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ED 038 287 

SE 008 137 





Majr, Kenneth 0. 

Prograaed Learning and Mathematical Education, A CEM 

Mathematical Association of America, Buffalo, N.Y. 
National Science Foundation, Washington, D. C. 



Mathematical Association of America, 1225 
Connecticut Ave. , N.W. , Washington, D.C. 20036 



EDKS Price MF-50.25 HC-S1.65 

Autoinstructional Methods, *College Mathematics, 
♦Instruction, Instructional Aids, ♦Instructional 
Materials, ♦Programed Instruction, Programed 
Materials, *Research 

Mathematical Association of America, National 
Science Foundation 


This study reports on the position of programed 
learning in the teaching of college mathematics. A historical survey 
is given describing the following programing styles: Skinner, 

Crowder, eclectic, hybrid, Pressey, and problem. Conclusions based on 
research are given to claims about the benefits of programing 
including those of: providing for individual differences, tutorial 
function, controlling the learning process, providing for greater 
motivation, importance of overt responses, and the saving of teacher 
time. The summary states that the teacher and text will continue to 
be of greatest importance, with programing material used only as an 
auxiliary study aid. The report is based on studies made by the 
Committee on Educational Media of the Mathematical Association of 
America in 1963. (RS) 

S£ 00? 13 7 

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matical Association off Amo ri co to inv ast igot a tho uso of now m a d i a in matha- 
moncai myncr Muconon. in# nofor pon or vn# wownnv## s won is corn#a on 
through four s u b projec ts ; Calculus, hogwaid Looming, I n dividual Laduras, 
and Prasarvice Training of Elementary Taochars. Tha m amb arsh ip of the Com- 
mittaa in 1964 



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The work of the Committee is supported by a grant from the 




EDO 38287 

Progr a med Learning 

Mathematical Education 

a cm Shidr 

This work is o CEM Study, prepared in 1964 at the request of the Programed 
Learning Panel. It represents the views of the author, and not necessarily those 
of the Programed Learning Panel, the Committee on Educational Media, or the 
Mat h ematical Assoc i ation of America. In 1964, the mentliership of the Pro* 
granted Learning Panel of CEM consisted of the following. 

KnmS W. Mi— iw 

Gorlina Col leg* 

Iww Wf H. 8m 
H — Eton Collage 

Mpt T. Mstmsr 

Tho Hondo Stole Univanity 

■abaft KoBa 

The Hondo Slot* Univafsity 

Kama* O. mar 

Codeton Collage 

Uaabr W. Smith 

School MolhcooHa Stody Groap 
Stanford Univanity 

This Stvdy has boon dislribwtad without charge to corront sabscribon to THE AMEtICAN MATHE- 
MATICAL MONTHLY. Singl* copras will b* soot without charge opoo reqeost to the Emarfivo 
Director, Coaaitto* on Edvcotionol Madia, P.O. Sox 2310, Son Francisco, California 94126. 

Progr a med Learning 

Mathematical Education 

A CfM Study 

by Kenneth O. May 

Professor of Mathematics 

Carleton College 

Committee on Educatio nal Media 
Mathematical Association of America 

San Francisco 


gy Mathematical Asto. of 
America ~ 


mom mm mm m am &m 


Copyright © The Mathematical Association of America (Incorporated), 1965. All 
rights reserved. 

Permission to reprint for educational and non-commercial purposes will be generously 
granted. Inquiries concerned with reproduction rights sh ou l d be directed to the As- 
sociation through the Executive Director, Committee on Educational Media, P.O. 
Box 2310, San Francisco, California 94126. 




This report is based on studies made during 1963-1964 while I was 
serving as consultant on programed learning for the Committee on Educa- 
tional Media of the Mathematical Association of America. However, the 
opinions expressed are my own and do not carry the endorsement of either 
the Committee or the Association. 

I have benefited from service on the Panel on Programed Learning of 
the C. E. M., from comments on ten memoranda written for internal use, 
and from participation in the C. E. M. summer 1964 writing project Pub- 
lishers, organizations, and individuals have been generous with materials 
and helpful suggestions. It would be impossible to acknowledge them all. 
However, I wish to thank Edwin E. Moise, Leander W. Smith, Marshall H. 
Stone, and the S. M. S. G. algebra programing group for helpful com- 
ments on a first draft of this report 

It should be emphasized that this report is concerned with programed 
learning in mathematics at the college level. I discuss broader issues only 
in that context It appears that effective educational methods are highly 
specific to subject matter and level, so that such a limitation would be 
desirable even if it had been possible to cover a wider area. Mathematics 
teachers at the high school and elementary level are particularly cautioned 
that my descriptions and opinions may or may not apply to their situation. 

For the sake of informality and easy reading, I have abjured footnotes 
and cut citations to a minimum. The bibliography represents only a small 
percentage of the hundreds of items examined. However, it is sufficient to 
provide substantial further information for those interested, and leads to 
the literature in which can be found verification of the statements made. In 
accordance with C. E. M. usage I have spelled “programing” and “pro- 
gramed” with one “m,” but the double letter is used where it appears in 
quotations and citations. 

Kenneth O. May 

Berkeley, California 
October 1964 

A Definition of Educational Programing 

Since the phrase “programed learning” has been applied to several kinds of 
instructional activity, a genetic approach may be helpful. The movement 
traces its ancestry to Sydney L. Piessey, who in 1926 observed that his 
multiple choice testing machine also performed a teaching function. Though 
he was well aware of the implications of this discovery, it had little impact 
in psychology or education until teacher shortages led industry and the 

military to experiment with teaching machines. 

The current movement was launched by B. F. Skinner and associates, 
who in the fifties devised teaching machines inspired by “operant condition- 
ing” exp e rim en ts on animals. There followed a ten year boom and bust m 
teaching machines and a still continuing boom in printed materials whose 
form suggests their origin in teaching machine tapes. 

Most programed materials follow the Skinner paradigm — a linear se- 
quence of “frames,” each consisting of a few words (stimulus), a Wank to 
fill (response), and a correct answer (reinforcement). Error rate is kept low 
(about 5%), and all students go through the same sequence. We call this 
Skinner programing. 

A second well known pattern is that of Norman Crowder. Here tne 
frames are longer, consisting of an expository passage followed by multiple 
choice questions, and the student is “branched” to other frames according 
to his answers. T has been dubbed “intrinsic” or “scrambled,” but we 
call it Crowder programing. 

After some years of controversy over the merits of these two schools, 
there has emerged a tendency toward eclecticism. One finds a wide varia- 
tion in styles and many of the 2“ possible combinations of particular pro- 
graming devices. Some so-called programed books are hardly more than 
conventional texts with a few blanks inserted. Others look like flow charts 
for computer programing. But typically one now finds the Skinner pattern, 
modified by some branching and by a few of the amenities of text books 
that bad to be a bando ned for machine presentation. We call this eclectic 


Notable for their high quality are the programed texts produced by the 
University of Illinois Committee on School Mathematics and by the School 
Mathematics Study Group. The latter has published three programed ver- 
sions of their ninth grade algebra, one in the Skinner mode (Form CR 
constructed response), one in the Crowder mode (Form MC— multiple 
choice), and one combining the two with conventional exposition and prob- 
lem sett (Form H— hybrid). We call this hybrid programing . 


Filially, we mention that Pressey, the putative father of the movement, 
has remained a thoughtful critic of his offspring. He favors conventional 
exposition followed by a set of multiple choice questions designed to point 
up essentials. We call this pattern Pressey programing. 

Publications that lay claim to being programed differ from previous edu- 
cational materials primarily in the degree to which they control -the stu- 
dent’s learning activity. The contrast is most marked in areas, such as the 
social sciences, where text books consist almost entirely of exposition, with 
very little space devoted to questions or other devices to help the student 
It is least marked in subjects, such as mathematics and foreign languages, 
where the material is presented in a form convenient for study and is ac- 
companied by a variety of exercises. The reader will not be surprised to 
learn that the overwhelming majority of programs have been written in 
these latter fields. 

The familiar college mathematics text bode follows a pattern that might 
claim to be called programing. The “frame” is an expository section fol- 
lowed by a graded problem set with (partial) answers. By working through 
this unit the student is supposed to master the exposition, increase his 
skills, and make new discoveries. This mode we call problem programing. 
It enforces response and provides feedback, characteristics often used to 
define programing, but it accomplishes this with a much higher degree of 
student initiative than Skinner or Crowder programs. It can be varied so as 
to be similar to the Skinner, Crowder, or Pressey modes by choice of prob- 
lems and accompanying instructions. 

The above discussion su g gests the following definition: ed ucational pro- 
graming is the scheduling and control of student be!:avior in the learning 
process. From this point of view a program specifies the steps in a learning 
process, and programed materials are those that include a detailed program. 
In game theoretic terms, a program is a strategy that gives the move to 
make at each point in the learning process. Programed materials provide 
both content and the strategy for learning it 

With this definition it follows that all educational materials are pro- 
gramed to some extent At one extreme is the treatise, w’Jch vaguely sug- 
gests a learning strategy by its organization and calls for a response only 
implicitly by its content and the gaps in its reasoning. The reader does his 
own programing. Intermediate is the familiar text through which the stu- 
dent can follow an enormous number of paths. Here programing is shared 
by book, teacher, ;md student At the other extreme is the S kinn er program 
which specifies a unique path. 

In this report we shall use “programing” in the sense of the above defini- 
tion, though wc are quite aware that it is more often defined so as to in- 
clude some particular feature dear to the heart of the definer. We refer to 


specific inodes whoever passible, but in this report “programed materials” 
means “fully" programed books in the Skinner, Crowder, or eclectic modes. 
Pressey and problem programing we consider in a different category be- 
cause they assist the student by presenting him with helpful problems but 
leave him largely free to determine his own learning program. Hybrid pro- 
graming is an integration of new and old devices. 

The issue is not whether learning should be programed, but rather how, 
when, and by whom. In how great detail should learning be programed? 
(Should the learning strategy be pure or mixed?) To what extent should it 
be done in advance? What share of programing should be done by die stu- 
dent, the teacher, and the materials? What role should prog raming play 
in the teaching system as a whole? In order to answer the questions just 
posed, we need to have in mind the systems by which mathematics is now 
taught at the college level. 

Currant Teaching Systems 

In the typical system a teacher plays the essential and controlling role. He 
determines both the general outline (course plan) and much of the detail 
(daily assignments) of the learning program. He imparts information in 
lectures and employs a variety of other procedures, involving give and take 
with his class as a group and with individuals. The text book is the second 
main component of current teaching systems. It serves foi exposition, re- 
view, and reference. Most important, it is a storehouse of problems. Its use 
is programed by student and teacher. Other components may be present: 
work book, reader, teaching assistant, and audiovisual aids. 

In this system the student learns by multiple exposure and activity in a 
repeated cycle of listening, reading, problem solving, writing, getting feed- 
back from answers and returned problem sets, etc. He is guided by the 
teacher’s example, the assignments, and instructions in the book. But his 
detailed path is self-determined. We know that students learn in such sys- 
tems. They even learn when possibilities are not fully exploited and when 
components are missing or of low quality. But we know also that many 
students do not learn and that the system at its best has some evident weak- 

The “straight lecture” has well known drawbacks. Even the best content 
is usually wasted through lack of student attention. Unless some interac- 
tion occurs, the lecturer might as well be talking to a multitude (in a large 
room, on film, or over TV), so that the procedure is wasteful of teacher 
time. The student could just as well (and often does) read the lecture in 
print. Of course we are not talking of the highly personalized lecture, in 
which the student sees “a mind at work,” nor of the group interaction 


achieved by the master teacher. But, generally yating, learning takes 
place rather inefficiently m existing leisures because of failure to involve the 

Text books have somewhat similar faults. They are most effective when 
they involve the student, Le. in doing problems. They are least efficient in 
exposition. Often the student does not read the text at all. He just does the 
problems and ^nores the verbiage. How familiar is the plaintive “I under- 
stood the lecture and die book, but I don’t see die connection with the 
problems.” We say then that the student has not understood, and we under- 
take to lead him step by step until he can attack the problems on his own. 

It seems that even when the familiar system functions well, the student 
at tim es needs additional detailed guidance (programing). And even in a 
complete system this guidance is often not available. When some com- 
ponents are in adequ ate or absent, the problem may become acute. The 
typical college student cannot learn from the typical college text on his 
own. Yet he is being put more and more on his own by the increasing 
teacher shortage. This suggests that programed materials may have a role 
to play even in the best existing systems, and that they, or some other 
device to replace the teacher’s detailed guidance, are essential in a seriously 
understaffed system. 

Tha Claims For P r ogramed Materials 

The programing movement has a very broad appeal. It offers to psychol- 
ogists a new means for controlled experimentation, to educational adminis- 
trators a lifeline in a sea of students, to audiovisual specialists an additional 
medium, and to academic or commercial promoters exciting possibilities. 
Programed learning has become fashionable, and a new specialist — the 
programer — has been created. It is natural thet the sober findings of ex- 
perimenters have been submerged by the confused verbiage of sciolists, the 
careless assertions of enthusiasts, and the unprincipled cries of hucksters. 
Since these claims are widely publicized, we must deal with them here, if 
only to sort out those worthy of serious consideration. We generally omit 
references to the literature, but the original sources can easily be found 
through the bibliography. Needless to say we do not try to assign claims to 
the categories mentioned above. The reader is reminded that "programed 
materials" means those in the Skinner, Crowder, or eclectic modes and 
does not include hybrid or other types described above. 

One essential fact has been established without a doubt: Students do 
learn from programed materials. On the other hand, there is no conclusive 
evidence that students learn significantly more or with greater efficiency. 
Programed materials have been used in a wide variety of situations and in 


comparison with an equal variety of so-called conv entional methods. The 
typical result is n. s. d. (no significant difference). 

Summarizing the results of using Skinner programed materials in place 
of texts in pre-calculus courses at the University of Buffalo, Sharpe [30]f 
writes: "To date, the experiment as it has been conducted indicates a prob- 
ability that programed materials may do an equivalent job, but pre sent s no 
evidence that programed materials are superior. . . . Students in this expert 

rnent had programs backed up by good instructors, yet no records were 

Of course one would expect if programs were added to existing systems, 

that more learning would take place, since additional exposure could hardly 

decrease results [21]. But as substitutes for text-book*, programs have not 
been impressive. This is not surprising. Programed materials offer more 
detailed g uid a nce th a n text?, but they have few of the many features *f»a t 
make texts so bandy for preview, summary, and review. Above aD, they 
lade e xt ended problem? and connected exposition. The n. s. d. result, here 
as elsewhere, is due in part to a balancing of many variations in the total 
teaching system. 

Moreover, it is evident that programed materials have inherent limita- 
tions as to the terminal behavior they can induce. Skinner and Crowder 
programing are compatible with objective testing and are not designed to 
elicit behavior that must be judged otherwise. Skinner programing gives no 
practice in reading or writing sustained discourse. Crowder prog ram s pro- 
vide slightly more extended reading, but limit student activity to checking 
multiple choice boxes, though this checking must sometimes be based on 
some scratch work. The following conclusion seems p lausible : Programed 
materials are incapable of eliciting the full range of behavior i nclud ed in 
the objectives of college mathematics. R. C. Buck has spoken eloquently to 
this point [11, 12], and we return to it below. 

Clearly programed materials are no panacea, but it still may be that they 
will find a place in the teaching system. With this in mind, we amnmc 
some specific claims. 

(a) Do programed materials perform a tutorial function ? A common 
daim is that a program "has all the advantages of a private tutor." 
(N. E. A. Journal, November 1961, p. 18) It seems to be based on no 
experimental evidence or analysis of the tutorial function, but on the mere 
fact that both tutors and programs ask questions. Of the assertion that 
Skinner sequences are like the Socratic dialogues one can say that the great 
teacher-philosopher would not be flattered to have his intensely thoughtful 
dialogues comp'**rd to the operant conditioning of pigeons and humans. 
The similarity exists only if one ignores content. 

t Numerals in square brackets refer to the bibliography beginning on page 22 . 


There Is a jr-; of truth in the comparison between tutoring and 
Crowder programing, since some flexibility is introduced in the latter by 
branching. But it is only a grain, because programed materials lack die 
essential feature of the tutoring situation— the interaction of two human 
beings in all its in tel lectual and emotional complexity. Can one imagm. | 
tutor who asks every student the identical sequence of questions and always 

gets the same answers (Skinner), or who offers the student only two or three 

pat explanations prepared in advance to correspond with anticipated an- 
swers to multiple choice questions (Crowder)? Comparison would be more 
* coach or catechist, but even here the programed miff pik would 
make a very poor showing. We conclude that programed materials do not 
perfotm the tutorial function, though they may perform the drill sometimes 
done in the name of tutoring. 

(b) Do programed materials provide for individual differences? Provi- 
sion for individual differences is usually cited as a hallmark of programed 
materials. It is true as claimed that “the student can go at his own pace,” 
but this is a characteristic of all printed materials for indiv idua l use. Sdf- 
pacicg or external pacing is determined by the teacher, 
schedules rather than by the kind of printed materials hwng used. The bd 
is that Skinner programing removes all individualization e xcep t in pacing. 
Crowder programing has a better claim to individualization because of 
branching, but the variety of paths is rather limited compared with that 
permitted by the usual text book. Indeed, sheer bulk limits branching, and 
the scrambled format prevents any significant departure from the alterna- 
tives anticipated in advance. We conclude that programed materials (Skin- 
ner, Crowder, eclectic) are less adaptable to individual differences than are 
hybrid, problem, and Pressey programs. 

When stressing the importance of self-pacing, programing enthusiasts 
seem to be contrasting the individual studying a program with the group 
listening to a lecture or studying together. A comparison with an individual 
studying a text and doing problems would be more appropriate. But in any 
case numerous studies have indicated that self-pacing is not as helpful as 
one might imagine. As long as the group pace is not too far from the 
average, learning is sot significantly impeded. In particular, group-paced 
work with Skinner programs yields good results. Advocates of self-pacing 
seem to have overlooked the advantages of group interaction. It appears 
that pacing is not a very important issue and that self-pacing has no neces- 
sary or unique connection with programed materials 
Tiw real possibilities for using programed materials to cope with in- 
dividual differences lie in different directions. One is the development of 
large libraries of brief units focused on narrow problems, beamed to specific 
student difficulties, and utilizing programing devices most appropriate to 


the audience and difficulty. Such libraries, stored in books, films, slides, 
tapes, and computer memories, could serve as valuable learning resources 
into which the student could branch on his own or teacher initiative. A 
second direction is the use of computers to exploit die brandling idea to the 
point where individual differences are really accommodated. We this 
below in the section on automation. 

Individual i za ti on is a great rallying cry in secondary education, but little 
is said about it in college. No doubt this is partly due to the greater homo- 
geneity of college students, but the college population is still quite hetero- 
genous enough. A more likely explanation is the general opinion that col- 
lege students are mature enough to provide individualization for them- 
selves. It can be argued that the best way to individualize education for 
the college student is to let him do his own programing after providing him 
with a variety of materials and assuring him of hdp if needed. In this light, 
programed materials are an obstacle to individualization unless their use is 
optional and tailored to individual needs. 

(c) Does programing provide greater control of the learning process? This 
clai m is certainly justi f ied. As we have seen, Skinner programing virtually 
determines every move. It allows variation only in pacing and is so de- 
signed that the student almost always gives the expected response. This has 
obvious advantages for research in the psychology of learning. It enables 
the prpgramer to locate poor frames and to improve the program. It like- 
wise helps the diagnostic work of the teacher. But it does not follow that it 
is best for the student Programed materials force greater involvement in 
die learning process, but they do it in a predetermined inflexible pattern * 
that excludes student responsibility for controlling the learning process. In 
contrast, hybrid and problem programing give the student wider latitude. 
He decides on his own pattern of reading and problem solving, and we 
know that the result is quite individual and complicated — usually involving 
a great deal of switching back and forth between reading, checking, cal- 
culating, thinking, and writing. 

One of the goals of college mathematics is to teach the student to “work 
on his own** — to “write his own program" in the professional jargon. He 
will certainly not learn to do this if he is fed ideas intravenously drop by 
drop instead of having to get out and grub for them. Yet programed ma- 
terials are explicitly designed to carry all students painlessly from ignorance 
to mastery, provided only that they follow directions. 

One of the standard boasts of Skinner programers is “the student writes 
the program." By this they mean that the program is revised until the error 
rate is low. The perfect program is supposed to produce learning without 
error. Student failure is abolished. There is only program failure. The boast 
overlooks that one of our goals is to teach students tnat errors are in- 



inevitable and instructive, to give them courage to take the chances with- 
out which great achievement is impossible. By etimmatmg sustained effect 
to overcome o b s t acl es and to correct errors we would be failing to prepare 
the student for the adult world. 

Crowder and Skinner supporters take sharp issue on the need for alow 
error rate. S kinn e r disciples consider errors dangerous because they may 
become fixed, while disciples of Crowder let them play a role in brandling. 
Research studies show n. s. d. when the error rate is varied, but the whole 
controversy is beside the point There are times in the cd.xational p nyf tt 
when errors are to be avoided, when accurate painstaking routine work is 
essential. T here are other times when errors are to be permitted, when 
freewheeling experimentation is ap propri at e. Ma th e mati c s requires both fol- 
lowing and b* making “the rules” No fixed programing pattern can take 
care of the full range of objectives in mathematical educa tion. 

One of the sellmg points for teaching machines was that they prevented 

cheating, Le., looking ahead or bade in the program. Programed books 
try to approximate this control by “scrambling” in a random pattern, by 
sequencing frames on successive pages, by slightly removing the answers, 
or by admonishing the student to cover the answer. A number of studies 
showed, however, that students learn just as much when they look at an- 
swers before responding! This is partly due to the trivial nature of the 
answers called for m most programs. But it suggests that tight control of 
the student is less in his, than in the experimenter’s interest Of course, stu- 
dents will not learn much if they thoughtlessly copy answers, as illustrated 
by students who “go through” programs without learning anything. But it 
may be helpful to look at answers, to say nothing of reviewing or checking 
over previous material— a procedure virtually impossible in most pro- 
gramed materials. Our current text books allow for much flexibility on this. 
Some problems have complete answers, some partial, and some not at 
all. The student has practice in working at various levels of independence. 
Moreover, the problems are designed to teach him how to use the exposi- 
tion, to which he k supposed to refer to for help. Perhaps the teacherand 
the book should provide more guidance in using these feedbacks, but pre- 
venting the student from getting information is not conducive to learning. 

We conclude that programed materials inhibit initiative, independence, 
and responsibility in the learning process, and do not contribute to the 
achievement of related educational objectives. 

To the above conclusion it may be objected that programed materials 
could accomplish such goals if only we could specify operationally 'he de- 
sired “terminal behavior.” “You tell me what you want in concrete op- 
erational terms, and 111 program it!” We can accept this challenge, but 
the desired terminal behavior cannot be tested immediately or objectively, 



and the appropriate program would take the form of a guide to the study 
of unprogramed materials and the working of unprogramed problems. 
Someone should try to write such a program, one that would help die stu- 
dent use a text book effectively and then go beyond it It should not follow 
any existing programing pattern. 

(d) Do programed materials provide greater motivation? As is well 
known, conditioning experiments on animals depend on a dose temporal 
linkage of des i re d responses with feedbacks of a rewarding or reinforcing 
diameter. In Skinner programing, immediate knowledge of the correctness 
of the lesponse is supposed to function in this way for human*. Of course 
it is true that immediate feedback increases learning by ruickly correcting 
errors, reducing anxiety, and giving encouragement On the other hand, 
research studies show that students learn as wdl or better when they are 
“prompted” (supplied with the answer in advance) as when they are ques- 
tioned and then reinforced. Animals are hungry for the pellets they get for 
correct responses. Apparently students are not as hungry for confirmation 
of an answer that is 95% sure to be right 
While students do get some satisfaction from always moving ahead, tiny 
typically get bored with programed materials. Perhaps this is because hu- 
man motivation is more complicated than that of pigeons, especially in such 
an abstract and aesthetic activity as mathematics. Experience at the col- 
lege level tells us that students have two prime drives in studying mathe- 
matics: their belief that mathematics will be useful to them, and the joy 
that comes from mathematical insight and accomplishment. The former 
propels them through even bad courses taught by incompetent teachers 
from miserable text books. The monotonous pat on the bock of Skinner 
programing makes little difference. The second drive is an addiction that 
the teacher tries to establish by getting the student to do mathematics and 
to appreciate the mathematics done by others. Both these drives call for 
connected exposition and non-trivial problems, which are precluded by the 
Skinner and Crowder patterns. We infer that programed materials cannot 
provide adequate motivation at the college level. 

(e) Does programing lead to better specification of content and objec- 
tives? One characteristic of programers has been their insistence on pre- 
cise specification of objectives in behavioral and testable terms. Before 
writing, they define “terminal behavior” and prerequisites, not in vague 
generalities, but in detail. Then they write “readiness-tests” and “post- 
tests” that pin things down further. Since the material is to be presented 
piecemeal it must be analysed and ordered carefully. This is all to the good, 
an example for every writer and teacher to heed. But there is the danger 
that when goals are difficult to define they may be abandoned. 

In mathematics we desire not merely rote responses on an objective 


examination, but the development of behavior patterns, both overt and 
covert, that are reflected fully only in substantial calculations, nontrivial 
expositions, originality, continued interest in mathematics, performance in 
later courses, and use of mathematics in later life. Above all we desire 
unexpected insightful responses, which, by definition, cannot be programed 
except by avoiding the overprograming that inhibits them. 

Programers typically suggest that “conventional text books” are not as 
carefully planned as are programed materials, though no evidence is given 
to support such claims. In fact, mathematics texts are not very different 
from programs in this respect, depending in both cases on the competence 

“ dUStry ° f ** authors Most Programs remind one of the 
cook book” for which the reform movement in mathematics is trying to 
substitute more literate materials. The best mathematics texts include mat- 
ter not related to any immediate behavioral goal, but nevertheless impor- 
tant for the education of the student The claimed superior specification^ 

programed materials covers an impoverishment resulting from the limita- 
tion of objectives. 

One characteristic of programing is the fragmentation of subject matter. 
Step size has been a favorite topic of debate among the specialists, 
diough all seem to agree that knowledge should be fed in small amounts 
As usual, research shows n. s. d. with variation in step size, but it is obvi- 
ous that people who learn solely from fragmented presentations will not 
learn to see the big picture, to read long passages, to analyze complex ideas 
wi hout guidance, or to express themselves in an extensive way. An essen- 
tial feature of mathematical thinking is to look at problems in both the 
small and the large, to master both detail and big ideas. Problem and hy- 
bndprogrammg provide practice in this; Skinner and Crowder programing 

Once again we find that programing has the faults of its virtues. The 
programing movement has made a contribution by emphasizing the im- 
portance of planning, but the Skinner and Crowder approaches have tended 
1 0 " ar [ OW °J>jecUves and to fragment subject matter to the impoverishment 
of both goals and content. The potential of programing for improving con- 
tent has not been realized. 6 

(0 Are programed materials better designed to achieve their objectives? 
One of the boasts of programers is that by revising on the basis of testing 
they produce a product guaranteed to achieve results. After objectives 
have been specified, a draft program is prepared, tried out on individuals 
and revised until it appears a good first approximation. Then it is “field- 

• J and rcv,sed untiI a PP ro pr«ate error rates and post-test results are 
obtained. Programers seem to imagine that this process is something quite 


new and in sharp contrast to the offhand way in which conventional texts 
are supposedly dashed off. Actually the only innovations are large scale 
statistical testing and the trial and error method of writing and revising. 
Both these procedures were required by the fact that many programs were 
written by people unfamiliar with the subject matter and without experi- 
ence teaching it It is doubtful if large scale statistical testing, can add 
much to the product of a competent mathematics teacher who has experi- 
mented, drafted, tried, revised, and benefited from the thoughtful com- 
ments of reviewers and colleagues who have used preliminary editions. The 
basic difficulty with “scientific” statistical methods of testing is that the 
meaning of the results depends so much on the original limitation of ob- 
jectives, the testing procedures and content, the population of students, and 
the teaching system in which the program has been tried. An unsound 
mathematics program could easily make high scores if the testing were 
done “right” Moreover, as we have pointed out, tests cannot measure 
some of the most important features of a course. Actually, testing seems to 
have played a bigger role in promotion than in improving quality beyond 
what one could expect from competent and experienced writers. There 
is no convincing evidence that better design is a concomitant of programed 

(g) What is the importance of overt response? Overt versus covert re- 
sponse has been a bone of contention. As one might expect, research 
studies show no significant difference between programs requiring “overt 
response” (writing in a blank) and “covert response” (choosing an alter- 
native). Pressey [27] condensed a Skinner type program in psychology ii to 
an expository passage followed by a few multiple choice questions. The 
result was better learning and an 80% saving in study time (and paper!). 
Students learn as much from Skinner programs with the blanks already 
filled in as they do from filling them in. All this is not surprising, since 
millennia of experience shows that what people learn depends on what hap- 
pens in their minds, and that overt conduct is important only in so far as 
it impinges on the central nervous system. 

On the other hand, we also know that certain kinds of overt action are 
required to condition the mind to produce certain overt acts. In particular 
we are convinced of the common sense idea that to learn any type of be- 
havior we have to practice it Programed materials cannot teach cer tain 
desirable overt responses because they give no opportunity for them to be 
practiced. Moreover, the kind of overt behavior we desire in mathematics 
cannot be manifest without very complex and extended covert activity 
(thinking — if the behaviorists will excuse the expression). It follows that 
we wish to elicit both covert and overt responses, even though we can only 


observe the latter. One of the great disadvantages of excessive prog raming 
is that it enables the student to “succeed” without extensive or intensive 

On this matter programing enthusiasts flagrantly overlook the actual role 
played by text bodes. For example, Markle wrote: “A program requires of 
the student mom than does a text book. In a program the text is more in- 
complete. The student himself completes the text by filling in key m nteinr* 
or answering significant questions.” [23, p. iii] Anyone who compares a 
program with the typical mathematics text bode will find this rather amus- 
ing. Our books are full of incomplete arguments, and the number of prob- 
lems (to say nothing of the number of steps required to solve them) is often 
larger than the number of frames in a linear program supposedly covering 
the same material. 

The point for college mathematics is that programed materi al s cannot 
teach the full repertoire of covert and overt behavior that is desirable for 
effective work in mathematics. 

(h) Is there a:i art of programing? The early years of the progr aming 
boom weie dominated by psychologists and others who took for granted 
that programing was an independent skill and that a programer need not 
know anything about the subject Many programs have been written by 
programed who “followed” text books with little more knowledge of the 
subject matter than that provided by a consultant It is amusing to read 
the early discussions of the qualifications of a good programer without 
seeing once any mention of knowledge of the subject or of experience in 
teaching it (to say nothing of experience in t itoring, the supposed model). 

These early claims based on the alleged superiority of the expert pro- 
gramer as opposed to the amateur text book writer are seldom heard now. 
It is recognized that knowledge of subject matter helps. But the heresy re- 
mains that there is a special profession of programing, independent of sub- 
ject matter competence. Psychologists and educators in the past have not 
written manuals on how to prepare textbooks, but they feel no inhibition 
in the new field of program writing. Their publications often include in- 
sights and specific examples that are illuminating, but they illustrate once 
again that teaching problems are specific to the subject matter and the stu- 
dent Generalities are of little use, and the best results come from a com- 
bination of imaginative teaching and deep mastery of content In so far as 
the techniques of programing are valuable, they should be utilized by writers 
of textual materials. But there is no more reason for the profession of pro- 
gramer than for that of text book writer. We opine that there should be no 
profession of program writing separate from the general art of writing 
educational materials as part of the profession of teaching particular sub- 


(i) Does programing reduce educational costs? In certain situations in 
industry and the military, where teachers are lacking and goals narrowly 
defined, programed materials and teaching machines have accomplished 
telct aot otherwise possible. But in the typical college situation there is 
no indication that programed learning is more economical. Existing pro- 
gramed materials are generally more costly per course than text books, in 
spite of much thinner content For example, a recently published program 
on vectors takes about 100 pages (scrambled) to cover material normally 
dealt with in about six or seven pages of an elementary calculus book. It 
cost about thirty times as much per idea. Part of such high costs are due 
to depending on formal procedures rather than knowledge and experience. 
Nevertheless, there seems no reason to expect that programed materials could 
be produced at a lower cost per page than other printed matter, and de- 
tailed programing inevitably multiplies bulk by an order of magnitude. The 
S. M. S. G. ninth grade algebra in Crowder form required 2357 pages 
bound in six volumes. An entire calculus course with as rich a content as 
the familiar voluminous texts, if programed in detail, would require a large 
pocketbook for its purchase and a sturdy wheelbarrow for its transporta- 
tion. Evidently programed materials are more expensive than text books, 
but they might still cut educational costs if they allowed economies e'se- 

(p Do programed materials save teacher time? There is no evidence 
that programed materials can take the place of a complete teaching system 
or perform the functions of a teacher. Serious trouble has been experienced 
by those who tried experiments of this kind. It is of course possible for in- 
dividuals to learn from programs without assistance just as they can learn 
from text books on their own, but there is no evidence that they learn 
better, and our previous discussion suggests that what they leam is narrower 
in scope and thinner in depth. On the other hand programed materials 
might, just as terts now do, save class time by providing drill and practice. 
Here they would compete with work books and problem sets, but at higher 
cost In so far as they take over routine classroom chores they may be 
used to allow larger classes or to shift the teacher’s activity in the direction 
of more individual tutoring. There is no reason to think that programed 
materials will displace teachers. As supplements to text books they may, 
however, bring about shifts in the teacher’s role by taking over some routine 

The above catalog of claims is fairly complete, though we have not in- 
cluded a number that are based on the fallacious attribution to programing 
of effects due to other causes. For example, writers and users of programs 
have benefited by their experiences, but they might have gained as much 
from similar experiences with unprogramed materials. 


In summary, the advantages and disadvantages of programed material* 
are just what one might deduce from their characteristics. They have no 
magic, and their claims to universality or general superiority are quite un- 
founded. On the other hand they are effective within their limitaHnn< < and 
the programed learning movement has developed techniques and concepts 
that may be valuable in college mathematics. Before discussing these pos- 
sibilities, we consider the related matter of teaching machines. 


Machines are usually designed to simulate existing operations or to 
mechanize the fabrication of existing products. Examples are the typewriter 
to simulate writing and the printing press for copying manuscripts. But 
as these examples illustrate, machines usually modify both process and 
output Pressey’s first machine was designed to give and score a multiple 
choice test It actually changed the manner of giving such tests, taught 
while exa mi nin g , and suggested the possibility of other teaching machines. 

It sometimes happens, also, that machines are designed to produce a 
new product by a new process. The first Skinner teaching machines were 
built for the automatic operant conditioning of pigeons. For human use, they 
were redesigned to use verbal stimuli, responses, and rewards. The result 
was a new kind of learning. But the machine package soon aopeared to be 
inessential. When printed stimuli were used on subjects capable of turning 
pages, the machine proved less efficient than the student in moving froir- 
frame to frame. What remained was the Skinner program, a machine tape 
in book form. 

Machines designed to present Crowder programs had more to do. But 
branching increased the cost, and students appeared quite willing and able 
to accomplish the same thing by page flipping in a scrambled book. That 
most ancient teaching machine, the book, proved superior to machines for 
presenting simple branching programs. 

The virtual demise of the teaching machine industry before it could 
even get in the black might have been anticipated. As long as material can 
conveniently be put in book form, a teaching machine presenting the same 
material is just a cumbersome, expensive (or unreliable), and tiresome page 
turner. There is no more reason to expect such machines to replace bound 
books for educational purposes than to expect people to buy equipment 
for presenting their other reading matter automatically. Books just aren’t 
that inconvenient, and t? *y don’t get out of order! A teaching machine 
must either get old results at lower costs or do something ih~t has not 
previously been possible. Moreover the gain must be substantial, since 
machines have disadvantages. For example, one educator has pointed out 


that they “place the habituation of the act of learning one step further 
from learning as it occurs in normal living situations.” [24] 

We have noted previously that Crowder programed materials have some 
claim to simulate a tutor because of the orancliing structure, which is too 
bulky in book form but quite well suited to an electronic computer. “Com- 
puter based instruction” clearly has possibilities worthy of exploration, and 
energetic experimentation is in progress. The main problems are com- 
munication with the machine and the actual design of a suitably rich and 
flexible branching program. So far the typewriter and visual displays are 
being tried for communication, and some programs have been devcloped. 
As one might expect, the computer behaves like a slightly deaf teacher 
with an enormous memory and little imagination, who has been coached 
by someone with quite a bit of knowledge and experience. Interestingly 
enough, the computer sometimes directs the student to read a book. At 
ether times it asks questions, displays material, and comments on student 
responses. It can take into account all past performance of the student and 
all information about him that has been fed in, provided someone has 
written a program sufficiently complex to involve all these factors. In effect 
the student is learning under the guidance of the teacher who programed 
the machine. Instruction is individualized and mimics the tutor just in so 
far as the programer anticipates all individual differences. 

Experience shows that students learn from computer based instruction. 
The use of a computer is justified, however, only in so far as it can do 
thing; not otherwise possible at comparable cost. A potential cost reducing 
factor is that the computer can tutor substantial numbers of students at 
the same time. Nevertheless, computers are expensive and awkward. Their 
advantages and costs have to be compared with those of other devices for 
achieving the same degree of individualization. It seems possible that the 
same effect could be achieved by libraries of the kind mentioned under 
(b) in the previous section. 

One kind of teaching machine that has proved itself is the simulator of 
environments in which the student needs practice. Examples are the well 
known simulated space vehicles in which the cosmonaut can gain skill 
without the expense or risk of actual flight Such teaching machines arc 
verv specialized. Their high cost is balanced by the still higher cost of the 
reai thing. The language laboratory seems to fit in this category, since it 
simulates the expensive experience of conversing with someone fluent in 
the language. There does not appear to be much application for such ma- 
chines in college mathematics, except possibly in the field of computer 

The devices discussed so far are all for automating individual instruction. 





They are directly in line with the first invention of this sort — writing They 
compete with text books, work books, problem assignments, and individual 
personal instruction. What are the possibilities for automating group in- 
struction, that prehistoric move to meet the teacher shortage? 

^*°» T.V., films, and tapes i mmedi a te ly come to mind. The firs' two 
mainly increase the size of the audience. The last two, l&e books, permit 
unbmitai duplication at other times and places. They simulate personal 
communication more closely than does a book. But it does not follow that 
they will replace books. On the contrary, for most subjects, and certainly 
for college mathematics, verbal communication is slower, less effective, and 
more costly than reading. Automating the lecture does not cfaange'this. 
It simply permits the lecturer to reach more people at die cost of ifttw* 
audience feedback- Clearly these media compete with the live lecture, not 
with books or the small class. (Of course, we are talking of “canned” lec- 
tures, not of the use of films to present visual materials uniquely possible 
m the medium, e.g. animation. Such films justify themselves by presenting 
something not previously possible.) There seems little likelihood that these 
media will replace the teac»-r of college mathematics, because he spends 
or ought to spend, only a small part of his time in “straight f xture n But 

such materials could be very useful to present the lecture portion of a class 

or to otherwise supplement live teaching. 

Would it be possible to simulate the typical small mathematics classroom 
with its two way communication and student participation? The first step 
m this direction was taken in connection with T.V. lectures. Equipment 
was developed to permit listeners to question the let rurer. with the entire 
exchange audible to all viewers. The result was not very different from a 
lecture in an enormous hall with provision for a few questions. Perhaps 
the effect is better because the equipment overcomes acoustical and visual 
problems. Still it is far from the live small classroom. 

It is certain that the real live teacher (as opposed to a lecturer) cannot 
be fully simulated by a machine, because the teacher is able to respond to 
unanticipated events. On the other hand, one could simulate much of the 
small class activity. Classroom communication systems exist for presenting 
films, film strips, slides, and sound tapes individually or in combination 
under the control of a teacher’s console. Students can communicate in- 
stantly from individual push button stations and have their answers evalu- 
ated and individually recorded. The console will also display frequencies 
of different responses to the teacher. The teacher can record the entire class 
presentation, including his own participation. Then, without any tea cher 
the console can reproduce everything, including questions to the class, de- 
lays for response, etc. The cost is relatively low. With such equipment a 
gifted teacher might extend to a very large audience some of the values of 


a small class. More important, his presentations recorded on tape could be 

reproduced and used without limitation- 

Because of its unlimited duplicability and cheapness, the book has for 
many centuries been the main medium of individually paced learning. The 
taped classroom presentation described above may possibly play a similar 
role for group instruction. Of course, a mechanized classroom is not die 
same as a live one. Neither is a book the same as individual person^ in- 
struction by the author. Vet books have some distinct advantages. First 
they make high quality content widely available. Second they communicate 
more quickly. Thirdly they are very portable and flexible in jse. Finally, 
they can be adapted by the individual to his own needs. Packaged class- 
room prese ntatio ns seem to have some erf these advantages, and they might 

take over a large part of group instruction. 

The historical record suggests that improvements in educational tech- 
nology increase the number of learners and the amount learned without 
in any way diminishing the need for human instruction. This is not sur- 
prising. Students need the personal touch of the teacher, group interaction 
with their peers, the unexpected, the humorous, as wdl as the routine. No 
matter how much students could learn on their own with the aid of various 
devices, their demand for human guidance and example will remain . A 
shortage of teachers may force the student-teacher ratio up and lower the 
quality of education, but the loss cannot be fully repaired by automation. 
Of course, students may still leant without much attention from other hu- 
man beings, and they may even learn as much in terms of narrowly con- 
ceived criteria, but their education will nevertheless have been impoverished 
in ways not easy to measure. Speculations about machines replacing teach- 
ers are based on arguments that would apply equally weU to book* or 
phonograph records. We may expect students to continue to demandlive 
teachers, and the most likely effect of educational automation is a shift of 
teacher activity away from routine tasks and toward the essentially human 
aspects of the teaching job. 

Automation clearly is not essential to programed instruction. On the 
other hand, automation requires detailed advanced programing when- 
ever it does not provide for human control on the spot. We are not speak- 
iag here of programed materials but simply of the obvious fact that a non- 
human presentation must be laid out in advance, whether it be a book, a 
computer controlled tutorial, or a taped class. 

Programing Collogo Mathematic* 

If the previous analysis has any merit it follows that programing and 
automation are not alternatives to familiar ways of teaching but rather two 


related aspects of any educational process. The questions posed on page 
3 cannot be given useful general answers. We have to enmW. our pres- 
ent system with a view to specific weaknesses and possible improvements. 

We look first at a classroom small enough to allow questions and dis- 
cussion. We know that even in such classes, attention lags and students 
get “lost” Often teachers force participation, reinforce ideas, and check 
understanding by verbal questions — the Pressey programing pattern. Feed- 
back to the teacher consists of a few verbal responses and/or more subtle 
clues. Certainly a more complete response would be better. One way of 
achieving it is to have students write brief responses that are collected. The 
writer can testify that this yields attention and encourages regular outside 
preparation, but the teacher does not see the results until later. A class- 
room communicator of the kind described in the previous section would 
seem to meet the need completely, except that it is limited to multiple 
choice questions. Would it really help and is it worth the cost? Only ex- 
periment can tell. 

Next we turn to the large class, where intimate personal acquaintance 
is lost and the teacher has to talk to the group as a whole with only token 
individual feedback. This is probably the typical situation in college mathe- 
matics today, since “small” classes have for years been too large for genuine 
small group interaction. As we have pointed out, such a class is not essen- 
tially different from a TV presentation which includes provision for ques- 
tions from the viewers. Unless the lecturer is very unusual, such classes 
arc largely a waste of time. Students cut when they can, or they come and 
think about other matters. Here a classroom communication system would 
permit a gifted live lecturer to use a variety of media, to get instantaneous 
observable feedback, and to record student participation. Some of tlie 
values of small classes could be incorporated in very large ones, and there 
would be advantages not present in even the best small class. Moreover, 
high quality presentations of this kind could be reproduced from tapes 
without a live teacher present. Automated presentations would not be 
sensitive to student reaction, but then neither is a lecturer in a large hall. 
And the teachers using such automated presentations could provide for 
flexibility by their own direct intervention as well as by modifying the tape. 

1 suspect that learning would be substantially increased because of student 
involvement, even if content were not improved. 

Packaged classroom presentations are certainly still in the future, but 
courses of filmed lectures are already available. Their main weakness is 
student boredom. This might be changed dramatically if they were accom- 
panied by carefully planned questions to be answered by the student on the 
spot. These could be included in the film, or (more cheaply) be presented 
by other media with the film stopped. Answers could be written or given 


through a communication system. The combination might be better than 
loany s mall classes. It is certainly worth trying. 

How we turn to the primary locus of learning college mathematics — the 
student studying alone. He is reading and doing problems. Often his trouble 
is that he can T t read the book with understanding or do problems that vary 
from the worked examples. Hopefully, the discuss i on so far has convinced 
the reader, if be was not already convinced, that Ute solution is not to 
throw away the book but rather to teach the student to use it effectively . 
Only by using books will the student be able to karn what he mus t in our 
present society. 

Imagine him reading the exposition. He finds some definitions, axioms, 
and theorems more or less motivated and explained. He is offered no help 
in mastering them, the idea being that he will do so by repeated r e a ding , 
by working problems, and by whatever devices he has picked up from past 
experience. Perhaps he does, but more often he makes little headway and 
gives up. Why not provide auxiliary materials to help the student; not just 
more exposition, not just general advice on how to study, but a program of 
activities to master the topic/ The student could use such materials as 
much or as little as required. True, if he “knows how to study," he “ought” 
not to need such help. But he does need it, and it is part of our job to see 
that he gets it Individual personal help is impossible on a co n tinuin g basis, 
and printed substitutes would be better than nothing. 

Now the student attacks the problems. But he gets stuck, and besides 
his work habits and exposition are regrettable. He needs help and guidance, 
but bo does not get it until much later in class or when his sloppy problem 
set is returned with justifiably caustic comments by the reader. Why not 
provide him with more detailed guidance in striving the problems, some- 
thing between the completely worked example in the book and the problem 
with on»y a final answer for checking? Such programed study aids might 
begin with completely worked problems and gradually require the student 
to do more himself. They could inculcate good form. They would be de- 
signed to teach the student how to attack problems 01* his own, even when 
he did not “know how to do them.” They might be called programed work 

Sometimes our student meets another difficulty. He finds himself ignorant 
of something the author and teacher have assumed as a prerequisite. The 
solution is, of course, to study this material in another book. But since 
there is no teacher for this project, the best book might be one that is highly 

Should not a good text book give the kind of help just described? To 
do so in sufficient detail for all students would lead to impossible bulk. 
The text book would no longer be as useful for the big picture, for review, 


and for reference. At the college level the hybrid style runs the risk of 
falling between two stools. 

Does die student really need such detailed guidance, and is it good for 
him? It depends on die student of course, and it is bad to give him any 
more or any less than needed for his maximum development. The guidance 
need not be limited to telling him what to do in each case. It may provide 
him general rules and guidelines. An example is Polya’s How to Solve it, 
which presents general rules for a tt ac k i ng problems. Students might gain 
from auxiliar y mate rials that guided them in applying such strategics to 
particular problems. Could not such programs lead the student by die 
hand thro u g h heuristic thinking, always permitting him to branch out on 
his own, but leading him as much as nece s sa r y in the process of fin d ing 
results new to him? Such a program would be more than a “hint" and less 
than a solution. It could not follow any of the standard programing styles. 

A famous way of programing mathematical t e a ching is co nn ected with 
the name of R. L. Moore. Roughly speaking, the teacher supplies the 
a xio m*, theorems, and some intuitive material. The students supply die 
proofs without benefit of books or other aids. The teacher and students 
act as critics. This method, or variations of it, has been used with great 
success to produce mathematicians. The teacher plays the central role, 
as important for what be refrains from doing as for what he does. Could 
such a procedure be programed for the individual student? Of course 
it can! Many mathematicians have programed their own study in this 
way by simply not looking at die proof of a theorem until they have worked 
out their own. Some new proofs have been found this way. Any stu den t 
could try to follow this pattern. But perhaps we might produce a program 
designed especially to help such a stu den t . 

In sum, we need a wider variety and larger quantity of auxiliary ma- 
terials to assist students in their individual study outside of class. These 
auxiliaries need not be programed according to any existing style, but 
they must offer more programing than die usual text. They should all 
have a double purpose: to help the student master the material at hand, 
and to help him learn to master such material with less outside assistance. 
Programs should program themselves out of the student’s life. Moreover, 
the amount of help should continue, as at present, to be tapered off as 
the student advances, until he can program his own learning from straight 
mathematical exposition. In order to accomplish this, the student must 
always do some thing* on his own. There should always be some material 
to be studied with no guidance and some problems without any hints and 
answers. There should be gentle but firm pressure on the student to work 
on his own. 


Existing Programsd Materials 

Of wdl over ooe hundred p ro gram s in mathematics [8], only a handful 
are usable at the college level None cover a full course of college mathe- 
matics, though some claim to deal with high school courses often taught 
in college. The evaluation of programed materials was very difficult for 
a time because of a failure to follow familiar standards and format It was 
hard to review a bode that lacked a table of contents, chapters, subhead- 
ings, index, or even pagination! (The writer has found ooe useful dodge: 
just look at the answers and tests; then spot check a few frames.) But these 
faults are being corrected, and reviews of programs are begi nn i ng to ap- 
pear in journals. Meanwhile, one can say in general that existing programs 
are much thinner in content, of a substantially lower quality, and much 
higher in cost than corresponding text books. Even if the student were to 
ma s te r everything in a typical programed text book now on the market, 
he would get only a part of a respectable college course. Available pro- 
grams should be used only for additional exposure, independent study, 
or remedial work. 

Summary and Conclusions 

The problem of education is not to deride whether one procedure is 
superior to another in isolation, but rather to build effective teaching 
systems. A teaching system for college mathematics must take into ac- 
count the full range of objectives, many of which cannot be tested easily 
or objectively. It must reflect an ap p aren t contradiction in mathematics 
itself: the fact that mathematics requires, on the one hand, accurate and 
even automatic application of existing rules, algorithms, and theories, 
a nd on tiie other hand, insight, imagination, originality, trial and error. 
These twin aspects are present at every level of mathematical education 
a nd practice. Our problem is to find ways to teach both in the context of 
increasing demands and a growing teacher shortage. The solution is not 
to narrow our goals to those compatible with some instructional device, 
but to experiment with a variety of devices without abandoning those 
that have proved themselves capable in the past In particular: 

1. Teacher and text book should and will remain the central compo- 
nents of college teaching of mathematics. 

2. College mathematics text books might well incorporate some de- 
vices developed by the programing movement, but they should main- 
tain the present exposition-problem pattern. 

3. Prog raming (the scheduling and control of student behavior in the 
le arning process) should remain primarily in the hands of the individual 


teacher and student, with the responsibility being shifted increasingly 
toward the student as he gets older. 

4. Printed programs should be used only as auxiliary study aids, not in 
place of text books or teachers. 

5. Adherence to any one style of programing should be avoided in 
favor of eclectic and hybrid styles determined by the particular teaching 
task. We should look for new patterns. 

6. Programs in mathematics should be written by mathematicians on 
the same basis as other materials, judged by the same standards, and sold 
at comparable prices. 

7. Vigorous experimentation should be undertaken in writing and 
using a variety of special purpose auxiliary materials. 

8. Experiments in automation should concentrate on devices that 
maximize individualization for the single student (e.g. computer based 
instruction) and, for group instn'-tion, that extend the teacher's range and 
the degree of student involvement (e.g., multi-media presentation systems 
and classroom communicators). 

General Introductions 

1. Hughes, J. L. Programed Instruction for Schools and Colleges. Chicago: Science 

Research Associates, 1962. 

2. Lysaught, J. P., and Williams, C. M. A Guide to Programed Instruction. New 

York: Wiley, 1963. 

Source Books 

3. Smith, W. I., and Moore, J. W., ed. Programmed Learning: Theory and Research. 

Princeton, NJ.: Van Nostrand, 1962. 

4. Lumsdaine, A. A., and Glaser, R., eds. Teaching Machines and Programmed 

Learning. Washington, D.C.: National Education Association, 1960. 

Programing Manuals 

5. Becker, J. L. A Programed Guide to Writing Auto-Instructional Programs. 

Camden, N. J.: Radio Corporation of America, 1963. 

6. Mager, Robert F. Preparing Objectives for Programed Instruction. San Francisco: 

Fearon Publishers, 1962. 

7. Markle, S. M. Good Frames and Bad. New York: Wiley, 1964. A comparison 

of this with [23] will indicate the trend toward eclecticism. 


8. Programs '< 63 . Washington, D.C.: U. S. Office of Education (OE-34015-63), 

1963. Lists 352 programs with description and sample pages, including 123 
programs in mathematics, almost entirely at the elementary and secondary 


9. Raddiffe, Shirley. "Teaching Machine? and Programed Instruction, An An- 

notated Bibliography,** Audiovisual Instruction, February, 1963. 

10. Srhramm, Wilbur. The Research on Programed Instruction, An Annotated 

Bibliography. Washington, D.C.: U.S. Office of Education (OE-34034), 1964. 
Detailed abstracts, mostly illustrating the no-significant-difference syndrome. 


Papers are very widely scattered, though there is some concentration in Audio- 
visual Instruction, Audiovisual Communication Review, National Society for Pro- 
gramed Instruction Journal, and Programed Instruction. Current articles are listed 
in the Education Index under Programed. . .; Teaching Machinii; and Programs, 

Selected Books and Papers 

1 1. Buck, R. C. "Some thoughts on teaching machines,** C.UJ’M. Bulletin, Number 

20, June 5, 1961. 

12. Buck, R. C. "Statement on Mathematics and Programmed Learning,** American 

Mathematical Monthly, Vol. 69, No. 6, June-July 1962, 561-564. 

13. Carpenter, C. R., and Greenhill, L. P. Comparative Research on Methods and 

Media for Presenting Programmed Courses in Mathematics and English. 
University Park, Pennsylvania: Penn. State University, 1963. 

14. Cartier, F. A. “After the Programing Fad Fades, then What?" Audiovisual 

Communication Review, Vol. 1 1, 1963, 3-9. 

15. Coulson, J. E. ed. Programed Learning and Computer Based Instruction. New 

York: Wiley, 1962. 

16. Criteria for Assessing Programed Instructional Materials. Interim Report of 

the Joint Committee on Programed Instruction and Teaching Machines of 
the American Educational Research Association, American Psychological 
Association, and Department of Audiovisual Instruction of the N.E.A. 
Widely published, including in Audiovisual Instruction , February 1963, 84-98, 
and in Programs, ’63 [8]. 

17. Delaney, A. A. “On Behalf of the Oldest Teaching Machine," High School 

Journal, Vol. 46, 1963, 185-188. 

18. Frazier, A. “More Opportunity for Learning— or Less,” Educational Leadership, 

February 1961. 

19. Froehlich, H. P. “What About Classroom Communicator.;,” Audiovisual Com- 

munication Review, Vol. 11, May-June 1963. 

20. Kvaraceus, W. C. “Future Classroom — an Educational Automat,” Educational 

Leadership, February 1961. 

21. Lane, Bennie R. “An experiment with programmed instruction as a supplement 

to teaching college mathematics by closed-circuit television," Mathematics 
Teacher, October 1964, 395-397. 

22. Mager, R. F. “A Method for Preparing Auto-instructional Programs,” IJI.E. 

Transactions on Education, Vol. E4, No. 4, December 1961. 

23. Markle, S. M., Eigen, L. D., and Komoski, P. K. A Programed Primer on 

Programing. New York: Center for Programed Instruction, 1961. 

24. Noall, M. S. “Automatic Teaching of Reading Skills in High School,” Journal 

of Education, February 1961. 

25. Phi Delta Kappan, Vol. 44, March 1963. A special issue on programing. 


26. Pres s ey , SL L. ‘Teaching Machine (and Leaning Theory) Crisis," Journal of 

Applied Psychology, VoL 47, 1963, 1-6. 

27. Pressey, S. L. “A Puncture of the Huge ‘Programing’ Boom?”, Teachers College 

Record, February 1964, 413-411. 

28. Quinn, A. K. “Hour to Program (In Ten Difficult Lessons),” Audiovisual In- 

struction, February 1963, 80*83. 

29. w. Programed Instruction Today and Tomorrow. New York: Fund 
for the Advancement of Education, 1962. 

30. Sharpe, B. Programed Mathematics, a Two Year Experiment 1962-1964. Buffalo, 

N.Y.: Department of Mathematics, State University of New York, 1964. 

31. Wohlwfll, J. F. “The Teaching Machine: Psychology’s New Hobby Horse," 

Teachers College Record, VoL 64, 1962, 139-146. 

Samples of programing 

The hstimOe pages in [8] illustrate the variety. Below we list a very few titles 
without any implications about quality. See also [7] and [23] 

32. i ackham, David C. Analytic Trigonometry. Chicago: Encyclopaedia Britannica 

Press. (Skinner mode.) 

33. Crowder, Norman A. Trigonometry. New York: Doubleday and Co. 

34. Colman, H. I~, and Smallwood, C. P. Computer Language. New York: Mc- 

Graw-Hill, 1963. (Eclectic, in the form of a flow chart) 

35. Sc hool Mathemati cs Study Group. Programed First Course in Algebra , Revised 

form H. 4 volumes. Stanford University, 1964. (Hybrid.) 

36. Adkins, D. C. Statistics. Columbus, Ohio: Charles E. Merrill, 1964. (Pressey 

mode. Answers to multiple choice questions have Mack dots to be moistened, 
the correct dot turning green.) *