Natural Sciences and Mathematics || The Teaching of Mathematics at the College and University Level

The Teaching of Mathematics at the College and University LevelAuthor(s): James H. ZantSource: Review of Educational Research, Vol. 34, No. 3, Natural Sciences and Mathematics(Jun., 1964), pp. 347-353Published by: American Educational Research AssociationStable URL: http://www.jstor.org/stable/1169410 .

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CHAPTER VIII

The Teaching of Mathematics at the College and University Level

JAMES H. ZANT

1 wo IMPORTANT requirements for the improvement of the teaching of mathematics at the college and university level are (a) the presence of well-trained mathematicians as teachers and (b) a modern program of courses. In the colleges and universities, it is well known that the supply of well-trained teachers is short of the demand. With college enrollments increasing for the immediate future, it is apparent that many departments of mathematics must utilize less-competent staffs than they should like to employ.

Wisner (1961) reported that the annual number of doctoral degrees awarded in mathematics in the United States rose very slowly during the 13 years from 1948 to 1960. Since the number for 1960 was only 303, Wisner was not able to predict any rapid rise in the immediate future. Lindquist (1961) presented similar data in an article in the American Mathematical Monthly.

In a recent report, Lindquist (1963) revealed that there had been in- creases of 26.8 percent and 13.5 percent, respectively, in the number of degrees awarded at the master's and doctor's levels during the 1960-61 academic year in comparison with the 1959-60 period. Although the percentage of increase in doctorates in mathematics was substantial, it was smaller than that noted in four other major fields. It was his opinion that the number of degrees awarded in mathematics would continue to rise significantly at both the undergraduate and graduate level in the foreseeable future.

Training of Mathematicians for College Teaching

The sort of training mathematicians need for college teaching is well recognized, and the efficiency of this procedure is well authenticated. Ref- erence may be made to the Ph.D. programs of American universities. After studying basic mathematics over most of the areas in the field, a candidate for the Ph.D. degree chooses a specialty, and through advanced courses and seminars obtains a thorough background in this specialty. He becomes familiar with material from textbooks as well as with the results of past and current research. Finally he chooses a dissertation topic and gains a familiarity with virtually all the research related to it. The report

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of the results of his individual, original research becomes his doctoral dissertation.

Such a mathematician is academically qualified to enter the field of college teaching. About one-half of mathematicians with doctorates select teaching as a career; the others select jobs with higher salaries in industry and government service. As mathematicians, they continue their interest in research as it is published in the journals. They continue to do research, and many of them publish papers on the results they obtain. Such activities keep the mathematician abreast of his field, give him advanced knowledge, and add knowledge to the growing field of mathematics itself.

Of the many hundreds of research papers in mathematics, many are useful only in advanced courses or for advanced students. Some, however, are pertinent to undergraduate courses. All are considered useful to the teaching mathematician for the reasons previously cited. From this point of view, current and past research being done by mathematicians in the universities, in industry, and in government service makes a vital contribution both to the training of students and to the subsequent teaching that many will do at the college and university level.

It is obviously not possible to give any comprehensive list of these research studies in mathematics. It is also not possible to say that a particular study is significant without knowing the background of a particular teacher, the courses he teaches, and the preparation level of the students in these courses.

One article by Margaris (1961) was chosen in a purely random fashion for citation. As is suggested by the title, "Successor Axioms for the Inte- gers," the article deals with a topic usually discussed in introductory courses in modern algebra or in the foundations of mathematics. The content of the paper would also enlighten teachers and students in a number of other areas of college mathematics. Although its contents are useful for a teacher of college mathematics, the same general ideas might be acquired from a number of other papers which could be listed.

Modern Curriculum in College Mathematics

The second vital need in mathematics teaching at the college and university level is a mathematics curriculum that includes a modern program of courses. The Committee on the Undergraduate Program in Mathematics (CUPM) of the Mathematical Association of America has probably made the most consistent and widespread contribution of any professional organization. However, their efforts are not necessarily superior to those made by many individual mathematicians who have developed textbooks and carefully organized courses for the colleges and universities. With liberal support from the National Science Foundation, CUPM has organized four panels for work in various areas of the improvement of the teaching of college mathematics: (a) Panel on Pre-Graduate Training, (b) Panel on

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Mathematics for the Biological, Management, and Social Sciences, (c) Panel on Mathematics for the Physical Sciences and Engineering, and (d) Panel on Teacher Training.

Although the work of CUPM probably cannot be classified as research, it has been significant. Through its Panels, CUPM has devoted its main efforts to rewriting the college curriculum in mathematics in outline form. This task has been done by holding many meetings and conferences in- volving not only mathematicians but also physical scientists, engineers, biological scientists, management personnel, social scientists, elementary and secondary school teachers, teachers of education, school administrators, and many others. The purpose of these conferences has been to decide on the content and structure of mathematics courses for the various areas of the discipline. Much effort has been spent, largely by mathematicians, in writing outlines and course guides for modern courses in mathematics. It is hoped that these activities will encourage competent mathematicians to write textbooks from this modern point of view.

This group effort, which is recognized by CUPM as a first but not complete step, is having the desired effect. Some colleges and universities are teaching modern courses in mathematics and are using textbooks stimulated by CUPM recommendations. Though these improvements are widespread, the percentage of colleges using such programs is still small. Much remains to be done.

It should be obvious at this point that there are many questions which could and should be answered with these programs in mind: for example, Do mathematics courses such as those proposed by CUPM prepare engineering students better for their future study and/or practice of engineering than do traditional programs? This question and many others cannot be answered until new courses have been organized and taught, or until students have progressed into more advanced courses. One college of engineering spent a year studying the CUPiv recommendations on mathematics for engineers. The staff are now convinced that CUPM has suggested a promising program for use with their students. This decision is based to a large extent on their stated objective that the aim of the college of engineering is to train engineers to handle competently the engineering problems that will face them 25 years hence. However, since an established institution cannot be reorganized quickly, their present timetable calls for a complete implementation of the mathematics program in September 1970.

This delay does not mean that all research must wait that long; it does mean that some of the research on teaching problems in college mathematics may not be significant. For example, a study designed to show that students learn traditional college algebra more adequately (or less adequately) in large than in small classes has no significance for a college that sees no particular advantage to teaching college algebra from this point of view. Such research studies are not included in this chapter.

A review of the literature does reveal a few recent investigations that

may be considered significant. These studies fall into several areas. Four

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areas will be discussed: (a) selection and enrollment of mathematics students, (b) prediction of success and achievement, (c) attitudes and opinions toward mathematics, and (d) curriculum and improvement of teaching.

Selection and Enrollment of Mathematics Students

Most studies concerned with selection and enrollment have been made of students in general rather than of mathematics students alone. There were two noteworthy exceptions.

Sparks (1960) made a comparison of students in Iowa schools that ranked high and ranked low in mathematical achievement. He found that students from schools ranking high in achievement, in contrast to those students from schools ranking low, were more likely to take additional mathematics, liked mathematics more, and had devoted more out-of-class time to its study. Teachers in these high-ranking schools, in comparison to those in low-ranking ones, had longer tenure and more college hours in mathematics, helped the students more, and taught their subjects better.

Coon (1963) identified a special group of beginning students who had had previous experience in the study of School Mathematics Study Group (SMSG) mathematics in high school. He studied the success of those students who subsequently enrolled in analytic geometry and calculus as their first college mathematics course.

These two studies illustrated the importance of enrolling students in the proper mathematics courses when they enter college. With the increasing number of high school students entering college with a background in the so-called modern courses in mathematics, additional research is needed regarding selection, placement, and enrollment as well as the types of courses they should study. This research is needed particularly because many colleges are just beginning to start high school graduates with the course in analytic geometry and calculus.

Prediction of Success and Achievement

Although most of the studies concerned with predictors of success in college are of a general nature and deal with total success in college study, some deal with the prediction of success in special curriculums closely related to mathematics. Because of marked changes in the nature of college mathematics curriculums and in the characteristics of college populations, the findings of most predictive studies rapidly become obsolete.

Harrington (1962) studied the relationships between attitudes toward mathematics and grades obtained in a freshman mathematics course. These attitudes, which are largely developed by former teachers, were reported as favorable to a significant degree. 350

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Coon (1963), whose work was previously mentioned, attempted to de- termine the relation of (a) previous experience with SMSG secondary mathematics to (b) success in the first semester course in analytic geometry and calculus. He used data based on a pretest and a posttest as well as on teachers' marks. While he was able to show that students with SMSG secondary mathematics experience performed better than those without such training, he was unable to identify a specific reason for this other than the somewhat vague factor of mathematical maturity.

One may expect to find the greatest number of future investigations concerning the prediction of college success in mathematics to be reported in educational and psychological journals that feature predictive validity studies.

Attitudes and Opinions Toward Mathematics

As in the instance of predictive validity studies, investigations concerned with attitudes and opinions toward mathematics risk rapid obso- lescence. Aiken and Dreger (1961) conducted an investigation requiring the construction of an attitude scale for college students. They tested hy- potheses concerning the relation of mathematics attitudes to achievement, to experiences with mathematics, and to characteristics of personality. They showed that mathematics attitudes were related to numerical ability, to intellective factors and achievement, and somewhat to experiences with former mathematics teachers, but were not related to temperament variables.

Phelps (1963) studied the attitudes toward mathematics of SMSG and traditional mathematics students. He noted that fifth grade SMSG students did respond significantly more positively in their measured attitudes than did those who had studied traditional mathematics. However, this finding was not true for eighth grade SMSG students. Since there was generally no significant difference in attitudes associated with IQ, Phelps concluded that SMSG materials could be presented to average students without risk- ing implantation of negative attitudes.

Although somewhat significant to mathematics education, these studies make it apparent that much research remains to be done concerning how attitudes of both students and staff members affect teaching, learning, and the curriculum.

Curriculum and Improvement of Teaching

The first investigation to be cited in this section illustrates methods of choosing subject matter or content for curriculums. Milligan (1961) employed operative sets of objectives and criteria for content selection for a course in freshman mathematics and showed that this procedure tends to lead to the selection of a modern, balanced body of subject matter for

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this course. He further concluded that there are relatively few tools in curricular theory that are available, operative, and effective for the individual teacher to utilize in the very difficult area of selection of subject matter.

From the standpoint of research in the improvement of teaching mathematics, studies which synthesize and add to certain mathematical topics are useful and necessary to a capable mathematics student and to mathematics teachers. A growing group of young mathematicians has undertaken this kind of research. Members of this group are interested in college teaching and in the fundamental knowledge that should be in the background of all mathematics teachers. Hight (1961) made a critical study of the limit concept as used in the SMSG revised sample textbooks. He was forced to add theorems and to suggest other approaches in order to make the concept clear, particularly for teachers who might be using the books. Van Ryzin (1960) studied the Arabic-Latin tradition of Euclid's elements in the twelfth century; and Evans (1960), who investigated teaching- machine variables, used learning programs in symbolic logic. Evans found that mode of response to programed items was not a significant factor in learning and that the procedure saved some time which could then be made available for other purposes and activities.

The area of curriculum and improvement of teaching, course content, and its organization for teaching and learning is probably the most critical one in mathematics education today. As indicated, some research has been done, but not much. The CUPM Panel on Pre-Graduate Training has at- tacked this broad problem of curriculum and instruction from the recognized mathematical method of constructing a sequence of courses that involve deductive systems and continuity of the basic concepts of mathematics. In time, sample course outlines will be suggested, and competent members of the mathematical community will be encouraged to write books incorporating these suggestions. These course outlines, which will become the basic curriculum or program of study for the colleges, and later the textbooks, have been and will be written from a broad background both in mathematics and in teaching experience. They illustrate the type of courses which the mathematics community considers suitable. They are presented as a way to teach mathematics. The research, already under way in a few places, will come from both (a) critical studies of small areas of the program and (b) experimental uses of the subject matter to deter- mine its teachability with varying types of students.

Conclusion

This paper has been written from the point of view that the improvement of mathematics teaching in the college and university depends on two basic factors: (a) a well-trained staff and (b) a modern program of study. The critical staff problem will probably become worse before it

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improves. Samples of a modern program are being developed, and experi- mentally oriented research will reveal the changes that should be made.

Although a little of this research has been done, much more must be undertaken in the future.

Bibliography

AIKEN, LEWIS R., JR., and DREGER, RALPH MASON. "The Effect of Attitudes on Per- formance in Mathematics." Journal of Educational Psychology 52: 19-24; Febru- ary 1961.

COON, LEWIS HULBERT. School Mathematics Study Group Mathematics as a Factor in Freshman Calculus. Doctor's thesis. Stillwater: Oklahoma State University, 1963. 107 pp. Abstract: Dissertation Abstracts (in press).

EVANS, JAMES LEE. An Investigation of "Teaching Machine" Variables Using Learning Programs in Symbolic Logic. Doctor's thesis. Pittsburgh: University of Pittsburgh, 1960. 142 pp. Abstract: Dissertation Abstracts 21: 680-81; No. 3, 1960.

HARRINGTON, LESTER GARTH. Attitude Towards Mathematics and the Relationship Be- tween Such Attitude and Grade Obtained in a Freshman Mathematics Course. Doc- tor's thesis. Gainesville: University of Florida, 1960. 47 pp. Abstract: Dissertation Abstracts 20: 4717; No. 12, 1961.

HIGHT, DONALD WAYNE. A Study of the Limit Concept in the SMSG Revised Sample Textbooks. Doctor's thesis. Stillwater: Oklahoma State University, 1961. 123 pp. Abstract: Dissertation Abstracts 23: 554-55; No. 2, 1962.

LINDQUIST, CLARENCE B. "Mathematics and Statistics Degrees During the Decade of the Fifties." American Mathematical Monthly 68: 661-65; August-September 1961.

LINDQUIST, CLARENCE B. "Numbers of Degrees in Mathematics Continue To Increase Significantly." American Mathematical Monthly 70: 665; June-July 1963.

MARGARIS, ANGELO. "Successor Axioms for Integers." American Mathematical Monthly 68: 441-44; May 1961.

MILLIGAN, MERLE WALLACE. An Inquiry into the Selection of Subject Matter Content for College Freshman Mathematics. Doctor's thesis. Stillwater: Oklahoma State University, 1961. 163 pp. Abstract: Dissertation Abstracts (in press).

PHELPS, JACK. A Study Comparing Attitudes Toward Mathematics of SMSG and Traditional Elementary School Students. Doctor's thesis. Stillwater: Oklahoma State University, 1963. 78 pp. Abstract: Dissertation Abstracts (in press).

SPARKS, JACK NORMAN. A Comparison of Iowa High Schools Ranking High and Low in Mathematical Achievement. Doctor's thesis. Iowa City: State University of Iowa, 1960. 225 pp. Abstract: Dissertation Abstracts 21: 1481-82; No. 6, 1960.

VAN RYZIN, SISTER MARY ST. MARTIN, O.S.F. The Arabic-Latin Tradition of Euclid's Elements in the Twelfth Century. Doctor's thesis. Madison: University of Wiscon- sin, 1960. 476 pp. Abstract: Dissertation Abstracts 21: 1210; No. 5, 1960.

WISNER, ROBERT J., editor. The Production of Mathematics Ph.D.'s in the United States. Mathematical Association of America, Committee on the Undergraduate Program in Mathematics, Report No. 3. Berkeley, Calif.: the Committee (P.O. Box 1024), 1961. Unpaged.

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Natural Sciences and Mathematics || The Teaching of Mathematics at the College and University Level