`Creative mathematics' – real or rhetoric?

The notion of creativity has its natural home in the fine arts, where the artist literally creates something that can be perceived by the senses. The products of mathematical activity are clearly not of this kind, yet some distinguished mathematicians have claimed that mathematics offers considerable scope for creativity. The title of the book under review, and some claims to be found in it, suggest that creativity can indeed be associated with mathematics, and that young children may experience it in the classroom. We suggest that the word creative is being used in rather different senses in these different contexts, yet the meanings associated with the arts, say, are in danger of being applied to mathematical situations for rhetorical purposes.This revised version was published online in September 2005 with corrections to the Cover Date.     

Beyond the comfort zone: A curriculum and assessment “immersion” experience in graduate early childhood teacher education

Abstract

A graduate course in Early Childhood Education assessment which combines coursework and a practicum with young children is designed and implemented using constructivist principles and practice. Teacher education is more effective when field work and coursework are combined, and when students are able to apply what they learn in class to real life experiences with young children [Journal of Early Childhood Teacher Education 21 (1) (2000) 65], learning is enhanced. This course was constructed in a way that incorporated theory, application, reflection, and reconstruction, allowing students an opportunity to rethink their teaching practice through the experience of an early childhood practicum. The key components of the model — the components that are thought to most effectively help students to apply theory to practice, change some of their ways of thinking about teaching and learning are the following: the relationship of the coursework to the setting, continuity from previous courses, the laboratory school environment, the intensive faculty involvement, the co‐constructivist, collaborative nature of the faculty team, the collaborative nature of student work, the use of the three curriculum models, the use of embedded assessment, formative nature of the assessment, informed teaching (on‐going use of student feedback), the tools provided to student that assist in the process, and the practical hands‐on assessment techniques.

Changing students’ ways of thinking about the teaching, learning and assessment process is considered a vital part of the course. [Journal of Early Childhood Teacher Education 17(1) (Marshall, 1996) 43] has identified constructivist principles to guide teacher education. She believes that students must experience first‐hand those behaviors that she hopes they would exhibit in their own teaching (Rand, 1999, p. 126). It is through the processes of the course that graduate students are challenged to experience the implementation of emergent curriculum, examine their own beliefs in relation to their practice, articulate why they do what they do and see how classroom assessment strategies help to support meaningful teaching and learning. Furthermore, collaborative practice is considered an important factor that encourages and facilitates reflection, supports a more multi‐dimensional view of teaching and assessing young children, and enables students to listen and observe more effectively and look for the impact of children making choices. Through participation in this course, students became more aware of the impact their view of children and respect for children has on their ability to implement effective and appropriate assessment practices in the classroom. Students gained more of a conceptual understanding of the role of assessment in informed teaching, and were better able to meet individual children's instructional needs.

Finlly, students learned how to use tools inquiry, collaboration and reflection that are likely to assist them in future personal professional growth.     

Representational guidance and student engagement: examining designs for collaboration in online synchronous environments

In this paper we report the results of a study which investigated the affordances of multi-user virtual environments (MUVEs) for collaborative learning from a design perspective. Utilizing a mixed methods approach, we conducted a comparative study of the effect of varying representational and interactional design features on a collaborative design activity in three online synchronous environments. We compared environments featuring multiple modes of interaction (MUVEs), shared representations (text chat and 2D still images) and text-only features. Sixty-one students enrolled in an undergraduate course on Child Development participated in the study. Participants were asked to design a theoretically-based, developmentally appropriate, preschool classroom setting. Students were randomly assigned to one of three online learning environments that provided varying levels of representation and interaction. Significant differences in collaborative problem solving interactions were found. Participants in the shared representations + text condition evidenced stronger learning outcomes as regards substantive discussion and integration of child development concepts; while participants in the MUVE condition reported the most enjoyment with the experience. These findings are explained by the concepts of representational guidance, representational bias, educational affordances and interface design metaphors. Suggestions for the design of MUVEs for collaborative learning are provided.     

Leveraging library trust to combat misinformation on social media

One reason librarians are confident they have a role to play in fighting misinformation is the level of trust in libraries as institutions. Exactly how they might leverage that trust remains unclear and untested. Building on recent work in correcting health misperceptions on social media, this study tests whether libraries can leverage trust to combat misinformation online. Using a misperception about the influenza vaccine as a test case, an experiment (n?=?625) was conducted in fall 2018 using Amazon's Mechanical Turk. Results suggest that the misperception can be reduced, but not by library institutions. An unsuccessful follow-up (n?=?600) suggests that the effectiveness of the correction is season dependent and opens the possibility that libraries may yet play a role, but not necessarily because they are trusted. Future library proposals for combating misinformation need to be developed and tested within a broader contemporary misinformation research program.     

Preservice teachers’ expressed awarenesses: emerging threads of retro-spection of learning and pro-spection of teaching

In this paper, we report an enquiry into elementary preservice teachers’ learning, as they engage in doing mathematics for themselves. As a group of researchers working in elementary Initial Teacher Education in English universities, we co-planned and taught sessions on growing pattern generalisation. Following the sessions, interviews of fifteen preservice teachers at two universities focused on their expressed awareness of their approach to the mathematical activity. Preservice teachers’ prospective planning and post-teaching evaluations of similar activities in their classrooms were also examined. We draw on aspects of enactivism and the notion of reflective “spection” in the context of teacher learning, tracing threads between preservice teachers’ retro-spection of learning and pro-spection of teaching. Our analysis indicates that increasing sensitivity to their own embodied processes of generalisation offers opportunities for novice teachers to respond deliberately, rather than to react impulsively, to different pedagogical possibilities. The paper contributes a new dimension to the discussion about the focus of novice elementary school teachers’ retrospective reflection by examining how deliberate retrospective analysis of doing mathematics, and not only of teaching actions, can develop awarenesses that underlie the growth of expertise in mathematics teaching. We argue that engaging preservice teachers in mathematics to support deliberate retrospective analysis of their mathematics learning and prospective consideration of the implications for teaching can enable more critical pedagogical choices.

     

The impact of incorrect examples on learning fractions: A field experiment with 6th grade students

Educational research indicates that error reflection, especially reflection on incorrect examples, has a positive effect on knowledge acquisition. The benefit of error reflections might be explained by the extended knowledge of incorrect strategies and concepts (negative knowledge) which fosters the learning of new content. In a field experiment with a pre-post-design we taught fractions to N = 195 6th grade students and compared two conditions that encouraged reflection on either incorrect or correct examples. We found that incorrect examples supported students’ negative knowledge more than correct examples. However, regarding the knowledge of fractions, only advanced students could benefit from incorrect examples; students with low prior knowledge learned more from correct examples. Even though negative knowledge showed a partial mediation effect for knowledge acquisition, it did not mediate the effect of error reflections on the acquisition of knowledge of fractions. The implications for school instruction are discussed.     

Testing elementary and secondary school students’ ability to perform historical perspective taking: the constructing of valid and reliable measure instruments

Historical reasoning competencies play an important role in history education. However, valid and reliable large-scale measurement instruments to assess these competencies are scarce. This study considers two instruments for measuring students’ ability to perform historical perspective taking (HPT) as a historical reasoning competency. The instruments have been tested for validity and reliability among 1,270 Dutch upper elementary and secondary school students, ranging in age from 10 to 17 years. One instrument offers effective validity and reliability and can map HPT performance among a large and heterogeneous student population. The results show that even upper elementary school students are capable of performing HPT. However, as students age, their ability to perform HPT increases. Differences regarding the ability to perform HPT were also found between educational levels. Pre-university students performed HPT more successfully compared to students at lower educational levels. The results of this study can be used to gain insight into the construct of HPT and into how historical reasoning competencies such as HPT can be measured. Furthermore, the results provide insight into how differences between students, such as age and educational levels, influence the performance of HPT.     

Social origins,school type and higher education destinations

To what extent and why do social origins matter for access to higher education, including access to elite universities? What is the role of private and selective schooling? This paper uses the 1970 British Cohort Study (BCS70) to analyse the trajectories of a generation currently in early middle age. We find that the influence of social origins, especially parental education, remains when both a wide range of cognitive measures and school attainment are controlled. Attending a private school is powerfully predictive of gaining a university degree, and especially a degree from an elite institution, while grammar schooling does not appear to confer any advantage.     

Promoting early career teacher resilience: a framework for understanding and acting

In this paper, we undertake a brief review of the ‘conventional’ research into the problems of early career teachers to create a juxtaposed position from which to launch an alternative approach based on resilience theory. We outline four reasons why a new contextualised, social theory of resilience has the potential to open up the field of research into the professional lives of teachers and to produce new insights into the social, cultural and political dynamics at work within and beyond schools. We then move from these theoretical considerations to explain how we used them in a recent Australian research project that examined the experiences of 60 graduate teachers during their first year of teaching. This work led to the development of a Framework of Conditions Supporting Early Career Teacher Resilience which we outline, promote and advocate as the basis for action to better sustain our graduate teachers in their first few years of teaching. Finally, we reflect on the value of our work so far and outline our practical plans to ‘mobilise’ this knowledge in ways that will make it available to a variety of audiences concerned with the welfare of this group of teachers.