College-level Mathematics Courses Research Articles

THE IMPACT OF UNDERGRADUATE MATHEMATICS COURSES ON COLLEGE STUDENT’S GEOMETRIC REASONING STAGES

The purpose of this study is to investigate possible effects of different college level mathematics courses on college students’ van Hiele levels of geometric understanding. Particularly, since logical reasoning is an important aspect of geometric understanding, it would be interesting to see whether there are differences in van Hiele levels of students who have taken non-geometry courses that emphasize or focus on logic and proofs (Category I) and those that don’t (Category II). We compared geometric reasoning stages of students from the two categories. One hundred and forty nine college students taking various courses from the two categories have been involved in this study. The Van Hiele Geometry Test designed to find out students’ van Hiele levels was used to collect data. After the collection and analysis of the quantitative data, the participants’ van Hiele levels are reported and the reasoning stages of two groups are compared. The results show that students taking logic/proof based courses attain higher reasoning stages than students taking other college level mathematics courses, such as calculus. The results may have implications that are of particular interest to teacher education programs. Finally, the results also confirm a previous assertion about correlation between van Hiele levels and proof writing.

What are We Developing? A Case Study of a College Mathematics Program

Over one‐third of all college mathematics enrollments are in courses considered to be developmental. While such courses have been the subject of a large body of research, one question that seems not to have been studied empirically is the alignment of the content of developmental and college level mathematics courses. This paper gives the results of such a study, conducted at a medium sized public liberal arts university. While the content of the developmental mathematics courses was used subsequently in the Precalculus course, most of the content of the Intermediate Algebra course was not used in other college level mathematics courses. It is hoped that the results of this paper will serve as a catalyst for other institutions to examine carefully and define the mission of both their developmental and college level mathematics courses.

Activities for Students: Unit Fractions and Their Basimal Representations: Exploring Patterns

Major foci of secondary mathematics include understanding numbers, ways of representing numbers, and relationships among numbers (NCTM 2000). This article considers different representations of rational numbers and leads students through activities that explore patterns in base ten, as well as in other bases. These activities encourage students to solve problems and investigate situations designed to foster flexible thinking about rational numbers. Preservice teachers in a college-level mathematics course carried out these activities. Their conjectures and ideas are incorporated throughout this article.

Proof schemes and learning strategies of above-average mathematics students

What patterns can be observed among the mathematical arguments above-average students find convincing and the strategies these students use to learn new mathematical concepts? To investigate this question, we gave task-based interviews to eleven female students who had performed well in their college-level mathematics courses, but who differed in the number of proof-oriented courses each had taken. One interview was designed to elicit expressions of what students find convincing. These expressions were categorized according to the proof schemes defined by Harel and Sowder (1998). A second interview was designed to elicit expressions of what strategies students use to learn a mathematical concept from its definition, and these expressions were classified according to the learning strategies described by Dahlberg and Housman (1997). A qualitative analysis of the data uncovered the existence of a variety of phenomena, including the following: All of the students successfully generated examples when asked to do so, but they differed in whether they generated examples without prompting and whether they successfully generated examples when it was necessary to disprove conjectures. All but one of the students exhibited two or more proof schemes, with one student exhibiting four different proof schemes. The students who were most convinced by external factors were unsuccessful in generating examples, using examples, and reformulating concepts. The only student who found an examples-based argument convincing generated examples far more than the other students. The students who wrote and were convinced by deductive arguments were successful in reformulating concepts and using examples, and they were the same set of students who did not generate examples spontaneously but did successfully generate examples when asked to do so or when it was necessary to disprove a conjecture.

Identifying Cyclic Patterns of Interaction to Study Individual and Collective Learning

Our analysis of a college level mathematics course for prospective secondary mathematics teachers revealed that each student developed, at least to some degree, a conceptual orientation for teaching mathematics (A. G. Thompson, Philipp, Thompson, & Boyd, 1994). This initial finding led to a more in-depth question: If we assume an emergent perspective (Cobb & Bauersfeld, 1995; Cobb & Yackel, 1996) in which the values, practices, and social motivations of the classroom are believed to play critical roles in students' conceptual development, what social aspects emerged that supported these individual constructions? To address this question, we documented the emergence of a collective conceptual orientation and then used this construct to explore the reflexivity between its emergence and individual students' development of conceptual orientations.

Attitudes toward Statistics Assessments

Although statistics instructors have attempted to improve the cognitive aspects of instruction, particularly with respect to assessments, relatively little attention has been paid to non-cognitive issues, including students' attitudes, feelings, beliefs, perceptions, motivations, and interests. Thus, the purpose of the present study was to determine (1) the methods of statistics assessments that students most prefer; (2) the methods of statistics assessment that students feel induce the least amount of anxiety; (3) the methods of statistics assessments that students most rate as inducing higher-order thinking; (4) the characteristics of students with the most negative overall attitudes toward statistics assessments; and (5) how students rate performance assessments and authentic assessments. Findings revealed that students appear most to prefer statistics examinations in which at least limited supporting material is permitted. Interestingly, assessments in which some form of supporting material is allowed appeared to be more popular than are examinations with no time constraints. However, examinations that are untimed and in which supporting material is allowed were regarded best as inducing the least amount of anxiety, as increasing levels of performance, and as promoting higher-order thinking. Overall, students tended to rate performance assessments the most highly. Unfortunately, authentic assessments were not rated as highly as were performance assessments. Canonical correlation analyses suggested that age, the number of college-level mathematics courses, the number of years elapsed since students' last statistics class, and levels of statistics anxiety are determinants of these attitudes. The implications of these findings are discussed, and recommendations for future research are presented.

Novices entering mathematics 2: The graphic calculator and distance learners

For generations of students, learning mathematics has been a difficult and unrewarding experience. Electronic technology offers the opportunity to change the situation: it has made mathematics accessible—not just to a few gifted learners, but to the majority. In this paper we report a case study of how distance learners in an introductory college level mathematics course were able to learn concepts and develop their mathematical skills using a graphic calculator, and that they enjoyed their experience.

A Classroom Experiment Involving Basic Mathematics and Women

In recent years much attention has been given to the inadequate mathematics training of women and their anxiety toward mathematics. Experimental efforts to rectify this situation have used various approaches. Included in these attempts is the still slightly controversial innovation of all-female mathematics classes. Disagreement over the implications of single-sex mathematics classes has motivated continued research in this area. In 1978, Carolyn MacDonald published an article in this journal which dealt with an experimental precalculus all-female mathematics class. Differences in format for the all-female section made it impossible to isolate the effect of having only women in the class. This unanswered question was the primary basis for the experiment discussed in this article. Specifically, the objective of the experiment was to compare the achievement of women in an all-female section of M014, a basic mathematics course, with the achievement of women in mixed-sex classes. Basic Algebra (MO14) is a noncredit algebra course taught to approximately 1000 students each year at Indiana University, Bloomington. Students arriving at IU with a weak math background take M014 to provide them with the mathematical skills needed in college level mathematics courses such as finite mathematics and a survey of calculus. M014 is taught entirely by graduate students who are assisted by undergraduate interns. Classes are lecture-lab oriented with a stress on individualization. To assist in the individual treatment of students, two instructors and one undergraduate intern are assigned to each section, which consists of about 30-50 students. M014 is graded on a pass-fail basis and there are several chances to retake each of the four major tests. The women in this study were all undergraduate women enrolled in six sections of M014 during the spring semester of 1980. The female-only section consisted of 25

DEVELOPMENTAL MATHEMATICS: AN EX POST FACTO STUDY

Virginia Western Community College (VWCC) conducted a two‐year follow‐up of its developmental mathematics students. The students left a statistical path that led to some important implications about the efficiency of VWCC's program and that identified some variables and relationships that other developmental educators may want to investigate. Some major findings were: (1) Blacks and males enrolled in and failed developmental mathematics at significantly higher rates than whites and females. (2) Female instructors exhibited significantly higher pass rates than male instructors, and there appeared to be an interaction between the sex of the instructors and the sex of the students. (3) The developmental program effectively prepared its graduates for all of the college level mathematics courses except business mathematics. (4) The pass rates of the developmental teachers varied significantly, but there was no correlation between their pass rates and the college level success rates of their graduates.