Confronting expectation in grade 4: Tossing two coins

This study focuses on the experiences of 91 Grade 4 students who had been introduced to expectation and variation through trials of tossing a single coin many times. They were then given two coins to toss simultaneously and asked to state their expectation of the chances for the possible outcomes, in a similar manner expressed for a single coin. This paper documents the journey of the students in discovering that generally their initial expectation for two coins was incorrect and that despite variation, a large number of tosses could confirm a new expectation.

Expectation and variation with a virtual die

By the time students reach the middle years they have experienced many chance activities based on dice. Common among these are rolling one die to explore the relationship of frequency and theoretical probability, and rolling two dice and summing the outcomes to consider their probabilities. Although dice may be considered overused by some, the advantage they offer is a familiar context within which to explore much more complex concepts. If the basic chance mechanism of the device is understood, it is possible to enter quickly into an arena of more complex concepts. This is what happened with a two hour activity engaged in by four classes of Grade 6 students in the same school. The activity targeted the concepts of variation and expectation. The teachers held extended discussions with their classes on variation and expectation at the beginning of the activity, with students contributing examples of the two concepts from their own experience. These notions are quite sophisticated for Grade 6, but the underlying concepts describe phenomena that students encounter every day. For example, time varies continuously; sporting results vary from game to game; the maximum temperature varies from day to day. However, there is an expectation about tomorrow’s maximum temperature based on the expert advice from the weather bureau. There may also be an expectation about a sporting result based on the participants’ previous results. It is this juxtaposition that makes life interesting. Variation hence describes the differences we see in phenomena around us. In a scenario displaying variation, expectation describes the effort to characterise or summarise the variation and perhaps make a prediction about the message arising from the scenario. The explicit purpose of the activity described here was to use the familiar scenario of rolling a die to expose these two concepts. Because the students had previously experienced rolling physical dice they knew instinctively about the variation that occurs across many rolls and about the theoretical expectation that each side should “come up” one-sixth of the time. They had observed the instances of the concepts in action, but had not consolidated the underlying terminology to describe it. As the two concepts are so fundamental to understanding statistics, we felt it would be useful to begin building in the familiar environment of rolling a die. Because hand-held dice limit the explorations students can undertake, the classes used the soft-ware TinkerPlots (Konold & Miller, 2011) to simulate rolling a die multiple times.

Introducing the practice of statistics: are we environmentally friendly?

The practice of statistics is the focus of the world in which professional statisticians live. To understand meaningfully what this practice is about, students need to engage in it themselves. Acknowledging the limitations of a genuine classroom setting, this study attempted to expose four classes of year 5 students (n=91) to an authentic experience of the practice of statistics. Setting an overall context of people’s habits that are considered environmentally friendly, the students sampled their class and set criteria for being environmentally friendly based on questions from the Australian Bureau of Statistics CensusAtSchool site. They then analysed the data and made decisions, acknowledging their degree of certainty, about three populations based on their criteria: their class, year 5 students in their school and year 5 students in Australia. The next step was to collect a random sample the size of their class from an Australian Bureau of Statistics ‘population’, analyse it and again make a decision about Australian year 5 students. At the end, they suggested what further research they might do. The analysis of students’ responses gives insight into primary students’ capacity to appreciate and understand decision making, and to participate in the practice of statistics, a topic that has received very little attention in the literature. Based on the total possible score of 23 from student workbook entries, 80 % of students achieved at least a score of 11.

The correlation between reading and mathematics ability at age twelve has a substantial genetic component

Dissecting how genetic and environmental influences impact on learning is helpful for maximizing numeracy and literacy. Here we show, using twin and genome-wide analysis, that there is a substantial genetic component to children’s ability in reading and mathematics, and estimate that around one half of the observed correlation in these traits is due to shared genetic effects (so-called Generalist Genes). Thus, our results highlight the potential role of the learning environment in contributing to differences in a child’s cognitive abilities at age twelve.

Perspectives on Australian, Indian and Malaysian approaches to STEM education

STEM education faces an interesting conundrum. Western countries have implemented constructivist inspired student centred practices which are argued to be more engaging and relevant to student learning than the traditional, didactic approaches. However, student interest in pursuing careers in STEM have fallen or stagnated. In contrast, students in many developing countries in which teaching is still somewhat didactic and teacher centred are more disposed to STEM related careers than their western counterparts. Clearly, factors are at work which impact the way students value science and mathematics. This review draws on three components that act as determinants of science education in three different countries – Australia, India and Malaysia. We explore how national priorities and educational philosophy impacts educational practices as well as teacher beliefs and the need for suitable professional development. Socio-economic conditions for science education that are fundamental for developing countries in adopting constructivist educational models are analysed. It is identified that in order to reduce structural dissimilarities among countries that cause fragmentation of scientific knowledge, for Malaysia constructivist science education through English medium without losing the spirit of Malaysian culture and Malay language is essential while India need to adopt constructivist quality indicators in education. While adopting international English education, and reducing dominance of impact evaluation, India and Malaysia need to prevent losing their cultural and social capital vigour. Furthermore the paper argues that Australia might need to question the efficacy of current models that fail to engage students’ long term interest in STEM related careers. Australian and Malaysian science teachers must be capable of changing the personal biographies of learners for developing scientific conceptual information. In addition both Malaysia and Australia need to provide opportunities for access to different curricular programmes of knowledge based constructivist learning for different levels of learner competencies.

Transition: Exchange establishing a visual arts practice based on personal pedagogy

There is a perceived tension in the relationship between the roles of art teacher and artist that led to the question: can an art teacher use their professional training and experience to establish an authentic artistic identity? This self-study tracked and analysed how the process of making her own art enabled an art teacher to also identify as an artist. Drawing on Lamina, the public exhibition of her multimedia artworks, the final exegesis proposes five conditions for art teachers in developing their own art practice: developing an identity as artist, using time and space mindfully, tolerating uncertainty, mentoring, and privileging the process.

A continuum to characterise and support teacher interpretation of an innovative curriculum

A continuum for describing the degree to which teachers interpret the various features of a curriculum is presented. The continuum has been developed based upon the observation of classroom practices and discussions with a group of teachers who are using an innovative junior secondary mathematics curriculum. It is anticipated that the ongoing use of the continuum will lead to its improvement as well as the refinement of the curriculum, more focussed support for the teachers,improved student learning, and the building of explanatory theory regarding mathematics teaching and learning.

Design thinking pedagogy: The educational design ladder

As global industries change and technology advances, traditional education systems might no longer be able to supply companies with graduates who possess an appropriate mix of skills and experience. The recent increased interest in Design Thinking as an approach to innovation has resulted in its adoption by non-design-trained professionals. This development necessitates a new method of teaching Design Thinking and its related skills and processes. As a basis for such a method, this research investigated 51 selected courses across 28 international universities to determine what Design Thinking is being taught (content), and how it is being taught (assessment and learning modes). To support the teaching and assessment of Design Thinking, this paper presents The Educational Design Ladder, an innovative resource/model that provides a process for the organisation and structuring of units for a multidisciplinary Design Thinking programme.

Conceptualising the challenge of integrating digital technologies in pedagogy

The integration of digital technologies in pedagogy is positioned as an important change in education, but widespread innovative use of digital technologies is yet to be truly realised. The gap between the potential and the reality of digital technology integration is commonly attributed to a range of challenging extrinsic and intrinsic influences. Activity Theory (Engeström, 2009) is used to analyse challenges created by extrinsic influences (Nielsen, Miller, & Hoban, 2012); a complementary theory is needed to conceptualise intrinsic influences. System 1 and System 2 thinking theory (Kahneman, 2011) will be advanced as a conceptual framework for understanding conscious and unconscious aspects of teacher practice, particularly the interaction between innovation and teacher routine, attitudes and beliefs. Transformative Learning Theory (Mezirow, 2009) will be positioned to comprehend the nexus of extrinsic and intrinsic influences. This paper will propose how, when faced with extrinsic and intrinsic influences on innovative practice, educators can use these theories to conceptualise the challenge of integrating digital technologies in pedagogy.

Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.

Future learning landscapes for architectural education: Linking space, place, pedagogy, technology + context

Diversification and expansion of global higher education in the 21st century, has resulted in Learning Landscapes for architectural education that can no longer be sustained by the traditional model. Changes have resulted because of surging student numbers, extensions to traditional curricula, evolving competency standards and accreditation requirements, and modified geographical and pedagogical boundaries. The influx of available new technology has helped to democratise knowledge, transforming when, where and how learning takes place. Pressures on government funded higher education budgets highlight the need for a critical review of the current approach to the design and use of learning environments. Efficient design of physical space contributes significantly to savings in provision, management and use of facilities, while also potentially improving pedagogical quality. The purpose of this research is to identify emerging trends in the design of future Learning Landscapes for architectural education in Australasia; to understand where and how students of architecture are likely to learn, in the future context. It explores the important linkages between space, place, pedagogy, technology and context, using a multi methodological qualitative research approach. An Australasian context study will explore the Learning Landscapes of 23 Schools of Architecture across Australia, New Zealand and Papua New Guinea. The focus of this paper is on the methodology which is being employed to undertake dynamic data collection for the study. The research will be determined through mapping all forms of architectural learning environments, pedagogical approaches and contextual issues, to bridge the gap between academic theory, and architectural design practice. An initial understanding that pedagogy is an intrinsic component imbedded within the design of learning environments, will play an important role. Active learning environments which are exemplified by the architectural design studio, support dynamic project based and collaborative connected learning models. These have recently become a lot more common in disciplines outside of design and the arts. It is anticipated, therefore, that the implications for this research may well have a positive impact far beyond the confines of the architectural studio learning environment.

Lionhearts of the Playworld : An ethnographic case study of the development of agency in play pedagogy

This is an ethnographic case study of the creation and emergence of a playworld – a pedagogical approach aimed at promoting children’s development and learning in early education settings through the use of play and drama. The data was collected in a Finnish experimental mixed-age elementary school classroom in the school year 2003-2004. In the playworld students and teachers explore different social and cultural phenomena through taking on the roles of characters from a story or a piece of literature and acting inside the frames of an improvised plot. The thesis takes under scrutiny the notion of agency in education. It produces theoretically grounded empirical knowledge of the ways in which children struggle to become recognized and agentive actors in early education settings and how their agency develops in their interaction with adults. The study builds on the activity theoretical and sociocultural tradition and develops a methodological framework called video-based narrative interaction analysis for studying student agency as developing over time but manifesting through the situational material and discursive local interactions. The research questions are: 1. What are the children’s ways of enacting their agency in the playworld? 2. How do the children’s agentive actions change and develop over the spring? 3. What are the potentials and challenges of the playworld for promoting student agency? 4. How do the teachers and the children deal with the contradiction between control and agency in the playworld? The study consists of a summary part and four empirical articles which each have a particular viewpoint. Articles I and II deal with individual students’ paths to agency. In Article I the focus is on the role of resistance and questioning in enabling important spaces for agency. Article II takes a critical gender perspective and analyzes how two girls struggled towards recognition in the playworld. It also illuminates the role of imagination in developing a sense of agency. Article III examines how the open-ended and improvisational nature of the playworld interaction provided experiences and a sense of ‘shared agency’ for the students and teachers in the class. Article IV turns the focus on the teachers and analyzes how their role actions in the playworld helped the children to enact agency. It also discusses the challenges that the teachers faced in this work and asks what makes the playworld activity sustainable in the class. The summary part provides a critical literature review on the concept of agency and argues that the inherently contradictory nature of the phenomenon of agency has not been sufficiently theorized. The summary part also locates the playworld intervention in a historical frame by discussing the changing conceptions of adulthood and childhood in the West. By focusing on the changing role of play and art in both adults’ and children’s contemporary lives, the thesis opens up an important but often neglected perspective on the problem of promoting student agency in education. The results illustrate how engaging in a collectively imagined and dramatized pretend play space together with the children enabled the teachers to momentarily put aside their “knower” positions in the classroom. The fictive roles and the narrative plot helped them to create a necessary incompleteness and open-endedness in the activity that stimulated the children’s initiatives. This meant that the children too could momentarily step out of their traditional classroom positions as pupils and initiate action to further the collective play. Engaging in this kind of unconventional activity and taking up and enacting agency was, however, very challenging for the participating children and teachers. It often contradicted the need to sustain control and order in the classroom. The study concludes that play- and drama-based pedagogies offer a unique but undeveloped potential for developing educational spaces that help teachers and children deal with the often contradictory requirements of schooling.

Representing the epistemic nature of teachers’ practical knowledge : The case of class teachers’ general pedagogy

The present study addressed the epistemology of teachers’ practical knowledge. Drawing from the literature, teachers’ practical knowledge is defined as all teachers’ cognitions (e.g., beliefs, values, motives, procedural knowing, and declarative knowledge) that guide their practice of teaching. The teachers’ reasoning that lies behind their practical knowledge is addressed to gain insight into its epistemic nature. I studied six class teachers’ practical knowledge; they teach in the metropolitan region of Helsinki. Relying on the assumptions of the phenomenographic inquiry, I collected and analyzed the data. I analyzed the data in two stages where the first stage involved an abductive procedure, and the second stage an inductive procedure for interpretation, and thus developed the system of categories. In the end, a quantitative analysis nested into the qualitative findings to study the patterns of the teachers’’ reasoning. The results indicated that teachers justified their practical knowledge based on morality and efficiency of action; efficiency of action was found to be presented in two different ways: authentic efficiency and naïve efficiency. The epistemic weight of morality was embedded in what I call “moral care”. The core intention of teachers in the moral care was the commitment that they felt about the “whole character” of students. From this perspective the “dignity” and the moral character of the students should not replaced for any other “instrumental price”. “Caring pedagogy” was the epistemic value of teachers’ reasoning in the authentic efficiency. The central idea in the caring pedagogy was teachers’ intentions to improve the “intellectual properties” of “all or most” of the students using “flexible” and “diverse” pedagogies. However, “regulating pedagogy” was the epistemic condition of practice in the cases corresponding to naïve efficiency. Teachers argued that an effective practical knowledge should regulate and manage the classroom activities, but the targets of the practical knowledge were mainly other “issues “or a certain percentage of the students. In these cases, the teachers’ arguments were mainly based on the notion of “what worked” regardless of reflecting on “what did not work”. Drawing from the theoretical background and the data, teachers’ practical knowledge calls for “praxial knowledge” when they used the epistemic conditions of “caring pedagogy” and “moral care”. It however calls for “practicable” epistemic status when teachers use the epistemic condition of regulating pedagogy. As such, praxial knowledge with the dimensions of caring pedagogy and moral care represents the “normative” perspective on teachers’ practical knowledge, and thus reflects a higher epistemic status in comparison to “practicable” knowledge, which represents a “descriptive” perception toward teachers’ practical knowledge and teaching.