Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

Teachers' and education students' perceptions of and reactions to children with and without the diagnostic label 'ADHD'

Thirty-four elementary school teachers and 32 education students from Canada rated their reactions towards vignettes describing children who met attention-deficit/hyperactivity disorder (ADHD) symptom criteria that included or did not include the label “ADHD.” “ADHD”-labeled vignettes elicited greater perceptions of the child's impairment as well as more negative emotions and less confidence in the participants, although it also increased participants' willingness to implement treatment interventions. Ratings were similar to vignettes of boys versus girls; however, important differences in ratings between teachers and education students emerged and are discussed. Finally, we investigated the degree to which teachers' professional backgrounds influenced bias based on the label “ADHD.” Training specific to ADHD consistently predicted label bias, whereas teachers' experience working with children with ADHD did not.

Data modeling in elementary and middle school classes : a shared experience

This paper argues for a renewed focus on statistical reasoning in the elementary school years, with opportunities for children to engage in data modeling. Data modeling involves investigations of meaningful phenomena, deciding what is worthy of attention, and then progressing to organizing, structuring, visualizing, and representing data. Reported here are some findings from a two-part activity (Baxter Brown’s Picnic and Planning a Picnic) implemented at the end of the second year of a current three-year longitudinal study (grade levels 1-3). Planning a Picnic was also implemented in a grade 7 class to provide an opportunity for the different age groups to share their products. Addressed here are the grade 2 children’s predictions for missing data in Baxter Brown’s Picnic, the questions posed and representations created by both grade levels in Planning a Picnic, and the metarepresentational competence displayed in the grade levels’ sharing of their products for Planning a Picnic.

Digital Storytelling within a Service-Learning Partnership: Technology as Product and Process for University Students and Culturally and Linguistically Diverse High School Youth

This chapter describes a university/high school partnership focused on digital storytelling. It also explains the multi-stage process used to establish this successful partnership and project. The authors discuss the central role that technology played in developing this university/high school partnership, a collaboration that extended the impact of a digital storytelling project to reach high school students, university students, educators, high school administrators, and the local community. Valuing a reflective process that can lead to the creation of a powerful final product, the authors describe the impact of digital storytelling on multiple stakeholders, including the 13 university students and 33 culturally and linguistically diverse high school youth who participated during the fall of 2009. In addition, the chapter includes reflections from university and high school student participants expressed during focus groups conducted throughout the project. While most participants had a positive experience with the project, complications with the technology component often caused frustrations and additional challenges. Goals for sharing this project are to critically evaluate digital storytelling, describe lessons learned, and recommend good practices for others working within a similar context or with parallel goals.

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.

Aspects of elementary number theory illustrated in the spreadsheet environment

Number theory has in recent decades assumed a great practical importance, due primarily to its application to cryptography. This chapter discusses how elementary concepts of number theory may be illuminated and made accessible to upper secondary school students via appropriate spreadsheet models. In such environments, students can observe patterns, gain structural insight, form and test conjectures, and solve problems. The chapter begins by reviewing literature on the use of spreadsheets in general and the use of spreadsheets in number theory in particular. Two sample applications are then discussed. The first, factoring factorials, is presented and instructions are given to construct a model in Excel 2007. The second application, the RSA cryptosystem, is included because of its importance to Science, Technology, Engineering, and Mathematics (STEM) students. Number theoretic concepts relevant to RSA are discussed, and an outline of RSA. is given, with example. The chapter ends with instructions on how to construct a simple spreadsheet illustrating RSA.

STEM learning through engineering design: fourth-grade students' investigations in aerospace

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.

School counsellors' and students' attitudes to online counselling: A qualitative study

An exploratory qualitative study was conducted to examine the perceptions and attitudes of both school counsellors and students to online counselling. Focus groups were conducted with two groups of school counsellors and six groups of secondary students. It was found that counsellors were hesitant to use online counselling because they were not convinced that it was effective and without the necessary online skills, they were concerned they would not be competent to deal with potential litigious and security pitfalls. Students were generally positive about the opportunity to access the school counsellor online. Implications for practice and future research are discussed.

Different countries, same science classes: Students’ experiences of school science in their own words

This paper reviews the remarkably similar experiences of school science reported by high school students in Sweden, England, and Australia. It compares student narratives from interpretive studies by Lindahl, by Osborne and Collins, and by Lyons, identifying core themes relating to critical contemporary issues in science education. These themes revolve around the transmissive pedagogy, decontextualized content, and unnecessary difficulty of school science commonly reported by students in the studies. Their collective experiences are used as a framework for examining student conceptions of, and attitudes to, school science more generally, drawing on an extensive range of international literature. The paper argues that the experiences of students in the three studies provide important insights into the widespread declines in interest and enrolments in high school and university science courses.

Career development and personal functioning differences between work-bound and non-work bound students

We surveyed 506 Australian high school students on career development (exploration, planning, job-knowledge, decision-making, indecision), personal functioning (well-being, self-esteem, life satisfaction, school satisfaction) and control variables (parents’ education, school achievement), and tested differences among work-bound, college-bound and university-bound students. The work-bound students had the poorest career development and personal functioning, the university-bound students the highest, with the college-bound students falling in-between the other two groups. Work-bound students did poorest, even after controlling for parental education and school achievement. The results suggest a relationship between career development and personal functioning in high school students.

Students' and teachers' perceptions of adolescents' problems and coping strategies

In an investigation of the problems and coping strategies of Australian high school students, comparisons were made between the responses of 1664 students enrolled in years 8 to 12 in 1988, 1620 students enrolled in the same year levels in 1993, and 178 high school teachers in 1993. The subjects completed the High School Stressors Scale and the Adolescent Coping Strategies Scale. Data analyses using MANOVAs, ANOVAs, and t- tests revealed close similarities between the responses of the 1993 students and those of the 1988 students, but a considerable amount of incongruence between the responses of the students and those of the teachers. In particular, the teachers generally seemed to regard their students' problems as being more serious than was acknowledged by the students, and the teachers generally seemed to project a less positive view of adolescents' coping strategies than did the students. These discrepancies are discussed in terms of the different orientations that students and teachers bring to the student- teacher relationship. It is suggested that teachers and counsellors need to take cognisance of the differences between adolescents' perspectives and their own if they are going to be effective in assisting students to develop positive coping strategies and in creating more positive learning environments.

School-related stressors in adolescents

The aim of this study was to investigate high school students' perceptions of school-related problems. Some 1583 high school students responded to the 35 item High School Stressors Scale, published by Burnett and Fanshaw in 1997, which measures nine areas of problems experienced by adolescents in schools. These are teaching methods, student-teacher relationships, school workload, school environment, feeling vulnerable, personal organisation, achieving independence, anxiety about the future, and relationships with parents. The results and implications for educators, guidance officers and school psychologists working in high schools are presented.

Learning support : what do students want?

Assumptions are often made about students' needs, especially in the area of learning support. In this study 89 students were asked 8 questions relating to receiving learning support. The results are presented both qualitatively and quantitatively, and indicate that all students have individual needs that cannot be assumed. The findings reveal that the most common area of perceived need was in literacy. There were some differences between primary and middle school students' responses to withdrawal from the classroom, but the majority of students in both groups indicated a preference for withdrawal because they could concentrate better in an environment that was less noisy and because they felt they might look 'stupid' if they remained in class.