Redesigning the calculus sequence at a research university: issues,implementation, and objectives

The paper discusses the progress and challenges of a new reformed calculus sequence for science, engineering, and mathematics students developed by the Institute of Technology Centre for Educational Programs and School of Mathematics, University of Minnesota. The main objective of the Initiative is to enable undergraduates to better learn calculus and the critical thinking skills necessary to apply it in a variety of science and engineering problems. Changes in content and pedagogy are emphasized, including instructional teamwork and student-centred learning, involving students working cooperatively in small groups and exploring mathematical ideas using appropriate technologies. Achievement and retention of Initiative students are compared with a control group from the standard calculus sequence. Student attitudes about the usefulness of the Initiative's curriculum, pedagogy, and its influence on learning are discussed. Future implications including new uses of distributed learning are also addressed.     

Polynomial calculus: rethinking the role of calculus in high schools

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.     

Examination of the dynamic software-supported learning environment in data analysis

The Fatih Project in Turkey has improved software in mathematics teaching such as data analysis software. As a result, the need to inquire into the efficiency of computer-supported learning environments has emerged. This study aims to examine the effect of dynamic data analysis software-supported learning environments on secondary school students’ achievement and attitude. The research method employs a quasi-experimental design with a pre-test, post-test control group. Basic topics related to data analysis were introduced through dynamic statistics software in the experimental group while the students were taught with the help of smart boards, course books and exercises in the control group. Data were collected with an achievement test, attitude scale and semi-structured interviews. Also, interviews were conducted with four students from the experimental group in order to get more detailed information from students. The data gained in the study were analysed both quantitatively and qualitatively. The findings revealed that statistics teaching through statistics software is more efficient than the one with the traditional method on achievement and attitudes. In accordance with this result, it is suggested that computer-supported statistics software should be used in statistics teaching.     

Integrating learning theories and application-based modules in teaching linear algebra

The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a) linear algebra and (b) learning theory as applied to the study of mathematics with an emphasis on linear algebra. The purpose of the ongoing research, partially funded by the National Science Foundation, is to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of individuals in both domains. The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted, in some students, a rich understanding of both domains and that had a mutually reinforcing effect. Furthermore, there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and, consequently, better learning by their students. The courses developed were appropriate for mathematics majors, pre-service secondary mathematics teachers, and practicing mathematics teachers. The learning seminar focused most heavily on constructivist theories, although it also examined socio-cultural and historical perspectives. A particular theory, Action-Process-Object-Schema (APOS) [10], was emphasized and examined through the lens of studying linear algebra. APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics. The linear algebra courses include the standard set of undergraduate topics. This paper reports the results of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has influenced the future direction of the project.     

The Effects of a Mathematics Infusion Curriculum on Middle School Student Mathematics Achievement

Increasing mathematical competencies of American students has been a focus for educators, researchers, and policy makers alike. One purported approach to increase student learning is through connecting mathematics and science curricula. Yet there is a lack of research examining the impact of making these connections. The Mathematics Infusion into Science Project, funded by the National Science Foundation, developed a middle school mathematics‐infused science curriculum. Twenty teachers utilized this curriculum with over 1,200 students. The current research evaluated the effects of this curriculum on students' mathematics learning and compared effects to students who did not receive the curriculum. Students who were taught the infusion curriculum showed a significant increase in mathematical content scores when compared with the control students.     

Revival of a classical topic in differential geometry: the exploration of envelopes in a computerized environment

Learning mathematics in a technology-rich environment enables us to revive classical topics which have been removed from the curriculum a long time ago. Both theoretical issues and applications can be studied with an experimental process. We present how envelopes of 1-parameter families of plane curves and some of their applications can be presented early in the curriculum either for pre-service teachers or for in-service teachers. This approach may be useful for students in an engineering curriculum. Working with technology yields important effects, such as reviving classical topics, broadening perspectives on already known topics, and enhancing the learner's experimental skills, where conversion between various registers of representation is an important issue.     

Using Prediction to Promote Mathematical Understanding and Reasoning

Research has shown that prediction has the potential to promote the teaching and learning of mathematics because it can be used to enhance students' thinking and reasoning at all grade levels in various topics. This article addresses the effectiveness of using prediction on students' understanding and reasoning of mathematical concepts in a middle school algebra context. In the treatment classroom, prediction questions were utilized at the launch of each algebra lesson, and in the control classroom such questions were not used. Both classrooms were taught by the same teacher and used the same curriculum. After completing each of the linear and exponential units, the two classrooms were compared in terms of their mathematical understanding and reasoning through unit assessments. Overall, the treatment classroom outperformed the control classroom on the unit assessments. This result supports that prediction is a valid construct with respect to enhanced conceptual understanding and mathematical reasoning.     

Integrating supplementary application-based tutorials in the multivariable calculus course

This article presents a study in which applications were integrated in the Multivariable Calculus course at the Technion in the framework of supplementary tutorials. The purpose of the study was to test the opportunity of extending the conventional curriculum by optional applied problem-solving activities and get initial evidence on the possible impact of the tutorials on students’ beliefs about the value of learning mathematics with applications. The study lasted three semesters and consisted of three experiments in which supplementary tutorials were offered in different forms: a weekly evening meeting for interested students, a weekly extra hour added to the conventional calculus class, and a workshop which introduces mathematics concepts from the application perspective. The study reveals: (1) significant positive effect of the tutorials on the students’ beliefs (in all three experiments); (2) statistically significant advantage of the group involved in the tutorials in relation to the group that learned in the conventional way (second experiment); (3) students’ positive evaluation of the workshops for better understanding the course lectures (third experiment). Grounding on the study experience, we propose for further discussion a stage model of the applied problem solving cycle.     

An Outlook in Teaching and Research in Mathematics in Educational Institutions

The present paper outlines some basic principles upon which an outlook in teaching and research in mathematics in educational institutions can be developed. Defining applied mathematics, the author analyses the basic topics of mathematics which permeate throughout the wider fields and diversified aspects of this branch of mathematical science.

For the institutions of learning, a course curriculum has been proposed which will gradually motivate, orient and prepare individuals through the process of learning to understand the methods and techniques, so as to apply them to physical problems. This has been illustrated in the fields of fluid dynamics and aerodynamics through the analysis of complex function theory and differential equations.

Because of the frustrating situations prevailing amongst students and teachers in the institutions of learning in the present day, a critical analysis has been made to identify the respective roles of the academicians and administrators, and with a view to improving the situation, some remedy has been suggested regarding teaching and research in educational institutions.     

The quality of non-textual elements in mathematics textbooks: an exploratory comparison between South Korea and the United States

As an exploratory investigation, this study aims to analyze non-textual elements in some Korean and US mathematics textbooks using a conceptual framework whose components include accuracy, connectivity, contextuality, and conciseness. By analyzing three US textbooks and three Korean ones, the study not only shows patterns in the use of non-textual elements in mathematics textbooks in different contexts but also provides insights into how to assess the quality of non-textual elements in mathematics textbooks, which I hope will contribute to the provision of more meaningful and productive learning opportunities to school children. Overall, the results from this study show that there is significant difference across topics and textbooks, which implies different opportunities to learn through non-textual elements. This study makes a unique contribution to the conceptualization of non-textual elements in mathematics education and has implications for textbook analysis and curriculum development.     

Mathematics on the Playground

Non‐traditional forms of instruction provide exciting and engaging opportunities for mathematics education. This article proposes the use of mathematicalfigures painted on the school playground as an environment to support mathematical teaching, learning, and understanding. Teachers plan mathematics lessons to take advantage of the playground figures and present new topics, reinforce current topics, and review previous topics. Figures were chosen to support mathematical concepts required by the curriculum and state and national standards. The intentional blank portions of the playground figures allowed for adjustment of lesson activities to meet different grade level and individual student needs, as well as making the figures interactive with the use of sidewalk chalk. A teacher handbook suggests activities for each figure by grade level, and teachers often create their own ideas for using the figures. Students like the change of perspective and teachers feel that such lessons help to raise standardized test scores because the information is better retained. The lessons addressed multiple learning styles, help with vocabulary for English Language Learners, and infuse higher thinking levels into lessons.     

Students’ experiences with mathematics teaching and learning: listening to unheard voices

This study documents students’ views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students’ views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students’ views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.     

Introducing Differential Calculus

In this paper we discuss three ways of introducing calculus all based on concepts which students would either already know or which can be introduced without much difficulty at this stage of their mathematical training. The aim of this paper is to persuade teachers of mathematics that topics in ‘higher mathematics’, specifically calculus, can and should be presented to students in terms of concepts with which they are familiar.     

Reorganizing freshman business mathematics I: background and philosophy

This article is the first of the two-part discussion of thedevelopment of a new Freshman Business Mathematics (FBM) courseat our college. Part I of the article describes the backgroundand history behind the course, and provides a theoretical frameworkfor the design of the course. This design involves studentsin learning and applying mathematics to real world problemsusing common business tools, in this case spreadsheets. Thisnew course is centered on the concept of building and interpretingmodels of data, but touches on many topics in statistics, pre-calculusand calculus.     

Pedagogical alternatives for triple integrals: moving towards more inclusive and personalized learning

This paper is based on the presumption that teaching multiple ways to solve the same problem has academic and social value. In particular, we argue that such a multifaceted approach to pedagogy moves towards an environment of more inclusive and personalized learning. From a mathematics education perspective, our discussion is framed around pedagogical approaches to triple integrals seen in a standard multivariable calculus curriculum. We present some critical perspectives regarding the dominant and long-standing approach to the teaching of triple integrals currently seen in hegemonic calculus textbooks; and we illustrate the need for more diverse pedagogical methods. Finally, we take a constructive position by introducing a new and alternate pedagogical approach to solve some of the classical problems involving triple integrals from the literature through a simple application of integration by parts. This pedagogical alternative for triple integrals is designed to question the dominant one-size-fits-all approach of rearranging the order of integration and the privileging of graphical methods; and to enable a shift towards a more inclusive, enhanced and personalized learning experience.     

The Impact of Enacted Mathematics Curriculum Models on Prospective Elementary Teachers' Course Perceptions and Beliefs

This paper communicates the impact of prospective teachers' learning of mathematics using novel curriculum materials in an innovative classroom setting. Two sections of a mathematics content course for prospective elementary teachers used different text materials and instructional approaches. The primary mathematical authorities were the instructor and text in the textbook section and the prospective teachers in the curriculum materials section. After one semester, teachers in the curriculum materials section (n= 34) placed significantly more importance on classroom group work and discussions, less on instructor lecture and explanation, and less on textbooks having practice problems, examples, and explanations. They valued student exploration over practice. In the textbook section (n= 19), there was little change in the teachers' beliefs, in which practice was valued over exploration. These results highlight the positive impact of experiences with innovative curriculum materials on prospective elementary teachers' beliefs about mathematics instruction.     

Curriculum Materials Supporting Problem-Based Teaching

The vision for school mathematics described by the National Council of Teachers of Mathematics (NCTM) suggests a need for new approaches to the teaching and learning of mathematics, as well as new curriculum materials to support such change. This article discusses implications of the NCTM standards for mathematics curriculum and instruction and provides three examples of lessons from problem-based curricula for various grade levels. These examples illustrate how the teaching of important mathematics through student exploration of interesting problems might unfold, and they highlight the differences between a problem-based approach and more traditional approaches. Considerations for teaching through a problem-based approach are raised, as well as reflections on the potential impact on student learning.     

Teachers' Conceptions of Integrated Mathematics Curricula

In this qualitative research study, we sought to understand teachers' conceptions of integrated mathematics. The participants were teachers in the first year of implementation of a state‐mandated, high school integrated mathematics curriculum. The primary data sources for this study included focus group and individual interviews. Through our analysis, we found that the teachers had varied conceptions of what the term integrated meant in reference to mathematics curricula. These varied conceptions led to the development of the Conceptions of Integrated Mathematics Curricula Framework describing the different conceptions of integrated mathematics held by the teachers. The four conceptions—integration by strands, integration by topics, interdisciplinary integration, and contextual integration—refer to the different ideas teachers connect as well as the time frame over which these connections are emphasized. The results indicate that even when teachers use the same integrated mathematics curriculum, they may have varying conceptions of which ideas they are supposed to connect and how these connections can be emphasized. These varied conceptions of integration among teachers may lead students to experience the same adopted curriculum in very different ways.     

School mathematics curriculum materials for teachers’ learning: future elementary teachers’ interactions with curriculum materials in a mathematics course in the United States

This report describes ways that five preservice teachers in the United States viewed and interacted with the rhetorical components (Valverde et al. in According to the book: using TIMSS to investigate the translation of policy into practice through the world of textbooks, Kluwer, 2002) of the innovative school mathematics curriculum materials used in a mathematics course for future elementary teachers. The preservice teachers’ comments reflected general agreement that the innovative curriculum materials contained fewer narrative elements and worked examples, as well as more (and different) exercises and question sets and activity elements, than the mathematics textbooks to which the teachers were accustomed. However, variation emerged when considering the ways in which the teachers interacted with the materials for their learning of mathematics. Whereas some teachers accepted and even embraced changes to the teaching–learning process that accompanied use of the curriculum materials, other teachers experienced discomfort and frustration at times. Nonetheless, each teacher considered that use of the curriculum materials improved her mathematical understandings in significant ways. Implications of these results for mathematics teacher education are discussed.     

The Application of a Computer Algebra System as a Tool in College Algebra

This study compared a traditional lecture-based college algebra course to an experimental algebra course. The experimental course stressed active student involvement and the use of the computer as a tool to explore mathematics. One hundred thirty-seven subjects were divided into an experimental group and a control group. Subjects in the experimental group scored significantly higher than the control group on a final measure of inductive reasoning, visualization, and problem solving while maintaining an equivalent level of manipulation and computation skills. However, the attitude of subjects in the experimental group towards the use of the computer in learning mathematics declined significantly.