Incorporating Mathematics into the Science Program of Students Labeled “At‐Risk”

Twenty three at‐risk high school female students who had failed mathematics and science in a traditional school setting were the subjects of this study that integrated mathematics and science lessons over a period of four weeks. Using a combination of direct instruction, calculations, graphing, hands‐on projects, and discussion, the topic of mechanical advantage was studied, to find out how well students understood the topic. The study found that these students who initially knew very little about mechanical advantage, and who did not see any need to use mathematics in the study of science, indicated an increased understanding of mechanical advantage, and also seemed to realize that integrating mathematics and science enhanced learning.     

“Then you can take half … almost” — Elementary students learning bar graphs and pie charts in a computer-based context

Forty Swedish elementary students, 7-12 years of age and working in pairs, constructed a series of bar graphs and pie charts using a graphing application software as an instructional tool under the guidance of the researcher. After successive withdrawal of help, each pair drew a small number of graphic displays manually at the end of the data collection period. Evidence is provided that children's engagement with the graphing application software enhanced their understanding of essential graphical ideas and that even the youngest students appropriated and talked insightfully about a number of critical aspects of graphing. The students’ gradual mastering of different aspects of graphing is argued to be movements within their “zones of proximal development” towards a more competent use of graphs.     

Three women's understandings of algebra in a precalculus course integrated with the graphing calculator

This study investigated the role of function in a precalculus classroom which incorporated the graphing calculator in the instructional process. Perspectives were taken from students, teachers, and textbooks. Emphasis was placed on choice of functional symbol system when thinking and problem solving, connections across symbol systems, the role of the instructor and the textbook in learning, affective components, and the effect of the graphing calculator.The study starts with a defination of the concept of structure as it relates to function. The account of a semester-long qualitative study on students' concept images of function and its role in problem solving follows. It was found that the students involved in the study entered the graph-intensive course with predominantly symbolic notions of algebra, in part due to prior instruction. The students also possessed highly procedural views of algebraic content. These preconceptions and expectations resulted in the students' inability to effectively coordinate graphic and symbolic notions of algebra, both in procedural and conceptual realms. Implications and curricular suggestions are provided.     

Building bridges within mathematics education: Teaching, research, and instructional design

Within mathematics education, classroom teachers, educational researchers, and instructional designers share the common goals of understanding and improving the teaching and learning of mathematics. Teachers work to help students learn; researchers study how people learn and teach mathematics; and designers develop instructional materials to support teachers and students. Each community (of teachers, of researchers, and of designers) develops its own perspectives, methods, and expertise. Too seldom, however, do practitioners have the opportunity to share their knowledge across communities. This first-person, retrospective case study speaks to the challenges and rewards of building bridges among these three communities by charting the evolution of an instructional activity (using graphing software to explore slope) through four cycles of teaching, research, and design. Initially separate, the three perspectives of teacher, researcher, and designer begin to interact as the worksite moves from the university laboratory to the author's classroom and then to other teachers’ classrooms. Many of these interactions are fruitful, resulting in new insights and strategies that strengthen the final product and inform the practitioner. At the same time, some tensions arise, particularly between teaching and research, highlighting fundamental differences between these fields. Lessons from this case study suggest implications for collaborations among teachers, researchers, and designers.     

Transformation!—A graphing calculator activity to practice transformations of functions

An understanding of function transformation is essential for mastering mathematics in high school and beyond. The classroom activity presented here and the game,Transformation!, are designed so that students can develop fluency in working with transformations of functions. Both take advantage of the technology of the graphing calculator and the method of cooperative learning.     

Locating starting points in differential equations: a realistic mathematics education approach

The paper reports on ongoing developmental research efforts to adapt the instructional design perspective of Realistic Mathematics Education (RME) to the learning and teaching of collegiate mathematics, using differential equations as a specific case. This report focuses on the RME design heuristic of guided reinvention as a means to locate a starting point for an instructional sequence for first-order differential equations and highlights the cyclical process instructional design and analysis of student learning. The instance of starting with a rate of change equation as an experientially real mathematical context is taken as a case for illustrating how university students might experience the creation of mathematical ideas. In particular, it is shown how three students came to reason conceptually about rate and in the process, develop their own informal Euler method for approximating solution functions to differential equations.     

Running to keep up with the lecturer or gradual de-ritualization? Biology students’ engagement with construction and data interpretation graphing routines in mathematical modelling tasks

Through a commognitive lens, we examine twelve first-semester biology students’. engagement with graphing routines as they work in groups, during four sessions of Mathematical Modelling (MM). We trace the students’ meta-level learning, particularly as they fluctuate between deploying graphs for mere illustration of data and as sense-making tools. We account for student activity in relation to precedent events in their experiences of graphing and as fluid, if not always productive, interplay between ritualised and exploratory engagement with graph construction and interpretation routines. The students’ construal of the task situations is marked by efforts to keep up with lecturer expectations which allow for changing degrees of student agency but do not factor in the influence of precedent events. Our analysis has pedagogical implications for the way MM problems are formulated and also foregrounds the capacity of the commognitive framework to trace de-ritualization and meta-level learning in students’ MM activity.     

Graphing in Groups: Learning About Lines in a Collaborative Classroom Network Environment

This article presents a design experiment in which we explore new structures for classroom collaboration supported by a classroom network of handheld graphing calculators. We describe a design for small group investigations of linear functions and present findings from its implementation in three high school algebra classrooms. Our coding of the problem-solving efforts of six student focus pairs in this environment over the course of several class sessions indicates that these students tended to move from exploratory and visual to more analytic means of establishing lines of a specified slope. As they adopted these analytic approaches, they were also more likely to enact their strategies jointly. In closer examination of emerging analytic strategies in episodes selected from the work of one of the pairs, we argue that the processes by which these students discovered the need for coordinated action on their respective points, and came to establish mathematical meaning for the relations between their coordinate locations as slope, were overlapping and intertwined.     

Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops

In a research project with one-day teacher education workshops for secondary-school mathematics teachers, our study explores the potential of tool-supported discussions in helping them to notice important and critical aspects of mathematics teaching talk. Mathematical practices of naming and explaining in teaching talk, students’ content learning challenges, and noticing processes of identifying, interpreting and deciding are the components of our framework and the tools that guided the design and implementation of three workshops on linear equations, fractions and plane isometries. The data was collected during the discussions with the seven teachers and the teacher educator throughout these workshops. The coding of the discussions allowed us to see discourse moves that reveal the teachers’ noticing of: (i) challenges in the identification of mathematical naming, (ii) mathematical explaining that voices the students’ learning, (iii) classroom practice in relation to mathematical naming and explaining.     

The Effect of Calculator‐Based Ranger Activities on Students' Graphing Ability

Middle school students can learn to communicate with graphs in the context of appropriate Calculator‐Based Ranger (CBR) activities. Three issues about CBR activities on graphing abilities were addressed in this study: (a) the effect of CBR activities on graphing abilities; (b) the extent to which prior knowledge about graphing skills affects graphing ability; (c) the influence of instructional styles on students' graphing abilities. Following the use of CBR activities, students' graphing abilities were significantly more developed in three components _ interpreting, modeling, and transforming. Prior knowledge of graphing skills on the Cartesian coordinate plane had little effect on students' understanding of graphs. Significant differences, however, were found in students' achievement, depending on instructional styles related to differentiation, which is closely connected to transforming distance‐time graphs to velocity‐time graphs. The result of this study indicates that the CBR activities are pedagogically promising for enhancing graphing ability of physical phenomena.     

Diversifying the Curriculum in a Mathematics Classroom Streamed for High-Ability Learners: A Necessity Unassumed

Gifted learners are considerably more diverse than they are generally believed to be ( Matthews & Keating, 1995 ). Consequently, curricula must provide a wide range of learning opportunities, not simply a lock-step program at an accelerated pace. In the present study, the authors developed an experimental functions curriculum in which everyone learned a core set of concepts, then explored different kinds of functions independently using computer graphing technology. This was tested against a more typical, text-based program. Two groups of high-ability eighth-grade boys participated. Experimental students scored higher on a posttest than control students and showed much more diversity in their answers.     

When time is an implicit variable: An investigation of students’ ways of understanding graphing tasks

In this study, we investigate students’ ways of understanding graphing tasks involving quantitative relationships in which time functions as an implicit variable. Through task-based interviews of students ages 14–16 in a summer mathematics program, we observe a variety of ways of understanding, including thematic or visual association, pointwise thinking, and reasoning parametrically about changes in the two variables to be graphed. We argue that, rather than comprising a hierarchy, these ways of understanding complement one another in helping students discover an invariant relationship between two dynamically varying quantities, and develop a graph of the relationship that captures this invariance. From these ways of understanding, we conjecture several mathematical meanings for graphing that may account for students’ behavior when graphing quantitative relationships.     

Assessing the Use of Graphing Calculators in College Algebra: Reflecting on Dimensions of Teaching and Learning

This article considers various aspects of teaching and learning stemming from the integration of graphing calculator use in a semester-long college algebra course. The project examined four class sections, in which two instructors each taught one section using graphing calculators and one section using a traditional approach. Achievement and attitude data showed no significant differences for treatment or instructor; a significant difference in achievement was found for gender. Students in the calculator sections responded to an open-ended questionnaire about their use of the calculator and were generally supportive of the technology. Students were in agreement about specific topics for which the technology was most useful. Overall, findings indicated that the technology use had a positive impact on various dimensions of student learning.     

Developing shift problems to foster geometrical proof and understanding

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often carefully constructed reinvention route, which do not fit the needs of teachers and students working from conventional curriculum materials. To help to narrow this gap, we developed an intervention—‘shift problem lessons’. The aim of this article is to discuss the design of shift problems and to analyze learning processes occurring when students are working on the tasks. Specifically, we discuss three paradigmatic episodes based on data from a teaching experiment in geometrical proof. The episodes show that is possible to create a micro-learning ecology where regular students are seriously involved in mathematical discussions, ground their mathematical understanding and strengthen their relational framework.     

Constructing Graphical Representations: Middle Schoolers' Intuitions and Developing Knowledge About Slope and Y‐intercept

Middle‐school students are expected to understand key components of graphs, such as slope and y‐intercept. However, constructing graphs is a skill that has received relatively little research attention. This study examined students' construction of graphs of linear functions, focusing specifically on the relative difficulties of graphing slope and y‐intercept. Sixth‐graders' responses prior to formal instruction in graphing reveal their intuitions about slope and y‐intercept, and seventh‐ and eighth‐graders' performance indicates how instruction shapes understanding. Students' performance in graphing slope and y‐intercept from verbally presented linear functions was assessed both for graphs with quantitative features and graphs with qualitative features. Students had more difficulty graphing y‐intercept than slope, particularly in graphs with qualitative features. Errors also differed between contexts. The findings suggest that it would be valuable for additional instructional time to be devoted to y‐intercept and to qualitative contexts.     

Covariational reasoning and invariance among coordinate systems

Researchers continue to emphasize the importance of covariational reasoning in the context of students’ function concept, particularly when graphing in the Cartesian coordinate system (CCS). In this article, we extend the body of literature on function by characterizing two pre-service teachers’ thinking during a teaching experiment focused on graphing in the polar coordinate system (PCS). We illustrate how the participants engaged in covariational reasoning to make sense of graphing in the PCS and make connections with graphing in the CCS. By foregrounding covariational relationships, the students came to understand graphs in different coordinate systems as representative of the same relationship despite differences in the perceptual shapes of these graphs. In synthesizing the students’ activity, we provide remarks on instructional approaches to graphing and how the PCS forms a potential context for promoting covariational reasoning.     

About the use of the TI-92 for an open learning approach to power functions

In this report we present the results of a teaching study introducing the concept “power function” using a graphing calculator. The focus of our attention is on the development of the understanding of 15–16 year-old mathematics students. In the centre of our interest is their learning through graphs of power functions by discovering the properties of graphs. Our report presents the mathematical and social constructivist background together with a new deliberately constructivist approach beginning the teaching experiment with an open question. The students' cognitive and intuitive strategies and their attitudes towards computer algebra are described.     

The role of the graphic calculator in mediating graphing activity

The PIGMI (Portable Information Technologies for supporting Graphical Mathematics Investigations) Project ¹ investigated the role of portable technologies in facilitating development of students' graphing skills and concepts. This paper examines the impact of a recent shift towards calculating and computing tools as increasingly accessible, everyday technologies on the nature of learning in a traditionally difficult curriculum area. The paper focuses on the use of graphic calculators by undergraduates taking an innovative new mathematics course at the Open University. A questionnaire survey of both students and tutors was employed to investigate perceptions of the graphic calculator and the features which facilitate graphing and linking between representations. Key features included visualization of functions, immediate feedback and rapid graph plotting. A follow-up observational case study of a pair of students illustrated how the calculator can shape mathematical activity, serving a catalytic, facilitating and checking role. The features of technology-based activities which can structure and support collaborative problem solving were also examined. In sum, the graphic calculator technology acted as a critical mediator in both the students' collaboration and in their problem solving. The pedagogic implications of using portables are considered, including the tension between using and over-using portables to support mathematical activity.     

Affect and graphing calculator use

This article reports on a qualitative study of six high school calculus students designed to build an understanding about the affect associated with graphing calculator use in independent situations. DeBellis and Goldin's (2006) framework for affect as a representational system was used as a lens through which to understand the ways in which graphing calculator use impacted students’ affective pathways. It was found that using the graphing calculator helped students maintain productive affective pathways for problem solving as long as they were using graphing calculator capabilities for which they had gone through a process of instrumental genesis (Artigue, 2002) with respect to the mathematical task they were working on. Furthermore, graphing calculator use and the affect that is associated with its use may be influenced by the perceived values of others, including parents and teachers (past, present and future).     

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.