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How might fourth graders address the problem of measuring the size of a leaf? And why might their classroom teachers choose such a shape to measure? One group of fourth-grade students in an urban Australian school were interested in knowing the sizes of leaves on a passionfruit vine growing at their school. The children were excited after a visit from an agricultural scientist, who shared his knowledge about photosynthesis. This article presents mathematical modeling explored in their classroom, as students explored ways to predict the surface area of a leaf (leaf size), within an agricultural science context involving fruit production. The first author and her co-teacher, who regularly shared teaching responsibilities, worked with an agricultural scientist in designing the inquiry for their fourth-grade class.

This article reports on a study in which third-grade students (8-9 years) were given a degree of agency in conducting chance experiments and representing the outcomes. Students chose their own samples of 12 coloured counters, ensuring all colours were represented. They predicted the outcomes of item selection, tested their predictions, explained the outcomes, quantified their chances of colour selections, and created two representations displaying the probabilities. Children displayed awareness of randomness and variation, together with proportional reasoning, as evident in their identification of one or more colours as having a greater chance of being selected, or equal chances when proportions of colours were equal. Evidence of children's metarepresentational competence appeared in their creation of two representations to display their probabilistic outcomes, with bar and circle graphs, as well as stacked bars, created. The inclusion of their own forms of inscription revealed a range of probability and statistics understandings. In selecting and justifying their preferred representations for conveying their outcomes, students favoured both bar and circle graphs, with a focus on how accurately, effectively, and efficiently their representation displayed the data, with the importance of the inscriptions highlighted.

An Australian longitudinal study of 319 Kindergartners developed, implemented and evaluated an intervention, the Pattern and Structure Mathematics Awareness Program (PASMAP). It comprised of repetitions and growing patterns, structured counting and grouping, grids and shapes, partitioning, additive and multiplicative structures, measurement and data, and transformations. Each component was implemented, evaluated and refined as a cyclic process in collaboration with co-operating teachers, replacing and extending the regular mathematics program. An innovative pedagogical approach was effective in engaging students in modelling and representing, visualising and generalising, and sustaining their learning. On an interview-based measure of mathematics achievement pre- and post-intervention, there were significant differences found between the intervention and comparison groups at the end of the intervention year (p < 0.026) and highly significant differences found at the follow-up assessment (p < 0.002) ten months later. A descriptive analysis showed students' progress through five levels of structural development where most students advanced through one or more levels as they progressed through the program. The upper third of students demonstrated emergent generalisations at the structural and advanced structural level which supported early algebraic thinking. Less-able students also showed impressive growth--half of these students moved from pre-structural to at least emergent level over the duration of the program. Students' growth in structural development is exemplified at five points during the intervention through fine-grained analyses of mathematical representations and explanations. The study demonstrates that Kindergartners are capable of developing mathematical patterns and structural relationships well beyond curriculum expectations.

As STEM education becomes more prominent at the school level, many possibilities could be suggested for beginning experiences to create a foundation for young students as they begin their journeys with respect to STEM learning. The research reported here is based on the premise that building statistical understanding will enhance young students' learning as they engage with STEM learning experiences. This paper hence reports on an activity created to introduce students in Year 3 to the concept of statistical variation with data in a STEM-related context where variation occurs in an easily measured and realistic fashion. Students' capabilities to appreciate the fundamental nature of statistical variation and use it for comparison in a STEM context were assessed through responses in student workbooks, to questions on an end-of-year survey, and in individual interviews. The results illustrate the beginning stages of student thinking about variation and the use of data when learning is embedded within a STEM context. They show that students can take on the idea of variation and use it in explaining their experiences with the hands-on activity, which involved comparisons of a hand-made and machine-made product.

Science, technology, engineering, and mathematics (STEM) education is an emerging initiative in Australia, particularly in primary schools. This qualitative research aimed to understand Year 4 students' involvement in an integrated STEM education unit that focused on science concepts (e.g., states of matter, testing properties of materials) and mathematics concepts (e.g., 3D shapes and metric measurements) for designing, making and testing a strong and safe medical kit to insulate medicines (ice cubes) at desirable temperatures. Data collection tools included student work samples, photographs, written responses from students and the teacher, and researcher notes. In a post-hoc analysis, a pedagogical knowledge practice framework (i.e., planning, timetabling, preparation, teaching strategies, content knowledge, problem solving, classroom management, questioning, implementation, assessment, and viewpoints) was used to explain links to student outcomes in STEM education. The study showed how pedagogical knowledge practices may be linked to student outcomes (knowledge, understanding, skill development, and values and attitudes) for a STEM education activity.

An educational priority of many nations is to enhance mathematical learning in early childhood. One area in need of special attention is that of statistics. This paper argues for a renewed focus on statistical reasoning in the beginning school years, with opportunities for children to engage in data modelling activities. Such modelling involves investigations of meaningful phenomena, deciding what is worthy of attention (identifying complex attributes), and then progressing to organising, structuring, visualising, and representing data. Results reported here are derived from the first year of a three-year longitudinal study in which three classes of first-grade children and their teachers engaged in activities requiring the creation of data models. The theme of "Looking after our Environment," a component of the children's science curriculum at the time, provided the context for the activities. Findings include children's abilities to focus their attention on qualities of items rather than the items themselves in identifying attributes, switch their attention from one item feature to another, and create a broad range of models in organising, structuring, and representing their data. Children's development of meta-representational knowledge facilitated their choice and nature of data representations. (Contains 2 figures.)

This article explores 6th-grade students' modelling with data in generating models for selecting an Australian swimming team for the (then) forthcoming 2016 Olympics, using data on swimmers' times at various previous events. We propose a modelling framework comprising four components: working in shared problem spaces between mathematics and statistics; interpreting and reinterpreting problem contexts and questions; interpreting, organising and operating on data in model construction; and drawing informal inferences. In studying students' model generation, consideration is given to how they interpreted, organised, and operated on the problem data in constructing and documenting their models, and how they engaged in informal inferential reasoning. Students' responses included applying mathematical and statistical operations and reasoning to selected variables, identifying how variation and trends in swimmers' performances inform model construction, recognising limitations in using only one performance variable, and acknowledging uncertainty in model creation and model application due to chance variation.

The teaching and learning of statistical thinking begins at a young age in Australia, with a focus on data representation and interpretation from Foundation Year (age 5), and the collection, sorting and categorising of items from the natural environment starting even earlier. The intangible concept of data, as part of statistical literacy, can be complex for children to grasp, especially when applying the notion of data to the everyday world or when data are explored in isolation to an investigation process. Authentic data modelling experiences present meaningful opportunities to apply statistical thinking although expert STEM knowledge is not always accessible to primary classroom teachers, nor is it always obvious how to implement such authentic problems within a classroom context. In this exploratory case study, we present data from a Year 4 classroom (age 9) statistical investigation addressing, 'How big is a leaf?' linking data to the real-life STEM context they represented. The authors were interested in how the teacher's communication processes supported her students' emerging understandings about data. Wit's (2018) cognitive tuning framework offered a way to capture how the communication processes in a group build to a commonly shared frame of reference. Findings revealed a pattern of communication between the teacher and students, supporting students' changing conceptions of data and related statistical thinking processes, throughout the investigation. [ABSTRACT FROM AUTHOR]

This study reports on a classroom activity for Grade 5 students investigating their reaction times. The investigation was part of a 3-year research project introducing students to informal inference and giving them experience carrying out the practice of statistics. For this activity the focus within the practice of statistics was on introducing two different ways of collecting data to answer a statistical question, in this case, "What is the typical reaction time of Grade 5 students?" Workbook entries were used to assess students' capacities to engage in the investigation. Results indicated that although the students were proficient with the procedures and measures introduced, they were less able to explain and apply the underlying concepts. The activity provides a suggestion and benchmarks for others wishing to follow student development of concepts related to the practice of statistics.

As part of a 3-year longitudinal study, 136 sixth-grade students completed an engineering-based problem on earthquakes involving integrated STEM learning. Students employed engineering design processes and STEM disciplinary knowledge to plan, sketch, then construct a building designed to withstand earthquake damage, taking into account a number of constraints. On testing, students redesigned to build an improved structure. Using a framework of design processes, we report on the students' capabilities in planning, creating annotated sketches, and transforming these into 3D models. An understanding of core STEM concepts was apparent in their responses, including shape properties, stability, rigidity, balance, strength, and engineering techniques. Group problem solving involved moving iteratively between design phases, frequently revisiting problem scoping, being cognizant of the problem goal, boundaries and constraints, and appreciating design sketches as guiding constructions. Students were also observed to consider a number of problem components simultaneously, indicating their capabilities in handling the complexity of the task.

This report is based on an activity completed by 89 Grade 6 students carrying out a statistical investigation with a focus on posing an initial question given a context and then refining it for a chosen data set. Having reached a conclusion based on evidence from their data analysis, students were given additional information on the context and asked to reconsider their conclusions and degree of certainty associated with them. Data from student workbooks were used to assess students' capacity to engage in the extensions of a statistical investigation not usually encountered at the school level. Results showed approximately 70% of students were able to carry out the complete investigation and justify a meaningful decision.

Nationally and internationally there have been calls for a focus on STEM (science, technology, engineering and mathematics) teaching and learning in schools to prepare students for the many future careers in the STEM fields. One way to do this is through engineering activities that provide opportunities for integrating STEM to solve problems using engineering design principles. Based on the Australian curriculum developed by the Australian Curriculum and Assessment Reporting Association (ACARA), the researchers designed an optical engineering activity for Year Five students (9-10 year olds) that was a culminating STEM activity in a unit of work focussing on light. The task required the application of mathematics, science and technology concepts to build an optical instrument that would enable students to see a hidden object or person. The study found that using an engineering design model that involved building, testing and re-designing, afforded students opportunities to demonstrate the application of core mathematics, science and technology concepts. This article describes firstly, the foundational work on light and optical engineering that preceded the engineering problem, and secondly, the engineering problem.

How do cross-bracing, geometry, and base isolation help buildings withstand earthquakes? These important structural design features involve fundamental geometry that elementary school students can readily model and understand. The problem activity, Designing an Earthquake-Resistant Building, was undertaken by several classes of sixth- grade students (10-11-year-olds) toward the end of a three-year STEM education program that began when they were in grade 4. Students learned about engineers, how they improve society, and how design processes help solve engineering-based problems.

As an extension to an activity introducing Year 5 students to the practice of statistics, the software "TinkerPlots" made it possible to collect repeated random samples from a finite population to informally explore students' capacity to begin reasoning with a distribution of sample statistics. This article provides background for the sampling process and reports on the success of students in making predictions for the population from the collection of simulated samples and in explaining their strategies. The activity provided an application of the numeracy skill of using percentages, the numerical summary of the data, rather than graphing data in the analysis of samples to make decisions on a statistical question. About 70% of students made what were considered at least moderately good predictions of the population percentages for five yes-no questions, and the correlation between predictions and explanations was 0.78.

Internationally there is a need for research that focuses on STEM (Science, Technology, Engineering and Mathematics) education to equip students with the skills needed for a rapidly changing future. One way to do this is through designing engineering activities that reflect real-world problems and contextualise students' learning of STEM concepts. As such, this study examined the learning that occurred when fifth-grade students completed an optical engineering activity using an iterative engineering design model. Through a qualitative methodology using a case study design, we analysed multiple data sources including students' design sketches from eight focus groups. Three key findings emerged: first, the collaborative process of the first design sketch enabled students to apply core STEM concepts to model construction; second, during the construction stage students used experimentation for the positioning of lenses, mirrors and tubes resulting in a simpler "working" model; and third, the redesign process enabled students to apply structural changes to their design. The engineering design model was useful for structuring stages of design, construction and redesign; however, we suggest a more flexible approach for advanced applications of STEM concepts in the future.

Statistical literacy is a vital component of numeracy. Students need to learn to critically evaluate and interpret statistical information if they are to become informed citizens. This article examines a Year 5 unit of work that uses the data collection and analysis cycle within a sustainability context.

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.

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 then 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.

Background: Internationally, there is a growing concern for developing STEM education to prepare students for a scientifically and technologically advanced society. Despite educational bodies lobbying for an increased focus on STEM, there is limited research on how engineering might be incorporated especially in the elementary school curriculum. A framework of five comprehensive core engineering design processes (problem scoping, idea generation, design and construction, design evaluation, redesign), adapted from the literature on design thinking in young children, served as a basis for the study. We report on a qualitative study of fourth-grade students' developments in working an aerospace problem, which took place during the first year of a 3-year longitudinal study. Students applied design processes together with their mathematics and science knowledge to the design and redesign of a 3-D model plane. Results: The study shows that through an aerospace engineering problem, students could complete initial designs and redesigns of a model plane at varying levels of sophistication. Three levels of increasing sophistication in students' sketches were identified in their designs and redesigns. The second level was the most prevalent involving drawings or templates of planes together with an indication of how to fold the materials as well as measurements linked to the plane's construction. The third level incorporated written instructions and calculations. Students' engagement with each of the framework's design processes revealed problem scoping components in their initial designs and redesigns. Furthermore, students' recommendations for improving their launching techniques revealed an ability to apply their mathematics knowledge in conjunction with their science learning on the forces of flight. Students' addition of context was evident together with an awareness of constraints and a consideration of what was feasible in their design creation. Interestingly, students' application of disciplinary knowledge occurred more frequently in the last two phases of the engineering framework (i.e., design evaluation and redesign), highlighting the need for students to reach these final phases to enable the science and mathematics ideas to emerge. Conclusions: The study supports research indicating young learners' potential for early engineering. Students can engage in design and redesign processes, applying their STEM disciplinary knowledge in doing so. An appropriate balance is needed between teacher input of new concepts and students' application of this learning in ways they choose. For example, scaffolding by the teacher about how to improve designs for increased detail could be included in subsequent experiences. Such input could enhance students' application of STEM disciplinary knowledge in the redesign process. We offer our framework of design processes for younger learners as one way to approach early engineering education with respect to both the creation of rich problem experiences and the analysis of their learning.

Incorporating engineering concepts into middle school curriculum is seen as an effective way to improve students' problem-solving skills. A selection of findings is reported from a science, technology, engineering and mathematics (STEM)-based unit in which students in the second year (grade 8) of a three-year longitudinal study explored engineering concepts and principles pertaining to the functioning of simple machines. The culminating activity, the focus of this paper, required the students to design, construct, test, and evaluate a trebuchet catapult. We consider findings from one of the schools, a co-educational school, where we traced the design process developments of four student groups from two classes. The students' descriptions and explanations of the simple machines used in their catapult design are examined, together with how they rated various aspects of their engineering designs. Included in the findings are students' understanding of how their simple machines were simulated by the resources supplied and how the machines interacted in forming a complex machine. An ability to link physical materials with abstract concepts and an awareness of design constraints on their constructions were apparent, although a desire to create a ''perfect'' catapult despite limitations in the physical materials rather than a prototype for testing concepts was evident. Feedback from teacher interviews added further insights into the students' developments as well as the teachers' professional learning. An evolving framework for introducing engineering education in the pre-secondary years is proposed.

Jane Watson and Lyn English use a chance activity exploring expectation and variation with coin tossing to highlight the importance of understanding the part-whole relationship embodied in percentage and its power to measure and compare for different wholes, in this case different sample sizes. The purpose of this article is to raise awareness of the opportunities to distinguish between the use of raw numbers and percentages when comparisons are being made in contexts other than the media. It begins with the authors' experiences in the classroom, which motivated a search in the literature, followed by a suggestion for a follow-up activity.

Implementing educational reform requires partnerships, and university-school collaborations in the form of investigative and experimental projects can aim to determine the practicalities of reform. However, there are funded projects that do not achieve intended outcomes. In the context of a new reform initiative in education, namely, science, technology, engineering and mathematics (STEM) education, this article explores the management of a government-funded project. In a university-school partnership for STEM education, how can leadership be distributed for achieving project outcomes? Participants included university personnel from different STEM areas, school teachers and school executives. Data collected included observations, interviews, resource materials, and video and photographic images. Findings indicated that leadership roles were distributed and self-activated by project partners according to their areas of expertise and proximal activeness to the project phases, that is: (1) establishing partnerships; (2) planning and collaboration; (3) project implementation; and (4) project evaluation and further initiatives. Leadership can be intentional and unintentional within project phases, and understanding how leadership can be distributed and self-activated more purposefully may aid in generating more expedient project outcomes. (Contains 1 table and 1 figure.)

Children live in a highly sophisticated world composed of interlocking complex systems. An appreciation and understanding of such systems is critical for making effective decisions about everyone's lives as individuals and as community members. This article addresses one approach to introducing children of all achievement levels to introductory complex systems, namely, through mathematical modeling. It highlights five main attributes of effective modeling activities. These include the exploration of a meaningful complex system, the elicitation of models comprising important mathematical ideas, the explicit documentation of children's thinking, the provision of criteria for self-assessment, and the construction of shareable and reusable models. With their multifaceted and multidisciplinary nature, these activities also serve as valuable learning experiences that link several disciplines in the school curriculum. (Contains 7 tables and 3 figures.)

Assesses children's structural understanding of combinatorial problems when presented in a variety of task situations. Provides an explanatory model of students' combinatorial understandings that informs teaching and assessment. Addresses several components of children's structural understanding of elementary combinatorial problems. (Contains 50 references.) (ASK)

Fifty children, ranging in age from 4 to 10, were individually administered a series of tasks involving different combinations of 2 items selected from a discrete set of items. Analyses of their performances revealed a series of six, increasingly sophisticated, solution strategies ranging from random number selection of items to a systematic patterning of choices. (23 references) (Author/JJK)