Mathematics, programming, and STEM

Learning mathematics is a complex and dynamic process. In this paper, the authors adopt a semiotic framework (Yeh & Nason, 2004) and highlight programming as one of the main aspects of the semiosis or meaning-making for the learning of mathematics. During a 10-week teaching experiment, mathematical meaning-making was enriched when primary students wrote Logo programs to create 3D virtual worlds. The analysis of results found deep learning in mathematics, as well as in technology and engineering areas. This prompted a rethinking about the nature of learning mathematics and a need to employ and examine a more holistic learning approach for the learning in science, technology, engineering, and mathematics (STEM) areas.

Development of probabilistic understanding in fourth grade

We analyzed the development of 4th-grade students’ understanding of the transition from experimental relative frequencies of outcomes to theoretical probabilities with a focus on the foundational statistical concepts of variation and expectation. We report students’ initial and changing expectations of the outcomes of tossing one and two coins, how they related the relative frequency from their physical and computersimulated trials to the theoretical probability, and how they created and interpreted theoretical probability models. Findings include students’ progression from an initial apparent equiprobability bias in predicting outcomes of tossing two coins through to representing the outcomes of increasing the number of trials. After observing the decreasing variation from the theoretical probability as the sample size increased, students developed a deeper understanding of the relationship between relative frequency of outcomes and theoretical probability as well as their respective associations with variation and expectation. Students’ final models indicated increasing levels of probabilistic understanding.

Making sense of a trial maths intervention program for teachers of students with disability in Australia: Interim report

The Disability Standards for Education (2005) and the Australian Curriculum, Assessment and Reporting Authority relevant standards underscore the right of students with disability to access the curriculum on the same basis as students without disability. Students with disability are entitled to rigorous, relevant and engaging learning opportunities drawn from the Australian curriculum content. Taking this context into account, this paper provides a work-in-progress report on a two-year mathematics intervention project conducted in 12 special schools (Preparatory-Year 12) in Queensland, Australia. The project aims to build the capacity of teachers to teach mathematics to their students and to identify and make sense of the intervention program’s impact. It combines two approaches—appreciative inquiry and action research to monitor schools’ change processes. The interim findings demonstrated that teachers were concerned about their students’ underachievement in mathematics and that the multi-sensory forms of teaching advocated in the program increased student engagement and performance.

Orals ain't orals : how instruction and assessment practices affect delivery choices with prepared student oral presentations.

Despite an ostensibly technology-driven society, the ability to communicate orally is still seen as an essential ability for students at school and university, as it is for graduates in the workplace. The need to develop effective oral communication skills is often tied to future work-related tasks. One tangible way that educators have assessed proficiency in this area is through prepared oral presentations. While some use the terms oral communication and oral presentation interchangeably, other writers question the role more formal presentations play in the overall development of oral communication skills. Adding to the discussion, this paper is part of a larger study examining the knowledge and skills students bring into the academy from previous educational experiences. The study examines some of the teaching and assessment methods used in secondary schools to develop oral communication skills through the use of formal oral presentations. Specifically, it will look at assessment models and how these are used as a form of instruction as well as how they contribute to an accurate evaluation of student abilities. The purpose of this paper is to explore key terms and identify tensions between expectations and practice. Placing the emphasis on the ‘oral’ aspect of this form of communication this paper will particularly look at the ‘delivery’ element of the process.

Methodologies for investigating relationships between concept development and the development of problem solving abilities

This paper is the second in a pair that Lesh, English, and Fennewald will be presenting at ICME TSG 19 on Problem Solving in Mathematics Education. The first paper describes three shortcomings of past research on mathematical problem solving. The first shortcoming can be seen in the fact that knowledge has not accumulated – in fact it has atrophied significantly during the past decade. Unsuccessful theories continue to be recycled and embellished. One reason for this is that researchers generally have failed to develop research tools needed to reliably observe, document, and assess the development of concepts and abilities that they claim to be important. The second shortcoming is that existing theories and research have failed to make it clear how concept development (or the development of basic skills) is related to the development of problem solving abilities – especially when attention is shifted beyond word problems found in school to the kind of problems found outside of school, where the requisite skills and even the questions to be asked might not be known in advance. The third shortcoming has to do with inherent weaknesses in observational studies and teaching experiments – and the assumption that a single grand theory should be able to describe all of the conceptual systems, instructional systems, and assessment systems that strongly molded and shaped by the same theoretical perspectives that are being used to develop them. Therefore, this paper will describe theoretical perspectives and methodological tools that are proving to be effective to combat the preceding kinds or shortcomings. We refer to our theoretical framework as models & modeling perspectives (MMP) on problem solving (Lesh & Doerr, 2003), learning, and teaching. One of the main methodologies of MMP is called multi-tier design studies (MTD).

Action research for contexts of change and inequality

Undoubtedly, the past half-century has witnessed an escalation of changes in the social, political, economic and educational structures in many societies around the world. Some have seen change as a challenge and hope while, for many others, it is a source of concern and worry. Some have adopted change with gusto, while for many it is something to be resisted. Some say we live in a world and times with an increasing awareness that “times are changing”, while for some “the more things change, the more they stay the same”.

Year 8, 9 and 10 students’ understanding and access of percent knowledge

This paper reports on Years 8, 9 and 10 students’ knowledge of percent problem types, use of diagrams, and type of solution strategy. Non- and semi-proficient students displayed the expected inflexible formula approach to solution but proficient students used a flexible mixture of estimation, number sense and trial and error instead of expected schema based methods.

Interdisciplinary modelling

This article examines one approach to promoting creative and flexible use of mathematical ideas within an interdisciplinary context in the primary curriculum, namely, through modelling. Three classes of fifth-grade children worked on a modelling problem, The First Fleet (Australia’s settlement), situated within the curriculum domains of science and studies of society and environment. Reported here are the cycles of development displayed by one group of children as they worked the problem, together with the range of models created across the classes. Children developed mathematisation processes that extended beyond their regular curriculum, including identifying and prioritising key problem elements, exploring relationships among elements, quantifying qualitative data, ranking and aggregating data, and creating and working with weighted scores. Aspects of Goldin’s (2000, 2007) affective structures also appeared to play an important role in the children's mathematical developments.

The ability of economic thinking : the “nature” versus “nurture” debate

In their studies, Eley and Meyer (2004) and Meyer and Cleary (1998) found that there are sources of variation in the affective and process dimensions of learning in mathematics and clinical diagnosis specific to each of these disciplines. Meyer and Shanahan (2002) argue that: General purpose models of student learning that are transportable across different discipline contexts cannot, by definition, be sensitive to sources of variation that may be subject-specific (2002. p. 204). In other words, to explain the differences in learning approaches and outcomes in a particular discipline, there are discipline-specific factors, which cannot be uncovered in general educational research. Meyer and Shanahan (2002) argue for a need to "seek additional sources of variation that are perhaps conceptually unique ... within the discourse of particular disciplines" (p. 204). In this paper, the development of an economics-specific construct (called economic thinking ability) is reported. The construct aims to measure discipline-sited ability of students that has important influence on learning in economics. Using this construct, economic thinking abilities of introductory and intermediate level economics students were measured prior to the commencement, and at the end, of their study over one semester. This enabled factors associated with students' pre-course economic thinking ability and their development in economic thinking ability to be investigated. The empirical findings will address the 'nature' versus 'nurture' debate in economics education (Frank, et aI., 1993; Frey et al., 1993; Haucap and Tobias 2003). The implications for future research in economics education will also be discussed.

Mentors report on their own mentoring practices

There are only two ways to implement reform in an education system, namely through inservice education of existing teachers and preservice teacher education. Implementing the Australian Curriculum will require targeting both teachers and preservice teachers. Classroom teachers in their roles as mentors have a significant role to play for developing preservice teachers. What mentors do in their mentoring practices and what mentors think about mentoring will impact on the mentoring processes and ultimately reform outcomes. What are mentors’ reports on their mentoring of preservice teachers in science and mathematics? This mixed-method study presents mentors’ reports on their mentoring of primary preservice teachers (mentees) in mathematics (n=43) and science (n=29). Drawing upon a previously validated instrument (Hudson, 2007), this instrument was amended to allow mentors to report on their perceptions of their mentoring. A questionnaire elicited extended written responses that focused on: (1) the mentors’ rapport with their mentees, (2) successful mentoring strategies, (3) aspects that may lead the mentee to feel unsuccessful, and (4) ways to enhance their mentoring skills. Mentors claimed they mentored teaching mathematics more than science. However, 20% or more indicated they did not provide mentoring practices for 25 out of 34 survey items in the science and 9 out of 34 items in the mathematics. Educational reform will necessity mentors to be educated on effective mentoring practices so the mentoring process can be more purposeful. Indeed, mentors who have knowledge of such practices may address the potential issues of more than 20% of mentees not receiving these practices. These mentors also claimed that professional development on effective mentoring can enhance their skills. To ensure the greatest success for an Australian Curriculum will require targeting mentors for professional development in order to assist mentees’ development into the profession.

Comparing mentors’ reports on their own mentoring of primary preservice teachers

Implementing the Australian Curriculum will require targeting both teachers and preservice teachers as enactors of reform. Classroom teachers in their roles as mentors have a significant role to play for developing preservice teachers. What mentors do in their mentoring practices and what mentors think about mentoring will impact on the mentoring processes and ultimately reform outcomes. What are mentors’ reports on their mentoring of preservice teachers for teaching science and mathematics? This quantitative study presents mentors’ reports on their mentoring of primary preservice teachers (mentees) in mathematics (n=43) and science (n=29). Drawing upon a previously validated instrument (Hudson, 2007), this instrument was amended to allow mentors to report on their perceptions of their mentoring. Mentors claimed they mentored teaching mathematics more than science. However, 20% or more indicated they did not provide mentoring practices for 25 out of 34 survey items in the science and 9 out of 34 items in the mathematics. Educational reform will necessity mentors to be educated on effective mentoring practices for mathematics and science so the mentoring process can be more purposeful. Indeed, mentors who have knowledge of such practices may address the potential issues of more than 20% of mentees not receiving these practices. To ensure the greatest success for an Australian Curriculum mentors may need professional development in order to assist mentees’ development into the profession.

Primary students’ interpretation of maps : gesture use and mapping knowledge

Maps are used to represent three-dimensional space and are integral to a range of everyday experiences. They are increasingly used in mathematics, being prominent both in school curricula and as a form of assessing students understanding of mathematics ideas. In order to successfully interpret maps, students need to be able to understand that maps: represent space, have their own perspective and scale, and their own set of symbols and texts. Despite the fact that maps have an increased prevalence in society and school, there is evidence to suggest that students have difficulty interpreting maps. This study investigated 43 primary-aged students’ (aged 9-12 years) verbal and gestural behaviours as they engaged with and solved map tasks. Within a multiliteracies framework that focuses on spatial, visual, linguistic, and gestural elements, the study investigated how students interpret map tasks. Specifically, the study sought to understand students’ skills and approaches used to solving map tasks and the gestural behaviours they utilised as they engaged with map tasks. The investigation was undertaken using the Knowledge Discovery in Data (KDD) design. The design of this study capitalised on existing research data to carry out a more detailed analysis of students’ interpretation of map tasks. Video data from an existing data set was reorganised according to two distinct episodes—Task Solution and Task Explanation—and analysed within the multiliteracies framework. Content Analysis was used with these data and through anticipatory data reduction techniques, patterns of behaviour were identified in relation to each specific map task by looking at task solution, task correctness and gesture use. The findings of this study revealed that students had a relatively sound understanding of general mapping knowledge such as identifying landmarks, using keys, compass points and coordinates. However, their understanding of mathematical concepts pertinent to map tasks including location, direction, and movement were less developed. Successful students were able to interpret the map tasks and apply relevant mathematical understanding to navigate the spatial demands of the map tasks while the unsuccessful students were only able to interpret and understand basic map conventions. In terms of their gesture use, the more difficult the task, the more likely students were to exhibit gestural behaviours to solve the task. The most common form of gestural behaviour was deictic, that is a pointing gesture. Deictic gestures not only aided the students capacity to explain how they solved the map tasks but they were also a tool which assisted them to navigate and monitor their spatial movements when solving the tasks. There were a number of implications for theory, learning and teaching, and test and curriculum design arising from the study. From a theoretical perspective, the findings of the study suggest that gesturing is an important element of multimodal engagement in mapping tasks. In terms of teaching and learning, implications include the need for students to utilise gesturing techniques when first faced with new or novel map tasks. As students become more proficient in solving such tasks, they should be encouraged to move beyond a reliance on such gesture use in order to progress to more sophisticated understandings of map tasks. Additionally, teachers need to provide students with opportunities to interpret and attend to multiple modes of information when interpreting map tasks.

Data modelling with first-grade students

This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling. Results are reported from the first year of a 3-year longitudinal study in which three classes of first-grade children (6-year-olds) and their teachers engaged in data modelling activities. The theme of Looking after our Environment, part of the children’s science curriculum, provided the task context. The goals for the two activities addressed here included engaging children in core components of data modelling, namely, selecting attributes, structuring and representing data, identifying variation in data, and making predictions from given data. Results include the various ways in which children represented and re represented collected data, including attribute selection, and the metarepresentational competence they displayed in doing so. The “data lenses” through which the children dealt with informal inference (variation and prediction) are also reported.

Supporting exceptional students to thrive mathematically

Providing an appropriate education for exceptional students in mathematics is mandated in educational policy in Australasia (Australian Curriculum, Assessment and Reporting Agency (ACARA), 2010; Ministry of Education, 2009, 2011) but a challenge for teachers and schools. ‘Exceptional students’ refer to two distinct populations, namely those who are gifted in mathematics and have the capability to perform very highly compared to age peers and those who experience learning difficulties in mathematics and may underperform (Diezmann, Lowrie, Bicknell, Faragher, & Putt, 2004).

The challenges of preparing pre-service teachers to embrace a digital pedagogy

The need for pre-service teachers to be proficient in the use of information and communication technologies (ICT) in the classroom once they graduate is essential, though this process is not a straightforward process (Zhang, 2008) and needs to go beyond pre-service teachers just being able to use ICT. Research suggests that for teachers to successfully use ICT in their classrooms they need to be specifically trained to do so (Markauskaite, 2007; Batane, 2004; Jacobsen, Clifford & Friesen, 2002). Pre-service teachers must also be able to embrace and use new and emerging ICT’s, often referred to as digital technologies, within their pedagogical approaches to teaching and learning. According to UNESCO, these “new technologies require new teacher roles, new pedagogies, and new approaches to teacher training” (2008, p.9). However, new approaches to teacher training have moved very slowly in many areas and preparing pre-service teachers to develop proficiency in embracing a digital pedagogy within their own classrooms can be a challenge for teacher training institutes. This paper reports on a case study of first year education students (N=667) and their experiences during their first semester of pre-service teacher education in a core ICT unit. It will report on the background ICT knowledge and skills that these students bring to the course as well as their expectations of the unit and ICT in their future teaching. The paper will then draw on the research results to identify challenges facing teacher training of pre-service teachers in using digital technologies in their future classrooms.