Landscapes of Investigation

According to many observations, traditional mathematics education falls within the exercise paradigm. This paradigm is contrasted with landscapes of investigation serving as invitations for students to be involved in processes of exploration and explanation. The distinction between the exercise paradigm and landscapes of investigation is combined with a distinction between three different types of reference which might provide mathematical concepts and classroom activities with meaning: references to mathematics; references to a semi-reality, and references to a real-life situation. The six possible learning milieus are illustrated by examples. Moving away from the exercise paradigm and in the direction of landscapes of investigation may help to abandon the authorities of the traditional mathematics classroom and to make students the acting subjects in their learning processes. Moving away from reference to pure mathematics and in the direction of real life references may help to provide resources for reflection on mathematics and its applications. My hope is that finding a route among the different milieus of learning may provide new resources for making the students both acting and reflecting and in this way providing mathematics education with a critical dimension.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

On the importance of mathematical play

A number of existing theories and proposals for the meaning and characteristics of ‘play’ are considered before the authors suggest six characteristics of mathematical play, including the idea that it is not confined to childhood. Previous studies provide evidence for relating play to cognitive gain while the place of mathematical play in research activities is illustrated by describing a mathematician's approach to a number investigation from the classroom-The Six Circles. The problem-solving process for the Six Circles and observations of students solving calculator and integration problems are analysed in relation to theories of play and cognitive gain and also considered from the perspective of the students' experience. Piaget's theory for the assimilation and accommodation of new information and Davis's view of play as ‘space to support learning’ are reflected in the authors' rationale for suggesting that open questions and mathematical play provide opportunities for students to develop their own conjectures, with no threat of failure, and provide a foundation for mathematical learning. Some difficulties of implementing a ‘play’ approach in the classroom are discussed and further research questions proposed.     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

What Can Be Learned From Informal Student Writings in a Science Context?

This exploratory study analyzed four informal science-related writing tasks produced by 374 seventh-grade students (172 boys and 202 girls) from two schools with different socioeconomic populations. The study demonstrates that students' informal writing in science contexts can provide a rich source of information regarding students' cognitive and attitudinal engagement with science. Students' writing reflects the level at which students understand previously learned science-related ideas and gives insight into themes and issues they would be interested in learning. This study further demonstrates how students organize and personalize science knowledge acquired inside as well as outside of school when given novel and unconventional (informal) science-related tasks. The study also demonstrates that informal writing tasks encourage students to express opinions, values, and attitudes associated with science and science learning. Examples are provided of similarities and differences in students' writing preferences and in the quality of writing produced by boys and girls. Suggestions for further studies for teachers and researchers are discussed.     

DISTRIBUTED COGNITION,COMPUTERS AND THE INTERPRETATION OF GRAPHS

This paper reoprts on one aspect of recent research on students' understanding of graphs in computer and in pencil and paper media. Over 200 Year 10 [age 14–15] students in two comprehensive schools in Leeds participated in a learning experiment, 100+ with and 100+ without the use of computers. Pre, post and delayed post-tests and interviews provide data for a comparison of learning with the two media.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Supporting disciplinary and interdisciplinary knowledge development and design thinking in an informal,pre‐engineering program: A Workplace Simulation Project

In this paper, we examined students’ engagement in an implementation of a Workplace Simulation Project (WSP). The WSP was designed to actively engage students in learning disciplinary content by inviting engineers from industry to have a physical presence within the school building to collaborate with teachers and students to complete projects which simulate the tasks authentic to their work. We focus on the first year implementation of the program that partnered a high school in the rural Midwest with an engineering unit of a government organization. Using a multiple methods study design, we analyzed disciplinary and interdisciplinary pre and posts test along with students’ interviews to determine learning gains as well as students’ interpretations of creative and critical thinking as experienced in the project and their knowledge of the engineering design process. Effect sizes showed that students in the WSP group had notable gains over the control group participants. Additionally, students’ knowledge of core elements of the design process were identified in inductive analyses of the interviews. Findings from this study will provide usable knowledge about effective ways to support systems and design thinking and ways to support expert‐novice collaboration to ensure success.     

A Model for Extending Hands-On Science to Be Inquiry Based

Many popular hands-on science activities, as traditionally implemented, fail to support inquiry-based science instruction, because the activities direct teachers to terminate lessons prematurely. This paper presents a model describing one approach for extending seemingly limited hands-on activities into full-inquiry science lessons. The strategy involves (a) discrepant events to engage students in direct inquiry; (b) teacher-supported brainstorming activities to facilitate students in planning investigations; (c) effective written job performance aids to provide structure and support; (d) requirements that students provide a product of their research, which usually includes a class presentation and a graph; and (e) class discussion and writing activities to facilitate students in reflecting on their activities and learning. The paper presents the model as a tool for facilitating science teachers' efforts to understand and implement the type of powerful, effective, and manageable inquiry-based science instruction called for in the National Science Education Standards.     

Measuring striving for understanding and learning value of geometry: a validity study

The current study aimed to construct a questionnaire that measures students’ personality traits related to striving for understanding and learning value of geometry and then examine its psychometric properties. Through the use of multiple methods on two independent samples of 402 and 521 middle school students, two studies were performed to address this issue to provide support for its validity. In Study 1, exploratory factor analysis indicated the two-factor model. In Study 2, confirmatory factor analysis indicated the better fit of two-factor model compared to one or three-factor model. Convergent and discriminant validity evidence provided insight into the distinctiveness of the two factors. Subgroup validity evidence revealed gender differences for striving for understanding geometry trait favouring girls and grade level differences for learning value of geometry trait favouring the sixth- and seventh-grade students. Predictive validity evidence demonstrated that the striving for understanding geometry trait but not learning value of geometry trait was significantly correlated with prior mathematics achievement. In both studies, each factor and the entire questionnaire showed satisfactory reliability. In conclusion, the questionnaire was psychometrically sound.     

Inclusive Science Education: Classroom Teacher and Science Educator Experiences in CLASS Workshops

Inclusion is the meaningful participation of students with disabilities in general education classrooms. The CLASS project (Creating Laboratory Access for Science Students) is a unique initiative offering training and resources to help educators provide students with a variety of physical, sensory and learning disabilities equal access in the science laboratory or field. To determine whether participants believed a 2‐week residential workshop sponsored by CLASS raised disability awareness and provided teacher training in inclusive science teaching practice, a multipoint Likert scale survey and questionnaire was completed by all participants (N= 20) in four workshops. Participants reported large gains in their preparedness to teach science to students with disabilities. Participants also reported gains in their familiarity with instructional strategies, curricula, and resources and their ability to design, select, and modify activities for students with disabilities. Finally, shifts in attitudes about teaching science to students with disabilities were noted.     

Nontraditional College Students and the Role of Collaborative Learning as a Tool for Science Mastery

This study investigates the way collaborative learning that occurs primarily outside the classroom affects college students' understanding of science. Collaborative learning is particularly important for the increasing number of nontraditional students who have limited time available for study groups and other peer learning activities occurring outside of class time. Using a national study of 4,644 college students of various academic majors, multiple linear regression was used to identify variables that enhance science learning. Time spent in peer learning settings, such as teaching science to peers and discussing science with peers, were the strongest predictors of understanding science; moreover, this finding was consistent even for nontraditional students who reported less frequency of engagement in such activities. The study suggests that science educators can enhance learning when they structure their courses to include peer learning that engages students with each other over science issues outside the classroom.     

Preparing K‐8 Preservice Teachers in a Content Course for Standards‐Based Mathematics Pedagogy

Many K–8 preservice teachers have not experienced learning mathematics in a standards‐based classroom. This article describes a mathematics content course designed to provide preservice teachers experiences in learning mathematics that will help build a solid foundation for a standards‐based methods course. The content course focuses on developing preservice teachers' mathematical knowledge, as well as helping them realize what it means to learn mathematics that is taught using the pedagogy in the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Furthermore, findings are presented from a study on this course that describe students' pre‐ and postcourse beliefs, attitudes, and perceptions of what it means to learn and teach mathematics. These findings provide evidence that the students in the study are beginning to understand what is meant by a standards‐based classroom. Data were collected from surveys and interviews. Quotes from the students who aspire to be elementary teachers are used throughout the article to support the points.     

How students use the ‘see similar example’ feature in online mathematics homework

Homework is one of students’ opportunities to learn mathematics, but we know little about what students learn from homework. This study employs the instructional triangle and didactic contract to explore how students used the ‘see similar example’ feature in an online homework platform and how that use reflected their learning goals. Findings indicate students used similar examples to troubleshoot, to check if they were on the right track, and to see the form of the answer. Students also sought to unpack the reasoning in solution steps, used solutions as templates for solving their own problems, and sometimes copied answers. One student did a ‘see similar example’ problem for more practice. Students’ goals included completing the homework, maximizing their score, and understanding the content. This research lays groundwork for future work characterizing what students learn from homework and how features that provide students with similar examples help or hinder their learning.     

A support learning programme for first-year mathematics*

ABSTRACT

This article is a follow-up to an earlier paper on the mathematics support learning tutorial programme (SLT), an intervention programme at The University of Queensland that targets students considered to be at risk of failing Calculus and Linear Algebra I, the first tertiary level mathematics subject at The University of Queensland. The first paper (Hillock, P., Jennings, M., Roberts, A., & Scharaschkin, V. (2013). Amathematics support programme for first-year engineering students. International Journal of Mathematical Education in Science and Technology, 44(7), 1030–1044) reported on the inaugural programme implemented in 2012. This article provides an update of the progress of the SLT since 2012. We provide statistics for the subsequent 12 semesters to Semester 2, 2018 and describe the evolution of the SLT since its implementation. Statistical analysis of the additional data and student feedback indicate that the SLT continues to have a positive impact on student learning, with weak students making significant gains from attending the programme.     

Action research to improve methods of delivery and feedback in an Access Grid Room environment

This article describes a qualitative study which was undertaken to improve the delivery methods and feedback opportunity in honours mathematics lectures which are delivered through Access Grid Rooms. Access Grid Rooms are facilities that provide two-way video and audio interactivity across multiple sites, with the inclusion of smart boards. The principal aim was to improve the student learning experience, given the new environment. The specific aspects of the course delivery that the study focused on included presentation of materials and provision of opportunities for interaction between the students and between students and lecturers. The practical considerations in the delivery of distance learning are well documented in the literature, and similar problems arise in the Access Grid Room environment; in particular, those of limited access to face-to-face interaction and the reduction in peer support. The nature of the Access Grid Room classes implies that students studying the same course can be physically situated in different cities, and possibly in different countries. When studying, it is important that students have opportunity to discuss new concepts with others; particularly their peers and their lecturer. The Access Grid Room environment also presents new challenges for the lecturer, who must learn new skills in the delivery of materials. The unique nature of Access Grid Room technology offers unprecedented opportunity for effective course delivery and positive outcomes for students, and was developed in response to a need to be able to interact with complex data, other students and the instructor, in real-time, at a distance and from multiple sites. This is a relatively new technology and as yet there has been little or no studies specifically addressing the use and misuse of the technology. The study found that the correct placement of cameras and the use of printed material and smart boards were all crucial to the student experience. In addition, the inclusion of special tutorial type sessions were necessary to provide opportunities to students for one-on-one discussion with both lecturer and other students. This study contributes to the broader understanding of distance education in general and future Access Grid Room course delivery in particular.     

Designing examples to create variation patterns in teaching

This paper uses a post-qualitative philosophical perspective to find new ways of understanding teaching and learning. The paper presents a series of examples that were used in a longitudinal study, with the aim of creating variation patterns that would make it possible for students to discern the use of the four basic arithmetic operations in different situations. The focus of this article is the potential of the examples to systematically create variation patterns that students need to perceive in order to make generalizations. The result demonstrates that well-thought-out examples help identify the correct arithmetic operation in different situations, and provide a basis from which students can discern the connection between text and the use of operation in mathematical example. The result also demonstrates that students develop rhizomatic thinking through the creation of new links between aspects of the object of learning, association and linking of different aspects to each other and the creation of a whole with unique and specific characteristics that cannot be explained by simply adding the characteristics of the individual parts.     

Academic achievement with cooperative learning using homogeneous and heterogeneous groups

Cooperative learning is a proven teaching strategy that teachers have been using for over 40 years. Teachers often group students heterogeneously so that students who are lower achieving are learning with and from higher achieving students. The purpose of this study was to test homogeneous and heterogeneous grouping while using cooperative learning teaching structure. This study aimed to answer the question of whether students be grouped homogeneously or heterogeneously while participating in cooperative learning. The primary research design for this study was a quasi-experimental nonequivalent control-group design. A convenience sample of fifth-grade students was drawn from a Georgia elementary school in the 2017–2018 school year. The data were analyzed using analysis of covariance (ANCOVA) and paired samples t tests. The ANCOVA was run to compare the scores from the FOSS Survey/Posttest, and the analysis showed no significant difference between the homogeneous and heterogeneous group posttest scores. Both groups made significant gains (as shown by the results of the paired samples t-tests).     

Active learning: a case study of student engagement in college Calculus

This paper presents a case study for strategic engagement of students in a Calculus course in order to produce increased learning in the classroom. Since it has been shown that active learning can promote greater comprehension for students in science, technology, engineering, and mathematics (STEM) courses, the researcher utilized many types of active learning techniques to enhance classroom instruction. The key components implemented are presented as a model of enhanced learning through developed classroom engagement. This course redesign model entitled, Strategic Engagement for Increased Learning (SEIL), has the potential to (1) contribute to the body of knowledge on ways to improve mathematics skills for college students, (2) identify successful teaching strategies and technologies that will promote the retention of STEM students, (3) increase the success rate of students taking Calculus, and (4) help produce more STEM graduates needed for the STEM workforce in the United States of America.