The Graphing Calculator: A Survey of Classroom Usage

The purpose of this study was to investigate secondary mathematics teachers' use of the graphing calculator in their classrooms, The study examined whether algebra teachers are currently using this technology in their classrooms, their perceptions toward the technology, and any changes in the curriculum or instructional practices. A survey methodology was used in this study. The findings indicated that the use of the graphing calculator is still controversial to many algebra teachers. Teachers of algebra I were using graphing calculators to a significantly lesser degree than teachers of algebra H. However, modifications of the algebra curriculum are beginning to appear in classes using graphing calculators. Finally, a majority of algebra teachers responded that the graphing calculator was a motivational tool.     

Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations

Digital resources offer opportunities to improve mathematics teaching and learning, but meanwhile may question teachers’ practices. This process of changing teaching practices is challenging for teachers who are not familiar with digital resources. The issue, therefore, is what teaching practices such so-called ‘mid-adopting’ mathematics teachers develop in their teaching with digital resources, and what skills and knowledge they need for this. To address this question, a theoretical framework including notions of instrumental orchestration and the TPACK model for teachers’ technological pedagogical content knowledge underpins the setting-up of a project with twelve mathematics teachers, novice in the field of integrating technology in teaching. Technology-rich teaching resources are provided, as well as support through face-to-face group meetings and virtual communication. Data include lesson observations and questionnaires. The results include a taxonomy of orchestrations, an inventory of skills and knowledge needed, and an overview of the relationships between them. During the project, teachers do change their orchestrations and acquire skills. On a theoretical level, the articulation of the instrumental orchestration model and the TPACK model seems promising.     

The powers and pitfalls of algorithmic knowledge: a case study

This study examines one child's use of computational procedures over a period of 3 years in an urban elementary school where teachers were using a standards-based curriculum. From a sociocultural perspective, the use of standard algorithms to solve mathematical problems is viewed as a cultural tool that both enables and constrains particular practices. As this student appropriated and mastered procedures for addition, subtraction, multiplication and division, she could solve problems that involved fairly straightforward computations or where she could easily model the action to determine an appropriate computation. At the same time, her use of these algorithms, along with other readily available tools, such as her fingers or multiplication tables, constrained her ability to reflect on the tens-structure of the number system, an effect that had serious consequences for her overall mathematical achievement. The results of this study suggest that even when not directly introduced, algorithms have such strong currency that they can mediate more reform-oriented instruction.     

The effect of remanufacturing on procurement decisions for resellers in secondary markets

The role of remanufacturing as a competitive tool for firms has been reflected in a number of studies to show that remanufacturing can reduce the unit cost of production by reusing components. However, the fact that remanufacturing can be used as a strategic tool for serving secondary markets as well has not been acknowledged in the literature. In this paper, we study the use of remanufacturing as a tool to serve secondary markets. Specifically, we model the case of a reseller who procures used products based on an older generation of technology from an advanced market and then uses one of two options: (a) she can either resell a small fraction of these used products in a developing market where the technology is acceptable, or (b) she can invest in the remanufacturing of these products and then sell them in the developing market at a higher price. The main result of the paper is that using remanufacturing to serve secondary markets reduces the number of units procured from the advanced market for the reseller. In addition, we show based on certain cost structures that the reseller is always better off if she uses remanufacturing to a certain extent.     

Reflections on “Multiplication as Original Sin”: The implications of using a case to help preservice teachers understand invented algorithms

This article describes the use of a case report, Multiplication as original sin (Corwin, R. B. (1989). Multiplication as original sin. Journal of Mathematical Behavior, 8, 223-225), as an assignment in a mathematics course for preservice elementary teachers. In this case study, Corwin described her experience as a 6th grader when she revealed an invented algorithm. Preservice teachers were asked to write reflections and describe why Corwin’s invented algorithm worked. The research purpose was: to learn about the preservice teachers’ understanding of Corwin’s invented multiplication algorithm (its validity); and, to identify thought-provoking issues raised by the preservice teachers. Rather than using mathematical properties to describe the validity of Corwin’s invented algorithm, a majority of them relied on procedural and memorized explanations. About 31% of the preservice teachers demonstrated some degree of conceptual understanding of mathematical properties. Preservice teachers also made personal connections to the case report, described Corwin using superlative adjectives, and were critical of her teacher.     

Sociocultural perspectives in research on and with mathematics teachers: a zone theory approach

Sociocultural theories view teacher learning as changing participation in social practices that develop their professional identities rather than as acquisition of new knowledge or beliefs that are internal to the individual. Although sociocultural research on mathematics teacher education has tended to focus on understanding teachers’ learning, this article argues that sociocultural perspectives can also guide more interventionist research involving changing classroom practice. The approach illustrated here uses an adaptation of Valsiner’s zone theory to analyse teacher learning and development in two separate research studies. In one study the aim was to understand how teachers incorporated digital technologies into their practice, while the other study helped teachers implement an investigative approach to working mathematically consistent with a new syllabus. In both studies, productive tensions between teachers’ beliefs, contexts, and goals were a trigger for learning and development.     

Controlling Choice: Teachers,Students, and Manipulatives in Mathematics Classrooms

This research study examines the instructional practices of 10 middle grades teachers related to their use of manipulatives in teaching mathematics and their control of mathematics tools during instruction. Through 40 observations of teaching, 30 interviews, and an examination of 67 written documents (including teachers' plans and records), profiles were developed that describe how teachers used and controlled manipulatives during instruction. Results showed that teachers used a variety of manipulatives and other mathematics tools over the course of the year‐long study. Teachers reported using a mathematics tool (manipulative, calculator, or measuring device) in 70% of their lessons, and this self‐report was verified by observations in which teachers used mathematics tools in 68% of their lessons. During a 3‐ to 4‐month period of “free access,” in which students had some measure of control in their selection and use of the mathematics tools, the students used manipulatives spontaneously and selectively. During free access, teachers exhibited various behaviors, including posting lists of items on containers, assigning group leaders to manage tools, and negotiating the control of the mathematics tools during instruction.     

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.     

The Border Crossings of a Multicultural Science Education Enthusiast

Jill was a preservice science education student who wanted to make science more accessible to all students. This study is an examination of the “borders” she encountered as she completed her student teaching in a cultural setting that was different from her own. Her student teaching experience was documented through interviews, participant observations, field notes, lesson plans, and a journal. An inductive analysis of the documents and a context chart of the coded data revealed that Jill encountered the (a) cultural border of her students, (b) cultural border of science instruction, and (c) cultural border of the school. While some borders were crossed, others were not. This study suggests that during field experiences, preservice teachers may encounter multiple cultural borders, some consistent and some inconsistent with their instructional philosophy. As student teachers work with diverse populations, supervisors and cooperating teachers need to recognize the borders student teachers will encounter and encourage student teachers to examine their beliefs about practice as a means to acknowledge and understand the encountered borders.     

Graphing Calculator Use in Algebra Teaching

This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed. Forty of the 75 schools (53%) returned a total of 109 individual surveys. Results indicated that: (1) While 78% of teachers have some access to the technology, only 28% use it regularly. (2) Statistically significant relationships exist between use and age, years of experience, teaching assignment, and teaching level. (3) Respondents view graphical solution methods as secondary to symbolic methods. (4) Teachers are more receptive to using technology to supplement rather than expand the curriculum.     

Olga Taussky,a torchbearer for mathematics and teacher of mathematicians

Olga Taussky-Todd's mathematical and personal life (1906-1995), her achievements and obstacles, her scientific reasoning and teaching all have served as inspiration to many mathematicians. We describe her role in the mathematics world of the previous century as a torchbearer for mathematics and mathematicians, bearing the “torch of scientific truth” that burns inside of mathematics and its applications. Besides her many deep math contributions – too many to elaborate – she excelled at distilling and presenting mathematical concepts and ideas in her work and gave us many visionary papers and math talks. By sharing her mathematical vision freely she has inspired many of us. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Understanding the role of the teacher in emerging classroom practices: searching for patterns of participation

The relationship between acquisitionism and participationism is a challenge in research on and with teachers. This study uses a patterns-of-participation framework (PoP), which aims to develop coherent and dynamic understandings of teaching as well as to meet the conceptual and methodological problems of other approaches. The paper presents PoP theoretically, but also illustrates its empirical use. It presents a novice teacher, Anna, who often engages with mathematics and with aspects of ‘the reform’ in ways that link well with how she builds relationships with her students and positions herself in her team of teachers. However, in other situations her engagement with mathematics is overshadowed by her involvement in other practices. The study suggests that there is some potential in PoP in spite of methodological difficulties.     

Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lesson plan study

“Lesson plan study” (LPS), adapted from the Japanese Lesson Study method of professional development, is a sequence of activities designed to engage prospective teachers in broadening and deepening their understanding of school mathematics and teaching strategies. LPS occurs over 5 weeks on the same lesson topic and includes four opportunities to revisit one's own ideas and the ideas of others. In this paper, we describe one prospective teacher's growth in understanding right triangle trigonometry as she participated in LPS. This study is part of a much larger study investigating how prospective secondary teachers learn to teach mathematics within the context of LPS. Results of this study indicate that Image Saying, an activity for growth in understanding from the Pirie-Kieren model [Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165-190], is critical to prospective teachers’ growth in understanding school mathematics. Multiple opportunities and contexts within which to share understanding of school mathematics led to significant growth in understanding of right triangle trigonometry which in turn led to growth in understanding of teaching strategies. That is, the results of this study indicate that growth in understanding school mathematics (what to teach) leads to growth in understanding teaching strategies (how to teach) as prospective teachers participate in LPS.     

Challenges of maintaining cognitive demand during the limit lessons: understanding one mathematician’s class practices

In this study, we examined five limit lessons using Mathematical Tasks Framework to understand students’ opportunities to learn cognitively challenging tasks and maintain cognitive demand during limit lessons. Our analysis of Dr A’s five lessons shows that students rarely had opportunities to maintain or increase cognitive demand. There are two main factors that shaped her instructional practices, students and time. These two factors greatly influenced how she selects and implements limit tasks in her classes. To serve her students’ needs of knowing more rules, formulas and procedures, she selected and discussed those simple tasks a lot. Although Dr A thinks challenging tasks and asking demanding questions can be potentially good instructional practices, she thinks these instructional practices would not serve her students well. With these factors, we made possible recommendations to have more student-centred teaching.     

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.     

A Preservice Secondary Teacher's Moves to Protect Her View of Herself as a Mathematics Expert

This paper discusses the experience of a preservice secondary mathematics teacher during lesson study. Although the preservice teacher was a strong undergraduate mathematics student, she used compensation “moves” to deflect attention away from her insecurities about her conceptual understanding of secondary mathematics. She feared being labeled as “dumb” and redirected conversations in order to protect her identity as a knower of mathematics. This paper investigates the culture in which preservice teachers develop confidence in their personal mathematics knowledge and how that confidence may influence behavior.     

A Study of STEM Assessments in Engineering,Science, and Mathematics for Elementary and Middle School Students

The purpose of this study was to develop, scale, and validate assessments in engineering, science, and mathematics with grade appropriate items that were sensitive to the curriculum developed by teachers. The use of item response theory to assess item functioning was a focus of the study. The work is part of a larger project focused on increasing student learning in science, technology, engineering, and mathematics (STEM)‐related areas in grades 4–8 through an engineering design‐based, integrated approach to STEM instruction and assessment. The fact that the assessments are available to school districts at no cost, and represent psychometrically sound instruments that are sensitive to STEM‐oriented curriculum, offers schools an important tool for gauging students' understanding of engineering, science, and mathematics concepts.     

The bit‐economy: An artificial model of open‐ended technology discovery

We describe and demonstrate an artificial model of technology discovery called the Bit‐Economy. The model is built from a minimal set of fundamental hallmarks of technology and develops under an open‐ended evolutionary operator which rewards new technology which is able to coordinate both spatially and temporally with the existing technology set. The Bit‐Economy, is able to replicate several features of real technology development including nonmonotonic growth, bunching of creation and destruction events, qualitative topologies of patent networks, and efficiency and waste‐management gains. In contrast to related works, we do not apply an exogenous fitness landscape and so are able to study the process of technology discovery as a self‐guided search toward more complex outcomes. © 2013 Wiley Periodicals, Inc. Complexity 18: 57–67, 2013