Σ)α.

Similarly, Ψ Cm (α) is the unique real number t = t(α, m) such that the topological pressure associated to the potential φ -t1 Cm , P φ-t1C m , equals P φ (Σ)α.

Let us pick some α > 0. We denote by m Bm,α the measure m S obtained when we have R = B m and S = P φ (Σ)α in subsection 3.1. This measure is the unique equilibrium state associated to the potential φ -Ψ Bm (α)1 Bm . The measure weights B m , hence C m and we can take the induced measure on C m . Therefore we have (omitting the subscribe m for convenience)

h mB,α (f ) + φ -Ψ B (α) dm B,α = P φ (Σ) -α h mB,α (f ) + φ -(P φ (Σ) -α) dm B,α = m B,α (B)Ψ B (α) m B,α (C) h µB,α,C (g C ) + S rC (φ -(P φ (Σ) -α))dµ B,α,C = m B,α (B)Ψ B (α),
where µ B,α,C is the conditional measure m B,α|C and g C is the first return map on C. This measure has a pressure in C lower than Ψ C (α); we thus get

(10) m Bm,α (B m ) m Bm,α (C m ) Ψ Bm (α) ≤ Ψ Cm (α).
Recall that for positive α, 0 < Ψ Cm (α) ≤ Ψ Bm (α) and are upper bounded (uniformly in every compact set in [0, α in [).

Proposition 8. There exists some α > 0 such that for every α ∈ (0, α),

lim m→+∞ m Bm,α (D m ) m Bm,α (C m ) = 0.
In particular, Ψ in = Ψ out on this interval.

The proof of the proposition is an immediate consequence of these two lemmas and Inequality (10). Since µ m is an equilibrium state we have (11)

h µm + φ dµ m -Ψ Bm (α)µ m (B m ) = P φ (Σ) -α.
By semi-continuity for the metric entropy and the continuity of φ we obtain

(12) P ν (φ) = h ν + φdν ≥ P φ (Σ) -α + Ψ in (α)L.
If ν(B j ) = 0 then this yields that ν is a σ-invariant measure for the dotted system Σ Bj . Hence, its φ-pressure must be smaller than P φ (Σ Bj ), which is by definition P φ (Σ)α(B j ). If this holds for every j then P ν (φ) ≤ P φ (Σ)α in (remember that (α(B j )) converges to α in ). On the other hand by (12) we had P ν (φ) ≥ P φ (Σ)α, and α < α in . This yields a contradiction. Therefore ν(B j ) > 0 for some j.

Additionally, whenever m ≥ j we get µ m (B m ) ≥ µ m (B j ), and the later converges to ν(B j ) by continuity of 1 Bj . This achieves the proof of the lemma since C m ⊃ B m for any m. Proof. Let us fix some α ∈ (0, α). Let us pick any accumulation point ν for the sequence of measures µ m (we keep the notation of the preceding lemma). We claim that ν(∂A) = 0. Assume for a contradiction that ν(∂A) > 0. Then let H = ∪ n∈Z f -n ∂A be the invariant hull of ∂A. Let ν 0 and ν 1 be the conditional measures of ν on H and Σ \ H. These two invariant probabilities are such that ν = pν 0 + qν 1 for some p > 0. Observe that by definition, any ergodic component of ν 0 gives mass to H. Therefore even if ν 0 is not ergodic, since the entropy is affine we still get that (13)

h ν0 + φ dν 0 ≤ P φ (∂A) = P φ (Σ) -α.
Copying the equality (11), we get for every integers m ≥ j that

h µm + φdµ m -Ψ Bm (α)µ m (B j ) ≥ P φ (Σ) -α.
Thus letting m → ∞ gives, since the entropy is semi-continuous and affine, (

p h ν0 + φ dν 0 -Ψ in (α)ν 0 (B j ) +q h ν1 + φ dν 1 -Ψ in (α)ν 1 (B j ) ≥ P φ (Σ)-α, Hence 14)
and ( 14) yield that for every j

h ν1 + φ dν 1 -Ψ in (α)ν 1 (B j ) ≥ P φ (Σ) -α + p q (α -α).
We now choose j large enough such that

h ν1 + φ dν 1 -Ψ Bj (α)ν 1 (B j ) > P φ (Σ) -α
holds. This is a contradiction because the measure ν 1 would have a φ -Ψ Bj 1 Bjpressure strictly larger than the associated equilibrium state. Thus we have ν(∂A) = 0.

To finish the proof let us fix some ε > 0 and consider any j such that ν(D j ) < ε. Such an integer j exists by outer regularity of the measure ν and because ∂A = ↓D n . Note that 1 Dj is continuous. Now, for any m ≥ j we have D m ⊂ D j , and then we get 0 ≤ lim sup

m µ m (D m ) ≤ ν(D j ) < ε.
This holds for every positive ε, which proves the lemma.

Remark 3. We remark that under the assumption in statement 1 of the theorem, we always have ν(A) = 0 for the measure ν constructed in Lemma 5.2, therefore α = α in .

5.3. Coincidence for negative values of α. We remark that the measure m C,α is a Gibbs measure with full topological support, thus it gives weight to B. Therefore we can copy the case α positive and induce on B (instead of C); we get similarly

(15) m Cm,α (C m ) m Cm,α (B m ) Ψ Cm (α) ≤ Ψ Bm (α).
Proposition 9. There exists some real α < 0 such that for every α ∈ (α, 0),

lim m→+∞ m Cm,α (D m ) m Cm,α (C m ) = 0.
In particular, Ψ in = Ψ out on this interval.

The proof of the proposition is an immediate consequence of these two lemmas and Inequality (15). Since µ m is an equilibrium state we have ( 16)

h µm + φ dµ m -Ψ Cm (α)µ m (C m ) = P φ (Σ) -α.
By semi-continuity for the metric entropy and the continuity of φ we obtain

(17) P ν (φ) = h ν + φdν ≥ P φ (Σ) -α + Ψ out (α)L.
Therefore L = 0 otherwise the right hand side would be larger than the topological pressure of φ.

Lemma 5.4. There exists α < 0 such that for any α ∈ (α, 0) we have lim

m→∞ m Cm,α (D m ) = 0.
Proof. We keep the notation of the preceding lemma. Let ν be an accumulation point of (µ m ). We first show that ν(∂A) = 0. By equality (16) we get that for any integers m ≥ j, since C j ⊃ C m and now Ψ Cm (α) < 0, we have

h µm + φdµ m -Ψ Cm (α)µ m (C j ) ≥ P φ (Σ) -α.
Letting m → ∞ gives, since the entropy is semi-continuous and 1 Cj is continuous, that

h ν + φdν -Ψ out (α)ν(C j ) ≥ P φ (Σ) -α.
Assume for a contradiction that ν(∂A) > 0 and decompose ν = pν 0 + qν 1 as in the case α positive. Let δ > 0. By definition of Ψ out , for any j sufficiently large we have -Ψ Cj (α)ν 1 (C j ) + δ ≥ -Ψ out (α)ν 1 (C j ). Since the entropy is affine, we get

p h ν0 + φdν 0 -Ψ out (α)ν 0 (C j ) +q h ν1 + φdν 1 -Ψ Cj (α)ν 1 (C j ) + δ ≥ P φ (Σ)-α.
This together with (13) gives

q h ν1 + φdν 1 -Ψ Cj (α)ν 1 (C j ) + δ ≥ P φ (Σ)-α-p (P φ (Σ) -α -Ψ out (α)ν 0 (C j )) .
Since the pressure P ν1 (φ -Ψ Cj (α)1 Cj ) ≤ P φ (Σ)α this implies that

q(P φ (Σ) -α + δ) ≥ P φ (Σ) -α -p (P φ (Σ) -α -Ψ out (α)ν 0 (C j )) .
By outer regularity of the measure ν 0 we have ν 0 (C j ) → ν 0 (A) ≤ max(A) as j → ∞. Since δ is arbitrary this gives p(-α-Ψ out (α)+α) ≥ 0, which is contradictory if p > 0 and α is small enough, since the function α → α -Ψ out (α) max(A) is continuous and vanishes for α = 0. Thus there exists α < 0 such that if α ∈]α, 0[ we have ν(∂A) = 0.

The conclusion of the lemma follows as in the positive case.

Remark 4. We remark that under the assumption in statement 1 of the theorem, we always have ν(A) = 0 for the measure ν constructed in Lemma 5.4, therefore α = -∞.

6.

A dynamical proof of the coincidence of inner and outer approximation in the case of totally negligible boundary

In this section we give an alternative and somewhat more direct proof of the statement 1 in our theorem. By Proposition 5 and Proposition 6 it suffices to show the equality Ψ in = Ψ out on the interval (-∞, 0) for the set A and its complement A c . The hypotheses on the boundary is completely symmetric if we replace A by A c , so it is sufficient to prove the equality on the interval (-∞, 0) for the set A only. However, we also prove that the equality holds some interval (-∞, α in ) for some α in > 0. This in turn not only implies that the rate function Φ A exists on the whole interval [0, +∞) (except at discontinuity points), but also shows that the formula ( 1) is satisfied on some interval [0, u) for some u > 1 µ φ (A) . 6.1. Infinite rate function for return times near the boundary. Recall that D m = C m \ B m is the m-cylindrical neighborhood of the boundary ∂A. For convenience, and for general computations, we remove the subscript "m" and just write D. Our aim is to show that, the probability that the successive return times into D m are small, is extremely small. We first prove a key lemma. Let us pick any accumulation point µ for the sequence of probabilities (µ m ). Recall that the map 1 Dm is continuous. Let us pick some integer m. For simplicity we write converging sequences instead of converging subsequences.

µ(D m ) = lim n→∞ µ n (D m ) ≥ lim inf n→∞ µ n (D n ) ≥ lim n→∞ max(D n ) - 1 n = ρ.
By outer regularity of the measure µ this yields that ρ ≤ lim µ(D m ) = µ(∂A) = 0.

Proposition 10. For every v > 0, there exists some

M = M (v) such that for every m ≥ M , Φ Dm (v) = -∞.
Proof. Let v > 0. By Lemma 6.1 we always can consider m large enough such that 1 µ φ (D) > v.

Note that D is a union of (-m, m)-cylinders. We thus can use the large deviations principle for (r k D ) (see Remark 1) which gives

Φ D (v) = lim n→∞ 1 n log µ φ r n D n ≤ v = inf α<α ′ {-v α + Ψ D ( α)} , (18)
where α ′ = α(D) > 0 (it thus depends on m).

We emphasize that the slope of α → Ψ D ( α) as α goes to -∞ is Letting ε go to 0 we get that Ψ in (α) = Ψ out (α).

6.3.

Coincidence for negative values of α. We now do the proof for a fixed α < 0. Here again we omit the subscript "m" when it is not necessary. We also pick some positive ε. Then, we have: Let us pick some positive real τ which will be chosen latter. We have for some α ′ > α in . Fix some j and some α ∈ (0, α(B j )). Choose then τ such that τ α + Ψ Bj ( α) < 2Ψ Bj (α) < 0.

e
Recall that on R + all the Ψ C are lower than all the Ψ B , and the converse holds on R -. Therefore we get for every m that (27) τ α + Ψ Cm ( α) < 2Ψ Cm (α) < 0.

For n large enough, (26) and ( 27) yield (28) µ φ r n(1+ε) C

> n(1 + ε) τ ≤ e n(1+ε)(ΦC( τ )+ε) ≤ e n(1+ε)(2ΨC (α)+ε) .

Following Proposition 10 we get (29) µ φ (r nε D ≤ n(1 + ε) τ ) ≤ e 2n(1+ε)ΨC (α) for every large enough m and for every large enough n. Therefore (24), (28), and (29) yield for every large enough m and for every large enough n: e αr n Bm dµ φ ≥ e n(1+ε)(ΨC m (α)-ε)e 2n(1+ε)ΨC m (α)e n(1+ε)(2ΨC m (α)+ε) ,

For fixed m, letting n go to +∞ and using Proposition 4 with α < 0 we get for every ε > 0 (30) Ψ in (α) ≥ (1 + ε)(Ψ out (α)ε).

Proposition 2 .

 2 If R and S are non-empty finite unions of cylinders, then Ψ R (α) = lim 1 n log S e αr n R dµ φ , in particular we have Ψ R (α) = log λ P φ (Σ)-α .

Figure 1 .

 1 Figure 1. Inner and outer approximation of the set A by m-cylinders.

Proposition 6 .

 6 Assume that the Large Deviations Principle of return times into A c holds for any u ≤ 1 µ(A c ) with a continuous rate function Φ A c (except possibly for u = 1 max(A c ) ). Then the Large Deviations Principle for return times into A holds for any u ≥ 1 µ(A) (except possibly for u = 1 min(A) ) with a rate function Φ A which satisfies

Lemma 5 . 1 .

 51 For any α ∈ (0, α in ) we have lim inf m→+∞ m Bm,α (C m ) > 0 Proof. Let us denote by µ m the measure m Bm,α , and pick any accumulation point ν of (µ m ) such that µ m (B m ) converges (up to the correct subsequence) to L := lim inf m→+∞ µ m (B m ). Let us show that L > 0 whenever α < α in .

Lemma 5 . 2 .

 52 Let us set α := min(P φ (Σ) -P φ (∂A), α in ) > 0. For every α ∈ (0, α) we have lim m→+∞ m Bm,α (D m ) = 0.

Lemma 5 . 3 .

 53 For any negative α we have lim inf m→+∞ m Cm,α (C m ) > 0. Proof. Let us denote by µ m the measure m Cm,α , and pick any accumulation point ν of (µ m ) such that µ m (C m ) converges (up to the correct subsequence) to L := lim inf m→+∞ µ m (C m ).

Lemma 6 . 1 .

 61 With the assumption on ∂A, lim m→+∞ max(D m ) = 0. Proof. Since D m is decreasing the limit ρ := lim m→+∞ max(D m ) exists. For any m there exists some probability µ m such that µ m (D m ) ≥ max(D m ) -1 m .

1 maxFigure 2 .

 12 Figure 2. The rate function Φ D (v) is the maximal distance between αv and Ψ D ( α) on the negative axis.

1 max(

 1 Dm) < v. This implies by (18) that Φ Dm (v) ≤ lim α→-∞ α v -1 max(D) = -∞ (See Figure2).

6. 2 .

 2 Coincidence for positive values of α. Fix some α ∈ (0, α in ) and δ such that α + δ < α in . For sufficiently large m, all the Ψ Bm are defined on [0, α + δ] and equicontinuous. then choose a uniform τ in Proposition 3 such that the mass concentration holdsBm dµ φ for all sufficiently large m.Let us pick some fixed positive ε. αr n B dµ φ .The first term in the right hand side of this equation is simply bounded by(21) e αr n(1+ε) C dµ φ ≤ e n(1+2ε)ΨC (α)provided n is sufficiently large. We turn to the second term. The condition rn(1+ε) C < r n B ≤ nτ implies that r nε D ≤ nτ , hence we get B dµ φ ≤ e αnτ µ φ (r nε D ≤ nτ ) = e αnτ µ φ (r nε D ≤ (nε) , if we consider m ≥ M ( τ ε ) for some fixed ε, for n large enough we get µ φ (r nε D ≤ nε τ ε ) ≤ e -2ατn , Therefore, (22) gives for n sufficiently large r n(1+ε) C <r n B ≤nτ e αr n B dµ φ ≤ 1. Recall that Ψ C (α) ≥ 0 for α ≥ 0. Then, (20) together with (21) and (19) yield that Ψ B (α) ≤ (1 + 2ε)Ψ C (α) It follows from Proposition 4 that (23) Ψ in (α) ≤ (1 + 2ε)Ψ out (α).

  1 + ε) τ + µ φ (r nε D ≤ n(1 + ε) τ ) . (25)The large deviations principle for r k C meansΦ C ( τ ) := lim α<α ′ {τ α + Ψ C ( α)} (26)

  αr n B dµ φ ≥

			r n B ≤r	C n(1+ε)	e αr n(1+ε) C	dµ φ
	(24)	≥	e αr n(1+ε) C	dµ φ -	r n B >r	C n(1+ε)	e αr n(1+ε) C	dµ φ .

In the same way the sequence of functions (Ψ Cm ) m is a non-decreasing sequence of non-decreasing convex functions on [0, α in [ (for any m !). It thus (simply) converges to some limit function Ψ out ; this function Ψ out is convex and continuous on ]0, α in [. Note that by [START_REF] Ya | Topological pressure and the variational principle for non-compact sets[END_REF] we have 0 ≤ Ψ out ≤ Ψ in .

As Ψ Bm (0) = Ψ Cm (0) = 0 for any m, Ψ out is continuous on [0, α in [, and the convergence is uniform.

We do the same work on ] -∞, 0] using (6) instead of [START_REF] Ya | Topological pressure and the variational principle for non-compact sets[END_REF]. Note that for α ≤ 0 we have 0

The two functions Ψ in and Ψ out are convex and non-decreasing. By Proposition 4 we have

We emphasize that the existence of the limit Ψ A (α) immediately follows from the equality Ψ in (α) = Ψ out (α). Moreover, it also implies, quite surprisingly, despite any knowledge about the differentiability of the function Ψ A , that Φ A exists and is the Legendre transform of Ψ A :

on an open interval (α, α) then (i) the cumulant generating function Ψ A exists, is equal to Ψ in = Ψ out on this interval and it is a convex, continuous, non-decreasing function. In addition Ψ A is differentiable for all but countably many points and left and right limits of Ψ ′ A exists everywhere on the closure of the interval.

(ii) for all u ∈ (Ψ ′ A (α+), Ψ ′ A (α-)), the rate function Φ A (u) exists and satisfies the relation

{Ψ A (α) -αu}.

(iii) if α = -∞ then for any u < Ψ ′ A (α), except possibly for u = Ψ ′ A (-∞), the rate function Φ A (u) exists and satisfies the relation

(ii) Using exponential Markov inequality we immediately get the upper bound

We now prove that for Φ A defined with the lim inf the lower bound also holds true. Fix u ∈ (Ψ ′ A (α+), Ψ ′ A (α-)). By convexity of Ψ A the function Ψ A (α)αu attains its infimum for some α * ∈ (α, α), and the limits at the endpoints of the interval are strictly larger. Let ε > 0 so small that there exists β < α * < β in the interval (α, α) such that for α = β and β we have Ψ A (α)αu > Ψ A (α * )α * u + 2ε.

Suppose α * > 0, i.e. u > 1 µ(A) . By equality of the outer approximation Ψ out with Ψ A , there exists a set C which is a finite union of cylinders such that A ⊂ C


domain: math shs

text:
Elizabeth Lake
email: elizabeth.lake@ucl.ac.uk

To err is human. The management and emotional implications of teacher error

Keywords: Error handling, emotions, classroom, expertise

Research into errors in mathematics classrooms is often centred on student error. Whilst investigating teacher emotional expressions, using data from experienced teachers and affective pathways, I encountered examples of teacher mathematical error occurring in association with expressed emotions. Similarly, I have observed instances of new teachers making errors and hence have begun exploring the implications of teacher error management. In this paper, after describing illustrative examples, I suggest a model based on a continuum of affectively driven strategies that are likely to be familiar to secondary mathematics teachers. Each affectively driven strategy has implications for teacher student relationships and for the learning of mathematics. I discuss some of these implications in terms of longer term affective impact on classroom climate and for students. I offer the model for further discussion in relation to developing growth mindsets.

Introduction

There is no doubt that we all make errors and human instinct is to avoid errors as unpleasant experiences. Yet brain analysis research suggests error, however unpleasant, is essential for effective learning; for creating neural connections. [START_REF] Moser | Mind your errorsevidence for a neural mechanism linking growth mind-set to adaptive posterior adjustments[END_REF] through studying the neural mechanisms when making numerical mistakes shows that this activity fires synapses. There seem to be two synaptic responses. Firstly, when the brain experiences conflict, without awareness, there is an increase in electrical activity. Secondly, brain signals act to draw conscious attention to the error. Conflict (such as recognising an error), also triggers emotions which means making a mistake triggers an observable emotional response. The emotions may be exaggerated as error provides not only a trigger for emotions, but also requires the person experiencing the error to engage in some form of minimising or regulation. Such a positive view of errors is supported by [START_REF] Boaler | Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching[END_REF] and others who suggest we should go further than just avoiding error; that errors should be attended to as part of addressing misconceptions and hence are necessary for learning. In mathematics teaching errors can take many forms, often numerical. Boaler suggests creating a culture where students are comfortable with handling error is beneficial for learning mathematics. Such a culture depends not only on how a teacher addresses student error, but also on how they model addressing their own errors. There seem to be international and cultural differences that merit discussion in terms of how errors are seen and addressed by teachers, from commonly using error as a teaching tool through to avoiding errors and associated discussion as damaging to self-esteem [START_REF] Santagata | Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons[END_REF].

Researchers investigating error focus on teacher responses to student error, and a few explore how students respond to teacher error (e.g. [START_REF] Borasi | Exploring mathematics through the analysis of errors[END_REF][START_REF] Heinze | Mistake-handling activities in the mathematics classroom[END_REF]Ingram, 2014b;[START_REF] Santagata | Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons[END_REF][START_REF] Steuer | Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate[END_REF][START_REF] Tainio | Emotion work and affective stance in the mathematics classroom: the case of IRE sequences in Finnish classroom interaction[END_REF]. There is less research on the role of error or how teacher error management occurs in the classrooms of experienced teachers. If error triggers an emotional response, this has implications for classroom relationships and for students who may not notice error. This is important in mathematics as there are perhaps, more opportunities for error.

The definition of 'error' used here, interchangeable with 'mistake', is that error is a mismatch. A mistake demarcates the distinction between norms (such as usual behaviours) and a deviation [START_REF] Borasi | Exploring mathematics through the analysis of errors[END_REF], (the unexpected or different) thereby defining what is false and what is correct [START_REF] Heinze | Mistake-handling activities in the mathematics classroom[END_REF]. Using the classic constructivist definition of emotion as a response evoked by recognition of a disparity [START_REF] Mandler | Affect and learning: Causes and consequences of emotional interactions[END_REF], it seems error cannot occur without an emotional association. This association implies that patterns of individual emotions in mistake situations is indicative of classroom culture. For this reason, [START_REF] Steuer | Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate[END_REF] asked students about their teacher's handling of mistakes. In addition to identifying 'mistakes friendly' environments, they found that perceived mistakes friendly environment resulted in increased effort. Yet as recently as 2014, research suggests that UK teachers still predominantly give the message that errors are to be avoided, often through a variety of teaching strategies that rarely indicate an incorrect solution. "These strategies all give the interactional message that errors are to be avoided, or that errors are undesirable even when the teacher does not explicitly say this, or in fact explicitly states the opposite" (Ingram et al., 2014, p.40). In terms of beliefs, a teacher's stance on error may be revealed by their response to incorrect answers, but more indicative is their response to public revealing of their own errors. Such responses model expected emotional responses to error for that class. [START_REF] Tainio | Emotion work and affective stance in the mathematics classroom: the case of IRE sequences in Finnish classroom interaction[END_REF] consider this in relation to emotional contagion where "Emotional contagion means that in interactions, emotions are usually shared by participants after one participant has offered public forms of emotion for others to attend to" (p.84). Further, students will mimic or synchronise [START_REF] Hatfield | Emotional contagion[END_REF] with their teacher's publically expressed affective pathway. According to [START_REF] Goldin | Affective pathways and representation in mathematical problem solving[END_REF], people experience a series of emotions as they pass through the process of problem solving in mathematics. The result is an error climate. [START_REF] Steuer | Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate[END_REF] see what I refer to as affective pathways as a predominantly positive (adaptive reaction) or negative (maladaptive) patterning which, if public, means displaying certain emotions enhances learning.

Primarily, an adaptive reaction pattern is distinguished from a maladaptive pattern: An adaptive reaction pattern following errors and failure maintains learning motivation and functional affects such as joy; a maladaptive pattern decreases learning motivation and increases feelings of shame and hopelessness. (Steuer et al., 2013, p.197) Examples of transcript and identified affective pathways (Table 1 & 2) from the classrooms of experienced teachers (> 10 years) illustrate encountering and managing of error. I examine the data from the lens of affective pathways [START_REF] Goldin | Affective pathways and representation in mathematical problem solving[END_REF] as a potentially useful model to examine the illustrative examples. A pathway structures the interpreted emotional journey by labelling emotions from identifying a problem through to either resolution or abandonment. Examining teacher modelling of how to deal with error in conjunction with how a teacher emotionally manages error may assist in interpreting the affective impression given to students. [START_REF] Mandler | Affect and learning: Causes and consequences of emotional interactions[END_REF] suggests that using and modelling emotional responses to error inculcates a tolerance for error that benefits learning mathematics. How a teacher frames the handling and recovery from error can shape student experiences, and indicated preferred attitudes to error management [START_REF] Santagata | Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons[END_REF] especially if we consider error as a stimulus to action [START_REF] Borasi | Exploring mathematics through the analysis of errors[END_REF].

It seems we need errors to learn, whilst teachers can support learning and affect classroom environments by modelling positive responses to error. The question addressed here is whether developing a model of error management examining how teachers model error is useful. To address this question, the data presented below is from a larger study on teacher expressions of emotion in the classroom. Both extracts are drawn from episodes of emotional expression, deemed as such through observation, measurement of galvanic skin response (GSR), (used to roughly indicate internal emotions) and confirmed in post observation discussions. They represent how experienced teachers might manage error. Other strategies, such as used by novice teachers, are drawn from the literature, collaborated by my own experience of working with teachers.

Modelling dealing with mathematical error (Adam and Bertha)

In these examples, both Adam and Bertha successfully address a small numerical error, one type of mathematical error. Both express emotions (the dominant emotion is determined by the observer, using standard emotion classifications [START_REF] Scherer | What are emotions? And how can they be measured?[END_REF]), yet differ in pathway from recognising discrepancy to resolution, showing the teachers' disparate ways of modelling error management.

In the first example (Table 1), Adam accidently writes 2 for the difference between 4.5 and 3.5 when demonstrating upper and lower bounds for 5 -4. As he reads the four possible answers aloud, '2,0,2,1', he slows, quietens his voice and movements, pauses with pen poised, whilst his head moves from side to side scanning. He then steps back and looks, appearing absorbed. Once he has identified the error, he utters the sound 'uhh', interpreted as 'never mind'. He then engages in an exchange of silly noises with a student, corrects the error, rewards a student who is quick to align, and continues in a faster pace, as at the start of the episode, quickly moving on from the error. Bertha uses a well-known mathematics website to produce questions on the area of a circle (Table 2). Following the students finding the area for a given radius, Bertha enters a volunteered answer, but the website rejects this answer. As Bertha has already calculated the answer herself, agreeing with the student, the website rejection brings an unforeseen problem; she thinks she has made an error but has not. The discrepancy is between using π or 3.14, so it relates to the degree of accuracy

A proposed model of teacher error

Drawing attention to teacher error informs discussions about how such error should and could be managed, as this is not the same as when a student makes an error. Firstly, that there are emotional implications of the choices made by a teacher, including the degree of emotional labour required. Secondly, that the choices are indicative of how a teacher perceives error, both for themselves and for their students. And thirdly, that repeated over time, modelling by the teacher of error management sets the climate for students in terms of their own error management. If it is desirable to challenge behaviourist views of learning mathematics [START_REF] Boaler | Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching[END_REF], then discussion of error management provides an accessible route to address teacher beliefs. To support such discussions, the following continuum of error management strategies is proposed. This is derived from both the literature on error and the above data. The continuum moves from negative (associated with a mistake unfriendly environment and maladaptive error patterns) through to positive, and similarly from strategies with perceived reduced benefits, through to those which may have longer term positive impact for learning (mistakes friendly environment and adaptive patterns).

To compare and contrast with the examples above, I draw from research on and observations of trainee teachers. In this cohort, I observed embarrassment at making an error, ignoring the error, even when noticed, and errors on the whiteboard left uncorrected. I also observed surreptitious correcting when the students were engaged in another task. There is ample anecdotal and observable evidence that teachers hide numerical errors. They may later notice and leave it, or notice and amend privately.

The communicated message is that errors are dispreferred [START_REF] Ingram | The interactional treatment of mathematical mistakes[END_REF], and should be hidden.

Similarly, public error can evoke teacher embarrassment or be correctly rapidly, where the teacher 'steps out of' rather than 'stepping into' learning through addressing the error. 'Stepping out' attaches a negative emotional association to error making. A teacher may respond to revealed error by faking, as in "I meant to do that" and publically amend. Although positive in avoiding misunderstanding resulting from the error, this choice can be negative emotionally, in that frequent repetition erodes trust in the teacher. A further response is to associate self with the error through self-depreciation. A teacher might say something like, 'What a silly error!' The effect may be that students view all error as silly, and to be laughed at. This may be positive or negative as context specific, but students may reduce contributions if they feel they may be laughed at.

There are suggestions in the literature on error that using deliberate error as a teaching strategy can be effective. This assumes a constructivist view, that the place of error is a learning opportunity, rather than a culturally located behaviourist view, that errors should be avoided [START_REF] Santagata | Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons[END_REF]. In constructivist terms, viewing error as a learning opportunity is an ideal, yet how a mathematics teacher might cultivate such an ethos needs further research. Research from [START_REF] Ingram | The interactional treatment of mathematical mistakes[END_REF] found examples aimed at this ethos, but that in most cases the errors are still managed as something to be avoided. They identify a role for further strategies when positioning error in this way. They suggest either distancing self from the error since the purpose is for students not to make the same errors themselves, or apologising for an error. A more positive management might dismiss the error, but go on to associate with an emotionally positive outcome, such as thanking the student for pointing out the error. The emotional message is that criticism will not follow from error identification. However, taking the error handling strategy further, the identification may be made into an event through provoking a discussion, or may be acted out. There are many possible options, such as expecting error due to rapid engagement in doing mathematics or that error is just to be expected. The event could show that corrections are fine, that corrections are neither good or bad, and are just a learning opportunity. The attribution of value to the error communicates a positive message.

In the examples from Adam and Bertha there is a mixture of the above that tend to the emotionally positive end of the continuum. What was not apparent either in the examples of data collection was a view that 'I never make mistakes'. Although both examples are public, Adam notices his own error as part of the next step, whilst Bertha's attention is claimed by rejection of her answer by an online website. What may be significant is the emotion work in both cases to turn the event to a positive learning experience. As for similar examples from [START_REF] Tainio | Emotion work and affective stance in the mathematics classroom: the case of IRE sequences in Finnish classroom interaction[END_REF], the teachers take affective stances that display affiliation and humour (p.73). They also evoke emotionally positive responses for the students. Both show happiness in resolution of error, and give praise albeit located differently.

Bertha tells in a later interview of an occasion when a formal observation that went wrong because the questions on the website changed. A repeat experience, when again being observed, albeit for research purposes, is likely to re-evoke emotions associated with insecurity, and a need to check solutions with a form of authority, in this case the website answer. Modelling a need for accuracy, for rechecking, reinforces a product orientated 'feel' in the example, where correctness takes precedence. There is also modelling of internal thought processes. Seeking confirmation of correctness perhaps indicates a belief that 'real' mathematicians do not make errors. This example shows an intention to model what she thinks a teacher should be doing, indicating a disparity between real and aim more commensurate with trainee teachers. In contrast, Adam models shifting ownership of error by repositioning from 'we' to 'I', modelling that gives an impression that it is ok to err. He positively manages the error, moving sequentially from a point of uncertainty into exhibiting positive emotions using humour via exchanging noises and giving praise, which acts to restore lesson balance. His modelling of how to deal with error includes distraction of attention and shifting attention via assigning a social reward for correction of error. In the episode, the error becomes an object unassigned, before it is quickly shifted into a positive outcome. As an observer, it felt as if he was pinning the error somewhere distant from self. However, he included students in his happiness at resolving the error, and hence was rewarded. Either interpretation is a modelling that downplays error. As he says afterwards, Oh, yeah, did I put a mistake on the board to start off with? (Yeah) Yeah...I'm not fussed with that. It happens quite a lot. I always say to the students... I'll make mistakes, and they'll make mistakes...and there it goes...(Adam)

Both teachers seem to experience cognitive conflict, observable in the lesson as uncertainty, and resolve the error for themselves. They both resolve positively for their students, stepping into the error again as part of positive modelling. The data confirms that it is not the error itself but how it is managed that has implications. Public teacher error has more impact, whereas perhaps students are expected to make errors. Underlying the difference between the examples and trainee teachers lie the issues of confidence and risk with the subsequent implications to learning climate of handling error after and during public exposure. We assume that expert teachers have confidence, but the examples show that the use of emotions is part of the restorative process, perhaps as a distraction. This seems to warrant further investigation, as does the disparity between the expert (who makes no mistakes) and Adam's declared and enacted position on error. [START_REF] Mandler | Affect and learning: Causes and consequences of emotional interactions[END_REF]

Conclusion and implications

In the above I have used two short episodes to illustrate teacher error management, since the research suggests that modelling of error responses plays an important role in construction of positive emotional climates in a mathematics classroom. Based on this initially small sample, the use of affective pathways seems to support exploration of the adaptive/maladaptive reaction patterns. There may be potential in exploring adaptive patterns with teachers as part of professional development. It may also be useful, in conjunction with ideas such as positive mindsets (Boaler, 2014), to consider how students might participate in error management to a greater degree, for example as springboards as suggested by [START_REF] Borasi | Exploring mathematics through the analysis of errors[END_REF]. The different responses of the teachers, although both successfully resolved in that student reward is given as restorative praise in both cases, have different longer term impressions. Adam says, 'gold star' for a student and quickly moves on, whilst Bertha draws attention to the correct answer, and to rewarding the student. The impact on students of repeated modelling of 'not my error, let's move on' (process interruption only) compared to 'we must get this right' (product orientation) may be significant. Shifting attention acts to distance the teacher from the ownership of error, modelling addressing error as positive. This distancing compares to a negative impact that models dealing with errors as an annoying problem, one owned by both teacher and students. There is inevitably a degree of uncertainty in relation to error management. These examples represent extremes of a management continuum from valuing error as a learning experience (modelling an expectation of error into learning), to a belief that errors are obstacles to avoid. Indeed, an emotional risk to a teacher may be in not using positive emotions to manage error.

T

  You can either do for 5..., 4.5 or 5.5, they're your two options, because that's the lower and the upper bound. So what I have written out on the board is all four different combinations of what can happen. Pause with pen ready, his head moves as scans writing and then steps back] Looks like 2. [Pause] … Which 2 is it going to be? Well here... [ pauses again, this is in a quieter voice]

							Confident
	Ok… [pause]						Uneasy
	I'm going to work out all of them. So I am going to get 2, 0, 2, 1.	[GSR	ERROR
	PEAK]
	S1 Blast-off…
	S2 It should be 1.5
	Right, think about this. [pauses]				Confused
	The question was saying upper bound, the upper bound for 5 minus 4. That number	Re-
	is going to be as big as possible. [emphasis and shift in pace into regular time beat	establishing
	as	big	as	Pos-si-ble	1-and-2-and-3-and-4-]	certainty Interested
	Hm. [Doubt
							Uncertainty
							Thoughtful
	Oh, that's right, I've done that wrong, that should be 1, uhh, [faster pace resumed]	Satisfaction
	Who picked up on that? [HANDS UP] Chris [S3]? Gold star[smiling]		Pleased
	S2 How about me? Oh. Uh
	Uh						Humorous
	S2 Uh I said one though sir
	Sorry, that's a 1. The biggest number is 2. How did I get 2? I took the biggest number

possible here for 5, but the smallest number possible for 4. That made the differenceas-big-as-possible. 2. [firmly stated] Upper bound.

Confident

Table 1 : Affective Pathway from Adam's lesson, where he makes an error

 1

Table 2 : Affective Pathway from Bertha's lesson, where she makes an error

 2 Oh, I don't think if we can... yeah, we have. [GSR PEAK] Uncertain Can anybody think of a reason why you might have different, very slightly different answers? [Terry among others raises his hand] Terry... Confident Terry Is it because like one of us...um...were like... we weren't... um.... I don't know if it's right or....

	.

T [Frowning]

Well just, Terry, just say it, have more confidence in yourself sweetheart... Hopeful Terry ...some people pressed the... the pi button and some people didn't.

  In Bertha's case public thinking out-loud for self, "[muttering quietly] ...2...5...4...point...oh. I see, point 34. Let's try it again." Possibly this indicates Bertha seeking mathematical correctness. This corresponds with what she says in interview, I don't see myself as a mathematician. I see myself as someone who is good at maths and you can teach me anything in maths and eventually get it, which does again sounds obnoxious but that's...you know... it might take me a lot longer with some of the things. (Bertha)

  suggests emotions activate other mental contents to deal with situations perceived as being a mismatch between what is intended and what occurs.

Proceedings of CERME10

Acknowledgments

Thanks to UEA studentship funding for PhD data collection, with thanks as always to Prof. Nardi and to Dr Iannone and to the UCL, IOE Mathematics Education team for valuable discussions.