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To Teach or to
Learn - the Adult Learner Model
Introduction
Listening
| Information
Listening | Social
Listening | Listening
for Appreciation | Critical
Listening
Critical
Reading: The
SQ3R Reading Method
Reading
Material, Journals and References Bloom's
Learning: Benjamin Bloom's Holistic Teaching: Affective
| Psychomotor | Cognitive Education
Links Getting
to Knowledge Threshold
of BIC
Introduction What
is the difference between teaching and learning? Teaching
is passive, whereas learning is active. Never confuse motion with action,
says an old adage. View teaching as motion, whereas learning is motion and
action. Management guru Dr.
Henry Mintzberg likes to say that teaching management to the uninformed is a
waste of time and diminishes management - referring to full time MBA students
with little, to no practical management experience. Then there are the
experienced professionals enrolled in non-traditional universities who
complained because they are too busy and need to know exactly what to read, in
order to make efficient use of time and complete assignments as quickly as
possible." - In this context, "knowledge" as defined in Benjamin Bloom's
Holistic Teaching, can best be described as the existing beliefs,
context and ideas, whereas in Dr. Bloom's terminology, evaluation is more
closely related to the transfer function attempting to overcome the threshold
of BIC.
To
change teaching, counseling, advising, or learning styles, it is necessary to
understand
adult
learning characteristics and their implications" say Maxine
E. Rossman, Mark H. Rossman in their piece titled
The Rossman Adult Learning
Inventory:
Creating Awareness
of Adult
Development
as part of a doctoral dissertation. Using a sixty-item
questionnaire, the
Rossman
Adult Learning Inventory (RALI), was conceptualized, developed, field-tested,
and revised (Rossman, 1977). The purpose of the study was
to determine the extent to
which
community college faculty were aware of adult-learning characteristics. The
study concludes that adult learners are better suited for non-traditional
universities as:
- Adults need to know why they are
learning.
- Adults need to be seen as competent.
- Adults want their knowledge and
skills acknowledged in the classroom.
- Adults need to be self-directed and
experientially involved in their learning.
- Adults
want learning to be life-centered and applicable
So, how do we
reconcile the apparent paradox? To teach as Dr.Mintzberg suggests, adapt
teaching methods following the RALI guidelines, or? What
Mr. Mintzberg clearly refers to is a student body with little to practical
experience in management. This implies that the BIC (Beliefs,
Ideas and Concepts) database is null. Sort of like elementary school
children, or even trying to describe the an elephant to a person who has never
had eyesight. Now compare this to the part-time university student who has
plenty of practical experience, who sees himself more than competent, yet when
shown flaws in his reasoning, he answers "no, it is not a flaw in my
reasoning, I am just stating my opinion". Or what about the student
who complaints because the questions being asked require him to think, rather
than find the answer in a specific page of the assigned reading material? For
these purposes, we make a distinction between data,
information and knowledge, where in order for information to become
knowledge, it must overcome the threshold of BIC - that is, the information
obtained must break through the preconceived notions and ideas, such as the
previous example of flawed reasoning. That is, as long as the student
fails to recognize that his reasoning is flawed, the new information he
receives, might as well be discarded.
Robert Lee Moore (1882-1974) was a
towering figure in twentieth century mathematics, internationally
recognized as founder of his own school of topology, which produced some
of the most significant mathematicians in that field. The Moore
Teaching Method, which he devised
virtually prohibit students from using textbooks during the learning
process, call for only the briefest of lectures in class and demand no
collaboration or conferring between classmates encouraging students to solve problems
using their own skills of critical analysis and creativity. Moore summed
it up in just eleven words: 'That student is taught the best who is told
the least.'
St. Augustine
provides an excellent illustration of the challenges found in overcoming the
threshold of BIC:
"There is no doubt that it is better to communicate knowledge in parables for we appreciate
more that which we get from hard work, for the seeking gives greater pleasure
in the finding. For those who seek but do not find, suffer from hunger.
Those again who do not seek at all because they have everything, often die.
Either of these two extremes is to be avoided."
Then
there is the old Chinese Proverb: Tell me, I'll
forget, show me, I may remember, involve me, I'll not forget.
The Adult Learner
Method is a combination of Moore method, St. Augustine's edict and the Chinese
proverb: Involve the adult learning in the learning process using a
minimalist approach, while identifying and overcoming the threshold of BIC (Threshold
of BIC)
Bloom's
Learning
Bloom's Taxonomy is considered to be a foundational and an essential element within the education community, consisting of a classification of the different learning objectives that educators set for students, first presented in 1956 through the publication The Taxonomy of Educational Objectives, The Classification of Educational Goals, Handbook I: Cognitive Domain, by Benjamin Bloom (editor), M. D. Englehart, E. J. Furst, W. H. Hill, and David Krathwohl. The Taxonomy divides educational objectives into three domains:
Affective, Psychomotor, and
Cognitive. Within the taxonomy learning at the higher levels is dependent on having attained prerequisite knowledge and skills at lower levels . A goal of Bloom's Taxonomy is to motivate educators to focus on all three domains:
Affective
Skills in the affective domain describe the way people react emotionally and their ability to feel another living thing's pain or joy. Affective objectives typically target the awareness and growth in attitudes, emotion, and feelings.
There are five levels in the affective domain moving through the lowest order processes to the highest:
Receiving
The lowest level; the student passively pays attention. Without this level no learning can occur.
Responding
The student actively participates in the learning process, not only attends to a stimulus; the student also reacts in some way.
Valuing
The student attaches a value to an object, phenomenon, or piece of information.
Organizing
The student can put together different values, information, and ideas and accommodate them within his/her own schema; comparing, relating and elaborating on what has been learned.
Characterizing
The student holds a particular value or belief that now exerts influence on his/her behaviour so that it becomes a characteristic.
Psychomotor
Skills in the psychomotor domain describe the ability to physically manipulate a tool or instrument like a hand or a hammer. Psychomotor objectives usually focus on change and/or development in behavior and/or skills.
Bloom and his colleagues never created subcategories for skills in the psychomotor domain, but since then other educators have created their own psychomotor taxonomies.
Cognitive
Skills in the cognitive domain revolve around knowledge, comprehension, and critical thinking of a particular topic. Traditional education tends to emphasize the skills in this domain, particularly the lower-order objectives.
There are six levels in the taxonomy, moving through the lowest order processes to the highest:
Knowledge
Exhibit memory of previously-learned materials by recalling facts, terms, basic concepts and answers
· Knowledge of specifics - terminology, specific facts
· Knowledge of ways and means of dealing with specifics - conventions, trends and sequences, classifications and categories, criteria, methodology
· Knowledge of the universals and abstractions in a field - principles and generalizations, theories and structures
Questions like: What are the health benefits of eating apples?
Comprehension
Demonstrative understanding of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions, and stating main ideas
· Translation
· Interpretation
· Extrapolation
Questions like: Compare the health benefits of eating apples vs. oranges.
Application
Using new knowledge. Solve problems to new situations by applying acquired knowledge, facts, techniques and rules in a different way
Questions like: Which kinds of apples are best for baking a pie, and why?
Analysis
Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations
· Analysis of elements
· Analysis of relationships
· Analysis of organizational principles
Questions like: List four ways of serving foods made with apples and explain which ones have the highest health benefits. Provide references to support your
statements.
Synthesis
Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions
· Production of a unique communication
· Production of a plan, or proposed set of operations
· Derivation of a set of abstract relations
Questions like: Convert an "unhealthy" recipe for apple pie to a "healthy" recipe by replacing your choice of ingredients. Explain the health benefits of using the ingredients you chose vs. the original ones.
Evaluation
Present and defend opinions by making judgments about information, validity of ideas or quality of work based on a set of criteria
· Judgments in terms of internal evidence
· Judgments in terms of external criteria
Questions like: Do you feel that serving apple pie for an after school snack for children is healthy? Why or why not?
Some critiques of Bloom's Taxonomy's admit the existence of these six categories, but question the existence of a sequential, hierarchical link.
The revised edition of Bloom's taxonomy has moved Synthesis in higher order than Evaluation, for example,
whereas as some consider the three lowest levels as hierarchically ordered, but the three higher levels as parallel. Others say that it is sometimes better to move to Application before introducing
concepts.
Education
Links
Reading
Material, Journals and References
- Bagnato, Robert A., "Teaching College Mathematics by
Question-and-Answer," Educational Studies in Mathematics
5(Jun., 1973), 185-192. On JSTOR.
- Finkel, Donald L., Teaching with Your Mouth Shut,
(Boynton/Cook, 2000).
"The
teacher, then, attempts to create a blueprint for learning by keeping
her mouth shut and instead designing an environment for her students
with the following three features: (1) The teacher presents an overall
problem-to-be-solved, which is broken down into smaller problems that
build on each other …. (2) The teacher creates a blueprint for a
‘whole’ experience …. (3) She reconfigures the classroom so that
she is ‘out of the middle’ ….” (p. 108).
"In sum, the
teacher refuses to govern the students in their inquiry because he wants
the students to learn how to govern themselves” (p.115).
- Kilpatrick, Jeremy, "Confronting
reform." Amer. Math. Monthly 104 (1997): 955-962. Also
on CD.
- John Selden, and Selden, Annie, "Unpacking the Logic of
Mathematical Statements," Educational Studies in Mathematics, 29
(Sep., 1995), 123-151. On JSTOR.
- Wilder, Raymond L., "The Role of Intuition," Science, NS
156(1967), 605-610. On JSTOR.
"Intuition
plays a basic and indispensable role in mathematical research and in
modern teaching methods."
- Woodruff, Paul, Reverence: Renewing a Forgotten Virtue,
(Oxford University Press, 2001).
"A
silent teacher need not treat lofty subjects. You think it a trivial
fact that seven plus five equals twelve, but one may stand in awe of it
nevertheless (as has more than one great philosopher). With awe or
without, a teacher is well advised to be quiet from time to time about
even the most ordinary facts, so that students may have the freedom to
make those facts their own." (On "The Silent Teacher," p.
189.)
- Bart, Jody, Mary Ellen Rudin is mentioned in Women Succeeding in
the Sciences: Theories and Practices across Disciplines, (Purdue
University Press, 2000).
- Albers, D. and Reid, C., "An interview with Mary Ellen
Rudin,"
More Mathematical People, D.Albers, G.Alexanderson, C.Reid (eds)
(Harcourt Brace Jovanovich, 1990), pp. 283-303.
- Anderson, Richard D., "I Led Three Lives in Mathematics", MER
Newsletter, (MER Forum, Fall 1998), 3-11. On CD.
- Bing, RH, (1973, October 5). Remarks
at the Dedication of R.L. Moore Hall.
- Bing, RH, et al., (1976, January 24). Remarks
at The University of Texas at Austin Mathematics Award Honoring the
Memory of Professor Robert Lee Moore and Professor Hubert Stanley Wall.
Also on CD.
- Corry, Leo, "A Clash of Mathematical Titans in Austin: Harry S.
Vandiver and Robert Lee Moore," The Mathematical Intelligencer
29(2007), 62-74. (First page preview
from the publisher.)
- Fitzpatrick, B., "The
Students of R.L. Moore". Also on CD.
- Fitzpatrick, B., and Sher,
R.B. "B.J. Ball: An
appreciation," Topology and Its Applications 94(1999), 3-6.
On CD.
- Green, John W., Presentation
at the 2001 Legacy of RL Moore Conference.
A
principal research biostatistician with DuPont corporation talks about
how he went from topology to statistics and "how Dr. Moore’s
influence continues in this new career." In the course of a
challenging and politically sensitive research position Dr. Green shows
how important qualities such as persistence, self-reliance, and clear
thinking, as well as a sense of humor, have proven to be valuable
lessons he took from Dr. Moore’s classes. (See also his interview with
B. Fitzpatrick above.)
- Hennon, Claudia, "Mary Ellen
Rudin," in Women in
Mathematics: The Addition of Difference (Indiana University Press,
1997).
- Kenschaft, Patricia C., Change is Possible: Stories of Women and
Minorities in Mathematics, American Mathematical Society, 2005.
Accounts
of Mary Ellen Rudin, Lida Barrett, and Lucille Whyburn are included.
- Lewis, Albert C., R.L. Moore entry in The
New Handbook of Texas.
- Lewis, Albert C., R.L.
Moore entry in the Dictionary of Scientific Biography,
(Charles Scribner's Sons, 1990): vol. 18, 651-653.
- McAuley, L., Dedication to
F.B. Jones, Proceedings of the Auburn
Topology Conference, March 1969. On CD.
- Moore, R.L., "A lineage of
R.L. Moore." On CD.
Moore's genealogical notes on his paternal grandmother's family.
- Moore, R.L., "Genealogy research of
R.L. Moore." Moore's
notes. On CD.
- Murray, Margaret A.M., Women Becoming Mathematicians: Creating a
Professional Identity in Post-World War II America, MIT
Press, 2000.
Interviews
are included with Mary Ellen Rudin and Lida Barrett.
- Parker, J., R. L. Moore: Mathematician and Teacher (Mathematical
Association of America, 2005).
This is the most comprehensive biography to date and, in contrast to D.
R. Traylor’s 1972 account, it was able to make extensive use of Dr.
Moore's papers and the oral history resource in the Archives
of American Mathematics.
- Rogers, J.T., "F.B. Jones -- An appreciation," 1999. On CD.
- Rudin, Mary Ellen, "The early works of
F.B. Jones," Handbook
of the History of General Topology, Vol. 1. C.Aull and R. Lowen (eds),
Kluwer Academic 1997, 85-96. On CD.
- Stephan, F.F., Tukey, J.W., et al.,, "Samuel S. Wilks" Journal
of the American Statistical Association 60(1965). On
CD.
Wilks's first advanced mathematics course was under R.L. Moore.
- Traylor, D.R., Creative
Teaching: The Heritage of R.L. Moore, University of Houston, 1972.
Written during Dr. Moore’s lifetime but not authorized by him, this
account by a member of the Moore school of mathematics is a major
resource for information about his life and career. The author
interviewed early students and colleagues. Chapters by W. Bane and M.
Jones list publications by Moore and his mathematical descendants.See
Paul Halmos's review in Historia Mathematica 1(1974), 188-192.
- Wilder, Raymond L., "Robert
Lee Moore 1882-1974," Bull. AMS 82, 1976, 417-427. Also
on CD.
- Young, G.S., "Being a student of
R.L. Moore," 1938-42, A
Century of Mathematics Meetings, AMS, 1996, 285-293.
- Zettl, A., "Research on differential equations." On CD.
Zettl
describes his research but includes a brief biographical account of his
family's harrowing escape from a Yugoslavian concentration camp and
eventual immigration to the US. Thanks to the Moore Method used in
classes that he took, he found his weak mathematics background to be no
great disadvantage since he was on the same initial footing as everyone
else.
- Zitarelli, David, and
Bartlow, Thomas L.,"Who was Miss Mullikin?"
American Mathematical Monthly 116(2009): 99-114. See The
Mathematical Tourist synopsis
by Ivars Peterson.
Little
had been known about Moore's third PhD student Anna Mullikin (1922) who
published only one research paper, her dissertation, and became a high
school teacher. This article sheds new light on her life and shows how
influential her mathematics and her teaching were.
- Bing, RH, "Notes for R H Bing's Plane Topology Course." On
CD.
A one-semester undergraduate course started by Bing at Wisconsin and
continued by R.E. Fullerton, S.C. Kleene, and R.F. Williams. The notes
were used by C.B. Allendoerfer at the University of Washington and by
W.L. Duren, Jr., at Tulane University.
- Center for American History Archives. Collection
Description. R.L. Moore Archives.
- Fitzpatrick, B., "Some aspects of the work and influence of
R.L.
Moore," Handbook of the History of General Topology, Vol. 1.
C.Aull and R. Lowen (eds), Kluwer Academic 1997, 41-61. On CD.
- Frantz, J.B., "The Moore Method," The Forty Acres Follies,
Texas Monthly Press, 1983, 111-122. On CD.
- Frantz, J.B., "One-of-a-kind alumni," Alcalde
7(1984), 20-24. On CD.
- Jones, F. Burton, "The
Beginning of Topology in the United States and The Moore School,"
in C.E. Aull & R. Lowen eds., Handbook of the History of General
Topology, Volume 1. Kluwer Academic Publishers: 97-103 (1997). Also
on CD.
- Lewis, Albert C., "The beginnings of the R. L. Moore school of
topology," Historia Mathematica, 31, 2004, 279-295. Preprint
(240 KB PDF file).
- Lewis, Albert C., "Reform
and Tradition in Mathematics Education: The Example of R.L. Moore."
April 1998.
- Moore, R.L., (1930, May 22-23). Letter
to G.T. Whyburn concerning the axioms of Foundations of Plane
Analysis Situs. Also on CD.
- Whyburn, Lucille S., "Letters from the
R.L. Moore Papers", Proceedings
of the 1977 Topology Conference I, Topology Proc. 2 (1) (1977):
323-338.
- Wilder, Raymond L., "The mathematical work of
R.L. Moore: its
background, nature, and influence," Arch. History Exact Sci.
26 (1982): 73-97.
- Zitarelli, David, "Towering figures in American
mathematics," Amer. Math. Monthly 108(2001): 606-635. On CD.
On JSTOR.
- Zitarelli, David, "The Origin and Early Impact of the Moore
Method", Amer. Math. Monthly 111(2004): 465-486.
- Ager, Tryg, A., “From interactive instruction to interactive
testing,” in Artificial Intelligence and the Future of Testing
ed. Roy O. Freedle, (Lawrence Erlbaum Associates, 1990).
From
the section “Example of finding-axioms: mathematical conjecture”:
“VALID has one other exercise type that I would like to discuss. This
is by far the most complex type of problem in the course. Based on an
idea of R. L. Moore and modified for interactive use in VALID, there are
seven ‘finding-axioms’ exercises, of which the following is the
simplest” (p. 40).
- Dreyfus, T., and Eisenberg, T., "On different facets of
mathematical thinking," in The Nature of Mathematical Thinking,
eds. R. J. Sternberg, T. Ben-Zeev (Lawrence Erlbaum Associates, 1996),
pp. 253-284.
Includes a basically sympathetic account of the Moore Method but finds
cooperative learning, viewed as "humanizing Moore," to be more
congenial, especially for less motivated students.
- Garrity, T.A. All the Mathematics You Missed: But Need to Know for
Graduate School, (Cambridge University Press, 2002), pp.
77-78.
Discusses significance of point-set topology and its changing role in
the curriculum over the last 70 years with a special mention of its use
by Dr. Moore at the University of Texas.
- Knuth, Eric J., "
Secondary School Mathematics Teachers' Conceptions of Proof," Journal
for Research in Mathematics Education 33, 2002, 379-405.
"As undergraduates, do prospective teachers have opportunities to
experience and discuss these roles of proof? The Moore Method of
teaching, for example, which is used by some mathematicians, provides
undergraduate students with just such an experience" (400).
- Lucas, J.R. The Conceptual Roots of Mathematics: An Essay on the
Philosophy of Mathematics. (Routledge, 2000).
The
author, a Fellow of Merton College, Oxford, argues for a form of
logicism in which much of mathematics is grounded in transitive
relations instead of natural numbers or set theory. After looking at
Alfred North Whitehead’s failed attempt to found geometry and topology
on a mereological basis, i.e. a theory of whole and part, he considers
the transitive relation of ‘being embedded in’ as utilized by R.L.
Moore.
- Milnor, John, “Growing up in the old Fine Hall,” in Prospects
in Mathematics
ed. Hugo Rossi (American Mathematical Society, 1999), p. 3
“The
person who was closest to me in the early years [at Princeton
University] was Ralph Fox. … I particularly enjoyed the course in
point set topology which he taught by a form of the R. L. Moore method:
He told us the theorems and we had to produce the proofs. I can’t
think of a better way of learning how to make proofs and how to learn
the basic facts of topology – it was a marvelous education.”
- Morrel, J.H. “Why lecture? Using alternatives to teach college
mathematics,” in Teaching in the 21st Century: Adapting Writing
Pedagogies to the College Curriculum, (Routledge (UK), 1999), pp.
29-48. Online purchase http://www.questia.com/
“In order to adapt this [the Moore method where students ‘understood
the topics extremely well and had a lot of practice in writing and
explaining mathematics’] to an undergraduate setting, in which the
time constraints and required syllabi mitigate against the use of such a
method, I have used a modification of this approach … ” (p. 38).
- Palombi, Fabrizio, and
Rota, Gian-Carlo, Indiscrete Thoughts, (Birkhäuser, 1997).
"The
core of graduate education in mathematics was Dunford's course in linear
operators. Everyone who was interested in mathematics at Yale eventually
went through the experience, even such brilliant undergraduates as Andy
Gleason, McGeorge Bundy, and Murray Gell-Mann. The course was taught in
the style of R.L. Moore ..." (p. 29).
- Raiffa, H. “Game theory at the University of Michigan, 1948-1952,”
in Toward a History of Game Theory ed. E. Roy Weintraub, (Duke
University Press, 1993).
“I
took a course called ‘Foundations of Mathematics’ with Professor
Copeland, who taught in the R. L. Moore style: students are challenged
to act like mathematicians, to convince themselves and others of the
veracity of some plausible conjectures, to concoct starkly simple
illuminating counterexamples, to generalize, to speculate, to abstract.
No books were used. All the results were proved by the students. … I
became hooked. Even though I didn’t know Leonard J. (Jimmy) Savage at
the time, he also became enthralled in the same type of teaching program
by being forced to act like a mathematician. I decided to become a pure
mathematician and pursue a Ph.D. degree” (p. 166).
- Ross, Arnold E., "Creativity: Nature or Nurture? A View in
Retrospect," in N. Fisher, et al., eds., Mathematicians and
Education Reform, 1989-1990, (American Mathematical Society, 1991),
pp. 39-84.
Ross's
early education in the USSR he likens to the Moore Method. Its spirit of
“explanation and justification” continued to characterize his own
approach to teaching mathematics. However, he mistakenly attributes its
origins in the USA to E.H. Moore. (More on this misunderstanding can be
found in the 1999 videotaped interview with Ross available at the Archives
of American Mathematics.)
- Samuelson, Paul A., Inside the Economist's Mind: Conversations
with Eminent Economists, (Blackwell, 2006).
Robert
Aumann on taking real variables from the logician Emil Post at City
College in the 1940s: "It's called the Moore method—no lectures,
only exercises. It was a very good course." (p. 329)
- Selden, A., and Selden, J. “Tertiary mathematics education research
and its future,” in Teaching and Learning of Mathematics at
University Level: An ICMI Study, ed. Derek Holton, (Springer,
2001), pp. 255-274.
A section on the Moore Method suggests that courses making use of it
“could provide interesting opportunities for research in mathematics
education.”
- Shier, D.R., and Wallenius, K.T. Applied Mathematical Modeling: A
Multidisciplinary Approach, (CRC Press, 1999).
“The
modeling approach in applied mathematics has much in common with the
discovery methods used in pure mathematics, such as the famous R. L.
Moore approach” (p. 14).
- Winkler, P. Mathematical Puzzles: A Connoisseur's
Collection, (A K Peters, Ltd., 2004).
The author comments on the “figure 8s in the plane puzzle" that
he “heard it attributed to the late, great topologist R. L. Moore.”
- Avers, Paul W., "A unit in high school geometry without the
textbook," in Mathematics in the Secondary School Classroom:
Selected Readings, ed. by G.R. Rising and R.A. Wiesen. (Thomas Y.
Crowell, 1972), pp. 231-234.
"Avers'
instructional program parallels the work of the nationally known R.L.
Moore of the University of Texas. Both are able to get their students
emotionally involved with their subject by taking away their
textbooks." (From the editorial introduction, p. 227.)
- Brown, J., "My
Experiences with the Various 'Texas Styles' of Teaching," 1996.
Also on CD.
- Daniel, Dale; Eyles, Joseph W.;
Mahavier, Wm.Ted; Pember, J.Craig,
"On using the discovery method in the distance-education
setting," 2001. Handout.
- Foster, James A., and Barnett, M., “Moore formal methods in the
classroom: A how-to manual,” in Teaching and Learning Formal
Methods, ed. by M. Hinchey, C. N. Dean. (Morgan Kaufmann, 1996),
pp. 79-98.
“One
of us (Barnett) took courses taught using the method in the departments
both of mathematics and of computer sciences while in Austin in the
1980s. We describe the method as it was experienced at that time” (p.
85).
- Foster, James A.; Barnett, M.;Van
Houten, K.; Sheneman, "(In)Formal
Methods: Teaching Program Derivation via the Moore Method," Computer
Science Education, 6(1), 1995, pp. 67-91.
"[T]he
students learned the underlying mathematics of program derivation and
learned to apply it, by presenting proofs and derivations on a daily
basis. Professorial intervention in the classroom was minimal. Our
experience has been that students learn otherwise difficult material
better, and are better able to put it into practice, with this teaching
technique than they would have been able to do in the typical
classroom."
- Good, Chris, "Teaching
by the Moore Method," MSOR Connections 6 No. 2(2006),
34-38.
The author, professor of mathematics at the University of Birmingham in
England, describes a first-year course, "Development of
Mathematical Reasoning," which has proven popular and effective for
mathematics students entering the university.
- Green, John W., Interview with Ben Fitzpatrick. 1999. On CD.
- Heath, J., "The
Discovery Method." Also on CD.
- Heath, R.W., "The discovery method of teaching mathematics."
On CD.
- Ingram, W.T., and Hall,
L.M., "Report on the Faculty Mentoring
Project: Foundations of Mathematics Course (Math 209)," 2002. On
CD.
- Kauffman, Robert, "Three variations on a theme." On CD.
- Kennedy, Judy, "My
Experiences with the Moore Method." 1996. Also on CD.
- Lemke, James, "Discovery
Learning." 1996. An expanded version is on CD as "The
Moore Method."
- Mahavier, Lee, "On three crucial elements of Texas-style teaching
as shown to be successful in the secondary mathematics classroom,"
1999. Handout.
- McClendon James W., On
R.L. Moore.
The
theologian James W. McClendon recalls his experience as a calculus
student with Dr. Moore in 1943.
- Ormes, N., "A beginner's guide to the Moore Method," 1999.
On CD.
- Reagan, Becky, "Testimonial on the influences of R. L.
Moore." On CD.
- Reed, M., "Comments
on Moore-method teaching." 1996. Also on CD.
- Siegel, Martha J., “Teaching mathematics as a service subject,” in
A. G. Howson, et al., eds., Mathematics as a Service Subject,
ICMI Study Series, (Cambridge University Press, 1988), pp. 75-89.
“I
have taught, or rather the students have taught themselves, a full
syllabus of the course … using a small group discovery method. … In
a sort of Moore method approach, the students are given examples to work
out with guidance, a form of programmed prodding towards a solution.”
(p. 85)
- Smith, M., "Learning and Teaching Mathematics via
Discovery," 1996. On CD.
- Stallman, C., "What I learned from my semester with William S.
Mahavier," and W.S. Mahavier, "Mentoring the Moore
Method," 2001. On CD.
- Steenrod, N., Letter to
R.L. Wilder, 1937, regarding Lefschetz's
trying out the Moore Method at Princeton University. On CD.
- Suppes, P., “Uses of artificial intelligence in computer based
instruction,” in Artificial Intelligence in Higher Education:
International Symposium Proceedings, eds. V. Marik, O. Stepankova,
Z. Zdrahal, (Springer, 1991).
“We
adopted a computer version, so to speak, of the famous R.L. Moore method
of teaching. ... It is easy to implement such a method in a computer
framework. A typical example would be a presentation of fifteen
elementary statements about the geometrical relation of betweenness
among three points. Students are asked to select no more than five of
the statements as axioms and to prove the rest as theorems” (p. 208).
- Transue, W.R.R., "Thoughts
on 'Discovery' Type Teaching." 1996. Also on CD.
- Vick, James W., Review of two topology texts, Amer. Math. Monthly
116(2009): 373-375.
"The evidence is clear that the discipline
and rigor learned through [the Moore method] have lasting effects. I can
still remember the thrill of discovery and the triumph of presentation
of a basic theorem as a freshman 47 years ago. ... I will fashion a
course [from the text] in which I will include as many of the
applications as possible, and on Fridays I will convert the class into a
Moore method setting, with students proving theorems from a separate
list I have generated." Vick, a professor at the University of
Texas at Austin, is a third generation doctoral descendant of Moore.
- Wadle, L., "My experience with the
R.L. Moore teaching
method," 2001. On CD.
- Woodruff, Edythe, "My
experiences with the Moore Method," 2005.
- Young, S.W., "Christmas
in Big Lake." 1998. Also on CD.
- Zenor, P., "Personal essay on the Moore Method," 1996. On
CD.
- Adamson, I., A General Topology Workbook, ( Springer, 1996).
“Highly
influenced by the legendary ideas of R. L. Moore, the author has taught
several generations of mathematics students with these materials,
proving again the usefulness and stimulation of the Moore method”
(cover). The author was a student at Princeton 1949-52 (Ph.D. under Emil
Artin) and was introduced to the Moore method there by Ralph Fox.
- Adamson, I., A Set Theory Workbook, (Springer, 1997).
“The main purpose of this approach is to encourage readers, in the
well known educational method of R.L. Moore, to try hard to prove
results for themselves” (cover). Review in American Mathematical
Monthly on JSTOR.
- Burger, Edward B., Extending
the Frontiers of Mathematics, (Key College Publications, 2006).
An
inquiry-based introduction to mathematical proof echoing the Moore
style.
- Donnell, William, "Discovering Divisibility Tests," Mathematics
in School (May 2008), 14-15.
- Dydak, Jerzy, and Feldman, Nathan, "Major Theorems on
Compactness: a Unified Exposition," Amer. Math. Monthly
99(1992), 220-227. On JSTOR.
"[O]ne
can easily convert this text to a collection of problems in classes
where the Moore Method is used."
- Halmos, Paul, Linear Algebra Problem Book, (Mathematical
Association of America, 1997).
- Henle, James, An Outline of Set Theory, (Springer, 1986).
"[T]his
is a Moore-style text whose proper use depends on the confluence of a
patient instructor open to the Moore technique ... with motivated and
intelligent students. ... Highly recommended." (Judith Roitman,
from review in The Journal of Symbolic Logic, 52(1987),
1048-1049. JSTOR
link.) See also The American Mathematical Monthly, 95, 1988, p.
844 (on JSTOR) for
Roitman's response to the "vituperative review" by C.
Smorynski.
- Hodge, Jonathan, and Klima, Richard E., The Mathematics of Voting
and Elections: A Hands-On Approach, (American Mathematical Society,
2005).
“From
a pedagogical standpoint this book was inspired by our involvement in
the Legacy of R.L. Moore Project …. When we set out to write this
book, we wanted to capture the spirit of a Moore method course, but we
also wanted to make sure that the resulting text was accessible to a
non-mathematical audience.” (p. x)
- Levine, Alan L., Discovering higher mathematics: four habits of
highly effective mathematicians, Harcourt Academic Press, 2000.
"In
an effort to show prospective mathematics majors that mathematics is a
vital and beautiful subject, Levine organizes his book around active
participation by students in a four-stage scheme for doing mathematics:
experimentation, conjecture, proof, and generalization." -- Review by
Robert A. Fontenot. American Mathematical Monthly 108, 2001,
179-182 (on JSTOR).
- Mahavier, Wm.Ted, "Interactive Numerical Analysis," Creative
Math Teaching 3, 1-2.
- Marshall, David C. / Odell, Edward /
Starbird, Michael, Number
Theory Through Inquiry, Mathematical Association of America, 2007.
A
text "designed to be used with an instructional technique variously
called guided discovery or Modified Moore Method or Inquiry Based
Learning (IBL). The result of this approach will be that students:
- Learn to think independently
- Learn to depend on their own reasoning to
determine right from wrong
- Develop the central, important ideas of
introductory number theory on their own."
- Moise, Edwin E., Introductory Problem Courses in Analysis and
Topology, (Springer, 1982).
"The
Moore Method is an idea with many fruitful aspects. Let us not throw out
the whole idea because it has some difficult points, rather let us
search for a wider application of the good aspects. Moise has written an
excellent book; it should make it easier for the problem course approach
to find a larger place in the undergraduate curriculum." (From the
review by Carl C. Cowen in the Amer. Math. Monthly, 91(1984):
528-530.)
- Nanzetta, Philip, and
Strecker, George E., Set Theory and Topology, (Bogden & Quigley, 1971).
"Those
experienced in the Moore method, I believe, will appreciate and be able
to use this book just on the basis of the first two chapters on set
theory, but they write their own sets of notes for topology and won't
need this one. To those not experienced in the Moore method, I recommend
this book as a means of introduction to the method and say, 'try it,
you'll like it.'" (R.R. FitzGerald, from review in Amer. Math.
Monthly 79(1972), 920-921.)
- Schumacher, Carol, Chapter Zero: Fundamental Notions of Abstract
Mathematics (Addison-Wesley, 1996).
From
the blurb: "Written in a modified R.L. Moore fashion, it offers a
unique approach in which students construct their own understandings.
However, while students are called upon to write their own proofs, they
are also encouraged to work in groups." Reviewed by Robert A.
Fontenot. American Mathematical Monthly 108, 2001, 179-182 (on JSTOR).
- Terry, Lawson, Topology: A Geometric Approach, (Oxford
University Press, 2003).
"These
chapters are written in a very different style, which is motivated in
part by the ideal of the Moore method of teaching topology combined with
ideas of VIGRE programs in the US which advocate earlier introduction of
seminar and research activities in the advanced undergraduate and
graduate curricula" (p. vi).
- Backlund, Ulf, and Leif
Persson, "Moore's teaching method," Normat
41, 1996, 145-149. (In Swedish)
- Brown, Stephen I., Reconstructing School Mathematics: Problems
with Problems and the Real World (Peter Lang, 2001).
"What
might be viewed as highly suspect (if not downright immoral and
debilitating) in the light of present-day prizing of cooperative
learning, was a life-enhancing source for an unusually large number of
students. In fact, R.L. Moore single-handedly turned out this century's
leading set-theoretic topologists." (p. 42) The author was a
student of E.E. Moise
(Moore PhD, 1947) at the Harvard Graduate School of Education.
- Buck, Robert E., "Conjecturing," PRIMUS; 16(June 2006),
97-104.
A
seminar course for junior-senior mathematics majors is described. The
topic is continued fractions, taught by a modified Moore Method, where
the focus is on students creating their own mathematics.
- Chalice, D., "How to teach a class by the modified Moore
method," Amer. Math. Monthly 102, 1995, 317-321. On CD.
- Clark, David M., "R.L. Moore and the learning curve." On CD.
- Cohen, D.W. "A modified Moore method for teaching undergraduate
mathematics," Amer. Math. Monthly 89, 1982.
- Coppin, Charles A.;
Mahavier, W. Ted; May, E. Lee; and Parker, G.
Edgar, The Moore Method: A Pathway to Learner-Centered Instruction, Mathematical
Association of America, 2009.
- Dancis, Jerome and Davidson, Neil, "The
Texas Method and the Small Group Discovery Method." (1970).
Also on CD
- Devlin, K. "The greatest math teacher ever," Devlin's
Angle at MAA Online. Also on CD.
- Eyles, Joseph, Introduction to Ph.D. thesis, R.L.
Moore's Calculus Course, (The University of Texas at Austin,1998).
Also on CD.
- Eyles, Joseph, "Discovering
R.L. Moore's calculus class,"
talk at the International Congress of Mathematicians 1998. On CD.
- Fitzpatrick, B. "The teaching method of
R.L. Moore," (in
Chinese, translated by Yang Shoulian) Higher Mathematics 1, 1985,
41-45.
- Forbes, D.R. The Texas System: R.L. Moore's Original Edition,
Ph.D. thesis, University of Wisconsin, 1971.
- Halmos, Paul R., "How to Teach," I
Want to be a Mathematician. (SpringerVerlag, 1985). pp. 254-264.
Also on CD.
- Halmos, Paul R., "What is teaching?" Amer. Math. Monthly
101 (1994): 848-855.
"I
will not, I told them, lecture to you .... They stared at me, bewildered
and upset--perhaps even hostile. ... They suspected that I was trying to
get away with something, that I was trying to get out of the work I was
paid to do. I told them about R.L. Moore, and they liked that, that was
interesting. Then I gave them the basic definitions they needed to
understand the statements of the first two or three theorems, and said
'class dismissed'. It worked."
- Halmos, Paul R., &
Moise, Edwin E. "The problem of learning
how to teach," Amer. Math. Monthly 82 (1975): 466-474. On
CD.
- Halmos, Paul R., with Renz, Peter, I Want To Be A Mathematician: A
Conversation with Paul Halmos. Film by George Csicsery. 2009.
Available through the Mathematical Association of America. Description
and trailer from Zala Films.
"The
1999 interview with Paul Halmos (1916-2006) that forms the backbone of I
Want To Be A Mathematician was initiated to gather some comments
from Halmos about R.L. Moore for the Educational Advancement
Foundation’s R.L Moore Legacy Project."
- Jones, F. Burton, "The
Moore Method," Amer. Math. Monthly 84 (Apr. 1977):
273-277. Also on CD.
- Kapur, J.N., "Moore method of teaching," in Current
Issues in Higher Education in India, (S. Chand, 1975), pp. 203-205.
"Today's
world needs creative minds and in India the need for such minds is
desperate. We should experiment with any method which has a promise of
enabling the students to face unfamiliar situations with confidence. The
world is so dynamic today that mastery of facts has become secondary to
mastery of techniques of acquiring knowledge." Kapur was a
professor at Indian Institute of Technology, Kanpur, and Vice-Chancellor
of Meerut University.
- Kauffman, Robert M., "Some remarks on the Socratic method in
mathematics." On CD.
- Kauffman, Robert M. et al., "A
Demonstration of the Moore Method", Quicktime movie.
- Legacy of R.L. Moore Project, "Master of the Game,"
10-minute video. Available through the EAF.
This is a sample reel, prepared by George Paul Csicsery, for a proposed
new video on Dr. Moore and the Moore Method. It includes excerpts from
interviews with mathematician Paul Halmos and theologian James W.
McClendon.
- Mahavier, William S., "What
Is the Moore Method?" Primus, 9 (December 1999):
339-254. Also on CD.
- Mahavier, W. Ted, "A gentle discovery method (the modified Moore
method)," College Teaching 45, 1997, 132-135. On CD.
- Mahavier, W. Ted, et al., "A
Demonstration of the Moore Method", Quicktime movie.
- Mahavier, W.Ted, E. Lee May, and G. Edgar Parker, A
Quick-Start Guide to the Moore Method.
- Moise, E.E. "Activity
and motivation in mathematics," Amer. Math. Monthly 72,
1965, 407-412. Also on CD.
- Moore, R.L. "Challenge in the Classroom." Video.
(Mathematical Association of America, 1967). Available through the EAF
in VHS or DVD format. Transcript.
This film features interviews with Moore and shows him in the classroom.
It is the only attempt he made to publicize his teaching method.
- Moore, R.L. Notes for the MAA film, "Challenge
in the Classroom." Also on CD.
- Moore, R.L. (1948, May 17). Letter
to Miss Hamstrom, a prospective student. Published in A Century
of Mathematical Meetings (American Mathematical Society, 1996), pp.
295-300. Also on CD.
- Parker, G.E. "Getting
more from Moore." Primus, vol. 2 (September 1992):
235-246. Also on CD.
- Renz, Peter, The
Moore Method: What discovery learning is and how it works. FOCUS:
Newsletter of the Mathematical Association of America. (1999,
August/September). PDF
Version (37KB).
- Spresser, Diane, "The
Moore Method: Viewpoint of a Department Chair," 2008.
- Whyburn, Lucille S., "Student-oriented teaching - the Moore
method," Amer. Math. Monthly 77, 1970, 351-359. On CD.
- Wilder, Raymond L., "Axiomatics
and the development of creative talent," The Axiomatic
Method with Special Reference to Geometry and Physics, L. Henken, P.
Suppes, and A. Tarski (eds), North-Holland, (1959), 474-488. Also on CD.
- Wilder, R.L., "Material and method," Undergraduate
Research in Mathematics: A Report of a Conference Held at Carleton
College, Northfield, Minnesota, June 16 to 23, 1961, Edited by
Kenneth O. May and Seymour Schuster, pp. 9-27.
- Yorke, J.A., and Hartl, M.D. "Efficient methods for covering
material and Keys to Infinity," Notices of the AMS
(June/July 1997): 685-687. PDF
version on the AMS web.
“Since
the roots of the problems described above run so deep, it is imperative
that potential solutions (such as the Moore method) be implemented early
in students’ careers—and not just for students planning to become
mathematicians” (p. 686).
-
Anderson,
L. W. Sosniak, L. A. (1994) Bloom's Taxonomy: A forty-year
retrospective.
-
Krathwohl,
D. R, Anderson, L. W. (2001) A Taxonomy for Learning, Teaching, and
Assessing: A Revision of Bloom's Taxonomy of Educational Objectives
Learning Domains or Bloom's
Taxonomy - Donald R. Clark
-
Paul,
R. (1993). Critical thinking: What every person needs to survive in a
rapidly changing world (3rd ed.). Rohnert Park, California: Sonoma State
University Press.
-
Taxonomy
of Educational Objectives: The Classification of Educational Goals; pp.
201–207; B. S. Bloom (Ed.) Susan Fauer Company, Inc. 1956.
-
A
Taxonomy for Learning, Teaching, and Assessing — A Revision of Bloom's
Taxonomy of Educational Objectives; Lorin W. Anderson, David R.
Krathwohl, Peter W. Airasian, Kathleen A. Cruikshank, Richard E. Mayer,
Paul R. Pintrich, James Raths and Merlin C. Wittrock (Eds.) Addison
Wesley Longman, Inc. 2001
-
"Taxononmy
of Educational Objectives. Handbook II: The affective domain; Krathwohl,
D. R., Bloom, B. S., Masia, B. B.; 1964.
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