Understanding graphicacy : students’ making sense of graphics in mathematics assessment tasks

The ability to decode graphics is an increasingly important component of mathematics assessment and curricula. This study examined 50, 9- to 10-year-old students (23 male, 27 female), as they solved items from six distinct graphical languages (e.g., maps) that are commonly used to convey mathematical information. The results of the study revealed: 1) factors which contribute to success or hinder performance on tasks with various graphical representations; and 2) how the literacy and graphical demands of tasks influence the mathematical sense making of students. The outcomes of this study highlight the changing nature of assessment in school mathematics and identify the function and influence of graphics in the design of assessment tasks.

Practical insights into curricula integration for primary science

As indicated in a previous Teaching Science article, effective planning for curricula integration requires using standards from two (or more) subject areas (e.g., science and English, science and art or science and mathematics), which also becomes the assessment foci for teaching and learning. Curricula integration of standards into an activity necessitates pedagogical knowledge for developing students’ learning in both subject areas. For science education, the skills and tools for curricula integration include the use of other key learning areas (KLAs). A balance between teacher and student-centred science education programs that draw on democratic processes (e.g., Beane, 1997) can be used to make real-world links to target students’ individual needs. This article presents practical ways to commence thinking about curricula integration towards using Australian curriculum standards.

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.

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.

Evaluation of the "reconceptualising early mathematics learning" project

The Pattern and Structure Mathematics Awareness Project (PASMAP) has investigated the development of patterning and early algebraic reasoning among 4 to 8 year olds over a series of related studies. We assert that an awareness of mathematical pattern and structure enables mathematical thinking and simple forms of generalisation from an early age. The project aims to promote a strong foundation for mathematical development by focusing on critical, underlying features of mathematics learning. This paper provides an overview of key aspects of the assessment and intervention, and analyses of the impact of PASMAP on students’ representation, abstraction and generalisation of mathematical ideas. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in the evaluation throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ assessment and a Pattern and Structure Assessment (PASA) interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students; those categorised as low ability developed improved structural responses over a relatively short period of time.

Data and measurement in Year 4 of the national curriculum : Mathematics

The activities introduced here were used in association with a research project in four Year 4 classrooms and are suggested as a motivating way to address several criteria for Measurement and Data in the Australian Curriculum: Mathematics. The activities involve measuring the arm span of one student in a class many times and then of all students once.

Reconceptualizing Early Mathematics Learning : The Fundamental Role of Pattern and Structure

The Pattern and Structure Mathematics Awareness Program (PASMAP) was developed concurrently with the studies of AMPS and the development of the Pattern and Structure Assessment (PASA) interview. We summarize some early classroom-based teaching studies and describe the PASMAP that resulted. A large-scale two-year longitudinal study, Reconceptualizing Early Mathematics Learning (REML) resulted. We provide an overview of the REML study and discuss the consequences for our view of early mathematics learning. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in an evaluation of the PASMAP intervention throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ standardized assessment and the PASA interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students. Implications for pedagogy and curriculum are discussed.

Looking back : students' perceptions of the relative enjoyment of primary and secondary school science

This paper reports and discusses a contentious result from an Australia-wide study of the influences on students' decisions about taking senior science subjects. As part of the Choosing Science study (Lyons and Quinn 2010) 3759 Year 10 students were asked to indicate which stage of their schooling (lower primary, upper primary, lower secondary, middle secondary) they had most enjoyed learning science. Crosstabulations of responses revealed that around 78% of students indicated that they had enjoyed learning science more in secondary than in primary school, and 55% enjoyed it the most during Years 9 and 10. The perception that school science was more enjoyable in high school was also found among students who did not intend taking science in Year 11, though to a lesser extent. These findings are unexpected and significant, challenging the prevailing view that enjoyment of school science steadily declines after primary school. The paper elaborates on the findings and suggests that the different conclusions arrived at by studies in this field may be due to the different methodologies employed.

Retooling Chinese primary school teachers to use technology creatively to promote innovation and problem solving skills in science classrooms

This paper reports on the initial phase of a Professional Learning Program (PLP) undertaken by 100 primary school teachers in China that aimed to facilitate the development of adaptive expertise in using technology to facilitate innovative science teaching and learning such as that 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. By the end of the initial phase of the PLP, 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 learning and assessment activities. The findings from the study indicate that those engaged in the development of PLPs for teachers in China need to take cognizance of certain cultural factors and traditions idiosyncratic to the Chinese educational system. A set of revised principles is then presented to inform the future design and implementation of PLPs for teachers in China.

Curricula integration : identifying and locating engineering education across the Australian curriculum

Science, technology, engineering and mathematics (STEM) has become an educational package emerging throughout the world (e.g. UK, China, US & Australia). Although science, technology and mathematics are taught in schools and engineering education occurs in universities, there appear to be few if any explicit engineering education programs in primary and junior secondary schools. A stronger inclusion of engineering education at these levels could assist students to make informed decisions about career opportunities in STEM-related fields. This paper suggests how engineering education can be integrated with other key learning areas such as English, mathematics, science, history and geography within the new Australian Curriculum.

Use of genetic decompositions to scaffold the development of a structurally sequenced curriculum for mathematics acceleration

The authors have collaboratively used a graphical language to describe their shared knowledge of a small domain of mathematics, which has in turn scaffolded their re-development of a related curriculum for mathematics acceleration. This collaborative use of the graphical language is reported as a simple descriptive case study. This leads to an evaluation of the graphical language’s usefulness as a tool to support the articulation of the structure of mathematics knowledge. In turn, implications are drawn for how the graphical language may be utilised as the detail of the curriculum is further elaborated and communicated to teachers.

Understanding preservice teachers’ development of pedagogical knowledge practices when co-teaching primary science to peers

Preservice teachers articulate the need for more teaching experiences for developing their practices, however, extending beyond existing school arrangements may present difficulties. Thus, it is important to understand preservice teachers’ development of pedagogical knowledge practices when in the university setting. This mixed-method study investigated 48 second-year preservice teachers’ development of pedagogical knowledge practices as a result of co-teaching primary science to peers. Data were collected through a survey, video-recorded lessons, extended written responses and researcher observations. The study showed how these preservice teachers demonstrated 9 of 11 pedagogical knowledge practices within the co-teaching arrangement. However, research is needed to determine the level of development on each pedagogical knowledge practice and how these practices can be transferred into authentic primary classroom settings.

Linking pedagogical knowledge practices and student outcomes in STEM education for primary schools

STEM education is a new frontier in Australia, particularly for primary schools. However, the E in STEM needs to have a stronger focus with science and mathematics concepts aligned to the presiding curricula. In addition, pedagogical knowledge practices such as planning, preparation, teaching strategies, assessment and so forth need to be connected to key concepts for developing a STEM education. One of the aims of this study was to understand how a pedagogical knowledge practice framework could be linked to student outcomes in STEM education. Specifically, this qualitative research investigated Year 4 students’ involvement in an integrated STEM education program that focused on science concepts (e.g., states of matter, testing properties of materials) and mathematics concepts (such as 3D shapes and metric measurements: millilitres, temperature, grams, centimetres) for designing, making and testing a strong and safe medical kit to insulate medicines at desirable temperatures. Eleven pedagogical knowledge practices (e.g., planning, preparation, teaching strategies, classroom management, and assessment) were used as a framework for understanding how teaching may be linked to student outcomes in STEM education. For instance, “planning” involved devising a student booklet as a resource for students to understand the tasks required of them, which also provided space for them to record ideas, results and information. Planning involved linking national and state curriculum documents to the STEM education activities. More studies are required around pedagogical knowledge frameworks to understand what students learn when involved in STEM education, particularly with the inclusion of engineering education.

Learning through modelling in the primary years

Many nations are highlighting the need for a renaissance in the mathematical sciences as essential to the well-being of all citizens (e.g., Australian Academy of Science, 2006; 2010; The National Academies, 2009). Indeed, the first recommendation of The National Academies’ Rising Above the Storm (2007) was to vastly improve K–12 science and mathematics education. The subsequent report, Rising Above the Gathering Storm Two Years Later (2009), highlighted again the need to target mathematics and science from the earliest years of schooling: “It takes years or decades to build the capability to have a society that depends on science and technology . . . You need to generate the scientists and engineers, starting in elementary and middle school” (p. 9). Such pleas reflect the rapidly changing nature of problem solving and reasoning needed in today’s world, beyond the classroom. As The National Academies (2009) reported, “Today the problems are more complex than they were in the 1950s, and more global. They’ll require a new educated workforce, one that is more open, collaborative, and cross-disciplinary” (p. 19). The implications for the problem solving experiences we implement in schools are far-reaching. In this chapter, I consider problem solving and modelling in the primary school, beginning with the need to rethink the experiences we provide in the early years. I argue for a greater awareness of the learning potential of young children and the need to provide stimulating learning environments. I then focus on data modelling as a powerful means of advancing children’s statistical reasoning abilities, which they increasingly need as they navigate their data-drenched world.

Exploring with Grammar in the Primary Years: Learning about Language in the Australian Curriculum: English

This new volume, Exploring with Grammar in the Primary Years (Exley, Kevin & Mantei, 2014), follows on from Playing with Grammar in the Early Years (Exley & Kervin, 2013). We extend our thanks to the ALEA membership for their take up of the first volume and the vibrant conversations around our first attempt at developing a pedagogy for the teaching of grammar in the early years. Your engagement at locally held ALEA events has motivated us to complete this second volume and reassert our interest in the pursuit of socially-just outcomes in the primary years. As noted in Exley and Kervin (2013), we believe that mastering a range of literacy competences includes not only the technical skills for learning, but also the resources for viewing and constructing the world (Freire and Macdeo, 1987). Rather than seeing knowledge about language as the accumulation of technical skills alone, the viewpoint to which we subscribe treats knowledge about language as a dialectic that evolves from, is situated in, and contributes to active participation within a social arena (Halliday, 1978). We acknowledge that to explore is to engage in processes of discovery as we look closely and examine the opportunities before us. As such, we draw on Janks’ (2000; 2014) critical literacy theory to underpin many of the learning experiences in this text. Janks (2000) argues that effective participation in society requires knowledge about how the power of language promotes views, beliefs and values of certain groups to the exclusion of others. Powerful language users can identify not only how readers are positioned by these views, but also the ways these views are conveyed through the design of the text, that is, the combination of vocabulary, syntax, image, movement and sound. Similarly, powerful designers of texts can make careful modal choices in written and visual design to promote certain perspectives that position readers and viewers in new ways to consider more diverse points of view. As the title of our text suggests, our activities are designed to support learners in exploring the design of texts to achieve certain purposes and to consider the potential for the sharing of their own views through text production. In Exploring with Grammar in the Primary Years, we focus on the Year 3 to Year 6 grouping in line with the Australian Curriculum, Assessment and Reporting Authority’s (hereafter ACARA) advice on the ‘nature of learners’ (ACARA, 2014). Our goal in this publication is to provide a range of highly practical strategies for scaffolding students’ learning through some of the Content Descriptions from the Australian Curriculum: English Version 7.2, hereafter AC:E (ACARA, 2014). We continue to express our belief in the power of using whole texts from a range of authentic sources including high quality children’s literature, the internet, and examples of community-based texts to expose students to the richness of language. Taking time to look at language patterns within actual texts is a pathway to ‘…capture interest, stir the imagination and absorb the [child]’ into the world of language and literacy (Saxby, 1993, p. 55). It is our intention to be more overt this time and send a stronger message that our learning experiences are simply ‘sample’ activities rather than a teachers’ workbook or a program of study to be followed. We’re hoping that teachers and students will continue to explore their bookshelves, the internet and their community for texts that provide powerful opportunities to engage with language-based learning experiences. In the following three sections, we have tried to remain faithful to our interpretation of the AC:E Content Descriptions without giving an exhaustive explanation of the grammatical terms. This recently released curriculum offers a new theoretical approach to building students’ knowledge about language. The AC:E uses selected traditional terms through an approach developed in systemic functional linguistics (see Halliday and Matthiessen, 2004) to highlight the dynamic forms and functions of multimodal language in texts. For example, the following statement, taken from the ‘Language: Knowing about the English language’ strand states: English uses standard grammatical terminology within a contextual framework, in which language choices are seen to vary according to the topics at hand, the nature and proximity of the relationships between the language users, and the modalities or channels of communication available (ACARA, 2014). Put simply, traditional grammar terms are used within a functional framework made up of field, tenor, and mode. An understanding of genre is noted with the reference to a ‘contextual framework’. The ‘topics at hand’ concern the field or subject matter of the text. The ‘relationships between the language users’ is a description of tenor. There is reference to ‘modalities’, such as spoken, written or visual text. We posit that this innovative approach is necessary for working with contemporary multimodal and cross-cultural texts (see Exley & Mills, 2012). Other excellent tomes, such as Derewianka (2011), Humphrey, Droga and Feez (2012), and Rossbridge and Rushton (2011) provide more comprehensive explanations of this unique metalanguage, as does the AC:E Glossary. We’ve reproduced some of the AC:E Glossary at the end of this publication. We’ve also kept the same layout for our learning experiences, ensuring that our teacher notes are not only succinct but also prudent in their placement. Each learning experience is connected to a Content Description from the AC:E and contains an experience with an identified purpose, suggested resource text and a possible sequence for the experience that always commences with an orientation to text followed by an examination of a particular grammatical resource. Our plans allow for focused discussion, shared exploration and opportunities to revisit the same text for the purpose of enhancing meaning making. Some learning experiences finish with deconstruction of a stimulus text while others invite students to engage in the design of new texts. We encourage you to look for opportunities in your own classrooms to move from text deconstruction to text design. In this way, students can express not only their emerging grammatical understandings, but also the ways they might position readers or viewers through the creation of their own texts. We expect that each of these learning experiences will vary in the time taken. Some may indeed take a couple if not a few teaching episodes to work through, especially if students are meeting a concept or a pedagogical strategy for the first time. We hope you use as much, or as little, of each experience as is needed for your students. We do not want the teaching of grammar to slip into a crisis of irrelevance or to be seen as a series of worksheet drills with finite answers. We firmly believe that strategies for effective deconstruction and design practice, however, have much portability. We three are very keen to hear from teachers who are adopting and adapting these learning experiences in their classrooms. Please email us on b.exley@qut.edu.au, lkervin@uow.edu.au or jessicam@ouw.edu.au. We’d love to continue the conversation with you over time. Beryl Exley, Lisa Kervin & Jessica Mantei