The Curriculum and the Art of Teaching

M. Wiles Keller

Professor Emeritus of MathematicsPurdue University

Lafayette, Indiana 47907

Not to be satisfied with the results we obtain in our classroomsis certainly natural and desirable. To be seeking constantly ways toimprove our effectiveness in the classroom is a mark of a good teacher.However, to think that changes in the methods being used, will increasethe amount students learn and understand is wishful thinking if weare to believe the evidence1. To a limited extent this is also truefor curriculum changes.

Dissatisfaction with the results being obtained by our schools isa recurring theme in our society. As a consequence, teachers andadministrators are exhorted to seek ways to increase the amountstudents learn. Either because they are not aware of past experienceor because, of the exhortations, the same reforms continue to betried again and again, often as if they were new ideas.

Accordingly, let us look at some of the reforms used in the teachingof mathematics at different times. Thus, on pages 2 and 3 of TheTeaching of Mathematics in Secondary Schools, (1912) A. Schultzewrites, "�nearly all teachers of Mathematics try to find remediesfor the present unsatisfactory conditions, and the cure recommendedby most of them is the introduction and study of applications, alongwith pure science. Apply mathematics they tell us, teach applicationsand pure mathematics side by side, make pure mathematics growout of its applications, and eliminate, as far as possible, those partsof algebra and geometry that have no immediate practical bearing.""Undoubtedly the general recognition of the fact that the concrete

must precede the abstract (essentially what Pestalozzi, F. Kranckes,C. Trapp, Tillich, et al were advocating a hundred years before),and that a great deal of time-honored mathematical subject matterhas but small value, means a great advance in mathematical pedagogy.It seems doubtful, however, whether this one principle, even if itbe carried out completely, would be sufficient to improve mattersthoroughly. It is even doubtful whether under present conditions anychange in the subject matter taught could produce a considerablebetterment: for the inefficiency of teaching is not confined to mathe-matics, but appears in nearly all other subjects. The average studentwithin a short time forgets so much of his history, physics, andeconomics, that it is no exaggeration to say his permanent knowledge

1. Begle, E. G. "Some Lessons Learned by SMSG," Mathematics Teacher, 66 (March 1973), 207-214.

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falls far short of the amount studied and the grades attained inexaminations. Hence, the inefficiency of mathematical teaching cannotbe due mainly to the selection of the mathematical subject matter:it must be due to the same general causes that are at work in theteaching of any subject."

Certainly since F. Kranckes (1819) employed number pictures anddeveloped rules of operation from observation and exercises, thediscovery method has been advocated from time to time as the wayto improve the learning of mathematics. Often involved with thediscovery method are variations of the laboratory method and individ-ualized instruction. As a matter of fact, about 1700, the Franck Instituteat Halle noted that classes in arithmetic should not be formed becauseof the diverse aptitude of the students in the subject. Rather, theteacher was advised to go around among the students as they workedon their assignments and give help when necessary.Such individualized programs were again popular in the early part

of the twentieth century. The better known of these are the SanFrancisco State Normal, the Dalton, the Winnetka, and the Denverplans. In the Denver plan, for example, the individualized materialsfor the students were to be based on objectives derived from "lifesituations." The subject matter was then selected to meet these lifesituations and then activities developed to meet these objectives.Laboratory methods and individualized instruction, in various forms,

are again with us. In many ways, the attendant learning centers areno more elaborate than those used in the Dewey Laboratory Schooland other places about the turn of the century. At that time D. E.Smith2 observed that the reforms being advocated�the laboratorymethod and individualized instruction�were a return to the methodsof the past which had long since been weighed in the balance andfound wanting.The different methods continue to be reintroduced in spite of the

observation made in 1960 in "New Dimensions in Higher Education,No. 2, Effectiveness in Teaching" (OE-50006) p. 10�"Stated some-what concretely, the consensus of studies made since 1920 is thatno one mechanical teaching device in and of itself, is better thananother. Teaching by the lecture, recitation, discussion, tutorial,reading-study, reading-quiz, correspondence, or several different labo-ratory methods has not been demonstrated to be intrinsically betterthan any other technique."Again in "Analysis of Research in the Teaching of Mathematics,"

Bulletin 1965, No. 28 (OE-29007-2) it states on pages 1 and 2�"Aids

2. Smith, C. E., The Teaching of Elementary Mathematics (1900), 76

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in effective learning were also a subject of inquiry and, similarly,the crucial question was, which and for which children and for whichtopics? If children differ and if they learn differently, some techniquesmust be more effective with one child than others. Indeed most ofthe aids that were advocated did seem to help some pupils. Multisensoryaids, for example, have been shown to be helpful to some students,but research has given little direction as to which students and withwhich concepts any given multisensory aid should be used."

"Similarly, grouping seems to increase learning of certain topicswith some pupils, but the question�when and how can groupingbe helpful?�remains unanswered.""The several research studies on the best method of teaching a

particular skill or concept belong to the type of research that hasnot yielded great returns. Perhaps there is no one best method forall pupils to estimate quotients or to learn fractions."The evidence suggests that:

(1) Experiments to learn which method is most effective for teaching a particularskill or concept to all students should be abandoned.

(2) Many of the changes in curricula and methods that have been tried in the pastare constantly being revived and recommended without any empirical evidenceto establish that the proposed reforms have any more merit for all studentsthan those they are replacing.

(3) All too frequently, as self-appointed "leaders" proclaim they have "the answer"for improving the quality and/or quantity of education, teachers and schoolsystems quickly adopt neatly packaged old remedies to establish they areprogressive and in the vanguard of progress with the latest innovations withouta look for any convincing evidence as to its success in pilot runs. Changeis made synonymous with progress and better education.

(4) The time is long past for getting off this merry-go-round. To learn, beforeintroducing any innovation in method or curriculum, whether there is anyreasonable evidence that the proposed change will produce better trained oreducated individuals before time, effort and money are expended to no avail.

In The Teaching of High School Mathematics (1911) George W.Evans observed�"The hope for success in the widespread attempt(to make instruction in mathematics better fit the needs of citizensand workers, to give more consideration to the student’s mentaldevelopment, and to give less attention to the logical developmentof the subject) rests on new conditions."

History establishes that the hope expressed by Evans was a piousone. It is this kind of hope that is ordinarily associated with eachproposed reform. Involved is the belief that circumstances are reallydifferent. As a consequence what did not succeed previously willnow, even though there is no real evidence to substantiate that itwill. Actually the differences in circumstances tend to be superficialbecause the essential ingredient in the learning process�the student�is basically no different when it comes to learning than his predecessors.

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Each teacher should study the history of the different movementsand reforms in education.Examine the attempts:

(1) To build curricula and methods based on the needs and interests of students.(2) To use an intuitive, a formalistic, an integrated, a logical approach, etc.(3) To have schools in which the students were never compared, but were judged

in terms of their own abilities and sometimes were given no intrinsic rewards,relying entirely on the "inner satisfaction" that comes from learning.

(4) To develop individualized instruction programs, including the use of variouslaboratory methods, etc.

Learn the strengths and weaknesses of the different methods ofinstruction and the different curricula as found by those who havetried them. Determine by trial and error which are most effectivefor you in your efforts to help the greatest number of your studentsand use these. Follow the current research as it relates to teaching.Don’t join every reform and innovation parade as history suggeststhey are only going in circles.

INSTANT DECISION-MAKING

A car turns left 50 feet ahead of you. A large semi-trailer truck is tailgatingyou at 65 miles per hour with a string of five other cars behind.With the car in front of you and the truck at your heels, you look to

the road shoulder. It looks soft, but you take a chance.How did you come to this decision? With a large number of cues and

multiple sources of information, you were forced to make an instantaneousdecision: to drive off the road to avert an accident.How do people leam to assess and weigh information to make judgments?This is a question N. John Castellan, professor of psychology at Indiana

University, is studying, under a $33,000 grant from the United States PublicHealth Service, Castellan is doing research on how people take a large numberof cues, as in driving, and arrive at a global decision about what is appropriateor safe at a given moment.

Castellan, who is engaged in basic research, not specifically safe drivingskills, said his research could also be applied to clinical diagnosis wherea physician must take a great deal of information from a number of sourcesand arrive at a diagnosis.

Castellan stressed the importance of such research in today’s complexworld where a large amount of information from many sources is availableon almost any issue."Many pieces of information are presented�some of which are relevant

to the policy being pursued and some irrelevant," Castellan said. "In addition,some pieces of information are consistent with each other, and some areinconsistent."

Castellan, who has done research in this general area for nine years, useshuman subjects in his experiments. The experiments are directed at under-standing how people deal with irrelevant information and integrate complexpatterns of information in making judgments.