DEPARTMENT OF MATHEMATICS Baccalaureate Study in Mathematics Goals and Assessment of Student Learning Outcomes

Transcription

1 DEPARTMENT OF MATHEMATICS Baccalaureate Study in Mathematics Goals and Assessment of Student Learning Outcomes The Department of Mathematics at the Catholic University of America grounds its undergraduate courses of study in the three fundamental themes of modern mathematics -- analysis, algebra and geometry -- in a way that also emphasizes the logical structure and unity of mathematics. It offers three baccalaureate-level degree programs leading to either the Bachelor of Arts or the Bachelor of Science degree in mathematics. Majors interested in graduate studies in mathematics can choose either concentration. Majors intending to teach at the secondary level are advised to choose the B.S. in Mathematics and Secondary Education program, although some past B.A. and B.S. students have also gone on to teach in secondary schools. A flexible placement procedure is in place to enable each entering student to begin the mathematics program at the most suitable level. The fundamental theme of analysis is central to the calculus sequence for undergraduates, which develops the theory of functions of one or more variables and differential equations (MATH 121, 122, 221, 222) and at the same time exposes majors to how these concepts apply to other fields in science and engineering. An advanced course (MATH 521) explores the concepts of analysis from the ground up more rigorously. Algebra is the focus of courses in linear (MATH 501) and abstract algebra (MATH 505, 506), which cover the fundamentals of vector spaces, linear transformations, groups, rings and fields and take a first look at field extensions and Galois Theory. All of the department s courses in analysis and algebra involve ideas and concepts in geometry, and this fundamental theme is the explicit focus of the upper-level course in Euclidean and non-euclidean geometry (MATH 503.) Other departmental offerings, all of which use concepts from analysis, algebra and geometry, offer majors experience with important, specific topics in mathematics that are subjects of substantial current mathematical research and are also heavily involved in the application of mathematics to other areas. These include elective courses in Graph Theory (MATH 507), Number Theory (MATH 508), Combinatorics (MATH 515), Probability (MATH 531), Statistics (MATH 532), Stochastic Processes (MATH 533), Numerical Analysis (MATH 561), Logic (MATH 551), Fuzzy sets and Fuzzy Logic (MATH 537), Ordinary Differential Equation (MATH 540) and Partial Differential Equations (MATH 541).

2 Bachelor of Arts in Mathematics Program Description The Bachelor of Arts in Mathematics program requires thirteen courses in mathematics and two in physics, as follows. Four of the thirteen course are elective requirements. Note: The timing is typical, not mandated. It is common for a student begin the sequence with some advanced placement in calculus. Freshman Year: Analytic Geometry and Calculus I; (MATH 121) Analytic Geometry and Calculus II; (MATH 122) University Physics I; (PHYS 215) Sophomore Year: Analytic Geometry and Calculus III; (MATH 221). Calculus IV Differential equations; (MATH 222). Fundamentals of Advanced Mathematics; (MATH 305) University Physics II; (PHYS 216) Junior Year: Linear Algebra; (MATH 501) Abstract Algebra I; (MATH 505) Abstract Algebra II; (MATH 506) One 500-level elective requirement Senior Year: Introductory Analysis I; (MATH 521) Three 500-level elective requirements The 500-level mathematics electives are chosen from among: Euclidean and Non-Euclidean Geometry; (MATH 503) Graph Theory; (MATH 507) Elementary Number Theory; (MATH 508) Combinatorics; (MATH 515) Topology; (MATH 520) Complex Variables; (MATH 524) Chaotic Dynamics; (MATH 527) Fractal Geometry; (MATH 528) Probability and Statistics with Applications I, II; (MATH 531, 532) Stochastic Processes; (MATH 533) Introduction to Fuzzy Sets and Fuzzy Logic; (MATH 537) Ordinary Differential Equations; (MATH 540) Introduction to Partial Differential Equations; (MATH 541) Introduction to Mathematical Logic; (MATH 551)

3 Numerical Analysis I; (MATH 561) A selection of these courses is offered each academic year. Each of the courses in the list has been offered in recent years. Able students may consider mathematically-related summer internships. Some recent students have participated in REU programs ( Research Experiences for Undergraduates ). These are sponsored by the National Science Foundation. All the Mathematics Department s degree programs prepare graduates for a variety of careers, including jobs in industry, government, business, the actuarial profession, teaching at the elementary or secondary levels and, with the appropriate advanced degrees, teaching at the community college, college or university levels. Goals for Student Learning A student graduating with a Bachelor of Arts in Mathematics will: 1. Demonstrate proficiency in the body of knowledge provided in the core undergraduate curriculum to achieve the basic mathematical background required to start a career or pursue a graduate degree in mathematics or a field related to mathematics; 2. Demonstrate proficiency in reading mathematical texts and articles, writing and thinking; demonstrate familiarity with the axiomatic approach to mathematics (also being adopted increasingly in other fields in the natural and social sciences); 3. Think independently and critically and communicate mathematical information and analysis clearly, orally and in writing. Student Assessment Outcome Measures 1. Acceptance into the major requires completion of the basic core introductory courses MATH 121, 122, 221, 222 and 305 with a minimum GPA of Measures of subsequent progress toward the goals of the mathematics program are satisfactory performance (grades of C- or higher) in each of the following courses: MATH 501, 505, 506 and 521, as determined from graded homework assignments and course examinations, which stress problem solving, proof techniques, mathematical rigor and writing skills. A grade average of at least 2.0 is required in these courses. 3. The Senior Comprehensive Examination evaluates majors assimilation of material learned in coursework and abilities to think mathematically. It stresses problem solving, proof techniques, mathematical rigor and writing skills. Students are asked to handle prove-or-disprove problems, which put them in the real-life mathematical situation of not knowing the result in advance. A detailed description of the Senior Comprehensive Exam is attached. Use of Results to Improve Student Learning

4 The Mathematics Department is developing a system of on-going analysis with the following components: An annual review by committee of the results of the Comprehensive Exam, for the purpose of determining the degree to which our students have acquired the breadth of understanding of the major areas of mathematics, and the degree to which they can employ that understanding to solve problems. An annual review by committee of course evaluations from Math 305 on up which form the advanced portion of the curriculum. The review allows us to correlate student perceptions of the curriculum with our judgments of its effectiveness. An organized system for requesting comments from recent graduates to ascertain the effectiveness of our programs in helping them develop their careers, and to seek their views as to how their mathematics major succeeded in helping them, and how our programs might improve in that regard. A quadrennial review of the core curriculum for our major programs conducted by committee, based in part on the combined data from Comprehensive Exam results, course evaluations and information from graduates, with a report for discussion by the whole department. The review would include, but not be limited to, an examination of individual course content, an examination of the role of each course in the curriculum as a whole, a consideration of trends in the curriculum s effectiveness, a study of any external trends which might lead to, or require, a constructive response from us (e.g. modifying some course content; adding or deleting a course from the curriculum; modifying the structure of our programs).

5 Bachelor of Science in Mathematics Program Description The Bachelor of Science in Mathematics program requires thirteen courses in mathematics, two courses in physics, two computer science courses and four elective requirements in natural science or computer science which make substantial use of mathematics, as follows. Note: The timing is typical, not mandated. It is common for a student begin the sequence with some advanced placement in calculus. Freshman Year: Analytic Geometry and Calculus I; (MATH 121) Analytic Geometry and Calculus II; (MATH 122) Computer Science I; (CSC 123) Computer Science II; (CSC 124) University Physics I; (PHYS 215) Sophomore Year: Analytic Geometry and Calculus III; (MATH 221). Calculus IV Differential equations; (MATH 222). Fundamentals of Advanced Mathematics; (MATH 305) University Physics II; (PHYS 216) Natural Science/Computer Science elective Junior Year: Linear Algebra; (MATH 501) Abstract Algebra I; (MATH 505) Abstract Algebra II; (MATH 506) 500-level mathematics elective Natural Science/Computer Science elective Senior Year: Introductory Analysis I; (MATH 521) Three 500-level elective requirements Two natural Science/Computer Science electives The 500-level mathematics electives are chosen from among: Euclidean and Non-Euclidean Geometry; (MATH 503) Graph Theory; (MATH 507) Elementary Number Theory; (MATH 508) Combinatorics; (MATH 515) Topology; (MATH 520) Complex Variables; (MATH 524) Chaotic Dynamics; (MATH 527)

6 Fractal Geometry; (MATH 528) Probability and Statistics with Applications I, II; (MATH 531, 532) Stochastic Processes; (MATH 533) Introduction to Fuzzy Sets and Fuzzy Logic; (MATH 537) Ordinary Differential Equations; (MATH 540) Introduction to Partial Differential Equations; (MATH 541) Introduction to Mathematical Logic; (MATH 551) Numerical Analysis I; (MATH 561) A selection of these courses is offered each academic year. Each of the courses in the list has been offered in recent years. The natural science/computer science electives are chosen in consultation with an adviser from the Mathematics Department. Able students may consider mathematically-related summer internships. Some recent students have participated in REU programs ( Research Experiences for Undergraduates ). These are sponsored by the National Science Foundation. All the Mathematics department s degree programs prepare graduates for a variety of careers, including jobs in industry, government, business, the actuarial profession, teaching at the elementary or secondary levels and, with the appropriate advanced degrees, teaching at the community college, college or university levels. Goals for Student Learning A student graduating with a Bachelor of Science in Mathematics will: 4. Demonstrate proficiency in the body of knowledge provided in the core undergraduate curriculum to achieve the basic mathematical background required to start a career or pursue a graduate degree in mathematics or a field related to mathematics; 5. Demonstrate proficiency in reading mathematical texts and articles, writing and thinking; demonstrate familiarity with the axiomatic approach to mathematics (also being adopted increasingly in other fields in the natural and social sciences); 6. Think independently and critically and communicate mathematical information and analysis clearly, orally and in writing. 7. Demonstrate proficiency in computer science and knowledge of the use of mathematics in one or more areas of the natural sciences. Student Assessment Outcome Measures 4. Acceptance into the major requires completion of the basic core introductory courses MATH 121, 122, 221, 222 and 305 with a minimum GPA of Measures of subsequent progress toward the goals of the mathematics program are satisfactory performance (grades of C- or higher) in each of the following courses: MATH 501, 505, 506 and 521, as determined from graded homework assignments and

7 course examinations, which stress problem solving, proof techniques, mathematical rigor and writing skills. A grade average of at least 2.0 is required in these courses. 6. The Senior Comprehensive Examination evaluates majors assimilation of material learned in coursework and abilities to think mathematically. It stresses problem solving, proof techniques, mathematical rigor and writing skills. Students are asked to handle prove-or-disprove problems, which put them in the real-life mathematical situation of not knowing the result in advance. A detailed description of the Senior Comprehensive Exam is attached. Use of Results to Improve Student Learning The Mathematics Department is developing a system of on-going analysis with the following components: An annual review by committee of the results of the Comprehensive Exam, for the purpose of determining the degree to which our students have acquired the breadth of understanding of the major areas of mathematics, and the degree to which they can employ that understanding to solve problems. An annual review by committee of course evaluations from Math 305 on up which form the advanced portion of the curriculum. The review allows us to correlate student perceptions of the curriculum with our judgments of its effectiveness. An organized system for requesting comments from recent graduates to ascertain the effectiveness of our programs in helping them develop their careers, and to seek their views as to how their mathematics major succeeded in helping them, and how our programs might improve in that regard. A quadrennial review of the core curriculum for our major programs conducted by committee, based in part on the combined data from Comprehensive Exam results, course evaluations and information from graduates, with a report for discussion by the whole department. The review would include, but not be limited to, an examination of individual course content, an examination of the role of each course in the curriculum as a whole, a consideration of trends in the curriculum s effectiveness, a study of any external trends which might lead to, or require, a constructive response from us (e.g. modifying some course content; adding or deleting a course from the curriculum; modifying the structure of our programs).

8 Bachelor of Science in Mathematics/Secondary Education Program Description Undergraduate students who major in Mathematics and who wish to be eligible for teaching licensure may complete a joint program with the Department of Education. Students in the Mathematics Secondary Education Program must successfully meet the goals and requirements of the major and, concurrently, complete coursework and field experiences through the Department of Education. Program goals, outcomes assessments, and use of results for student learning for joint programs in secondary education are outlined in the description of Department of Education programs. The program requires eleven mathematics courses, two physics courses, two computer science courses, two statistics courses and seven courses in education. Note: The timing is typical, not mandated. It is common for a student begin the sequence with some advanced placement in calculus. Freshman Year: Analytic Geometry and Calculus I; (MATH 121) Analytic Geometry and Calculus II; (MATH 122) University Physics I; (PHYS 215) Computer Science I; (CSC 123) and Computer Science II; (CSC 124) or any two of: Introduction to Computers I; (CSC 104) Introduction to Computers II; (CSC 105) Computer Programming; (CSC 113) Sophomore Year: Analytic Geometry and Calculus III; (MATH 221). Calculus IV Differential equations; (MATH 222). Fundamentals of Advanced Mathematics; (MATH 305) Introduction to Statistics I; (ECON 323) Statistics II Intro Econometrics; (ECON 324) University Physics II; (PHYS 216) Foundations of Education; (EDUC 251) Psychology of Education; (EDUC 261) Junior Year: Linear Algebra; (MATH 501) Abstract Algebra I; (MATH 505) Abstract Algebra II; (MATH 506) One 500-level mathematics elective Curriculum and Methods in Adolescent Education; (EDUC 586) Teaching Mathematics in the Secondary Schools; (TRED 250)

9 (note: This course is taken at George Washington University) Senior Year: Introductory Analysis I; (MATH 521) One 500-level mathematics elective Supervised Internship and Seminar: Secondary Education; (EDUC 597, 598, 599) Goals for Student Learning A student graduating with a Bachelor of Science in Mathematics and Secondary Education will: 8. Demonstrate proficiency in the body of knowledge provided in the core undergraduate curriculum to achieve the basic mathematical background required to start a career or pursue a graduate degree in mathematics or in teaching mathematics at the secondary level; 9. Demonstrate proficiency in reading mathematical texts and articles, writing and thinking; demonstrate familiarity with the axiomatic approach to mathematics; 10. Think independently and critically and communicate mathematical information and analysis clearly, orally and in writing. 11. Demonstrate proficiency in teaching mathematics at the secondary level. Student Assessment Outcome Measures 7. Acceptance into the major requires completion of the basic core introductory courses MATH 121, 122, 221, 222 and 305 with a minimum GPA of Measures of subsequent progress toward the goals of the mathematics program are satisfactory performance (grades of C- or higher) in each of the following courses: MATH 501, 505, 506 and 521, as determined from graded homework assignments and course examinations, which stress problem solving, proof techniques, mathematical rigor and writing skills. A grade average of at least 2.0 is required in these courses. 9. The Senior Comprehensive Examination evaluates majors assimilation of material learned in coursework and abilities to think mathematically. It stresses problem solving, proof techniques, mathematical rigor and writing skills. Students are asked to handle prove-or-disprove problems, which put them in the real-life mathematical situation of not knowing the result in advance. A detailed description of the Senior Comprehensive Exam is attached. 10. The Praxis II Series of exams in Content Knowledge, Proofs, Models and Problems and Pedagogy are required for licensure to teach secondary level mathematics. The students take this series of exams prior to student teaching. 11. Under supervision of the Department of Education, students are placed in a local secondary school to develop their teaching skills during the semester of supervised student teaching (EDUC 597, 598 and 599). During this time, students create a portfolio which is evaluated by the Department of Education. The Department of Mathematics also sends a representative to observe the student in the secondary school. This representative provides feedback and commentary concerning the mathematical content and

10 organization of the student s classroom work. Use of Results to Improve Student Learning The Mathematics Department is developing a system of on-going analysis with the following components: An annual review by committee of the results of the Comprehensive Exam, for the purpose of determining the degree to which our students have acquired the breadth of understanding of the major areas of mathematics, and the degree to which they can employ that understanding to solve problems. An annual review by committee of the performance of students on external licensing exams. An annual review by committee of course evaluations from Math 305 on up which form the advanced portion of the curriculum. The review allows us to correlate student perceptions of the curriculum with our judgments of its effectiveness. An organized system for requesting comments from recent graduates to ascertain the effectiveness of our programs in helping them develop their careers, and to seek their views as to how their mathematics major succeeded in helping them, and how our programs might improve in that regard. A quadrennial review of the core curriculum for our major programs conducted by committee, based in part on the combined data from Comprehensive Exam results, external licensing exams, course evaluations and information from graduates, with a report for discussion by the whole department. The review would include, but not be limited to, an examination of individual course content, an examination of the role of each course in the curriculum as a whole, a consideration of trends in the curriculum s effectiveness, a study of any external trends which might lead to, or require, a constructive response from us (e.g. modifying some course content; adding or deleting a course from the curriculum; modifying the structure of our programs).

11 Bachelor of Science in Mathematics and Physics Program Description The Bachelor of Science in Mathematics and Physics program requires ten courses in mathematics, two courses in computer science, and twelve courses in physics, as given below. One of the ten mathematics courses is an elective requirement. Note: The timing is typical, not mandated. It is common for a student begin the sequence with some advanced placement in calculus. Freshman Year: Analytic Geometry and Calculus I; (MATH 121) Analytic Geometry and Calculus II; (MATH 122) University Physics I; (PHYS 215) Introductory Mechanics Laboratory; (PHYS 225) Computer Science I; (CSC 123) Computer Science II; (CSC 124) Sophomore Year: Analytic Geometry and Calculus III; (MATH 221). Calculus IV Differential equations; (MATH 222). Fundamentals of Advanced Mathematics; (MATH 305) University Physics II; (PHYS 216) Introductory Electricity Laboratory; (PHYS 226) Introduction to Modern Physics; (PHYS 506) Junior Year: Linear Algebra; (MATH 501) Abstract Algebra I; (MATH 505) Abstract Algebra II; (MATH 506) Mathematical Physics I; (PHYS 511) Mathematical Physics II; (PHYS 512) One 500-level mathematics elective Senior Year: Introductory Analysis I; (MATH 521) Thermodynamics and Statistical Physics; (PHYS 525) Quantum Theory I; (PHYS 531) Quantum Theory II; (PHYS 532) Analytical Mechanics; (PHYS 535) Electricity and Magnetism; (PHYS 536) The 500-level mathematics elective is chosen from among: Euclidean and Non-Euclidean Geometry; (MATH 503) Graph Theory; (MATH 507) Elementary Number Theory; (MATH 508) Combinatorics; (MATH 515)

12 Topology; (MATH 520) Complex Variables; (MATH 524) Chaotic Dynamics; (MATH 527) Fractal Geometry; (MATH 528) Probability and Statistics with Applications I, II; (MATH 531, 532) Stochastic Processes; (MATH 533) Introduction to Fuzzy Sets and Fuzzy Logic; (MATH 537) Ordinary Differential Equations; (MATH 540) Introduction to Partial Differential Equations; (MATH 541) Introduction to Mathematical Logic; (MATH 551) Numerical Analysis I; (MATH 561) A selection of these courses is offered each academic year. Each of the courses in the list has been offered in recent years. Able students may consider mathematically-related summer internships. Some recent students have participated in REU programs ( Research Experiences for Undergraduates ). These are sponsored by the National Science Foundation. All the Mathematics department s degree programs prepare graduates for a variety of careers, including jobs in industry, government, business, the actuarial profession, teaching at the elementary or secondary levels and, with the appropriate advanced degrees, teaching at the community college, college or university levels. Goals for Student Learning A student graduating with a Bachelor of Science in Mathematics and Physics will: 12. Demonstrate proficiency in the body of knowledge provided in the core undergraduate curriculum to achieve the basic mathematical background required to start a career or pursue a graduate degree in a field related to mathematics or physics; 13. Demonstrate proficiency in reading texts and articles in mathematics and physics, writing and thinking; demonstrate familiarity with the axiomatic approach to mathematics; 14. Demonstrate proficiency at a level appropriate for beginning graduate study in understanding the mathematical relationships which exist among the physical entities of the world and an understanding of the general principles of physics at the microscopic and macroscopic level. 15. Think independently and critically and communicate mathematical information and analysis clearly, orally and in writing. Student Assessment Outcome Measures 12. Acceptance into the major requires completion of the basic core introductory courses MATH 121, 122, 221, 222 and 305 with a minimum GPA of Measures of subsequent progress toward the goals of the mathematics program are

13 satisfactory performance (grades of C- or higher) in each of the following courses: MATH 501, 505, 506 and 521, as determined from graded homework assignments and course examinations, which stress problem solving, proof techniques, mathematical rigor and writing skills. A grade average of at least 2.0 is required in these courses. 14. The Senior Comprehensive Examination evaluates majors assimilation of material learned in coursework and abilities to think mathematically. It stresses problem solving, proof techniques, mathematical rigor and writing skills. Students are asked to handle prove-or-disprove problems, which put them in the real-life mathematical situation of not knowing the result in advance. A detailed description of the Senior Comprehensive Exam is attached. Use of Results to Improve Student Learning The Mathematics Department is developing a system of on-going analysis with the following components: An annual review by committee of the results of the Comprehensive Exam, for the purpose of determining the degree to which our students have acquired the breadth of understanding of the major areas of mathematics, and the degree to which they can employ that understanding to solve problems. An annual review by committee of course evaluations from Math 305 on up which form the advanced portion of the curriculum. The review allows us to correlate student perceptions of the curriculum with our judgments of its effectiveness. An organized system for requesting comments from recent graduates to ascertain the effectiveness of our programs in helping them develop their careers, and to seek their views as to how their mathematics major succeeded in helping them, and how our programs might improve in that regard. A quadrennial review of the core curriculum for our major programs conducted by committee, based in part on the combined data from Comprehensive Exam results, course evaluations and information from graduates, with a report for discussion by the whole department. The review would include, but not be limited to, an examination of individual course content, an examination of the role of each course in the curriculum as a whole, a consideration of trends in the curriculum s effectiveness, a study of any external trends which might lead to, or require, a constructive response from us (e.g. modifying some course content; adding or deleting a course from the curriculum; modifying the structure of our programs).