Theories of learning mathematics

According to Karl Popper, widely regarded as one of the greatest philosophers of science in the 20th century, falsifiability is the primary characteristic that distinguishes scientific theories from ideologies – or dogma. For example, for people who argue that schools should treat creationism as a scientific theory, comparable to modern theories of evolution, advocates of creationism would need to become engaged in the generation of falsifiable hypothesis, and would need to abandon the practice of discouraging questioning and inquiry. Ironically, scientific theories themselves are accepted or rejected based on a principle that might be called survival of the fittest. So, for healthy theories on development to occur, four Darwinian functions should function: (a) variation – avoid orthodoxy and encourage divergent thinking, (b) selection – submit all assumptions and innovations to rigorous testing, (c) diffusion – encourage the shareability of new and/or viable ways of thinking, and (d) accumulation – encourage the reuseability of viable aspects of productive innovations.

Surviving an avalanche of data

Open the sports or business section of your daily newspaper, and you are immediately bombarded with an array of graphs, tables, diagrams, and statistical reports that require interpretation. Across all walks of life, the need to understand statistics is fundamental. Given that our youngsters’ future world will be increasingly data laden, scaffolding their statistical understanding and reasoning is imperative, from the early grades on. The National Council of Teachers of Mathematics (NCTM) continues to emphasize the importance of early statistical learning; data analysis and probability was the Council’s professional development “Focus of the Year” for 2007–2008. We need such a focus, especially given the results of the statistics items from the 2003 NAEP. As Shaughnessy (2007) noted, students’ performance was weak on more complex items involving interpretation or application of items of information in graphs and tables. Furthermore, little or no gains were made between the 2000 NAEP and the 2003 NAEP studies. One approach I have taken to promote young children’s statistical reasoning is through data modeling. Having implemented in grades 3 –9 a number of model-eliciting activities involving working with data (e.g., English 2010), I observed how competently children could create their own mathematical ideas and representations—before being instructed how to do so. I thus wished to introduce data-modeling activities to younger children, confi dent that they would likewise generate their own mathematics. I recently implemented data-modeling activities in a cohort of three first-grade classrooms of six year- olds. I report on some of the children’s responses and discuss the components of data modeling the children engaged in.

Problem solving in the primary school (K-2)

This article focuses on problem solving activities in a first grade classroom in a typical small community and school in Indiana. But, the teacher and the activities in this class were not at all typical of what goes on in most comparable classrooms; and, the issues that will be addressed are relevant and important for students from kindergarten through college. Can children really solve problems that involve concepts (or skills) that they have not yet been taught? Can children really create important mathematical concepts on their own – without a lot of guidance from teachers? What is the relationship between problem solving abilities and the mastery of skills that are widely regarded as being “prerequisites” to such tasks?Can primary school children (whose toolkits of skills are limited) engage productively in authentic simulations of “real life” problem solving situations? Can three-person teams of primary school children really work together collaboratively, and remain intensely engaged, on problem solving activities that require more than an hour to complete? Are the kinds of learning and problem solving experiences that are recommended (for example) in the USA’s Common Core State Curriculum Standards really representative of the kind that even young children encounter beyond school in the 21st century? … This article offers an existence proof showing why our answers to these questions are: Yes. Yes. Yes. Yes. Yes. Yes. And: No. … Even though the evidence we present is only intended to demonstrate what’s possible, not what’s likely to occur under any circumstances, there is no reason to expect that the things that our children accomplished could not be accomplished by average ability children in other schools and classrooms.

Modeling as a bridge between real world problems and school mathematics

This article argues for an interdisciplinary approach to mathematical problem solving at the elementary school, one that draws upon the engineering domain. A modeling approach, using engineering model eliciting activities, might provide a rich source of meaningful situations that capitalize on and extend students’ existing mathematical learning. The study reports on the developments of 48 twelve-year old students who worked on the Bridge Design activity. Results revealed that young students, even before formal instruction, have the capacity to deal with complex interdisciplinary problems. A number of students created quite appropriate models by developing the necessary mathematical constructs to solve the problem. Students’ difficulties in mathematizing the problem, and in revising and documenting their models are presented and analysed, followed by a discussion on the appropriateness of a modeling approach as a means for introducing complex problems to elementary school students.

Looking back and looking forward : exploring distributed leadership with Queensland Prep teachers

There is an increasing expectation that early childhood teachers will be pedagogical leaders, particularly in a global context of curriculum reform. This paper reports on the distributed leadership experiences of early childhood teachers’ during the 2003 Preparatory Year (Prep) trial in Queensland, Australia. In 2010, 13 of the first Prep teachers participated in interviews to discuss their definitions of leadership and reflect on the opportunities they had to lead curriculum development during and since the 2003 trial. Data were examined using a conceptual framework based on the work of Woods et al. (2004), with a focus on the structural and agential aspects of distributed leadership. Participants identified a range of contextual influences, challenges, skills and enabling strategies that illustrate the complexities in leading curriculum.

Successful outcomes in vocational education and training courses : how pedagogy and expectations influence achievement

This paper reports on a four year Australian Research Council funded Linkage Project titled Skilling Indigenous Queensland, conducted in regional areas of Queensland, Australia from 2009 to 2013. The project sought to investigate vocational education, training (VET) and teaching, Indigenous learners’ needs, employer cultural and expectations and community culture and expectations to identify best practice in numeracy teaching for Indigenous VET learners. Specifically it focused on ways to enhance the teaching and learning of courses and the associated mathematics in such courses to benefit learners and increase their future opportunities of employment. To date thirty-nine teachers/trainers/teacher aides and two hundred and thirty-one students consented to participate in the project. Nine VET courses were nominated to be the focus on the study. This paper focuses on questionnaire and interview responses from four trainers, two teacher aides and six students. In recent years a considerable amount of funding has been allocated to increasing Indigenous Peoples’ participation in education and employment. This increased funding is predicated on the assumption that it will make a difference and contribute to closing the education gap between Indigenous and non-Indigenous Australians (Council of Australia Governments, 2009). The central tenet is that access to education for Indigenous People will create substantial social and economic benefits for regional and remote Indigenous People. The project’s aim is to address some of the issues associated with the gap. To achieve the aims, the project adopted a mixed methods design aimed at benefitting research participants and included: participatory collaborative action research (Kemmis & McTaggart, 1988) and, community research (Smith, 1999). Participatory collaborative action research refers to a is a “collective, self-reflective enquiry undertaken by participants in social situations in order to improve the rationality and justice of their own social and educational practices” (Kemmis et al., 1988, p. 5). Community research is described as an approach that “conveys a much more intimate, human and self-defined space” (p. 127). Community research relies on and validates the community’s own definitions. As the project is informed by the social at a community level, it is described as “community action research or emancipatory research” (Smith, 1999, p. 127). It seeks to demonstrate benefit to the community, making positive differences in the lives of Indigenous People and communities. The data collection techniques included survey questionnaires, video recording of teaching and learning processes, teacher reflective video analysis of teaching, observations, semi-structured interviews and student numeracy testing. As a result of these processes, the findings indicate that VET course teachers work hard to adopt contextualising strategies to their teaching, however this process is not always straight forward because of the perceptions of how mathematics has been taught and learned historically. Further teachers, trainers and students have high expectations of one another with the view to successful outcomes from the courses.

Mentoring pre-service teachers on school students’ differentiated learning

This Australian study explores the mentoring of pre-service teachers in selecting and implementing teaching strategies to meet students‟ learning needs. Two case studies involving 28 mentor teachers in a professional development program and a mentor-mentee partnership during a four week practicum provided data about mentoring teaching strategies for differentiated learning. Findings showed that contexts for learning about differentiation occurred at the pre action, in-action, and post-action stages. Central to each stage were pedagogical knowledge practices such as planning, preparation, classroom management, assessment, and problem solving (reflection-in-action to present solutions to problems) as key to in-action strategising and the mentoring processes. Mentoring pre-service teachers on how to devise teaching strategies for differentiated learning needs to be researched with a wider range of mentors and pre-service teachers, including those at different stages of development.

Education for social responsibility: Ethics and imagination for engaging students and teachers in educational reform

In this chapter we make assumptions about the primary role of education for the life of its beneficiaries and for society. Undoubtedly, formal education plays an important role in enhancing the likelihood for participation in future social life, including enjoyment and employment, by the student as well as the development of the well being of society in general. Similarly, education is often seen as a main means for intergenerational transmission of knowledge and culture. However, as Dewey (1916) argues, in liberal societies, education has the capacity of enhancing democratic participation in society that goes beyond passive participation by its members. One can argue that the achievement of the ideals of democracy demands a free and strong education system. In other words, while education can function as an instrument to integrate students into the present society, it also has the potential to become an instrument for its transformation by means of which citizens can develop an understanding of how their society functions and a sense of agency towards its transformation. Arguably, this is what Freire (1985) meant when he talked about the role of education to “read and write” the world. A stream of progressive educators (e.g., Apple (2004), Freire, (1985), Giroux (2001) and McLaren (2002)) taught us that the reading of the world that is capable of leading into writing the world is a critical reading; i.e., a reading that poses “Why” questions and imagines “What else can be” (Carr & Kemmis, 1987).

Conceptualising teachers’ assessment literacies in an era of curriculum and assessment reform

Teacher assessment literacy is a phrase that is often used but rarely defined. Yet understanding teacher assessment literacy is important in an international curriculum and assessment reform context that continues to challenge teachers’ assessment practices. In this article situated examples of classroom assessment literacies are analysed using Bernstein’s (Pedagogy, symbolic control and identity: Theory, research and critique, Taylor and Francis, London, 1996; Br J Sociol Educ 20(2):157–173, 1999) theoretical tools of vertical and horizontal discourses, classification and framing. Drawing on a sociocultural view of learning, the authors define teacher assessment literacies as dynamic social practices which are context dependent and which involve teachers in articulating and negotiating classroom and cultural knowledges with one another and with learners, in the initiation, development and practice of assessment to achieve the learning goals of students. This conceptualisation of assessment literacy aims to make explicit some underpinning theoretical constructs of assessment literacy to inform dialogue and decision making for policy and practice to benefit student learning and achievement.

Adaptive teachers embracing new ways of learning with robotics in Chinese schools

In July 2010, China announced the “National Plan for Medium and Long-term Education Reform and Development(2010-2020)” (PRC 2010). The Plan calls for an education system that: • promotes an integrated development which harnesses everyone’s talent; • combines learning and thinking; unifies knowledge and practice; • allows teachers to teach according to individuals’ needs; and • reforms education quality evaluation and personnel evaluation systems focusing on performance including character, knowledge, ability and other factors. This paper discusses the design and implementation of a Professional Learning Program (PLP) undertaken by 432 primary, middle and high school teachers in China. The aim of this initiative was to develop adaptive expertise in using technology that facilitated innovative science and technology teaching and learning as envisaged by the Chinese Ministry of Education’s (2010-2020) education reforms. Key principles derived from literature about professional learning and scaffolding of learning informed the design of the PLP. The analysis of data revealed that the participants had made substantial progress towards the development of adaptive expertise. This was manifested not only by advances in the participants’ repertoires of Subject Matter Knowledge and Pedagogical Content Knowledge but also in changes to their levels of confidence and identities as teachers. It was found that through time the participants had coalesced into a professional learning community that readily engaged in the sharing, peer review, reuse and adaption, and collaborative design of innovative science and technology learning and assessment activities.

Are Malaysian Teachers Ready to Assume the Duties of Reporting Child Abuse and Neglect?

Mandatory child abuse and neglect reporting laws apply to teachers in many countries of the world. However, such laws have not yet been introduced for teachers in Malaysia, and there is debate about whether the laws should be extended to teachers at all. This paper aimed to investigate the level of support among teachers to assume mandatory reporting duties and to identify factors determining this support in Malaysia. A total of 668 teachers from 14 randomly selected public primary schools completed an anonymous self-administered questionnaire. Results showed that 44.4 per cent of the respondents supported legislation requiring teachers to report child abuse. Teachers of Indian ethnicity, those with a shorter duration of service in teaching (< 5 years), the availability of knowledgeable and supportive school staff and a higher level of commitment to reporting were significant factors affecting teachers' support for mandatory reporting. This study provides important insights into factors influencing teachers' support for the introduction of mandatory reporting legislation for teachers in Malaysia. Teachers do not unanimously support these laws and there is a lack of clarity about what such laws will mean for teachers. The data highlight the need for specific training programmes to raise teachers' awareness, build their confidence and enhance their willingness to report child abuse.