Mathematics

Learning Outcomes/Goals:

For our majors two primary goals are:

  1. Students should have a working knowledge of calculus, including the ability to solve a wide range of problems and an understanding of the principles of differentiation and integration.
  2. Students should also have a working knowledge of linear algebra and develop a competency to understand and write clear, logical, mathematical arguments, including formal proofs.

Activities in Support of Goal:

Nearly all math majors take MTH 133 (Calculus II) and MTH 234 (Calculus II) as freshmen and sophomores and all majors take MTH 309 (Linear Algebra I) by the second semester of their sophomore year. We intend to assess these three courses in depth.

Assessment Method:

  1. A time series analysis, over a multi-year period, using data on student performance on Gateway Exams (MTH 133 only), final examination items analyzed by topic, and course grades.
  2. Summary statistics on course grade data from subsequent mathematics coursesusing departmental tracking data.

Assessment Results:

  1. At the end of the Fall 2004 semester, we examined MTH 309 final exam results forcore topics that had been agreed upon by all MTH 309 instructors. The exam consisted of 10 total questions, in 2 parts, in which all 4 questions in Part I were to be answered and any 4 out of 6 in Part II. There were a total of 81 exam scores. The results for Part II were deemed inconclusive due to the fact that there were too many variations of Part II due to the 15 possible ways of choosing 4 out of 6 exam questions. Therefore, we confined our attention to just Part I.
  2. For Part I, the mean score for all exams was 64%, which is acceptable, but indicates that more improvement is needed, especially in dealing with proof-type or theory-type problems as compared to more computational-type core problems.
  3. We found that of the 4 mandatory problems, two were computational and students averaged around 73%, while for the two core theory questions, students averaged about 58%, indicating clearly that more time should be spent dealing with theory and writing proofs; among which proofs by contradiction and "if-then" type proofsappear challenging, while proofs by induction or simple proofs by counterexample are somewhat easier for a student to deal with.
  4. At the end of Fall 2004, the 200 point MTH 133 final exam questions were categorized into core topics; namely, sequences and series, differentiation, and integration. The mean score for all student sections of MTH 133 was 112.86 or 56.4%. Of the 17 sections in which students are enrolled, 5 are small classes consisting solely of freshmen who have AP Calculus credit for MTH 132 (Calculus I). Students in these special AP Calculus sections averaged 131.6 on the final, while the 12 non-AP calculus sections averaged 105.7 on the final. This difference in section averages for the AP Calculus compared to the non-AP Calculus sections is due in part to the fact that "on track" students take MTH 132 or MTH 234 in the fall semester, not MTH 133. However, accelerated students who take AP Calculus AB in high school can start with MTH 133 in the Fall if they get a high enough AP Calculus score the preceding May.
  5. The MTH 133 final exam was further examined and key problems in the core areas wereidentified and comprised a total of 127 points, 54 for sequences and series, 27 for differentiation, and 46 for integration. In the most difficult topic, sequences and series, AP Calculus students averaged 64.9% while the non-AP students averaged just 47.2%. In the easiest of the three topics, differentiation, AP Calculus students averaged 74.8%, while the 12 non AP Calculus sections averaged 68.2%. Finally, in the third core area, integration, AP Calculus students averaged 74%, while the non-AP Calculus students averaged 58.6%
  6. Similarly, the Fall 2004, 200-point MTH 234 final exam questions were categorized into 4 core topics, namely, continuity and differentiability, integration, 2 and 3 dimensional geometry, and the Green and the Gauss theorems. The mean score for all 17 student sections of MTH 234 was 121.42 out of 200 or 60.7%. Of these 17 sections, 2 were special AP Calculus sections consisting solely of freshmen who received AP calculus credit for both Calculus I (MTH 132) and Calculus II (MTH 133). Students in the AP Calculus sections averaged 151.17 while the 15 non-AP Calculus sections averaged just 118.94.
  7. This difference in averages between the special AP and regular sections is interesting in that non AP calculus students who take MTH 234 in the fall semester are "on track" if they started with MTH 132 the previous Fall. However, the two special AP sections of MTH 234 are comprised of students who have taken a rigorous BC Calculus sequence in high school and are clearly very well prepared for Calculus III in the Fall. As in the MTH 133 final exam, key problems were identified in each of the 4 areas of the MTH 234 final and comprised a total of 172 points, 29 for continuity and differentiability, 24 points to Green, Gauss problems, 96 points to 3 dimensional integration, and 23 points to 2-3 dimensional geometry.
  8. As expected, the AP students fared very well in each of the 4 core areas with average scores of 82% in continuity and differentiation, 83.5% in Green-Gauss problems, 76.8% in integration in 3-dimensions, and 68% in 2-3 dimensional geometrywhile the 15 non-AP sections didn't do nearly as well, scoring 63.4% in continuity and differentiation, 59.4% in Green-Gauss problems, 61.l5% in integration, and 54.2% in 2-3 dimensional geometry.

Action Taken:

  1. For Spring 2005, MTH 309, we decided to make the following changes:
  2. We would have students answer all questions on the exam instead of giving them a choice of answering x out of y questions as was done in Part II of the Fall 2004 final exam.
  3. More attention would be paid to the wording of exam questions
  4. More attention would be given to writing of proofs, since MTH 309 is a course in linear algebra with an emphasis on proofs.
  1. As regards to both MTH 133 and MTH 234, initial data would indicate that studentswere performing at expected levels and no changes are anticipated in both of thesecourses until at least after the Spring 2005 data has been collected and analyzed.

Action Results:

As noted earlier, no action was taken for either MTH 133 or MTH 234. We continue to monitor both courses, and after Spring 2005 results are in, we will analyze them along with Fall 2004 results to see what changes, if any, are warranted. As far as Action Results for MTH 309 are concerned, we must await analysis of Spring 2005 data.

Future Steps:

To expand our assessment to MTH 132 (Calculus I) and to MTH 310 (Abstract Algebra I) and MTH 320 (Analysis I) since MTH 132, 133, and 234 is our basic calculus sequence for our majors and MTH 309, MTH 310, and MTH 320 is our basic entry-level sequence for senior-level (400 and above) courses.

Appendix (Another Goal that is Diversity Related)

Learning Outcomes/Goal:

To provide an environment that supports all learners and respects the diversity they bring.

Activities in Support of Goal:

The goal was assessed by means of three mathematics enrichment programs: Charles Drew Science Enrichment Laboratory (DREW), Mathematics Enrichment (ME): and the Emerging Scholars Program (ESP).

DREW is an enrichment program that is aimed at retaining underrepresented minorities in the sciences and mathematics. The ME programs is an enrichment program that focuses on at-risk students who need extra help preparing them to complete the University mathematics graduation requirement. The ESP program is an integrated, calculus-level, minority support program, integrated with rural students who have a similar under-performing profile as under-represented minority students. It was developed at Berkeley in the late 1970's by Uri Treisman and has now spread to some 100 schools in the U.S.

Assessment Method:

In DREW, the goal was assessed by means of graded problem sets, class attendance, quizzes, cumulative exams, comparison with expected MSU GPAs,DREW math placement test, oral presentations, collaborative learning, tutorial feedback, terms projects, inductive tasks that lead to generalizations for real real-life problems, and final exams.

Also, new DREW undergraduate and graduate teaching assistants are required to attend a ResourceCenter for People with Disabilities (RCPD) workshop during orientation week just before classes start (as is the case with all new undergraduate and graduate teaching assistants). In the RCPD workshop, new assistants are informed of ways to accommodate the needs of RCPD students such as providing extra time for exams and, in some cases, even a separate room to take exams.

In ME, the goal was assessed by graded problem sets, quizzes, exams, a final exam, class attendance, course grade comparison between ME and the course as a whole, and comparison of the percentage of students with 2.0 or above between ME and the course as a whole. During a workshop that takes place before classes start in the fall, someone from the Office of Supportive Services speaks with the ME TAs about the reasons that a student would be placed in a ME course. This helps the instructors understand our students better.

In ESP, the goal was assessed by quizzes, hour exams,a final exam, and class attendance in the enrichment sections.

Assessment Results:

We found thatDREW students did as well as or better than the regular University’s classes on the uniform final exam in MTH 1825 (pre-college algebra), MTH 103 (college algebra), and MTH 114 (trigonometry). We found that DREW students overall perform average on the MTH 132 uniform final exam. The DREW students also come out with higher freshman GPA’s than their predicted MSU GPA’s, based on academic history and geographical location( ie. high school district). The RCPD workshops experience works well, in general, to prepare assistants to deal with special accommodations for RCPD students.

In general, we find it best to let RCPD students initiate special accommodation requests with their instructors, since announcing special accommodation requests for examsto the entire classleads to a flood of requests from students who are not registered with RCPD yet feel they deserve special accommodations for exams.

For ME, we found that ME students usually do better than the course as a whole in both categories. The exception for the last three years (starting in FS01) has been MTH 103 during the Fall semester, which has performed a bit under the average for the course as a whole.Enrollment in ME courses has increased over time, which would indicate that the program has a good reputation among advisors and students. We almost always have a waiting list for the courses we offer. Many of our students sign up as a result of knowing a former ME student who recommended the program to them.

For ESP, we found several years after the ESP program started (in Fall 1992) that the average Calculus I course grades for ESP students was over 3.0 (out of a possible 4.0) whereas, the average grade for Calculus I as a whole averaged 2.3.

In 1996, ESP was expanded to Calculus II. The expansion to Calculus II has worked out well. However, the expansion downward to Precalculus (College Algebra and Trigonometry) has had some problems which we do not completely understand as yet. As compared to students who place directly into calculus, students who place into precalculus are far more fragile, seeming less willing and unprepared to handle the demands of university mathematics. They seem to require far more individual attention.

Action Taken:

For DREW,we decided to change by: A.) adding the use of technology (MyMathLab by Addison-Wesley) to provide additional tutorial, problem sets, and visuals for the calculus class. B.) We also added peer teaching to the calculus class. Peer teaching can occur in three ways: 1.) One way is to have students enrolled in the course present their solutions to problems that they are working on during the CURRENT class period. They present these problems to the whole class while at the board. 2.) The second way to do peer teaching is to give a student the responsibility for teaching a mathematical concept or a section in the book. They prepare a formal mini-lesson and present it to the other students in the NEXT class period. 3) The third way is by means of students tutoring other students on a one-to-one basis during the class or during tutorial sessions. C.) Since course supervisors must be informed of RCPD accommodation requests for exams, weplan to sendthe current list of course supervisors, which may change each semester, to the RCPD so they know who to contact to arrange for accommodations for exams.

In ME, MTH 103, for Fall 2004,we are using more worksheets and review materials in hopes of better preparing our students for exams.

Since Calculus I and Calculus II ESP students are doing well compared to non-ESP students, no significant changes have been made. We are, however, working on different ways to help precalculus students do better in their mathematics courses, such as giving precalculus ESP students more individualized helpas soon as they show any signs of having difficulty with the mathematics.

Action Results:

The changes for DREW were made in Fall 2004. MyMathLab was used both as a teaching resource and assessment tool for the MTH 132 (DREW section) course in Fall 2004.

Lesson plans were put on transparencies using the multimedia textbook. Thus, most of the class time was used in explaining the mathematical concepts instead of writing extensive notes. Students received the notes the day before as e-mail attachments, and brought their printed notes to the class.

Part of their assessment was done through two online quizzes in MYMathLab. The DREW section's mean on the uniform final exam was 134/200 and the median was 143. Even though this was a higher mean and median than in past semesters (Fall 2003, mean was 122, median 113; and Spring 2004, mean was 118, median 118), there are too many factors involved to directly attribute this rise to the teaching effectiveness of MyMathLab.

This semester MyMathLab will serve also as a learning resource for the students. They will be encouraged to use this technology beyond just the taking of on-line quizzes. Some of the concepts will be reviewed by them through self-generated practice exercises and by watching the videos of lectures in MyMathLab.

No significant changes are planned for dealing with RCPD test accommodations, since the current process works well as a whole.

Fall 2004 Math Enrichment results for MTH 1825 and MTH 103 are summarized as follows.

For Fall 2004, the Math Enrichment students had a higher average course grade in both MTH 1825 and MTH 103 when compared to the respective course as a whole. For MTH 1825, the average course grade was 2.53 for Math Enrichment and 2.14 for MTH 1825 as a whole. For MTH 103, the average course grade was 2.75 for Math Enrichment and 2.49 for the MTH 103 as a whole. Math Enrichment also had a higher percentage of the students earn a 2.0 or better.

For MTH 1825, the percentage of students who earned a 2.0 or better was 77.8% for Math Enrichment and 68.4% for the course as a whole. For MTH 103, the percentage of students who earned a 2.0 or better was 84.3% for Math Enrichment and 78.4% for the course as a whole.

As mentioned earlier, no significant changes have been made for College Algebra-Trigonometry (MTH 116), Calculus I (MTH 132), or Calculus II (MTH 133) students in ESP. This was justified by the very promising Fall 2004 results, whenwe compared course grades for ESP and non-ESP students taking MTH 116, MTH 132, or MTH 133.

The 751 non-ESP MTH 116 students had an average grade of 2.46, while the 45 ESP students averaged 3.26. In this group, the non-ESP underrepresented minority averaged 1.89, while the ESP underrepresented minority averaged 3.26. As for MTH 132, the 629 non-ESP students averaged 2.50, while their ESP counterparts averaged 2.73. When comparing non-ESP underrepresented minority MTH 132 students, we found they averaged 1.73 while their ESP counterparts averaged 3.25.

Finally, Fall 2004 grade data for MTH 133 showed that144 non-ESP, first time freshmen averaged 3.08, while ESP (all first time freshmen) averaged 3.40. Within the first time freshmen group, non ESP underrepresented minority students averaged 2.72, while their 22 ESP counterparts averaged 3.25.

Finally, non-first time MTH 133 students averaged 1.82 and within this group underrepresented minority students averaged 1.08.

Future Steps:

For DREW, we plan to continue using these markers given continued success.

For ME, we plan to continue changing our attendance policy in order to find an effective

one. For example, during the 2003-2004 academic year, we allowed a maximum of 12 unexcused events ( ie. being absent, tardy, or leaving early). During the 2004-2005 academic year we lowered the maximum to 7. We plan to talk to the teaching assistants at the end of the semester and ask them if they think we should lower the maximum number of unexcused events even further. Before deciding this, we must also analyze this year's attendance data. We also plan to collect more data concerning what happens to Math Enrichment students after they leave our program.

For ESP, we plan no significant changes for Calculus I and Calculus II ESPstudents. For precalculus, we are trying to devise a more natural and unobtrusive "early warning" procedure for finding those students who are beginning to have problems with their mathematics.