At a large, regional university in the Midwest a specific course (Fundamentals of Advanced Mathematics) at the sophomore level provides a transition for student majors from the more computationally-based aspects of the first year courses to the more abstract upper-division courses. Surveys have been developed to measure the effects of this course on upper level courses in abstract algebra and advanced calculus. In addition, these surveys provide information about student learning in the major.
Background and Purpose
The University of Akron is the third largest public university in the state of Ohio. The departmental goals for the mathematics major state that each student should be able to read and write mathematics, to communicate mathematical ideas verbally, and to think critically. In addition, each student should acquire a body of fundamental mathematical ideas and the skill of constructing a rigorous argument; each student should achieve an understanding of general theory and gain experience in applying theory to solving problems. Several years ago the mathematics faculty became concerned about the difficulty experienced by some students in making the transition from the traditional calculus courses, where few written proofs are required, to more advanced work in mathematics. The abrupt transition to rigorous mathematics often affected even those who had done superior work in the calculus courses and was targeted as an important issue involved in a student's decision to leave the study of mathematics. To address the problem we designed the course Fundamentals of Advanced Mathematics (FOAM) to help bridge the gap of the students' understanding and preparation for continued mathematical training.
The prerequisite for FOAM is the second semester of calculus, so students can take the course as early as their sophomore year. FOAM is required for mathematics majors and for those preparing to be secondary mathematics teachers. Other students with diverse majors such as physics, engineering, computer science, and chemistry also enroll in the course (especially those who desire a minor in mathematics). The topics included in FOAM were designed to introduce the students to the goals we have for our majors. Elementary symbolic logic is the first topic of the course and sets the foundation for later work. A study of standard proof techniques, including the Principle of Mathematical Induction, requires the application of symbolic logic to the proof process. The proof techniques are then applied to topics such as sets (including power sets and arbitrary families of sets), relations, equivalence relations, and functions. Discussion of these major topics shows students the interconnections of ideas that explain and give meaning to mathematical procedures. Students can also see how mathematicians search for, recognize, and exploit relationships between seemingly unrelated topics. Throughout the semester students write many short proofs. For nearly all, this is a first experience at writing a paragraph of mathematical argument. The proofs are graded so that each student has plenty of feedback. The students also read proofs written by others and evaluate them for approach, technique and completeness. Students are required to make periodic oral presentations of their own proofs at the board. Once FOAM had been taught for a couple of years, an assessment was necessary to see if it achieved the broad goal of "bridging the gap." We needed to know if the mathematical topics covered were suitable preparation for more advanced courses. In addition we wanted the course to alleviate some of the difficulty in making the transition to proof-oriented courses.
Method
We decided to assess the goals for the course by developing a survey in two parts. In one part we would assess the perceived value to the student of the course content in preparation for more advanced work. The second part would evaluate the confidence level of the students in doing mathematics in the later courses. Before constructing a survey instrument, we interviewed students who were currently enrolled in FOAM to hear their reactions to the course. We also reviewed the literature for other attitude surveys, especially those done in mathematics. (See, for example, [1] and [2].) From the student responses and the literature, we selected statements to include in an attitude survey.
The content of FOAM was analyzed as viewed from the attitudes of students who took the course and then continued on to take advanced calculus or abstract algebra. Over the past five years, surveys were distributed to students near the end of the semester in the advanced course. Because the FOAM course is required, we do not have a control group of students who did not take FOAM before these advanced courses. Students were asked to rate each major topic in FOAM with respect to its value in their current courses. The topics and a frequency count of their responses are listed in Table 1. (Not all students responded to every question.) The arbitrary coding — extremely helpful = 3, moderately helpful = 2, slightly helpful = 1, not helpful = 0 — is used, and a mean is calculated.
The second phase of the evaluation process was to determine if this initial experience in conceptual topics increased the confidence of students in advanced mathematics courses. Confidence in learning mathematics was found to be related to student achievement in mathematics and to students' decisions to continue or not continue taking mathematics courses. (See, for example, the attitude studies in [1, 3].) To gauge this, the same students responded to a sequence of statements on an attitude survey. The statements, and the frequencies and means of the responses are shown in Table 2.
Findings
A quick inspection of the frequency counts in Table 1 suggests that students feel the studies of proof techniques, logic, induction and set theory were helpful. They were divided on the study of functions, perhaps because this is a final topic and may not have been covered completely in a given term.
| Evaluate how helpful the following topics from FOAM have been in Advanced Calculus and/or Abstract
Algebra. | ||||||
| MH=moderately helpful | ||||||
| NH=not helpful | ||||||
| EH | MH | SH | NH | Mean | ||
| A.Advanced Calculus | ||||||
| Proof techniques | 24 | 22 | 8 | 1 | 2.25 | |
| Set theory | 18 | 18 | 10 | 9 | 1.82 | |
| Functions | 9 | 24 | 14 | 8 | 1.62 | |
| Induction | 24 | 21 | 6 | 4 | 2.18 | |
| Logic | 20 | 21 | 13 | 1 | 2.09 | |
| B.Abstract Algebra | ||||||
| Proof techniques | 24 | 34 | 10 | 5 | 2.05 | |
| Set theory | 20 | 33 | 14 | 7 | 1.89 | |
| Functions | 16 | 27 | 20 | 7 | 1.74 | |
| Induction | 23 | 25 | 11 | 12 | 1.83 | |
| Logic | 22 | 29 | 11 | 10 | 1.88 | |
Note: EH=3, MH=2, SH=1, NH=0.
Respond to these statements using the given code |
||||||
D=disagree |
A=agree Sd=strongly disagree |
N=neutral/no opinion | ||||
| Sa | A | N | D | SD | Mean | |
| 1. The experience of taking FOAM |
21 | 37 | 20 | 15 | 8 | 2.48 |
| 2. The experience of taking FOAM |
13 | 31 | 27 | 24 | 6 | 2.21 |
| 3. I understood how to write mathematical |
0 | 16 | 13 | 50 | 22 | 1.23 |
| 4. After taking FOAM I am |
37 | 46 | 8 | 9 | 1 | 3.08 |
| 5. After taking FOAM I can |
36 | 43 | 12 | 9 | 1 | 3.03 |
6. I am still confused about |
4 | 27 | 20 | 43 | 7 | 1.60 |
7. I like proving mathematical |
6 | 35 | 24 | 27 | 9 | 2.02 |
8. Taking FOAM has increased my general |
21 | 52 | 12 | 13 | 2 | 2.77 |
9. If I had known about the abstract nature of |
2 | 8 | 21 | 46 | 24 | 1.19 |
Note: SA=4, A=3, N=2, D=1, SD=0.
From Table 2 we note that the students reported an increase in confidence and in general reasoning skills. Most noticeably, they feel they are better able to read and write mathematical proofs after taking FOAM. They were divided on whether their mathematical interest had increased. It is encouraging to see that most are still interested in continuing a mathematical career!
Use of Findings
We can conclude that we are teaching correct and relevant topics to help prepare the students for later work. Students reported that their confidence and skills in mathematics increased with their experience in FOAM. The boost in confidence should help retain students in the field.
The surveys conducted also indicate that in certain areas the attitudes of mathematics education majors (prospective secondary mathematics teachers) differ from mathematics majors. The graphs in Table 3 report the percentages of strongly agree/agree responses on selected questions by the two groups of students. (The question numbers refer to the questions on the attitude survey in Table 2.) Both groups feel they are better able to read and write mathematical proofs after taking FOAM. However, only about half (49%) of the mathematics education majors reported an increase in confidence for later courses and fewer than a third reported an increase in interest (29%). From conversation with these students we infer that they do not recognize where they will use this type of material in their future classrooms and, at times, resent the requirement of the courses in their degree program. We are trying to address these issues now in FOAM, and also in the advanced courses, by introducing examples of the actual material in the secondary school setting.
This project has led to additional questions and studies. Using multiple linear regression, we built a model to predict a student's grade in FOAM based on grades in the first two semesters of calculus. The model predicts that a student's grade will drop about 0.5 (on a 4-point scale) in FOAM. However, for a certain collection of students, grades drop dramatically when going from the calculus sequence to FOAM. These students are the subjects of a separate survey to determine where they attribute the cause of this decline in their performance. Finally, the original study concentrated on students who complete FOAM and continue into later courses. We have collected data on the students who do not complete the course, and thus decide to quit the study of mathematics at this point. Analyzing the perceptions of these students will be important to the extended project.
Success Factors
A reliable assessment study is a slow process. The sample is small since there are not many mathematics/ mathematics education majors, even at a large university. Furthermore, individual classes may vary in size (in our case, from 12-30). Also student attitudes and reactions to a course are undoubtedly dependent on the instructor. Thus the data is best collected and evaluated over an extended time period.
Note: In [4], Moore reports on a study of cognitive difficulties experienced by students in a course similar to FOAM. He discusses experiences with individual students in learning proof techniques and includes a related bibliography.
References
[1] Crosswhite, F.J. "Correlates of Attitudes toward Mathematics," National Longitudinal Study of Mathematical Abilities, Report No. 20, Stanford University Press, 1972.
[2] Fennema, E. and Sherman, J. "Fennema-Sherman mathematics attitudes scales: Instruments designed to measure attitudes toward the learning of mathematics by females and males,"JSAS Catalog of Selected Documents in Psychology 6 (Ms. No. 1225), 1976, p. 31.
[3] Kloosterman, P. "Self-Confidence and Motivation in Mathematics," Journal of Educational Psychology 80, 1988, pp. 345-351.
[4] Moore, R.C. "Making the transition to formal proof,"Educational Studies in Mathematics 27, 1994, pp. 249-266.
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* Research supported in part by OBR Research Challenge Grant, Educational Research and Development Grant #1992-15, and Faculty Research
Grant #1263, The University of Akron.
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