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.     

Improving student learning in calculus through applications

Nationally only 40% of the incoming freshmen Science, Technology, Engineering and Mathematics (STEM) majors are successful in earning a STEM degree. The University of Central Florida (UCF) EXCEL programme is a National Science Foundation funded STEM Talent Expansion Programme whose goal is to increase the number of UCF STEM graduates. One of the key requirements for STEM majors is a strong foundation in Calculus. To improve student learning in calculus, the EXCEL programme developed two special courses at the freshman level called Applications of Calculus I (Apps I) and Applications of Calculus II (Apps II). Apps I and II are one-credit classes that are co-requisites for Calculus I and II. These classes are teams taught by science and engineering professors whose goal is to demonstrate to students where the calculus topics they are learning appear in upper level science and engineering classes as well as how faculty use calculus in their STEM research programmes. This article outlines the process used in producing the educational materials for the Apps I and II courses, and it also discusses the assessment results pertaining to this specific EXCEL activity. Pre- and post-tests conducted with experimental and control groups indicate significant improvement in student learning in Calculus II as a direct result of the application courses.     

Prevailing educational practices for area measurement and students’ failure in measuring areas

The present research was carried out with the participation of 106 students in their last grade in Elementary School and revealed certain problems that these students faced in understanding the concept of area measurement. The students in the sample persisted on using measurement strategies that often led to failure.Our research plan comprises a comparison between the strategies for area measurement strategies used by two groups: the experimental group (E.G.) and the control group (C.G.). The experimental group attended a special teaching course, which stressed the conceptual characteristics of the area measurement process.The present research aims at revealing the students’ understanding, strategies, and misconceptions regarding area measurement. In addition to that, we examine whether the special teaching course and the use of different measurement tools may lead the two research groups to adopt different measurement strategies.     

Computers in abstract algebra

The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the availability of multi-representational tools and appropriate software that can be used to bridge the existing gap between the concrete and the abstract. This paper focuses on the use of the computer program, Excel, in teaching an introductory group theory course at the university level. Here the author describes the use of the Excel ‘fill-handle’ and ability to colour blocks of cells as cosets quickly in constructing and completing Cayley tables for abstract groups. The relation of the approach to constructivist philosophies and cooperative learning is discussed.     

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.     

Experimental determination of the concentration of stresses and strains around a circular hole in a glass-reinforced plastic specimen

The results of experimental studies on the concentration of stresses and strains at a circular hole in a glass-reinforced plastic specimen, involving the use of the photoelastic coating method, are given. The effects of the anisotropy of the material, of ratio of the hole diameter to the width of the model, and of the load level on the magnitude of the stress and strain concentration factor are indicated.Translated from Mekhanika Polimerov, No. 1, pp. 159–163, January–February, 1972.     

Developing statistical modelers and thinkers in an introductory, tertiary-level statistics course

While models are an important concept in statistics, few introductory statistics courses at the tertiary level put models at the core of the curriculum. This paper reports on a radically different approach to teaching statistics at the tertiary level, one that uses models and simulation as the organizing theme of the course. The focus on modeling and simulation??along with inference??was facilitated by having students use TinkerPlots? software for all modeling and analysis. Results from a 3-month teaching experiment suggest that a course focused on modeling and simulation through randomization and resampling methods in which students learn to think using a powerful and conceptual modeling tool can foster ways of thinking statistically. Furthermore, such an approach seems to help students develop experiences with and appreciation for the science and practice of statistics.     

An Experiment in Teaching College Mathematics†

Kac has observed that the ideal preparation in mathematics, especially for non‐mathematicians, should focus not on acquiring skills but on acquiring certain attitudes. We administered a special attitude questionnaire to a sample of graduate students in mathematics and undergraduate speech majors. We found significant differences on 10 of 27 items on this test. We then administered this test to a mixed group of undergraduates at the beginning and at the end of a special experimental mathematics ‘course’ designed to modify and shape attitudes. We found changes in attitudes in the intended direction. The primary aims of the experimental course were to:

1. Get students without any prior acquaintance with mathematics or a fear thereof to approach their studies more analytically.

2. Acquire orientation to and acquaintance with 25‐75 basic concepts and methods covering sets, algebra, logic, computers, analysis, probability, math‐statistics and topology in an over‐all map of how they logically fit together and how they relate to problems of modern life.

3. Read, with appreciation, mathematical literature previously incomprehensible to them. These aims were met.

     

Comparisons of Success and Retention in a General Chemistry Course Before and After the Adoption of a Mathematics Prerequisite

This study of the relationships between mathematical ability and success and retention in a general chemistry course was conducted at an open‐enrollment university whose mission is to provide a quality education to a culturally and economically diverse student body. We studied the correlation between the demonstrated level of mathematical ability and success in chemistry and the correlation between the demonstrated level of mathematical ability and retention in chemistry. After the chemistry department implemented a mathematics prerequisite for the chemistry course, data were examined to compare success and retention prior to and after the adoption of the prerequisite. Analysis showed that success and retention in chemistry increased after the adoption of the mathematics prerequisite.     

Using interactive algebra software to support a discourse community

In this work we studied the impact of using NuCalc, an interactive computer algebra software, on the development of a discourse community in a college level mathematics class. Qualitative and quantitative data were collected over the course of 3 weeks of instruction. We examined the influence of the software on: group interactions; the mathematical investigations of learners; and the teacher’s interactions with students. Data points to four distinct ways in which the presence of NuCalc positively impacted the learning community we studied: (1) it served as a tool for extending students’ mathematical thinking, (2) it motivated students’ engagement in group discourse, (3) it became a tool for mediating discourse, (4) it became a catalyst for refining the culture of classroom, shifting the patterns of interactions between the teacher and learners.     

Chance and probability: What do they mean to university engineering students?

The great interest aroused by the incorporation of Statistics and Probability into curricular projects has been accompanied by considerable evidence of significant difficulties in the meaningful learning and application of the concepts. These difficulties have been the subject of many studies, mostly concerning secondary school students. This study seeks to investigate the level of understanding of random phenomena, in a group of university students, on the second year of a Technical and Industrial Engineering Degree at the University of the Basque Country, in Spain. The students, who had all undertaken an introductory course in Probability Theory, were tested through both written questionnaires and one-to-one interviews, where they were presented with situations that demanded the ability to apply the knowledge they had learnt, at a high level, to a range of situations and to justify their reasoning. The results show that the vast majority of students had a poor understanding of random phenomena, applying alternative ideas to the ones taught on their course in Probability. The need to improve teaching strategies in Probability and Statistics, so that students can develop skills in constructing probabilistic models, is highlighted.     

Modeling partial harvesting in intensive shrimp culture: A network-flow approach

Despite the promising laboratory results and theoretical arguments on the potential of partial harvesting in enhancing the productivity and profitability of intensive shrimp growout facilities, the implementation of partial harvesting in practical operation is rather difficult due to its complex nature. In this paper, we developed a practical model of partial harvesting using the network-flow approach so that it can be readily implemented and solved in MS EXCEL. We demonstrated the use of this spreadsheet model with data from a commercial shrimp farm in Hawaii. The results indicate that the model is capable of identifying the efficient harvest policy as well as assessing the viability of partial harvesting under a variety of managerial conditions and objectives.     

Decentralized optimal control of a group of dynamical objects

The problem of optimal control of a group of coupled dynamical objects is considered. The cases are examined in which the centralized control of a group of objects is impossible. Fast real-time optimal control algorithms of each of the dynamical systems are described that use information exchanged between group members in the course of control. The proposed methods supplement the earlier developed real-time optimal control methods for an individual dynamical system. The results are illustrated using optimal control of two coupled mathematical pendulums as an example.     

The role of computer simulations in learning analytic mechanics towards chaos theory: a course experimentation

We present the results of an experimental project, made at the University of Calabria, which uses computer simulations and scientific computational systems in teaching and learning the fundamentals of analytic mechanics, in particular the classical Lagrangian and Hamiltonian formulation of the matter. Starting with examples of classical dynamical systems we also introduced students to appreciate chaos concepts going step by step to more complex settings. The success obtained with the use of computer-based methodologies has been very clear and the final students' tests confirm it. In this paper, in particular, we compare the results obtained by four groups of students: the first two attended such an experimental course and employed computer support for their individual work at home, while the learning environment of the second two groups (the control ones) was based only on traditional classroom lectures with paper and pencil home activities. If on the one hand a better overall performance of the first two groups was evident, it is very interesting (and not so easy) to investigate how much in quantity and quality, the learning of the experimental groups differs from that of the other groups, and why. For example, we illustrate some relevant differences between them in the comprehension of the studied phenomena, in dealing with and solving the proposed problems, and also in the individual development of further personal ideas.     

Integrating design problems in mathematics curriculum: an architecture college case study

This paper reports an attempt to improve results in the mathematics course in one of the architecture colleges in Israel through practise in applications. The effect of integrating structure design problems in the calculus curriculum on students' achievements and attitudes was examined. The applied topics in the curriculum were connected to calculus topics and studied through problembased learning activities. The integrated curriculum was implemented and the learning results in experimental and control groups were assessed by means of achievement tests, attitude questionnaires and student interviews. The learning achievements in the experimental group proved to be significantly higher than in the control group. The positive impact of learning applications on motivation, understanding, creativity and interest in mathematics is indicated.     

Characteristics of Second Graders' Mathematical Writing

This study compared the characteristics of second graders' mathematical writing between an intervention and comparison group. Two six‐week Project M2 units were implemented with students in the intervention group. The units position students to communicate in ways similar to mathematicians, including engaging in verbal discourse where they themselves make sense of the mathematics through discussion and debate, writing about their reasoning on an ongoing basis, and utilizing mathematical vocabulary while communicating in any medium. Students in the comparison group learned from the regular school curriculum. Students in both the intervention and comparison groups conveyed high and low levels of content knowledge as indicated in archived data from an open‐response end‐of‐the‐year assessment. A multivariate analysis of variance indicated several differences favoring the intervention group. Both the high‐ and low‐level intervention subgroups outperformed the comparison group in their ability to (a) provide reasoning, (b) attempt to use formal mathematical vocabulary, and (c) correctly use formal mathematical vocabulary in their writing. The low‐level intervention subgroup also outperformed the respective comparison subgroup in their use of (a) complete sentences and (b) linking words. There were no differences between groups in their attempt at writing and attempts at and usage of informal mathematical vocabulary.     

Probing interactions in exploratory teaching: a case study

This study investigates an exploratory teaching style used in an undergraduate geometry course to help students identify an ellipse. We attempt to probe beneath the surface of exploration to understand how the actions of teachers can contribute to developing students’ competence in justifying an ellipse. We analyse the complex interactions between student, content, and teacher, and discuss explicit pedagogical strategies that help students develop a higher level of geometric reasoning. The findings indicate that students engaged in guided explorations by the teacher and in group discussions with peers were able to identify an ellipse and justify their reasoning.     

Pre- and in-service teachers’ perceived value of an experimental real analysis course for teachers

Prospective secondary mathematics teachers are usually required to complete several university advanced mathematics courses before being certified to teach secondary mathematics. However, teachers usually do not find these courses to be valuable for their teaching. We designed an experimental real analysis course with the goal of making real analysis content useful and relevant to teaching. Our approach was to ground the real analysis content in pedagogical situations that problematized a secondary mathematics topic, where the nuances of teaching secondary mathematics could be informed by the real analysis that was covered. The experimental course was implemented in a graduate teacher education programme with 32 pre- and in-service teachers (PISTs). After the course, we conducted focus group interviews with 20 of these PISTs to get feedback on how the course was valuable to their teaching practice. Many PISTs found the course to be valuable for teaching secondary mathematics, as well as for their understanding of secondary mathematics and real analysis.