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At a mid-sized, regional university in the East each student must complete a one-semester senior seminar in which a variety of assessment methods are used to assess student learning in the major. These methods include: a traditional final exam, a course project, an expository paper, reading and writing assignments, a journal and a portfolio — all of which are designed to assess a variety of skills acquired throughout a student's four years.
Background and Purpose
Kutztown University is one of fourteen universities in the State System of Higher Education in Pennsylvania. Of the approximately 7,000 students, four to six mathematics majors graduate each year. (This excludes those who plan teaching as a career.) As in many undergraduate institutions, a student with a mathematics major at Kutztown University is required to take a wide variety of courses that include a calculus sequence, several proof-based courses like abstract algebra and advanced calculus, and several application courses like linear algebra, and probability and statistics. It is typically the case, too, that a student's grade for each of these courses depends heavily (and sometimes entirely) on objective, computational exams. Because of options given to the students, rarely (if ever) do two students who finish our cafeteria menu of mathematics courses experience identical mathematics programs. How, then, does one measure the quality of the major? How can we ensure that each major is equipped to face a future in graduate school or in the work force?
One requirement for all of our mathematics majors is the successful completion of the Senior Seminar in Mathematics. It is designed as a culminating experience. It used to be taught using the lecture method on a mathematical topic of the instructor's choice, and was assessed using only traditional examinations. I was first assigned to teach the course in 1992. Having already begun to incorporate oral presentations in some of my other classes, I decided to experiment with nontraditional teaching methods and alternative assessment techniques. The course has evolved to the point that no lectures are given. Students now study a variety of mathematical topics, keep a journal, participate in class discussions, write summaries of readings, write a resume, do group problem solving, give oral presentations, and complete a project. (The project consists of writing an expository paper and of delivering an oral presentation, both of which are results of an exploration of a mathematical topic of each student's choice.) Such activities enable students to more successfully achieve these course objectives:
Method
Due to the small class size (about 10), considerable individual attention can be paid to each student. Moreover, assignments can be adapted to meet the needs of a particular group of students. The course grade is determined by a collective set of assessment techniques that include holistic grading, traditionally graded problems, instructor evaluations based on preselected criteria, faculty-student consultations, and portfolios. In the paragraphs that follow, assignments are described along with the assessment techniques used for them. Objectives that are addressed by the assignments are identified by number at the end of each paragraph.
Early in the course, each student is given a different mathematical article to read and is asked to organize and deliver a five minute talk about it. The content is elementary in nature in order that the student focuses on the mechanics of communication and is not intimidated by the content. While each classmate identifies (in writing) at least one positive and one negative attribute, the instructor uses a set of criteria known to the students. Both forms of evaluation provide immediate feedback to the speaker. Using the same mathematical article, students give a second talk, implementing the assessment input from the first talk. This pair of talks serves as preparation for the 25-minute oral presentation later as part of the course project. This longer talk is assessed two ways: the instructor uses written evaluations based on preset criteria and a team of three members of the department faculty uses a pass/fail evaluation based on the student's apparent level of understanding, and on the cohesion of the talk. (Objectives 1, 2, 3, 4, 5, 6)
To indicate how the major program is complemented by the general education curriculum, several class discussions of preassigned readings are held. Topics for discussion include historical, ethical, philosophical, and diversity concerns and how they are related to the mathematics profession. Assessing performance in a class discussion is based on two criteria: Did the student understand the main idea of the assigned readings? Did the student convey individual interpretations of the issues at hand? (Objectives 1, 3, 9)
There are a variety of writing assignments. These include journal writings, summaries of readings from books and professional journals, the writing of a resume, and the composition of a 20-page expository paper. Students keep a journal that reflects their thoughts on mathematics-related topics. Three to five entries are made each week. The journals are holistically graded three times during the semester. In addition, each student uses the journal to provide a partial self-assessment on the final examination for the course. (Objective 9)
Students read articles from books and journals and then write a two-page summary of each one. These are graded holistically, as they are used primarily for diagnostic purposes. Students are also required to write their resume for this course. They are urged to obtain assistance from the Office of Career Services. (Objectives 1, 3, 8, 9)
As part of the course project, each student writes a 20-page expository paper on a mathematical topic of his/her choice. The student is guided through several steps of the writing process throughout the semester. A draft of the paper is highly scrutinized based on preselected criteria (knowledge of the subject, breadth of research, clarity of ideas, overall flow of the paper, etc.). No grade is assigned to the draft. However, the same criteria are used in the final assessment of the paper. (Objectives 1, 2, 3, 4, 5, 6, 8)
Every college graduate should possess resource skills: library skills; internet skills; professional networking skills; and skills needed to find career-related information. To develop skills, students are given sets of specific questions, often in the form of a scavenger hunt. For example, they may be asked to find the number of articles published by Doris Schattschneider in a given year and then conjecture about her field of expertise, or they may be asked to find out as much as possible about Roger Penrose from the World Wide Web. These assignments are graded on correctness of answer. (Objectives 1, 2, 5, 6, 7, 8, 9)
The content portion of the course includes a wide variety of mathematical topics that are typically not covered in other courses required of our majors. These include graph theory, complex variables, continued fractions, fractals, non-Euclidean geometry, as well as topics in more classical areas of mathematics. (I have used books by William Dunham, The Mathematical Universe and Journey Through Genius: Great Theorems of Mathematics, as stepping stones into a vast range of topics.) This portion of the course is designed so that students both learn and tackle mathematics problems in small groups. The grouping of students is done by the instructor and is changed with each new topic and set of problems. Students earn both individual grades for correct solutions, and class participation points for team interaction. Groups are changed frequently in an effort to develop some degree of leadership skill in each student, and to allow students to learn of each others' strengths and knowledge base. (Objectives 1, 2, 3, 5, 6, 7, 8)
The final examination for the course consists of two parts: a set of questions, and a portfolio. The set of questions are principally content-based problems, to measure the student's mathematical knowledge based on the problems that were worked in small groups. There are a few open-ended questions that ask students to compare and contrast perspectives found in the assigned readings, written summaries, and class discussions. Students also provide an assessment of their attitudes and study habits that is based on their own journal entries. (Objectives 2, 3, 5, 6, 9)
A portfolio is prepared by each student and is focused on the individual's perceived mathematical growth. Students are asked to describe and document the growth that they made during the past three months. Content of the portfolio need not be restricted to materials obtained from the Senior Seminar. They write an essay unifying the various items that they choose to include. Assessment of the portfolio is based upon the breadth of perceived learning and how it is unified. Preselected criteria are used in the assessment. (Objectives 3, 8, 9)
Findings
Results of assessments of the oral components of the course indicate that examinations are not always a true indicator of a student's level of understanding. Mathematical knowledge can often be conveyed more effectively by oral discourse than by written computations or expositions.
Assessment results from writing assignments indicate that our mathematics majors need more than basic composition skills prior to enrolling in the Senior Seminar in Mathematics.
Prior to the reading assignments, few (if any) connections were recognized between the knowledge gained through general education courses and the discipline of mathematics.
Although students are more technologically prepared as they enter college than those who entered four years ago, results from the resource skills assignments indicate that little demand is made on them to develop further their information-retrieval skills during their undergraduate years.
A significant amount of bonding occurs as a result of working in small groups. Consequently, classmates are often used as resource people while the course project is being prepared. Students are forced to work with a variety of personality combinations, which each of them will have to do in the future.
Use of Findings
The oral presentations illustrate that students often can better demonstrate understanding by presenting a topic orally. Other mathematics courses should (and have begun to) incorporate oral assignments to facilitate learning.
Our major programs already require that students complete a scientific writing course and it is strongly recommended that a student complete this course prior to enrolling in the Senior Seminar. More writing needs to be incorporated in other mathematics courses as well as remain in the capstone course. Ideally, every mathematics course should incorporate some form of nontechnical writing component; realistically, post-calculus applications-based courses such as differential equations, numerical analysis, probability and statistics, and operations research appear to be those most suited for such assignments.
Students with severe writing difficulties (grammar, punctuation, run on sentences, etc.) or with reading deficiencies are identified early in the semester and advised to get extra help. Such assistance can be obtained without cost to the student from the Writing Lab and from the Department of Developmental Studies.
Information gleaned from the journals is used to identify an individual's strengths and weaknesses and to identify questions and concerns that confront the student. The instructor is able to provide relevant, timely feedback and is better able to serve as a facilitator of learning.
As a result of producing a resume, students become aware of the skills that may be required for a career that uses mathematics. The academic advising of our majors should be improved so that such information is realized by students earlier in the program. Then appropriate coursework could be recommended to help develop these skills. We also need to demand that more technological and resource skills be used in lower-level mathematics courses.
Class discussions indicate the need to guide the students in making connections: those among the many aspects of the discipline of mathematics; and those between mathematics and the general education component of the university degree. This type of assignment should remain part of the Senior Seminar course.
Results from the content-based portion of the course indicate that the assessment of previous coursework is reliable inasmuch as it measures computational skills and elementary critical thinking skills. Existing performance standards in required mathematics courses appear to be consistent.
Information gleaned from student portfolios is sometimes enlightening. For example, one student wrote that she felt that the amount of writing done in this course far surpassed the total amount of writing in all of her other classes combined! This reinforces our conclusion that more writing needs to be incorporated in other courses, whether mathematics courses or not. The portfolio also provides us with an early warning system for potential problems. For example, one student complained that our mathematics program did not prepare him for entrance into the actuary profession. (He took and did not pass the first actuary exam three times.) This has alerted us to try to better explain the difference between an undergraduate degree in mathematics and an individual's ability to score well on standardized tests.
Success Factors
By using a variety of assignments and assessment techniques, the Senior Seminar in Mathematics has become a place in which many skills can be developed. Assignments are all related to the successful pursuit of a career in mathematics, most of which require skills that lie outside the traditional scope of a mathematics content course. The course works because of its small enrollment, and the instructor's knowledge of a wide variety of writing techniques, mathematical fields, and assessment techniques, and the willingness of colleagues in the department to maintain quality programs. Faculty colleagues who have assessed an oral presentation very quickly gain a perspective of the strengths and weaknesses of our major programs. Continued support by them is an invaluable asset when changes in the programs are recommended.
However, it takes time and patience to build
quality assignments and develop meaningful assessment
devices, and this should be done gradually. The strengthening of
the major that results is worth the effort.
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