A Successful Experience With MathematicsFor Prospective Elementary Teachers

Robert CatanzanoBerry College

Mt. Berry, Georgia 30149

In designing a mathematics course (or courses) for elementaryteachers, student attitude, course content, student performance, andteacher process seem to be important variables to maximize. Of no lessimportance as a factor that must be resolved is the order of importanceplaced by the designer on these variables.

I have found more often than not during the first few days of classesprospective elementary teachers have a tendency to make one or more ofthe following comments about mathematics: "Math has always been mypoorest subject," or "I will never pass this course." Comments such asthese indicate to some degree the feelings and emotions of many of ourprospective elementary teachers towards mathematics. Some of thesestudents could even be classified as honor students. Unless theseprospective teacher attitudes about mathematics are changed, thenstudents taught by teachers with these feelings are likely to reflect thesame attitudes and to achieve correspondingly (Phillips, 1970). Further-more, improving teacher attitudes toward mathematics should result inmore positive student attitudes toward the subject (Aiken, 1972). Thusthe prospective elementary teacher attitude variable must be given a high,perhaps the highest, value when designing the course.A second variable considered by most mathematicians to be the most

important is course content. The standards set by the 1961 and the 1971CUPM reports have influenced the quantity and quality of under-graduate mathematics for the elementary teacher; in particular, theyhave increased the number of hours required for graduation and haveupgraded the content offered (Dubisch, 1970, and Watson, Moore, andPitts, 1971). The Watson, Moore, and Pitts (1971) study indicates thatmost of the programs reported by the respondents to the study include alltopics recommended by CUPM under the general classification�Struc-ture of the number system; that some of the programs include topicsfrom the classification�Algebra; and that a few include topics fromgeometry. The number of topics included in the programs seemed to behighly related to the number of semester hours required, which in mostinstances was six. However, it is questionable whether increasing thenumber of required hours (or increasing the number of topics offered)will in fact improve the teaching of mathematics to elementary students.What prospective elementary teachers need is mathematics that can be

6 School Science and Mathematics

used immediately by them at the level at which they will be teaching; thatis, useful mathematics (Van Engen, 1972).The third variable involves what is expected of the prospective teacher

in terms of performance and achievement. For students to settle for a"C" or "B" level of understanding of mathematics does not appear tohelp maximize this variable (Dubisch, 1970). Thus, testing proceduresand philosophies concerning testing must be changed in order to improvethe quality (and attitude) of the prospective teacher. Many prospectiveelementary teachers exhibit excellent reading competency but poormathematics competency. Thus, improving the quality of mathematicsteaching must be done but not at the expense of losing prospectiveteachers with other talents.

Teachers of prospective teachers ought to integrate as much as possiblemethod with content into the course sequences. Very early in the begin-ning course students should be shown the difference between how a topicis taught at different grade levels. Emphasis on the "why" of mathe-matics should be stressed (Brousseau, 1971). Many topics in the firstcourse should be presented to the prospective teacher in the same teach-ing-learning situation as the young child might experience (Leblanc,1970). The prospective teacher must learn to identify the scope andsequence of elementary school mathematics; in particular, when given aconcept, he should be able to identify the prerequisite knowledge forlearning that concept.

ONE TERMINAL COURSE FOR ELEMENTARY TEACHERS-TWO COURSES REQUIRED

In the fall of 19701 was assigned the task of teaching an undergraduatemathematics course. Course 1, for the elementary teacher at BerryCollege. The prerequisite for this course was a general mathematicscourse, Course 0, required of all students. Feelings and attitudes aboutCourse 1 derived from student questionnaires administered by theDepartment of Education were quite negative; namely, the text was toohard; the content meaningless; there was no relationship between whatthe course objectives were and what the prospective teacher was expectedto teach. However, students did score well in the course - usually a B orbetter.

In order to change student attitudes about his course, to not lose sightof the CUPM recommendations, and to maintain a high level of per-formance, a new text was chosen (the SMSG Studies in MathematicsVolume IX), a basic sequence of topics was selected (see Table 1), andthe testing procedure was altered by requiring all students to meet a mini-mum level of performance, 80%, on all major tests given during thequarter. Students not meeting this minimum performance level wouldcontinue to take tests over the specified content until they earned a grade

School Science and Mathematics 7

of at least 80%. The grade so earned would then become their score forthat test. The teaching procedures were heuristic in nature; that is,presenting problem situations, asking guiding questions, indicating thedifference, if any, in teaching small children the same concepts, and ask-ing why algorithms work.

TABLEiA SEQUENCE OF TOPICS AND CORRESPONDING NUMBER

OF SESSIONS FOR COURSE 1

Number of sessions Topics

7 Sets, set notation, set relations, set operations, number property ofsets, equivalence class, counting numbers, whole numbers.

3 Numeration systems, names for numbers, digits, place value,addition.

6 Meaning of addition of whole numbers, subtraction and itsinterpretations, number properties, addition and subtraction tech-niques and their justification.

6 Meanings of multiplication and division, models used for each, multi-plication and division algorithms and their justification.

1 Use of the number line, number sentences, solution set.4 Non-metric geometry, point, line, plane, space, properties of each,

simply closed curves, polygons, union and intersection of point sets.4 Metric geometry, congruence, greater than, convex polygons, circles,

measure of segments and angles, similarity.3 Non-metric geometry, solids, prisms, pyramids, cylinders, cones,

surfaces, spheres.5 Measures of plane and solid regions, meaning of formulas used to

determine areas and volumes.2 Factors, primes, L.C.M., G.C.D.6 Positive rational numbers, models for introducing rational numbers

and their operations.

Course 1 was taught three times during the 1970-1 academic year -winter quarter, spring quarter, and summer quarter. Students rated thecourse in all categories listed on a student evaluation form betweenexcellent and superior on the average. Thus, student attitude about thecourse was changed. Six of nine topics listed under the category, struc-ture of the number system, and nine of fifteen listed under the category,Geometry (Watson, Moore, and Pitts, 1971) were covered. Approxi-mately eight, 10-20 minute weekly quizzes, two, 50 minute tests, and oneexamination were administered for evaluation purposes each quarter.Tests and examinations counted 50% and 25% of each student’s grade,respectively. Thus, since each student had to make at least 80%(minimum grade) on each test (or retest), then each was assured that hewould have at least a B average for this portion of his grade before thefinal examination. The number of retests and average test scores beforeand after retests for the lowest scoring student are listed in Table 2 foreach quarter.

School Science and Mathematics

TABLE 2NUMBER TAKING RETESTS AND AVERAGE SCORES OF

THE LOWEST SCORING STUDENT

Number of Number tak- Number tak- Maximum Average of Average ofstudents ingretests ingretests number of re- tests before tests after

for test 1 for test 2 tests given retests retestsper student

Quarter 1

73 83

Quarter 2

8

Quarter 3

17

3

5

4

6

4

4

71

59

86

83

A cursory view of the findings revealed that (1) positive student atti-tudes toward the course were developed; (2) not nearly enough of theCUPM recommendations on content had been met by this course, and(3) the test-retest component seemed (a) to place the responsibility forlearning on the student, (b) to improve student grades, (c) to reducestudent anxiety about testing and (d) to improve student attitude aboutlearning. However, whether students taking retests achieved more than ifthey had not taken them is questionable. Furthermore, all retests wereprepared by the teacher and administered outside of class time whichmeans that more teacher input is required for such a program.

A THREE COURSE SEQUENCE - Two COURSES REQUIRED

The results derived from teaching the terminal mathematics course 1for elementary teachers suggested that at least two courses werenecessary (1) to meet a "bare" minimum of the CUPM requirements and(2) to prepare the prospective teachers for the teaching of mathematics aswell as for the content of mathematics. A third reason for consideringchanging the status quo was the introduction of an early childhoodcurriculum which required a course in early childhood mathematics.Thus Course 1 as described in the preceeding section became the firstcourse of the required two course sequence beginning with the 1971-2academic year. The second course of the sequence, either course 2a(grage N-3) or course 2b (grade 4-8) would be selected by the student onthe basis of his projected grade level of teaching. Topical outcomes aregiven in Table 3 for each course.

School Science and Mathematics 9

TABLESA SEQUENCE OF TOPICS AND

CORRESPONDING NUMBER OF SESSIONS

Number ofSessionsTopics

Course 2a

16 Sets, comparison of sets, joining sets, remainder set, number, addi-tion, subtraction; instructional strategies, aids, and pedagogy forgrades N-3.

11 Points, lines, segments, rays, angles, triangles, interiors, exteriors,plane regions, separation, surfaces, solid regions; instructionalstrategies, aids, and pedagogy for grades N-3.

13 Factors, primes, G.C.F., L.C.M., positive rationals, equivalent frac-tions, ordered fractions, models for positive rationals; definitions andproperties of addition, subtraction, and multiplication of the positiverationals; models used to interpret each operation; models, definition,and properties for division; decimals, ratios, percents, rates, negativerationals; addition and multiplication of negative rationals; inverse

_

operations; non-rationals within the real number system.

Course 2b

3 Factors, primes, G.C.F., L.C.D., positive rationals, equivalent frac-tions, ordered fractions, models for positive rationals.

2 Definitions and properties of addition, subtraction, and multiplica-tion of the positive rationals, models used to interpret each operation.

2 Models, definition, and properties for division.2 Decimals.1 Ratios, percents, and rates.4 Negative rationals, addition and multiplication of negative rationals,

inverse operations, non-rationals within the real number system.8 Mathematics and pedagogy for grade 4.8 Mathematics and pedagogy for grade 5.7 Mathematics and pedagogy for grade 6.7 Mathematics and pedagogy for grade 7.2 Mathematics and pedagogy for grade 8.

Approximately 30% of each course (2a and 2b) pertains essentially tocontent, the remainder to the teaching of mathematics. Some aspects ofthese courses which are usually not considered a part of the programoffered by mathematics courses for elementary teachers (or even perhapsfor general or specialized methods courses) are: (1) the required text forthe course is a set of texts that are written for the elementary schoolchild; namely, SMSG Elementary School Math, preschool - grade 3 forcourse 2a and SMSG Elementary School Math, grade 4 - grade 8 forcourse 2b; (2) each student teaches and is critiqued by the instructor twoor three times until a minimum performance level is reached on seveninstructional categories; namely, objectives, achievement goal, strategy,relationship between previous and following lessons, content errors,mode of instruction, and participation of students; and (3) approxi-mately eight weekly quizzes with the option of taking a requiz to improvegrades (and understanding) and two unit tests with a 80% minimum per-

10 School Science and Mathematics

formance level requirement were administered. Students were givenmany experiences in listing the prerequisite knowledge for a given mathe-matical concept, in examining the sequences of topics presented at eachgrade level, and in examining different sequences of topics at differentgrade levels.

FINDINGS

In connection with student attitude the following findings seemimportant: (1) student ratings on a five option Likert type campus widequestionnaire administered on the last day of class approached thesuperior category, the highest possible for each item; (2) during the1972-3 term twenty-five percent of those taking one of the two courses,2a or 2b, chose to take the other course also even though the secondcourse did not count toward graduation requirements; (3)approximately twenty percent of thirty-one students who had the time toselect and complete an area of concentration selected mathematics as hisarea of concentration. An area of concentration consists of a twentyhour sequence (usually courses in algebra, probability and statistics,geometry, and calculus) above the ten hour requirement, a program thatis well above the Level I and slightly below the Level II recommendationsoftheCUPM.Completion of the two required courses of the sequence meets most of

the Level I recommendations of the CUPM. Several of the algebra topicsare not covered, but several topics are included that are not listed underCUPM recommendations; namely, metric system and symmetry.Furthermore, the program introduces many mathematical models (andgames) that CUPM has not formally recognized.A completion type questionnaire was administered to students in

Courses 1, 2a, and 2b to ascertain attitudes and reasons students mayhave had concerning the retesting and requizzing program. Some of thefindings are listed in Table 4.

CONCLUSIONS AND SUGGESTIONS FOR TEACHER TRAINING

The following conclusions pertaining to the objectives of this studyappear relevant: (1) student attitude toward the three course sequence ispositive; (2) the mathematical content of the two course sequence isminimal with respect to the CUPM recommendations; (3) the minimumlevel of performance provided by the retesting and requizzing programimproves student attitudes about the courses (and may improve achieve-ment).The third conclusion indicates a method whereby operationally a

teacher of mathematics for the prospective elementary teacher canimprove his course. Other ways showng promise are:

Mathematics/or Prospective Elementary Teachers 11

TABLE 4FINDINGS FROM THE SEVEN ITEM QUESTIONNAIRE CONCERNING

RETESTS AND REQUIZZES FOR COURSES 1 AND 2

Question

How many retests didyou take?

How many requizzes didyou take?

Did you like beingrequired to continuetaking retests over thesame content materialuntil you reached themaximum level ofprefiency?

Did you like having thechoice of taking asecond quiz over thesame content material?

Do you think thattaking a retest (orrequiz) helped you learnthe content materialbetter than if you hadnot been given the re-quirement (or op-portunity)?

Did you like taking aretest or requiz over thegiven material?

Do you think that itwould have been betterfor you if you had notbeen given the oppor-tunity for a requiz or ifyou had not been re-quired to meet a mini-mum performance levelfor the tests?

Percent ofstudentstaking

requizzes

90%

Percent ofstudentstakingretests

83%

Percent ofyes

responses

97%

97%

100%

90%

.5%

Some reasons

Learn from your mis-takes

Felt you knew thematerial and was glad toget to show it

To improve my grade

Learn from mistakes

Repeated material instudying

Not as much pressure

Did not enjoy it but itwas well worth it

Otherwise would havefailed course

Felt retests were forassistance, not penalty

Made me keep tryinginstead of giving up.Did not know what Ididn’t understand until Imissed it on the tests.

12 School Science and Mathematics

(1) To include in the course sequence student teaching experiences with constructiveand objective criticism. It is important that a) a set of expected behaviors bedetermined, b) performance levels for each behavior be established, and c) eachpresentation be evaluated according to these behaviors.

(2) To require, as the course text, an elementary text series such as the SMSG materialand to have students question the sequence and prerequisites of selected topics.

(3) To train teachers to be sensitive to the content and the "art of questioning" duringtheir presentations.

REFERENCES

AIKEN, L. R. JR. Research on attitudes toward mathematics. In C. C. Riedsel (Ed.), Usingresearch in teaching. The Arithmetic Teacher, 1972, 19(3), 229-234.

BROUSSEAU, A. Conceptual mathematical methodology for prospective elementary schoolteachers. In F. J. Mueller (Ed.), Forum on teacher preparation. The Arithmetic Teacher,1971,18(4), 265-267.

DUBISCH, R. Teacher education. In E. G. Begle (Ed.), The Sixty-ninth yearbook of thenational society for the study of education. Chicago, 111.: The National Society for theStudy of Education, 1970.

LEBLANC, J. F. Pedagogy in elementary mathematics education-time for a change. TheArithmetic Teacher, 1970, 17(7), 605-609.

PHILLIPS, R. B. JR. Teacher attitude as related to student attitude and achievement inelementary school mathematics. Dissertation Abstracts International, 1970, 30(10),4316-A-4317-A. (Abstract)

VAN ENGEN, H. The mathematical education of an elementary teacher. The ArithmeticTeacher, 1972, 19(7), 517-518.

WATSON, C., MOORE, V., & PITTS, C. T. A study of the required mathematics contentcourses for undergraduate elementary teachers in the United States. Presented at IndianaState University, Terre Haute, Indiana 1973. Unpublished manuscript.

METRIC EDUCATION

Teachers are one of the most important links in getting Americans to "thinkmetric," according to a University of Idaho professor who has been named adirector of the Idaho-Montana-Utah-Wyoming Metric Consortium.Gwen Kelly, assistant professor of education, said the four-state effort re-

cently received a $75,000 grant from the U.S. Office of Education to train 300leaders in each state in metric education methods."These leaders will be chosen from all fields of education, not just math,"

Ms. Kelly said. "They will teach, in turn, teams of educators in each schooldistrict, helping them develop and present metric education workshops in localareas for service organizations, adult education classes and other publicgroups."

"It may be easier to set up programs with the larger school districts, some ofwhich have already begun metric education programs," Ms. Kelly remarked."Because of this, smaller districts may receive relatively more attention in theinitial phases of the consortium’s program.?’A project of the Montana Council of Teachers of Mathematics (MCTM), with

headquarters at Columbus, Mont., the metric consortium brings together ninecolleges and universities and four state departments of education. Daniel T.Dolan, MCTM director, will head the metric education project�one of four inthe nation receiving funding this year.