Problem Solving in the Mathematics Curriculum. A Report, Recommendations, and an Annotated ...

142 Pages·2007·2.23 MB·English

DOCUMENT RESUME.
SE 041 449
ED 229 248
Schoenfeld, Alan H.
AUTHOR
Problem Solving in the Mathematics Curriculum. A
TITLE
Report, Recommendations, and an Annotated
Bibliography. MAA Notes, Number 1.
Mathematical Association of America, Washington,
INSTITUTION
D.C.
PUB DATE
83
of
142p.; Prepared by the Committee on the reaching
NOTE
Undergraduate Mathematics.
.
1529
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Mathematic;
Annotated Bibliographies; *College
DESCRIPTORS
Curriculum Development; Educational Research;
Guidelines; Higher Education; *Mathematics
Curriculum; Hathematics Education; *Mathematics
Instruction; *Problem Solving; *Teaching Methods
Mathematical Assoc.iation of America; Mathematics
IDENTIFIERS
Education Research
ABSTRACT
This report, prepared for and published by the
the Teaching of
Mathematical Association of America's Committee on
of the state of the
Undergraduate Mathematics, includes a description
and makes
art on problem solving, lists available resources,
solving in the college
recommendations regarding the place of problem
recommends (1) an
curriculum and ways to teach it. The report
alert and
approach to teaching mathematics that fosters an
them in
questioning attitude in students and that actively engages
series of problem-soliring
the process Of doing mathematics, (2) a
sophistication as regular offerings in
courses at various levels of
series of texts for
the standard college curriculu, and (3) a
problem-solving courses at all levels to be developed and
how to teach problem
disseminated. Specific suggestions are given on
teacher and ways of
solving, especially pertaining to the role of the
class discussions are
organizing the-class. Some typical problems and
bibliography of
provided. Then follows an extensive annotated
type of
problem-solving resources, with characterizations of the
appropriate, its focus or subject
course for which each appears most
.
,and articles are listed
matter, and its level. Journals, books,
questionnaire and responses
separately. Finally, the problem-solving
are briefly presented:
(MNS)
t
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MATHEM:ATICS
COMMITTEE .ON THE,JEACHINGOr UNDWRADUATE
Alan H. Schoenfeld
PROBLEM SOLVING IN THE MATHEMATICS CURRICULUM:
A-REPORT, RECOMMENDATIONS, AND AN ANNOTATED BIBLIOGRAPHY
1983
The Mathematical Association of America
-/
Committee on the Teaching of Undergraduate Mathematics
0,
Acknowledgments
Contributions to this report and bibliography came Crom many people.
It is a pleasure to acknowledge their help,.and to thank them for it.
The
M.A.A. Committee on the Teaching of Undergraduate Mathematics and its Problem
Solving Subcommittee developed the Survey of Problem Solving Courses
(pp. 134-137), and the M.A.A. distributed it nationwide.
After the data were
compiled, the Committee suggested that I write the suggestions for teaching
problem solving and compile the bibliography.
Henry Alder first suggested
that the report, recommendations, and bibliography be combined into the
volume that you are now reading.
He provided encouragement and helpful
suggestions throughout its development. Tom Butts wrote the first draft of
A large number of people provided lists of "favorite" sources
section 3D.
Murray Klamkin provided a long
in response to question 17 of the Survey.
Johanna Zecker spent endless hours in -the.library
list of books and articles.
Jerry Alexanderson annotated many of the
checking bibliographic data.
Members of the Committee and Subcommittee, most notably Don
references.
I am solely
Of course,
Bushaw, vigilantly tracked down flaws in manuscript.
responsible for the flaws that remain.
AHS
- -
i
PROBLEM SOLVING IN THE MATHEMATICS CURRICULUM:
A REPORT, RECOMMENDATIONS, AND AN ANNOTATED-BIBLIOGRAPHY
Contents
Acknowledgments.
i
ii
Contents
Introduction
1
2
Recommendations
Suggestions for teaching problem solving: ,
5
Background and rationale
8
Some.issues in teaching problem solving
26
Class format
37
Some "typical" problems and class discussions
Annotated bibliography:
52
Overview
54
Journals
63
Books
109
Articles
State of the Art:
on\the
Report
130
Problem Solving Questionnaire & Responses
Introduction
If he fills
A Teacher of Mathematics has a great opportunity.
his allotted time with drilling his students with routine operations
inteligctual development, and
he kills their interest, hampers their
But if he challenges the curiosity of his-
misuses Kis opportunity.
knowledde,
students by setting them problems proportionate to their
and helps them to solve their problems with stimulating questions,
he may give them a taste for, and some means of, independent thinking.
G. POlya, How to Solve It
In March 1980 the Mathematical Association of America's Committee on the
Teaching of Undergraduate Mathematics formed a Subcommittee on Problem Solving,
with the following charge:
to gather information from undergraduate programs, analyze
the current literature, and produce a report which
describes the "state of the art,"
.
lists available resources, and
3. makes recommendations regarding the place of
problem solving in the curriculum and ways to
teach it.
In early 1981 a "Survey of Problem Solving Courses" was mailed to a group of
faculty inclUding all college level mathematics departmentchairmen in the
Of those,
A total of 539 departments responded.
United States and Canada.
195 indicated that they currently offer problem solving courses, and provided
In addition, there were 86 responses like the following:
descriptions of them.
"We do not have a problem solving course at present but are interested in
Please send a copy of your report and any other useful material."
developing one.
The responses to the questionnaire provide the basis for our description
There were many suggestions and much enthusiasm
of the state of the art.
There were also many requests for help.
for teaching problem solving.
a
SpecificallY, we received repeated requests for two kinds of information:
a bib-
collection of suggestions for teaching problem solving
course's', and
We are pleased to offer this volume
liography of resources for such courses.
in response to those requests.
RECOMMENDATIONS
The full rationale for offering problem solving courses is given in
To
"suggestions for teaching mathematical problem solving," which Follows.
w believe that the primary responsibility of mathematics
put things briefly,
faculty is to teach their students to think: to question and to probe, to get
rather
to the'mathematica1 heart of the matter, to be able to employ ideas
As P.R. Halmos argues in "The Heart of
than simply to regurgitate them.
Mathematics,"
'he major part of every'meaningful tife is the solution of
problems; a considerable part of the professional life of
technicians, engineers, scientists, etc. is the solution of
It is the duty ci all teachers, and of
mathematical problems.
teachers of mathematics in particular, to expose their students
to problems much more than to facts.
The "problem approach" to teaching mathematics is valuable for all
those who will simply "appreciate" it, those who will use it,
students:
and those who will live it (solving problems is, in essence, the life of
)
In particular,
the professional mathematician!).
We endorse any approach to teaching mathematics that fosters an
1.
alert and questioning attitude in students, and that actively engages students
We encourage the use of a "problem based
in the process of doing mathematics.
approach" wherever possible in standard course offerings, including the par.:
ticipation of students in discussing, solving, and presenting their solutions
(Those worried about subject matter coverage should see section
to problems.
We similarly encourage "problem of the week"
2D of the teaching suggestions.)
contests, informal problem seminars, etc.
We recommend that a series of problem solving courses at various
2.
levels of sophistication be eeveloped and made regular offerings in the
3
In particular,
standard curriculum.
welcome and meaning-
Elementary problem solving courses serve as
a.
take a college Math course but have
ful alternatives for students who wish to
bad"
replacements for the typical "math isn't so
no need for the calculus; as
to the calculus for students who
liberal arts courses; and as supplements
mathematics at an elementary level.
wish to be introduced to substantive
specific subject matter
Upper division problem courses, either on
b.
students
problem solving topics, can introduce
or covering a range of general
in a substantive way long before they
to the spirit of mathematical inquiry
in professional careers or in doing
would encounter it on their own, whether
mathematical rese'arch.
modeling, in general literacy, etc.
Special courses for teachers, in
c.
to the mathematical experience
(as in the survey results), all provide access
experience it.
for students who might not otherwise
of Recommendation 2, we
In order to foster the implementation
3.
problem solving courses at all levels
recommend that a series of texts for
be developed and widely disseminated.
The
step in that direction.
We hope that this volume serves as a
for teaching a problem solving course.
following section offers some suggestions
"College
spirit as recommendations in CTUM's
These are put forth in the same
We offer them for your
Suggestions on How to Teach It."
Mathematics:
The section on teaching is
consideration, and hope you find them useful.
problem solving and the results of
followed by an annotated bibliography .on
indicates, there are a great
As the scope of the bibliography
the survey.
Whatever the particular nature of the course
variety of available resources.
8
4
you might like to offer, you will find ample collections of problems appro-
.
.
priate for it and you will find a wide variety of ideas about teaching it.
In a sense, the most difficult aspect of giving a problem solving course
We encourage you to do so, and believe
is making the decision to offer it.
4
that you and your students will benefit from the experience.
Members of CTUM During the Preparation
of/this Manuscript
Alan H. Schoenfeld
Henry L. Alder
Donald W. Bushaw
Martha J. Siegel
Elmer Tolsted
Ronald M. Davis
James W. Vick, Chair
Gloria F. Gilmer
James E. Ward
Leon W. Rutland
.
June P. Wood
David I. Schneider
Sol%;iiig Subcommittee
Members of the Problein
1
and-Contributors to the Report
Gerald L. Alexanderson
Thomas R. Butts
Mary Grace Kantowski
Murray S. Klamkin
Donald G. Saari
Alan H. Schoenfeld, Chair
John E. Wetzel
c;.
9
\\\
PROBLEAOLVING
SUGGESTIONS FOR TEACHING MATHEMATICAL
be
Thei4 is no one "right" way to teach problem solving, and it would
there are as many effective ways
presumptuous to consider recommending one:
Moreover,
talented teachers.
to teach mathematical thinking as there are
-
What "works" for .one teacher
style.
classroom methods are a matter of personal
'
for another teacher to use it comfortably,
may have to be modified in order
For that
understanding.
These suggestions are offered with that
if at all.
The
informally and in the first person.
reason they are written somewhat
Please treat them as you would
suggestions have worked well in the classroom.
Consider them, try the'ones
colleague.
treat the suggestiOns from a close
then tailor them so that you feel
that seem.approp.riate oh fot size, and
comfortable witO them.
Background and rationale
1.
that we do mathematics-and
There is a huge difference.between the way
Doing mathematics is a vital, ongoing
the way that'our students see it.
understand the nature of particular math-
process of discovery, of coming to
..
As
witti an .area.
First we becalm familiar
ematical objects or systems.
thattsomething ought
We begin to suspect
wp do, our intuitions develop.
try to get
We test it with examples, look for counterexamples,
to be true.
When we think we know what makes it,
be true.
a sense of why it ought to
There ;ley
The attempt may or may not succeed.
work, we try to prove it.
retrenchments, and modifications.
be any number of false starts, reverses,
Few experiences
into place.
With perseverance and luck, the result falls
we have chartedAinknown territony, and
areso gratifyiog or exciting:
,
40