Self-efficacy is a conceptualisation of self-belief, developed by Albert Bandura . It is the belief an individual has in their capacity to be successful in a domain. It is a self-assessment of skills, knowledge and dispositions in a context. It is domain specific in that self-efficacy is contextualised, with demanding but related sets of challenges. The activities cannot be so trivial that the action required is relatively routine or straightforward. We are talking about problem solving in contexts, where there are complex decisions which may have multiple solutions and multiple means by which outcomes might be achieved.

Mathematics self-efficacy is the belief an individual has in their capacity to solve mathematics problems. It is a belief that they can achieve a level of success when undertaking maths-related work. Mathematics self-efficacy correlates with mathematics performance. Mathematics self-efficacy is an important predictor of mathematics performance .

According to Bandura, there are four sources of self-efficacy: *enactive mastery experience*, *vicarious experience*, *verbal persuasion* and *physiological and affective states*. I will explain each of these in turn:

### Enactive mastery experience

Self-efficacy is developed through experience, through working on and solving problems; if we we are to limit our discussion to mathematics self-efficacy. This can be easily understood from our own experience, if we practice and get positive results, i.e. we are successful, then we become more confident. However, Bandura, takes a more profound view of success, a broader view, and allows the possibility of acquisition of self-efficacy even when we fail.

Mathematical self-efficacy is developed not just as a consequence of getting questions right or simply by finding solutions to problems. Self-efficacy is developed through reference to the strategy that we took in solving problems. Effectively, we assess the the approach we took and how it led to the outcome. In developing self-efficacy, we do not assess the outcome in absence of the method we used. This explains why, even though our final result might be wrong, we can develop self-efficacy. The essence is in being be able to connect our actions to the outcome and understand, rationally, how that led to the result.

#### Vicarious experience

A second but weaker source of self-efficacy is through vicarious experience. We can develop self-efficacy by observing others carry out activities. If the modelled behaviour is self-efficacious then it can provides a source of self-efficacy for the observer. This is especially true if the observer identifies with the individual modelling the behaviour. If, as observer, we see ourselves as similarly, having similar capacities and potentialities, then we are likely to improve our self-efficacy by observing them model actions, and in mathematics, by modelling mathematical problem solving.

If the observer perceives the person modelling the action as considerably different – they might feel that they are more intelligent or more able – then it is less likely that the observer will develop self-efficacy vicariously.

#### Verbal pursuasion

A third, but still weaker source of self-efficacy, is verbal pursuasion. We can use encouragement to persuade learners of their capacity to be successful. If the encouragement is misplaced and we try and persuade learners that they will be successful and they ultimately fail, there is the possibility that self-efficacy will be undermined. Encouragement must be based on accurate assessment of the individual’s capabilities and potential. Furthermore, if the more knowledgeable other is not trusted by the learner, then it is unlikely that self-efficacy will be developed.

#### Physiological and affective states

Illness, tiredness and stress all undermine self-efficacy. It is important that learners are challenged and are set challenging objectives. But if the demands becomes overwhelming, then there can be negative effects. Equally, if a learner is unwell or if there are external stressors then self-efficacy is undermined and there will be a noticeable effect on mathematical performance.

### Cultural differences in the effects of vicarious experience and verbal pursuasion

consider cultural differences in the extent to which the social sources of self-efficacy impact on self-efficacy and mathematical performance overall. It has previously been suggested that social effects are different in cultures that are predominantly *individualist*, like the US and Western Europe, to cultures that are *collectivist, *as in South East Asia.

undertook a quantitative study in the US, the Philippines and in Korea.

The important results are as follows:

**Mathematics self-efficacy has a significant positive correlation with students’ mathematics achievement.**This is consistent with previous research, both theoretical and empirical. It also provides evidence that this relationship is independent of culture.**Mathematics anxiety is negatively correlated with mathematics self-efficacy.**Again this is an expected result and previous research has suggested this also. In more vernacular terms it means that the more confident a learner is in mathematics the less anxious they are.**Students in individualistic cultures report stronger mathematics self-efficacy compared with collectivist cultures.**This is often in spite of superior performance by learners in collectivist cultures. This could be because people in collectivist cultures refrain from higher ratings because of a cultural desire to express humility.**Vicarious sources of self-efficacy tend to be from teachers and verbal pursuasion comes from family and peers.**

### Concluding remarks

This research confirms the importance of self-efficacy in mathematics learning. It challenges the view that mathematics learning should be predominantly rote learning and practice. Problem solving is necessary to develop self-efficacy. Learners need chance to explore and examine non-routine problems. Importantly they need to develop the capacity to assess the strategies they use, themselves and with the support of teachers and peers.

Not only does self-efficacy correlate with mathematics performance, it is also important in respect to mathematics anxiety. The more self-efficacious the individual the less anxious they become.

Finally, although this research considers self-efficacy sources in different cultures, it draws attention to the importance of social sources and within what contexts this might develop. The research shows vicarious sources tend to be from the teacher, verbal pursuasion comes from family and peers. This is important in understanding multiple social roles in learning mathematics.

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