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Engineering-based modeling activities provide a rich source of meaningful situations that capitalize on and extend students' routine learning. By integrating such activities within existing curricula, students better appreciate how their school learning in mathematics and science applies to problems in the outside world. Furthermore, modeling problems are open to multiple approaches and solutions; thus, they encourage the creative, critical, and flexible thinking that is often lacking in more traditional textbook examples.

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.

Background: This study was based on the premise that variation is the foundation of statistics and statistical investigations. The study followed the development of fourth-grade students' understanding of variation through participation in a sequence of two lessons based on measurement. In the first lesson all students measured the arm span of one student, revealing pathways students follow in developing understanding of variation and linear measurement (related to research question 1). In the second lesson each student's arm span was measured once, introducing a different aspect of variation for students to observe and contrast. From this second lesson, students' development of the ability to compare their representations for the two scenarios and explain differences in terms of variation was explored (research question 2). Students' documentation, in both workbook and software formats, enabled us to monitor their engagement and identify their increasing appreciation of the need to observe, represent, and contrast the variation in the data. Following the lessons, a written student assessment was used for judging retention of understanding of variation developed through the lessons and the degree of transfer of understanding to a different scenario (research question 3). Results: The results were based either on the application of the hierarchical SOLO model or on non-hierarchical clustering of responses to individual questions in the student workbooks. Students' progress throughout the lessons displayed a wide range of explanations for the estimate of a single student's arm span, general surprise at the variation in measurements, and a large variety of hand-drawn representations based on the values or frequencies of measurements. Many different representations were also created in the software for the single student measurements and for the comparison of measurements for the two scenarios. Although the students' interpretations of their plots were generally more basic than sophisticated, the results of the assessment indicated that many students had developed the ability to transfer their appreciation of variation to another context and could clearly explain the meaning of variation. Conclusions: The findings highlight the importance of an early focus on variation and distribution, with meaningful activities that motivate students to conduct and observe measurements, together with creating both hand-drawn and software representations to relate their experiences.

The authors developed The Paper Plane Challenge as one of a three-part response to The Aerospace Engineering Challenge. The Aerospace Engineering Challenge was the second of three multi-part activities that they had developed with the teachers during the year. Their aim was to introduce students to the exciting world of engineering, where they could apply their learning of mathematics (measurement and geometry), science (force), and technology (design) to solve real-world problems. Students' appreciation and application of design processes underpinned the problem-solving activities throughout their STEM program. This article describes how the authors introduced design processes, worked with the students to develop the problem (explore; design, build, and test a paper airplane; redesign, rebuild, and retest), and challenges they encountered along the way.

An exciting event had occurred for the grade 3 classes at Woodlands State School. A new play space designated for the older grades had now been opened to the third graders. In sharing their excitement over this "real treat, real privilege," the teachers invited the children to find out more about playgrounds and, in particular, their new play area. Playgrounds thus formed the context for a set of third-grade activities designed to extend the children's understanding and appreciation of statistics. In the first activity, spanning six 40-minute lessons and described briefly in this article, the students created a survey to learn more about their play area. In the second activity (four 40-minute lessons described briefly in this article), they gathered further data as a basis for improving the context of their playground. The increasing importance of dealing with data from the earliest grades has been emphasized consistently by the National Council of Teachers of Mathematics (NCTM 2000) and more recently by past NCTM president Michael Shaughnessy (2010). Three core components in developing children's understanding and appreciation of data--establish a context, pose and answer statistical questions, represent and interpret data--lay the foundation for the fourth component: use data to enhance existing context. These components are explained while describing the lessons.

The National Council of Teachers of Mathematics (NCTM) continues to emphasize the importance of early statistical learning; data analysis and probability was the Council's professional development "Focus of the Year" for 2007-2008. Such a focus is needed, especially given the results of the statistics items from the 2003 NAEP. As Shaughnessy (2007) noted, students' performance was weak on more complex items involving interpretation or application of items of information in graphs and tables. Furthermore, little or no gains were made between the 2000 NAEP and the 2003 NAEP studies. One approach the author has taken to promote young children's statistical reasoning is through data modeling. Having implemented in grades 3-9 a number of model-eliciting activities involving working with data (e.g., English 2010), she observed how competently children could create their own mathematical ideas and representations--before being instructed how to do so. She thus wished to introduce data-modeling activities to younger children, confident that they would likewise generate their own mathematics. The author recently implemented data-modeling activities in a cohort of three first-grade classrooms of six-year-olds. In this article, she reports on some of the children's responses and discusses the components of data modeling the children engaged in. (Contains 5 figures and 2 tables.)

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.)

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.

This paper reports on some findings from the first year of a three-year longitudinal study, in which seventh- to ninth-graders were introduced to engineering education. Specifically, the paper addresses students' responses to an initial design activity involving bridge construction, which was implemented at the end of seventh grade. This paper also addresses how students created their bridge designs and applied these in their bridge constructions; their reflections on their designs; their reflections on why the bridge failed to support increased weights during the testing process; and their suggestions on ways in which they would improve their bridge designs. The present findings include identification of six, increasingly sophisticated levels of illustrated bridge designs, with designs improving between the classroom and homework activities of two focus groups of students. Students' responses to the classroom activity revealed a number of iterative design processes, where the problem goals, including constraints, served as monitoring factors for students' generation of ideas, design thinking and construction of an effective bridge. (Contains 2 figures and 1 table.)

A recent study in the United Kingdom provides strong evidence that well-organized activities outside the classroom contribute significantly to the quality and depth of children's learning, including their personal, social, and emotional development. Outdoor math trails supply further evidence of such enhanced learning. This article focuses on the creation, implementation, and the benefits of math trails to students. The math trail concept was first developed by Australian educator Dudley Blaine to promote active, meaningful mathematics learning beyond the classroom. Math trails are meaningful, stimulating, challenging, and exciting for children. Most important, these trails invite "all" students, irrespective of their classroom achievement level, to participate successfully in the problem activities and gain a sense of pride in the mathematics they create. As youngsters discover real-world shapes, patterns, numbers, data, symmetry, and reflections--to name just a few examples--their eyes open to the mathematics in their world. They become math detectives--posing questions and solving problems as well as documenting and communicating their discoveries in multiple ways. The authors present an excerpt from a student-generated math trail. (Contains 2 figures.) [Additional trail examples accompany the online version of this article in an appendix at www.nctm.org/tcm.]

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.)

This paper examines 6th grade children's local conceptual development and mathematization processes as they worked a comprehensive mathematical modeling problem (creating a consumer guide for deciding the best snack chip) over several class periods. The children and their teachers were participating in a 3-year longitudinal teaching experiment in which sequences of mathematical modeling problems were implemented from the 5th grade (10 years of age) though to the 7th grade. In contrast to traditional problem solving, mathematical modeling requires children to generate and develop their own mathematical ideas and processes, and to form systems of relationships that are generalizable and reusable. Reported here is a detailed analysis of the iterative cycles of development of one group of children as they worked the problem, followed by a summary of the mathematization processes displayed by all groups. Children's critical reflections on their models are also reported. The results show how children can independently develop constructs and processes through meaningful problem solving. Children's development included creating systems for operationally defining constructs; selecting, categorizing, and ranking factors; quantifying quantitative and qualitative data; and transforming quantities.

We report on how two middle-grade teachers supported their students' mathematical reasoning within the context of a novel modeling task in data analysis. We examine how the task features supported the development of teachers' knowledge as their students engaged with the task. Analyses of the teachers' practices suggest that the task features enabled teachers: (a) to develop new understandings of the mathematical content and the ways in which student ideas develop and are represented; (b) to adopt new roles in their interactions with the students, including a focus on listening and observing, and on asking questions for understanding and clarification; and (c) to engage in forms of interpretative listening that shifted the role of evaluation from the teacher to the student.

Mathematical modeling is explored as both problem posing and problem solving from two perspectives, that of the child and the teacher. Mathematical modeling provides rich learning experiences for elementary school children and their teachers.

Implications and suggestions for using combinatorial mathematics in the classroom through a survey and synthesis of numerous research studies are presented. The implications revolve around five major themes that emerge from analysis of these studies.

Discusses the nature of tasks used to elicit the development of such systems by middle school students. Analyzes mathematical reasoning development of students across tasks and the diversity of thinking patterns identified on problem tasks. Discusses student reasoning about the relationships between and among quantities and their application in related situations. (Author/KHR)

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)

Investigates 10-year-old children's abilities to reason by analogy in solving addition and subtraction comparison problems involving unknown compare sets and unknown reference sets. Children responded in a consistent manner to the tasks involving the basic addition problems, indicating substantial relational knowledge of these but responded in an erratic manner to the subtraction cases. Contains 49 references. (Author/ASK)

Examines elementary students' perspectives on the engagement potential of particular mathematical problems and students' views on classroom problem-solving activities in general. Nonroutine examples that focused on important reasoning processes and did not involve computation had the greatest engagement potential, whereas computational problems had the least appeal. (Author/CCM)