Teaching methods comparison in a large calculus class

We report findings from a classroom experiment in which each of two sections of the same Calculus 1 course at a North American research-focused university were subject to an “intervention” week, each for a different topic, during which a less-experienced instructor encouraged a much higher level of student engagement, promoted active learning (answering “clicker” questions, small-group discussions, worksheets) during a significant portion of class time and built on assigned pre-class tasks. The lesson content and analysis of the assessments were informed by existing research on student learning of mathematics and student interviews, though the interventions and assessments were also intended to be compatible with typical course practices in an attempt to appeal to practitioners less familiar with the literature. Our study provides an example of active learning pedagogy (including materials and assessment used) for students at this level of mathematics in a classroom of over one hundred students, and we report improved student performance—on conceptual items in particular—with a switching replication in that each section outperformed the other on the topic for which it received the intervention.     

Understanding the European policy landscape and its impact on change in mathematics and science pedagogies

Effecting an increase in the use of inquiry-based learning pedagogies in mathematics and science at a large scale across Europe has been promoted within the European education policy space. The primary goal is greater numbers of young people having more positive dispositions towards future careers in science, engineering and technology (Rocard in EUR22845—science education now: a renewed pedagogy for the future of Europe, http://www.eesc.europa.eu/resources/docs/rapportrocardfinal.pdf, 2007). We report on policy research as part of a European funded project that aims to support professional development that might bring about such changes. Our analysis points to policy activity that, to a greater or lesser degree, appears to be ineffective in meeting the desired outcomes in respect of teacher change. We identify why, although there is much policy activity, initiatives struggle to make an impact on teaching practices and learners’ experiences. Overall we point to misalignment in what policy defines in terms of input to learning, what is measured and valued in terms of outcomes, and what is desired in the teaching process. We also point to how policy fails to support development of the individual teacher and the teaching profession as a whole in ways that might effect the desired change.     

Exploring Images of Parental Participation in Mathematics Education: Challenges and Possibilities

In this article, we draw on research within a large project on parental involvement in mathematics education in working-class Latino communities. Our research is situated within a sociocultural framework and, in particular, the concept of funds of knowledge. We also draw on research on parental involvement in education, particularly that which critically examines issues of power and perceptions of parents. We build on the concept of dialogic learning and on the characterization of parents as intellectual resources and present a model for parental involvement in mathematics in which parents engage as (a) parents, (b) learners, (c) facilitators, and (d) leaders. In particular, in this article, we focus on the third component—parents as facilitators of mathematics workshops for the community at large—centering on some of the challenges as parents and teachers engage in this type of collaboration. We also look at the possibilities afforded by a model for parental involvement that views parents as intellectual resources. By looking at examples of interactions among parents and teachers, and among parents and children, in mathematics workshops, we challenge conventional notions about parental involvement—in particular, as they apply to working-class, language/ethnic "minority" parents.     

Principal and coach as partners

This paper explores the roles and responsibilities of mathematics coaches and principals. It suggests that principals need to equilibrate power by treating the coach as a partner, even if the coach is subordinate in the educational hierarchy. It posits that coaching would benefit all parties if they worked together across roles to enhance the capacity of educators at every level in the school hierarchy—teacher, coach, principal—to design, analyze, and implement mathematics instruction that results in deeper student learning. It briefly describes the model of Content Coaching. It questions some of the prevailing norms of interaction among educators, and it offers coaching scenarios that demonstrate that the initial design and implementation of coaching initiatives can lead to resistance and misunderstanding of the power and benefits of coaching for all educators and administrators. It goes on to challenge some of our assumptions regarding potential barriers (e.g. confidentiality) in relationships among coaches, teachers, and principals, and suggests that teacher learning goals be public and collaboratively engaged. It posits that positional authority is not a particularly strong lever for improving teacher practice, particularly if the principal lacks facility or interest in mathematics. It argues that if we want to enhance mathematics education in this country (as evidenced by robust student understanding of mathematics and its application in the world), there is great power in working collaboratively and inclusively, using coaching to support educators at every level.     

Informed integration of IWB technology,incorporated with exposure to varied mathematics problem-solving skills: its effect on students’ real-time emotions

Technology has an inseparable role in contemporary educational systems. Recent demands consider its informed integration in classrooms. The current study aims to investigate whether informed use of Interactive Whiteboard (IWB) technologies affects students’ real-time emotions in mathematics classes, through an integrative research model that combines two components: varying levels of technology integration and varied levels of problem-solving skills. Participants included 64 sixth-grade students who were assigned to one of two comparison groups who have been progressively exposed to different implementations of the research model. The study used a real-time judgment of emotions scale, and a reflective questionnaire. Findings indicate that when being exposed to informed IWB integration, more positive emotions arose in real-time during learning; specifically, toward understanding-performance, and when being required to demonstrate applying problem-solving skills. The study provides a holistic view that reinforces the interplay between the two components of the integrative research model, particularly as it supports the advantage of an informed IWB technology integration in a way that most successfully incorporated with the cognitive demand required from students.     

A model for a widespread implementation of inquiry-based learning

Innovative teaching practices such as inquiry-based learning (IBL) have long been topics of discussion amongst mathematics and science educators. However, it is not easy to change day-to-day teaching on a large scale. The relevant question of how to promote a widespread uptake of IBL in day-to-day teaching therefore needs more consideration. In order to ensure such uptake of IBL in a variety of different contexts, a model including dissemination and implementation strategies needs to be designed. In this paper, we present the design of a focused and flexible model for dissemination and implementation as developed within the international project PRIMAS, funded by the EU under Framework 7. The design of this model is rooted in design research. We will outline and explain the complexity of the model, including its theoretical basis, its iterative approach for evaluation and refinement, and its intended contributions to research.     

Urban Elementary STEM Initiative

The new standards for K–12 science education suggest that student learning should be more integrated and should focus on crosscutting concepts and core ideas from the areas of physical science, life science, Earth/space science, and engineering/technology. This paper describes large‐scale, urban elementary‐focused science, technology, engineering, and mathematics (STEM) collaboration between a large urban school district, various STEM‐focused community stakeholders, and a research‐focused private university. The collaboration includes the development of an integrated STEM curriculum for grade K–5 with accompanying teacher professional development. This mixed‐methodology study describes findings from focus group interviews and a survey of teachers from Title I elementary schools. Findings suggest the importance of the following critical features of professional development: (a) coherence, (b) content focus, (c) active learning, (d) collective participation, and (e) duration to the success of large‐scale STEM urban elementary school reform     

Integrating technology into mathematics teaching at the university level

The emergence of new computing technologies in the second half of the twentieth century brought about new potentials and promised the rapid transformation of the teaching and learning of mathematics. However, despite the vast investments in technology resources for schools and universities, the realities of schooling and the complexities of technology-equipped environments resulted in a much slower integration process than was predicted in the 1980s. Hence researchers, together with teachers and mathematicians, began examining and reflecting on various aspects of technology-assisted teaching and learning and on the causes of slow technology integration. Studies highlighted that as technology becomes increasingly available in schools, teachers’ beliefs and conceptions about technology use in teaching are key factors for understanding the slowness of technology integration. In this paper, I outline the shift of research focus from learning and technology environment-related issues to teachers’ beliefs and conceptions. In addition, I highlight that over the past two decades a considerable imbalance has developed in favour of school-level research against university-level research. However, several changes in universities, such as students declining mathematical preparedness and demands from other sciences and employers, necessitate closer attention to university-level research. Thus, I outline some results of my study that aimed to reflect on the paucity of research and examined the current extend of technology use, particularly Computer Algebra Systems (CAS) at universities, mathematicians’ views about the role of CAS in tertiary mathematics teaching, and the factors influencing technology integration. I argue that due to mathematicians’ extensive use of CAS in their research and teaching, documenting their teaching practices and carrying out research at this level would not only be beneficial at the university level but also contribute to our understanding of technology integration at all levels.     

Designing and using professional development resources for inquiry-based learning

This paper describes an attempt to design, analyse and refine professional development (PD) resources that encourage the implementation of inquiry-based learning (IBL). We describe the iterative development of the resources in England with over 100 mathematics teachers from secondary, tertiary and adult education and then analyse the impact these resources had on teachers’ beliefs and practices and the issues arising. This evaluation revealed that teachers had moved away from transmission-based orientations, encouraged by the use of less structured tasks and sample lesson plans, but some found it difficult to adopt IBL pedagogies. The most significant issues for teachers may be summarised as: confusing IBL with ‘discovery’ learning; developing and managing collaborative cultures within the classroom; and planning lessons that adapt to the emerging needs of learners.     

Evolution of microstructure in unstable porous media flow: A relaxational approach

We study the flow of two immiscible fluids of different density and mobility in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a mesoscopic scale and the mixing zone grows in time—an example of evolution of microstructure. A simple set of assumptions on the physics of this two‐phase flow in a porous medium leads to a mathematically ill‐posed problem—when used to establish a continuum free boundary problem. We propose and motivate a relaxation of this “nonconvex” constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well‐posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient. © 1999 John Wiley & Sons, Inc.     

A quadratic growth learning trajectory

This paper introduces a quadratic growth learning trajectory, a series of transitions in students’ ways of thinking (WoT) and ways of understanding (WoU) quadratic growth in response to instructional supports emphasizing change in linked quantities. We studied middle grade (ages 12–13) students’ conceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic growth as phenomenon of constantly-changing rate of change. We elaborate the duality, necessity, repeated reasoning framework, and methods of creating learning trajectories. We report five WoT: Variation, Early Coordinated Change, Explicitly Quantified Coordinated Change, Dependency Relations of Change, and Correspondence. We also articulate instructional supports that engendered transitions across these WoT: teacher moves, norms, and task design features. Our integration of instructional supports and transitions in students’ WoT extend current research on quadratic function. A visual metaphor is leveraged to discuss the role of learning trajectories research in unifying research on teaching and learning.     

Some Undergraduate Students'' Perceptions of Using Technology for Mathematics: Tales of Resistance

We describe and report some results from a project designed to evaluate the use of information technology (IT) for teaching and learning on a range of undergraduate mathematics courses at a U.K. university. The project involved qualitative methods of enquiry (diaries and seminars). We emphasise that many of the students were positive about much of their use of IT; however in this article we focus particularly on their tales of resistance. These tales were told both by students whose overall perspective about the use of IT in the learning of mathematics was very positive as well as by those who were reluctant users. We have organised this article around two themes: the desire for understandingand technology and power — who is in control? We suggest that these tales spring from non-trivial concerns and that we are likely to be better able to support the development of authoritative learners of mathematics if we heed them.This revised version was published online in September 2005 with corrections to the Cover Date.     

Innovation and learning performance implications of free revealing and knowledge brokering in competing communities: insights from the Netflix Prize challenge

Firms increasingly use open competitions to extend their innovation process and access new diverse knowledge. The Netflix Prize case we study in this paper is a multi-stage repeat-submission open competition involving the creation of new knowledge from across knowledge domains, a process which benefits from knowledge sharing across competing communities. The extant literature says little about the effects of different types and levels of knowledge sharing behavior on the learning and innovation outcomes of such a competitive system, or what the performance boundaries may be for the system as a result of such differences. Our research explores those boundaries unveiling important tradeoffs involving free revealing behavior—defined as voluntarily giving away codified knowledge and making it into a ‘public good’—and knowledge brokering behavior—defined as using knowledge from one domain to innovate in another—on the learning performance of competing communities. The results, analyzing the system-level average and volatility of learning outcomes, lead to three conclusions: (i) greater knowledge sharing, as portrayed by greater free revealing and knowledge brokering, helps achieve better average learning for the system as a whole, however, (ii) achieving the best overall outcome possible from the system actually requires controlling the amount of knowledge brokering activity in the system. The results further suggest that (iii) it should not be possible to simultaneously achieve both the best overall outcome from the system and the best average learning for the system. The tradeoffs that ensue from these findings have important implications for innovation policy and management. This research contributes to practice by showing how it is possible to achieve different learning performance outcomes by managing the types and levels of knowledge sharing in open competitive systems.     

A teacher’s learning process in dual design research: learning to scaffold language in a multilingual mathematics classroom

In this paper, we argue that dual design research (DDR) is a fruitful way to promote and trace the development of a mathematics teacher’s expertise. We address the question of how a teacher participating in dual design research can learn to scaffold students’ development of the language required for mathematical learning in multilingual classrooms. Empirical data were collected from two teaching experiments (each with 8 lessons, and 21 and 22 students, aged 11–12 years), for which lesson series about line graphs were co-designed by the researchers and the teacher. The teacher’s learning process was promoted (e.g. by conducting stimulated recall interviews and providing feedback) and traced (e.g. by carrying out 5 pre- and post-interviews before and after the teaching experiments). An analytic framework for teachers’ reported and derived learning outcomes was used to analyse pre- and post-interviews. The teacher’s learning process was analysed in terms of changes in knowledge and beliefs, changes in practice and intentions for practice. Further analysis showed that this learning process could be attributed to the characteristics of dual design research, for instance the cyclic and interventionist character, the continuous process of prediction and reflection that lies at its heart, and the process of co-designing complemented with stimulated recall interviews.     

Challenges students identified with a learning disability and as high-achieving experience when using diagrams as a visualization tool to solve mathematics word problems

This article addresses a much understudied topic and concern regarding how students of varying ability levels employ visualization as a strategy in mathematics learning. The importance of this topic can be found in its connection to students’ ability to solve mathematical word problems. Many students, particularly students with learning disabilities, often struggle to use visualization as a strategy and this impacts their mathematics performance. The purpose of this article is to present findings from a study that examined the challenges that students—those identified as learning disabled and high-achieving—displayed when using one visualization form, a diagram, to solve mathematics problems. Overall, nine challenges related to the use of diagram proficiency to solve problems were identified. Further, students with learning disabilities were found to be more likely than their high achieving peers to experience these challenges. Implications for practice are provided.     

Humans-with-media and continuing education for mathematics teachers in online environments

This paper begins by situating online mathematics education in Brazil within the context of research on digital technology over the past 25?years. I argue that Brazilian research on technology in mathematics education can be divided into four phases, and then present an example that ??blends?? aspects of the second and third phases. Phase two can be characterized by research with software designed to address traditional mathematics topics, such as functions, while the third phase is characterized by online courses. The data presented show creative solutions for a problem designed for collectives of humans-with-function-software. The paper is analyzed from a perspective that emphasizes the role of different technologies as teachers and professors collaborate to produce knowledge about the use of mathematical software in regular face-to-face classrooms. A model of online education is presented. Finally, the paper discusses how technology may change collaboration and teaching approaches in continuing education, as it allows for greater integration of online learning with teachers?? classroom activities in schools. In this case, the online platform plays an active role in the learning collective composed of humans-with-media.     

Materials by design and the exciting role of quantum computation/simulation

It is now well-recognized that we are witnessing a golden age of innovation with novel materials, with discoveries important for both basic science and device applications—some of which will be treated at this Workshop. In this talk, we discuss the role of computation and simulation in the dramatic advances of the past and those we are witnessing today. We will also describe the growing acceptance and impact of computational materials science as a major component of materials research and its import for the future. In the process, we will demonstrate how the well-recognized goal driving computational physics/computational materials science—simulations of ever-increasing complexity on more and more realistic models—has been brought into greater focus with the introduction of greater computing power that is readily available to run sophisticated and powerful software codes like our highly precise full-potential linearized augmented plane wave (FLAPW) method, now also running on massively parallel computer platforms.We will then describe some specific advances we are witnessing today, and computation and simulation as a major component of quantum materials design and its import for the future, with the goal—to synthesize materials with desired properties in a controlled way via materials engineering on the atomic scale. The theory continues to develop along with computing power. With the universality and applicability of these methods to essentially all materials and properties, these simulations are starting to fill the increasingly urgent demands of material scientists and engineers.     

Drawing inferences from learners’ examples and questions to inform task design and develop learners’ spatial knowledge

Examples that learners generate, and questions they ask while generating examples, are both sources for inferring about learners’ thinking. We investigated how inferences derived from each of these sources relate, and how these inferences can inform task design aimed at advancing students’ knowledge of scale factor enlargement (i.e. scaling). The study involved students in two secondary schools in England who were individually tasked to generate examples of scale factor enlargements in relation to specifically designed prompts. Students were encouraged to raise questions while generating their examples. We drew inferences about students’ thinking from their examples and, where available, from their questions. These inferences informed our design and implementation of a set of follow-up tasks for all students, and an additional personalised task for each student who raised any questions. Students showed increased knowledge of, and confidence with, scale factor enlargement independently of whether they asked questions during the exemplification task.     

Computational and mathematical organization theory: Perspective and directions

Computational and mathematical organization theory is an interdisciplinary scientific area whose research members focus on developing and testing organizational theory using formal models. The community shares a theoretical view of organizations as collections of processes and intelligent adaptive agents that are task oriented, socially situated, technologically bound, and continuously changing. Behavior within the organization is seen to affect and be affected by the organization's, position in the external environment. The community also shares a methodological orientation toward the use of formal models for developing and testing theory. These models are both computational (e.g., simulation, emulation, expert systems, computer-assisted numerical analysis) and mathematical (e.g., formal logic, matrix algebra, network analysis, discrete and continuous equations). Much of the research in this area falls into four areas: organizational design, organizational learning, organizations and information technology, and organizational evolution and change. Historically, much of the work in this area has been focused on the issue how should organizations be designed. The work in this subarea is cumulative and tied to other subfields within organization theory more generally. The second most developed area is organizational learning. This research, however, is more tied to the work in psychology, cognitive science, and artificial intelligence than to general organization theory. Currently there is increased activity in the subareas of organizations and information technology and organizational evolution and change. Advances in these areas may be made possible by combining network analysis techniques with an information processing approach to organizations. Formal approaches are particularly valuable to all of these areas given the complex adaptive nature of the organizational agents and the complex dynamic nature of the environment faced by these agents and the organizations.This paper was previously presented at the 1995 Informs meeting in Los Angeles, CA.     

Training Master's‐Level Graduate Students to Use Inquiry Instruction to Teach Middle‐Level and High‐School Science Concepts

Through the GK‐12 program of the National Science Foundation, graduate student fellows in a coastal marine and wetland studies program were trained to present targeted science concepts to middle‐ and high‐school classes through their own research‐based lessons. Initially, they were taught to follow the 5‐E learning cycle in lesson plan development, but a streamlined approach targeting the three attributes of science concepts—macroscopic, model, and symbolic—was found to be a better approach, while still incorporating key facets of the 5‐E model. Evaluation of the level of inquiry in the classrooms was determined using an inquiry scale from 0 to 4, differentiated by the relative number of actions that are student‐centered. The graduate fellows consistently delivered lessons at the targeted levels 2 or 3, guided inquiry. In order to assess student learning, the GK‐12 fellows were trained to develop single‐item pre‐ and post‐assessments designed to probe middle‐level and high‐school students' understanding of the macroscopic, model, and symbolic attributes of targeted science concepts. For the lessons based on the research of the fellows, about 80% of the students showed statistically and practically significant learning gains. The GK‐12 fellows positively impact the classroom and are effective science ambassadors.