Evaluating Mathematics Courses for ProspectiveElementary School Teachers

William B. MoodyUniversity of Delaware, Newark, Delaware

Grayson H. WheatleyPurdue University, Lafayette, Indiana

The much publicized revisions of the mathematics curriculum forthe elementary school have initiated changes in the mathematicsrequirements for elementary education majors at many institutionsof higher education. Until recently it was not uncommon to find col-leges and universities offering no mathematics content in their ele-mentary education programs and a few requiring a course or two inbasic college algebra. The change at the university level has been inthe form of courses in mathematics especially designed for the pre-service and in-service teacher of elementary school children. Someinstitutions require their elementary education majors to include intheir programs as many as three or four courses in mathematics con-tent, while many more offer at least one or two such courses.

There has been concern over the amount of exposure to mathe-matics content desirable, the topics to be included in such a programand the philosophy with which the mathematical concepts should bepresented to the propsective elementary school teacher. Recommen-dations have been made by such organizations as the Committee onthe Undergraduate Program in Mathematics1 and the NationalCouncil of Teachers of Mathematics2 concerning possible curriculumfor the programs. The former has recommended a program consistingof at least four college courses including two dealing with a develop-ment of the number systems, one concerned with algebraic contentand one course concentrating on geometric concepts. Various text-books have been written which are especially designed for thesecourses and institutions have been seeking instructors qualified tohandle the programs.A critical evaluation of mathematics courses for prospective ele-

mentary school teachers is long overdue. There have been some super-ficial attempts to measure the effectiveness of some of these mathe-matics programs. However, most of them have relied on the techniqueof testing a teachers ability to recall isolated facts and procedures

1 Mathematical Association of America, Recommendations for the Training of Teachers of Mathematics: ASummary (Berkeley, California: Committee on the Undergraduate Program in Mathematics, 1961), pp. 15.

2 National Council of Teachers of Mathematics, Mathematics for Elementary School Teachers (Washington,D. C.: National Council of Teachers of Mathematics, 1966), pp. 211.

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after he has been exposed to so-called ^modern mathematics" con-tent. One such attempt was conducted by Melson3 in Philadelphia.A sample of first year teachers from various institutions were ad-ministered a test which required the recall of mathematical content,most of which was dependent upon particular symbolism or defini-tions. When the teachers scored low on this test the author concludedthat colleges were not preparing teachers to handle the mathematicalconcepts present in the elementary school curriculum. Harper4 con-ducted a similar survey by administering a list of "basic conceptsand symbols7^ to teachers from 100 elementary schools in Colorado.He compared the performance of teachers who had a course in"modern mathematics" with that of teachers who had not been ex-posed to such a course. He concluded that most teachers profit fromthis exposure since the teachers who had the course scored signifi-cantly higher than those who did not have the exposure.There have been other attempts to evaluate mathematics training

of elementary school teachers, but most of them are similar to thetwo reported in that they employ the measurement of a teacher’sability to recall mathematics content. It would seem reasonable tointroduce some techniques of evaluation which explore a teachersbehavior at the types of tasks which he is likely to encounter in hisrole as a teacher of elementary school arithmetic. One such task wasdescribed by Helen Garstens of the University of Maryland. Shestates that a suitable background in mathematics for an elementaryschool teacher "should enable the teacher to read with discrimina-tion the material being prepared by the various experimental groupsand the texts being written for the forward-looking publishers."5 Itwould seem reasonable to include literature which appears in theprofessional journals published for the teacher of mathematics. Thefollowing is a report of a study which was designed to measure theeffect of a particular program in mathematics on the ability of pro-spective elementary school teachers to comprehend articles in profes-sional journals.

PROBLEMThe University of Delaware requires that the elementary educa-

tion major include in his program three semester courses in mathe-matics content and one course in elementary mathematics methods.These courses are especially designed to present a development of thenumber systems, from the whole numbers to the reals, geometric

« Ruth Melson, "How Well are Colleges Preparing Teachers for Modern Mathematics?" The ArithmeticTeacher, XII (January, 1965), 51-53.

< E. Harold Harper, "Elementary Teacher’s Knowledge of Basic Arithmetic Concepts and Symbols," TheArithmetic Teacher, XI (December, 1964), 543.

6 Helen L. Garstens, "Mathematics and Elementary Education Majors," The Arithmetic Teacher, XI (Decem-ber, 1964), 540.

Evaluating Pre-Service Courses in Mathematics 705

concepts and other aspects of mathematics felt desirable as back-ground for elementary school teachers. The textbook used for thefirst two of these courses is Basic Concepts of Mathematics6One of the objectives of this program is to provide a background

for the prospective teacher which will enable him to read with com-prehension the literature in the field of mathematics written specifi-cally for elementary school teachers. The attainment of this objectivewas investigated by testing the following hypothesis:Students who have been exposed to mathematics courses especially designed forelementary education majors will read mathematically oriented literature,written for elementary school teachers, with a greater depth of comprehensionthan students who have not had this exposure.

Depth of comprehension is used as defined by Tinker and McCul-lough and was measured by performance on a test of the materialpresented in a particular article.

Depth of comprehension refers to the degree of intellectual penetration achievedby the readers, i.e. the accuracy and completeness with which the pupil graspsthe meaning intended by the author.7

PROCEDURE

An article written by Brumfiel8 and appearing in The ArithmeticTeacher was selected as the vehicle for this investigation. BrumfiePsarticle dealt with a novel numeration system which did not employthe use of a symbol for the number zero. The topic of numerationsystems other than that which is based on ten is presented in many ofthe more recent elementary arithmetic programs. It is assumed thatthe mathematics courses for elementary school teachers would pro-vide a background for performance involving this topic. It should benoted that the particular system presented in this article was differ-ent in many respects from the usual numeration systems and that itwas unfamiliar to all subjects employed in this study.The investigators constructed a fifteen item multiple-choice test

covering the various aspects of the numeration system presented inthis article. A reliability measure of 0.74 was obtained by applyingthe Kruder Richardson Formula 20.

Copies of the article were presented to 138 undergraduates and 74inservice teachers at the University of Delaware in the spring of 1967.Each subject was given ample time to read the article and then ad-ministered the test. Subjects retained the copy of the article while

6 Cuthbert G. Webber and John A. Brown, Basic Concepts of Mathematics (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1963), pp. 328.

7 Miles A. Tinker and Constance McCullough, Teaching Elementary Reading (New York, N. Y.: Appleton-Century-Crofts, Inc., 1962), pp. 171.

8 Charles Brumfiel, "Zero Is Highly Overrated," The Arithmetic Teacher, XIV (May, 1967), 377-378.

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they completed the tasks on the test and the duration of the activitywas approximately 50 minutes. This activity was conducted duringclass periods which were not assigned to a methamatics course. Thiswas done in an attempt to control possible increase of motivation onthe part of students enrolled in mathematics courses.

Subjects were placed in one of the following four groups accordingto their University records.

1) Elementary education undergraduate major having completedthe first course in the mathematics program. (EMY).

2) Elementary education major who had not been exposed to amathematics course in college. (EMNO)

3) Liberal arts undergraduate major who had completed a basiccollege mathematics course. (LA)

4) Inservice elementary school teacher who had not taken a recentmathematics course. (IS)

Performance on the numeration test taken following the reading ofthe article was compared for the four groups of subjects. The follow-ing statistical hypothesis was examined using a t test for differencebetween mean scores.There is no significant difference between the mean score of the elementary edu-cation majors (EMY) exposed to the elementary mathematics course and themean scores of the other three groups of subjects employed in the study.

RESULTSThe results of the t test employed to compare each of the mean

scores of the EMNO, LA and IS groups with the mean score of theEMY group are reported in the following table. The table also pre-sents the number of subjects in each group, theic respective meanscores and standard deviations.

,-. Number ,..^ Standard .Group of Students Mean score Deviation t

EMYEMNOLAIS

71422574

11.810.39.79.3

2.682.452.413.27

3.05*3.42*4.95*

* Significant at the 0.01 level.

The elementary education majors (EMY) scored significantlyhigher on the criterion test than each of the other three groups at the0.01 level. These data, which lead to the rejection of the statisticalhypothesis stated previously, support the research hypothesis statedas follows:

Evaluating Pro-Service Courses in Mathematics 707

Students who have been exposed to mathematics courses especially designed forelementary education majors will read mathematically oriented literature,written for elementary school teachers, with a greater depth of comprehensionthan students who have not had this exposure.

DISCUSSIONAlthough this study was limited to the reading of a particular

article the results do suggest that there is value in exposing prospec-tive teachers to mathematics courses especially designed to providebackground for the tasks which they are to encounter. The readingwith comprehension of materials concerning mathematical topicssuitable for the elementary school curriculum is a task which teachersmust encounter when preparing for their role in the classroom. Onemight expect that students exposed to these elementary mathematicscourses would perform this task better than those who have not hadany college level mathematics courses. However, it is especially inter-esting to note that there was a significant difference between theirperformance and that of the students who had completed the basiccollege mathematics course. Although more support is needed, it isencouraging to find some evidence which indicates that these coursesfor elementary school teachers are performing a function which isnot served by a basic mathematics course. This is especially significantwhen taking into account the attention which has been given to thedevelopment of such programs.

Additional research should be carried out in an attempt toevaluate mathematics programs which are purported to serve aspecific role for elementary education programs. Too much emphasishas been placed on the measurement of student achievement in thecontent area as the sole indication of the value of the programs.Notenough attention has been given to the effect of mathematics courseson the particular types of behavior required of classroom teachers.

ARTIFICIAL INTELLIGENCEThree University of Wisconsin electrical engineers hope to develop an elec-

tronic system which can function much like the gray matter of the human brain.With their system, true artificial intelligence may some day be possible.The basis of the proposal by Alwyn C. Scott, Robert D. Parmentier and James

E. Nordman is a device called a superconductive tunnel junction neuristor. Aneuiistor is an electronic device which propagates electrical impulses much as anerve cell does.The Wisconsin neuristor is a long, narrow sandwich of insulating material

between two thin layers of metal. When the device is cooled to the temperatureof liquid helium, the metal layers offer almost no resistance to electric impulses.If an electrical impulse is applied at one end of the sandwich, it is propagated tothe other end much like an impulse in a nerve cell.The engineers state that with this device it may be possible to produce a mass

of neuristors with a packing density of a billion or more per cubic foot. This ap-proaches the neuron density of the human brain. Such a mass may also have thecapability of "learning," as does the human brain.