Mathematics and the unexpected pdf

Confidence in the own math abilities was positively related to reasoning abilities across samples, and with perceptions of gender equality in math, but the correlation was only statistically significant among High School girls. Math self- confidence related negatively with hostile sexist ideologies among women majoring STEM. The perception that women are as good as men in math was positively correlated to math self- confidence among High School girls and women majoring in STEM.

These perceptions were negatively correlated to hostile and benevolent sexist ideologies among High School girls, and to benevolent sexist ideologies among women attending STEM majors. Finally, data show a significant positive correlation between both types of sexist ideologies across samples.

For descriptive purposes, averages of indicators are presented here. The inspection of the data revealed that the proposed model exhibits a better fit among High School girls than among all university women. The more participants believed that girls and boys are equally good in math, the more they believe in their own mathematical abilities, after controlling for their scores on abstract reasoning abilities. The more participants believed in gender equality in math abilities and the more confident they felt about their own mathematic abilities, the better they performed in standardized math tests, after controlling for individual differences in reasoning abilities.

In this sample, only three of the seven postulated structural relations were statistically significant. The more participants believed in their own math abilities, the better they performed in standardized math tests, after controlling for their scores on reasoning abilities. The model explained The more participants believed in gender equality regarding math abilities, the more confident they felt about their own mathematic abilities, and the more confident they felt about their math capacities the better they performed in standardized math tests, after controlling for individual differences in reasoning abilities.

Again, the expected indirect effects of sexist ideologies on mathematic performance via perceptions of gender equality in math contexts and math self-confidence were not supported by the data in this site, which made mediation tests unnecessary.

In summary, data show that the proposed model was only partially supported across samples. Particularly in university contexts, our model did not describe the indirect effects of sexist ideologies and gender stereotypes on math performance well.

Testing an alternative model Given the inconclusive evidence of the expected indirect effects of sexist ideologies on math performance, we tested an alternative model, including additional direct paths from HS and BS to MATH. Table 3 presents the standardized parameter estimates for the measurement model for the alternative model across samples.

Table 5 shows fit indices and standardized parameter estimates for the direct and indirect paths of the alternative model across samples; while Diagrams B, D and F in Fig.

Across samples, all original structural coefficients in the alternative model remained similar in magnitude. In this sample, endorsement of hostile sexism was significantly associated with better performance in standardized math tests, after controlling for individual differences in reasoning. This model explained In summary, testing an alternative model adding direct effects of sexism on performance showed no better fit to the data than the original model of indirect effects.

Therefore, the more parsimonious model of indirect effects seems to better reflect the complex relationship of the variables. However, the surprising direct and positive relationship between Hostile Sexism with math performance among women majoring in STEM should not be neglected.

Standardized parameter estimates for the original and the alternative models across samples. Diagrams B, D, and F depict parameter estimates for the alternative models, in these environments, respectively. Our data partially support this complex chain of effects among High School girls, but also highlighted substantial differences between High School environments and university contexts, and revealed some unexpected effects of sexist ideologies on performance among university women majoring in STEM.

In light of this, we discuss our results taking into account the specificity of each educational environment. High-school environments Among high school girls, data showed that the more they endorsed sexist ideologies, the less convinced they were that girls are as good as boys in math. On the other hand, data showed that the more they believed in gender equality regarding mathematical abilities, and the more confident they felt about their mathematical capacities, the better they performed in standardized math tests.

These findings are notable in several ways. More research is needed to understand the ways in which self-confidence affects performance. Second, our findings show that even after controlling for their actual and perceived math abilities, stereotypical beliefs about gender equality in math abilities remained a significant predictor of performance.

Our data are consistent with research based on Eccles' model Eccles, , showing that the cultural transmission of gender-role stereotypes influences individuals' goals and general self-schemata, which in turn influence specific behaviors and performances. These findings are also consistent with research on Stereotype and Social identity Threat e. Although our study does not directly address issues of carrier choices, our results provide relevant information for teachers, school authorities, and policymakers in the context of the worldwide debate around the gender gap in STEM fields.

Rose, Third, our results suggest that sexist ideologies have a marginal distal influence on performance, primarily through gender stereotypes. Although not statistically significant, the sign and magnitude of the relationship between Hostile and Benevolent Sexism and perceptions of gender equality in math capacities should not be neglected, highlighting the need to continue examining the role of sexism in stereotypes in the academic domain.

The low magnitude of the observed effects and the lack of their significance are likely due to the fact that the content of HS and BS focus on competitive and complementary gender differentiation along with the interpersonal domain, rather than direct comparisons of men and women capacities in the academic domain. The putative effects of sexism on stereotypes were therefore found here to be only indirect and marginal. More research is needed in order to capture those aspects of sexism that directly influence gender stereotypes and performance in academic domains.

These results, together with our data, highlight multiple paths in which sexism negatively affects performance in academic domains which deserve more research in the future.

University environments Among university students, the expected indirect effect of sexist ideologies and stereotypes on performance was, overall, not fully supported. Rather than gender ideologies or cultural beliefs about gender differentiation, the most important predictors of their performance were their sense of math self-confidence and their reasoning abilities. Perhaps these findings relate to the typical self-selection processes involved in the admission to any university.

Previous research by Correll has shown that beliefs about gender differences in mathematics impact individuals' assessments of their mathematical competence, which in turn leads to gender differences in decisions to persist on a path toward a STEM career. This, of course, can only be adequately tested with longitudinal designs and indicates the need for more longitudinal studies in this field.

These results also point to the need for the inclusion of different populations in the empirical studies, so as to acknowledge the moderating role of different educational and socio-demographic contexts in the relationships under consideration and to avoid sweeping generalizations. Perhaps the most surprising finding in university contexts is the positive direct coefficient between hostile sexist ideologies and math performance among women majoring in STEM, that is, those who exhibit higher levels of math performance also exhibit more hostile sexism.

This complex process might account for the negative attitudes toward the specific "types" of women pictured in the HS measure.

However, our data are not conclusive in this regard and should be taken with caution, given the potential suppression effect suggested by the differences between the zero-order correlations and the beta weights in this sample. Nevertheless, this unexpected result indeed suggests interesting research paths to tackle these issues and to continue studying women in STEM environments.

As a final methodological note, it is worth pointing out that the reasoning abilities test used as a control variable for math performance proved to accomplish a very good job in all the models, presenting standardized path coefficients higher than. It shows the importance of employing controls such as this for models with endogenous variables that involve intellectual skills. Since attitudes and other psychological traits might correlate with reasoning abilities in observational studies, our recommendation is to always consider the use of this kind of control variable when working with observational data.

In this way, the possible confounding effects generated by the association between psychological traits and basic reasoning abilities are neutralized. Limitations Our observational data inhibit us from establishing reliable causal inferences, especially in university samples, in which performance data were retrospectively collected.

Nevertheless, longitudinal data are still necessary in order to test the causal relationship between the variables correctly.

Likewise, there might be some concern regarding the low reliability of some of the measures. In other words, measurement issues cannot fully account for the pattern of associations observed here. However, attempts to improve our measures should be a constant goal in our research. Finally, one might have questions as to the elevated percent of missing data, especially among High School students.

In this regard, it is important to notice that in order to take the Exit Exams and the Admission Tests, High School students are required to complete the eleventh grade. In this scenario, a portion of missing data is due to the fact that some students either dropped out of school before taking the standardized tests, or could not successfully complete their eleventh grade. In such circumstances, missing data rates in studies like ours are expected to be relatively high.

The effect of ideologies and stereotypes on math self-efficacy and math performance was greater among high school girls. In university environments, on the other hand, math self-efficacy showed a substantive effect on performance, after controlling for individual differences in abstract reasoning.

The unexpected positive relationship between hostile sexism and math performance among women majoring in STEM might reflect adaptation mechanisms in a male-dominated learning environment that should be further studied.

The use of a reasoning test to neutralize the possible confounding effects of basic intellectual ability on the relations between math performance and the socio-affective traits sexism, gender equality in math contexts, and math self-confidence proved to be a fortunate decision, since basic reasoning abilities explain an important part of math performance variance. Personality and Social Psychology Bulletin, 23, Bian, L.

Contemplating the randomness of nature, Ekeland extends his consideration of the catastrophe theory of the universe begun in Mathematics and the Unexpected, drawing upon rich literary sources and current topics in math and physics such as chaos theory, information theory, and particle physics.

Line drawings. Gathers paradoxes, logic puzzles, number problems, geometric problems, gambling puzzles, optical illusions, string, word, and chess problems featured in Scientific American. Guide to Information Sources in Mathematics and Statistics. The discussion below is structured around these features. The names of the students who drew the circles have been deleted to preserve anonymity class.

The Different Teacher Responses Here I address the first research question, concerning ways in which an experienced teacher of mathematics exploits unexpected situations. It may be that here I felt more under pressure, with older, less motivated students who were at risk of downing tools, and I wished to rescue the situation by directing students quickly to an alternative task—coping with the unexpected situation rather than exploiting it.

It seems likely that the fact that, as referred to above, the source of the disturbance was external to the classroom community played a part in this. Tanner et al. However, although the experienced jazz musician will on some occasions produce better improvisations than on others, they will almost never dry up completely or play an unintended clashing chord.

Similarly, it may be that the mathematics teacher does not need to have total certainty about the direction in which they are travelling but merely a reasonable belief that they will be able to engage fairly mathematically with whatever arises.

Implicit in several of the accounts of the unexpected situations given above are strategies I instigated to make this way of working safer. A repeated gambit seems to have been to find a way, as soon as practicable, of giving students something to do e. There was good evidence of flow situations Csikszentmihalyi, , arising within at least some parts of three of the four lessons described. Students came alive relative to their manner in more ordinary circumstances, talking more loudly and more animatedly, making active suggestions and arguing with one another about mathematics.

Further research would be necessary to attribute this to particular causes, although it seems likely that the personal validation of something that the student has said or done is motivating. Students may also enjoy witnessing a situation in which the teacher is clearly out of their comfort zone and improvising.

Rowland et al. One fairly consistent feature of my preparedness was reading about mathematics and mathematics teaching. As Coker comments: Improvisation, like composition, is the product of everything heard in past experience, plus the originality of the moment.

This study shows ways in which unexpected situations can be exploited mathematically and highlights factors that may predispose the teacher to do this more effectively. However, it does not address the powerful cultural and institutional factors that push teachers away from using their professional judgment to adapt classroom learning to the contingent needs and interests of the students as they present themselves there and then Foster, a.

However, responding to interruptions to lessons by deviating from the intended lesson to exploit an unexpected situation can lead students into intense and enjoyable mathematical engagement, such as that characterised by Csikszentmihalyi , as flow. How mathematics teachers might learn to do this skilfully in the moment remains a crucial area for further research. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author s and the source are credited.

Teaching as stand-up comedy: The metaphor of scripted and improvised performance of teaching. Davidson, D. Piette Eds. Available at www. Atkinson, T. The intuitive practitioner: On the value of not always knowing what one is doing. Maidenhead: The Open University. Berliner, D. Learning about and learning from expert teachers.

International Journal of Educational Research, 35 5 , — The skillful teacher: On technique, trust, and responsiveness in the classroom 2nd ed. San Francisco: Wiley. Bryant, J. How round is your circle?

Oxford: Princeton University Press. Chick, H. Teachers of mathematics as problem-solving applied mathematicians.

Claxton, G. The anatomy of intuition. Claxton Eds. Coker, J. Improvising jazz. Englewood Cliffs: Prentice-Hall. Cryer, B. Butterfly brain. London: Orion Books Ltd. Csikszentmihalyi, M. The flow experience and its significance for human psychology. Csikszentmihalyi Eds. Cambridge: CUP. Flow: The classic work on how to achieve happiness. London: Rider. Optimal experience: Psycholog- ical studies of flow in consciousness.

Davis, B. Inventions of teaching: A genealogy. London: Routledge. Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Educa- tion, 34 2 , — Becoming more curious about learning. Journal of Curriculum and Pedagogy, 1 1 , 26— Complexity science and educational action research: Toward a pragmatics of transformation.

Educational Action Research, 13 3 , — Engaging minds: Changing teaching in complex times 2nd ed. Fernandez, C. Lesson study: A Japanese approach to improving mathematics teaching and learning. London: Lawrence Erlbaum. Foster, C. Mathematical behaviour. Mathematics Teaching, , 12— Resisting reductionism in mathematics pedagogy. Curriculum Journal, 24 4 , — Geertz, C. Thick description: Toward an interpretive theory of culture. McIntyre Eds. London: MIT Press. George, M. How mathematics teaching can be like improv theater.

Griffiths, J. Mathematics Teaching Incorporating Micromath, , Hoftstadter, D. Thoughts on geometrical thinking. Mathematics in School, 36 4 , LeCompte, M. Ethnography and qualitative design in educational research. San Diego: Academic Press.