**ANNOTATED BIBLIOGRAPHY**

In shaping my thinking about this course I have been influenced by numerous readings and sources. These include books and journal articles on the theory of mathematical problem solving as well as many collections of problems. Below is an annotated bibliography of those books and articles that I found most helpful.

Brown, Stephen I. and Walter, Marion I.(1990)
*The Art of Problem Posing*, *Second Edition*. Hillsdale, New Jersey.
Lawrence Erlbaum Associates, Publishers.

To my mind the best problems are those that involve several steps to a solution. Finding that path generally involves formulating a number of intermediate problems which then each must be solved. Consequently, a good problem solver is someone who is experienced at asking questions and posing problems of their own. Problem posing often also arises at the point at which one has solved a problem and one examines what has been obtained and what more might be achieved, that is, in the process of generalization. This book is about the process of problem posing. Its purpose is to encouage a shift in control from external authority to the student of mathematics. Thus, the authors suggest that through problem posing students may be able to attain a new way of viewing the standard topics of mathematics and thereby achieve a deeper understanding. Also, they suggest, by shifting the locus of control it may be possible to help struggling students overcome math anxiety. The main goals of the book are to understand: What problelm posing consists of and why it is important; what strategies exist for engaging in and improving problem posing; and how problem posing relates to problem solving. The strategy of the book is to initiate different problem posing issues through an activity and then to reflect on its significance.

Cofman, Judita. (1990) *What to Solve? Problems
and Suggestions for Young Mathematicians*. Oxford. Clarendon Press.

This book is a compilation of problems and solutions from several international camps for young mathematicians (ages 13-19) given by the author. The difficulty level of the problems vary but there are a number which are definitely appropriate to a college level problem solving course. In addition to the collection of problems for independent work there is a section on approaches to problem solving which develops the main technqiues in the context of specific problems, including: hte use of alternative representation, generalization, the search for invariants, the extremal method, mathematical induction and others.

Cofman, Judita (1995) *Numbers and Shapes
Revisited: More Problems for Young Mathematicians*. Oxford. Clarendon Press.

Essentially a sequel to the previous book with a problems drawn from a greater variety of mathematical areas, including number theory, combinatorics, abstract algebra, geometry, sequences of numbers. In the first part of the book each chapter begins with the introduction of new concepts and a derviation of fundamental properties followed by problems which build on this foundation.

Dreyfus, Tommy (1999) "Why Johnny Can't
Prove (with Apologies to Morris Kline)," In *Educational Studies in Mathematics*
**38**: 85-109.

This paper investigates the quality and rigor of students' written explanations given as responses to instructor or textbook questions such as homework assignments. It includes examples of student explanations, research on students' conceptions of proof, an examination of the experiences students have, either through the reading of textbooks or classroom instruction, with mathematical proof, and theoretical considerations. Since many, if not most, of the problems which I assigned were of the "prove" type, the research summarized in this paper was of interest to me.

Engel, Arthur (1998) *Problem-Solving Strategies*.
New York. Springer-Verlag.

This book contains an enormous number of excellent problems organized in 14 chapters by the types of strategies most appropriate for a solution. These include: the invariance principle, coloring, the extremal principle, the pigeonhole principle, enumerative combinatorics, number theory, inequalities, induction, sequences, polynomials, geometry, games and others. The discussion of methods is generally short and mosty illustrated through the solution of several examples at the beginning of each chapter.

Fomin, Dimitri, Genkin, Sergey, and Itenberg,
Ilia (1996) *Mathematical Circles (Russian Experience)*. Rhode Island.
American Mathematical Society.

A large collection of problems that were used with school children in connection with the "mathematical circles" in the former Soviet Union, groups of students, teachers and mathematicians. As stated in the Foreword, the book is "predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport, without necessarily being competitive." These particular problems come from the circles in the former Leningrad, now St. Petersburg. Though primarily used with school age students (middle school and high school) there are nonetheless some worthwhile problems even for beginning college problem solvers. The problems are divided according to either the mathematical content from which the problem is drawn or alternatively the type of method or technique used to solve them. These include combinatorics, pigeonhole principle, elementary number theory, graph theory, geometry and invariants.

Gardiner, A. (1987)* Discovering Mathematics:
The Art of Investigation*. Oxford. Clarendon Press.

This book could be a good source for more extended problem - investigations - that might be given to students in a problem solving course after they have had experiences with novel but shorter problems or else serve as a model for what is desireable in longer, more complex investigatory problems. The particular problems in this book are all drawn from elementary mathematics and does not require anything deeper than a rudimentary knowledge of the integers and geometry and a modicum of mathematical sophistication. It teaches both problem solving techniques and heuristics in the context of the several short and two longer investigations that are developed in detail.

Hornsberger, Ross. (1996) *From Erdos to Kiev*:*
Problems of Olympiad Caliber*. The Mathematical Association of America.

A collection of problems with solution drawn from the highest level secondary school mathematics competitons.

Hornsberger, Ross. (1997) *In Polya's Footsteps:
Miscellaneous Problems and Essays*. The Mathematical Association of America.

Another collection of problems, with solutions, drawn from high level secondary school mathematics competitions.

Kadesch, R.R. (1997) Problem Solving Across the Disciplines, Preliminary Edition. Upper Saddle River, NJ. Prentice Hall.

This book is intended to instruct students in problem solving, in particular, how to develop the intellectual skills of recalling approporiate knowledge and tools, comprehension, analysis, synthesis, application and evaluation. Some emphasis is placed on developing mental control and application of heuristics (referred to as commands) and identified as "those things problem solvers tell themselves to do in seeking solutions to problems before actually doing them." It has a recommended list of such commands and also ideas for how students can develop their own. While the book does contain problems drawn from many disciplines, the primary source is mathematics. The first chapter is an introduction to the art and skill of problem solving and contains 26 short warmup problems. Chapter II deals with probablistic problems, Chapter III with decision making strategies. Chapter IV deals with games and game theory, Chapter V with graph theory. Patterns are the subject of Chapter VI and logic and reasoning that of Chapter VII. Chapter VIII deals with the difficulties that arise in problems solving, Chapter IX with problems in ethics. Chapter X is devoted to four advanced problem solving topics. Each chapter begins with some introductory problems that provide a context for a discussion of content and techniques, followed by additional, more difficult problems. As mentioned, there is an appendix with a lengthy list of commands an aspiring problem solver can use to get started.

Kisacanin, Branislav. (1998) Mathematical Problems and Proofs: Combinatorics, Number Theory, and Geometry. New York and London.

A source of problems of moderate difficulty by content area, specifically, as the title suggests, in combinatorics, number theory and geometry.

Kranz, Steven G. (1997) *Technqiues of Problem
Solving*. Rhode Island. American Mathematical Society.

A treatment of the basic principles of problem solving. Some emphasis is given to forming representations of problems but the central focus is on specific methods and techniques as well as some heuristics such as: generalization, specialization, reformulation, decompostion, recombination, the introduction of auxilary elements and problems. The content areas from which the book draws its material are: elementary combinatorics, probability, plane and solid geometry, graph theory, and game theory. The book contains many problems, some challenging ones as well as solutions to nearly all the odd numbered exercises.

Larson, Loren C. (1983)* Problem-Solving Through
Problems*. New York Berlin. Springer-Verlag

The first chapter of this book has a good abbreviated account of heuristics which are each illustrated by several examples. Subsequent chapters are devoted to methods of problem solving (induction and pigeonhole principle) or problems in content areas such as number theory, polynomials, series, real analysis, inequality and geometry. Some good problems can be found here.

McKnight, C., Magid, A., Murphy, Teri J., and
McKnight, Michelynn. (2000) *Mathematics Education Research: A Guide for the
Research Mathematician*. Rhode Island. American Mathematical Society.

This book is intended for the research mathematician who may wish to consult the growing literature of education research on undergraduate mathematics in effort to improve their own teaching practice. It is well written and short enough to finish in a couple readings. In part its purpose is to help the research mathematician understand the difference between research in mathematics and in mathematics eduation and to be able to discriminate between good and poor educational research. It suggests that mathematics education research is "inquiry by carefully deveoped research methods aimed at providing evidence about the nature and relationships of many mathematics learning and teaching phenomena." It suggests necessary criteria for categorizing mathematics education work as serious research. These include: clear research questions, well-chosen variables, a statement of methods, analysis procedures appropriate level of generality, and honesty in separating conclusions from speculations. In the first chapter, on evidence based pedagogy there is an excellent typography of the different types of educational research, such as experimental versus observational, the contrast between prospective and retrospective research as well as quantative and qualitative. Amongst the 12 chapters there are ones dealing with critiquing quantative research, reliability and validity in quantitative research, a survey of statistical methods, critiquing qualitative research, reliability and validity in quantative research, a survey of qualitative methods, teaching experiments, evaluation and assessment and how to do literature searches. After reading this book I learned about all the things I was doing wrong in terms of framing the questions that I did (overly broad) and collecting data. However, for this project it came far too late, since I only obtained the book after most of this course portfolio was complete and long after the course was first given and all the data collected. Nonetheless it will inform subsequent offerings of the course and be reflected in revisions to this portfolio.

Michalewicz, Z. and Fogel, D.B. (2000) *How
to Solve It: Modern Heuristics*. New York. Springer-Verlag.

This book is intended as a text for a class in modern heuristics for students in science, business and engineering. It assumes a basic knowledge of discrete mathematics and computer programming. Its subject matter which provides the context for the book is primarily algorithms but it is not a book about algorithms. It teaches both special problem solving techniques as well as how to frame new problems and think creatively as well as general heuristics. Despite an emphasis on algorithms there are many worthwhile mathematical problems throughout the text that require formal proofs with worthwhile accompanying discussion.

Polya, G. (1957) *How to Solve It: A New Aspect
of Mathematical Method, Second Edition*. Princeton, NJ. Princeton University
Press.

First published in 1945, this is the first of the many books written by Polya on problem solving and heuristics (and described by Schoenfeld as the book which inspirted his own investigations in problem solviing). The first part sets out a general framework for problem solving which I used in my class: begin by understanding the problem (I referred to this as orientation), devise a plan, execute the plan, look back, check results and evaluate the solution. About 80% of this little book is devoted to a "dictionary" of heuristics with nearly seventy concepts introduced and each described briefly. Among the entries are: auxilary elements, auxilary problem, decomposing and recombining, do y ou know a related problem, generalization, separate the various parts of the condition, and working backwards. The book concludes with some twenty problems, hints and solutions.

Polya, G. (1968)* Induction and Analogy in
Mathematics. Volume I of Mathematics and Plausible Reasoning*.Princeton,
NJ. Princeton University Press. *Patterns of Plausible Inference. Volume II
of Mathematics of Plausible Reasoning*. Princeton, NJ. Princeton University
Press.

These two books, first published in 1954, are
a continuation of *How to Solve It*. Plausible reasoning is distinguished
from demonstrative reasoning (rigorous, logical proof). The demonstrative aspect
is only one aspect of mathematics, insightful "guessing" is another,
and this involves plausible reasoning. While Polya decries the idea that there
is a foolproof method to learn guessing in mathematics, he nonetheless argues
here that it can be a practical skill which can be learned and honed by imitation
and practice. Most of the chapters of volume I discuss mathematical discoveries,
some of consequence, others more pedestrian, each accompanied by a lively narration
and uncovering the motives for the discovery. Volume I is also devoted to a
particular form of plausible reasoning, namely the use of induction - moving
from an accumulation of evidence to a possible mathematical proposition. The
problems which form the context of volume I are drawn from solid geometry, the
theory of numbers, problems of maxima and minima, physical mathematics and the
isoperimetric problem. The book also contains many problems for the reader as
well as solutions. Volume II is more philosophical and general and attempts
to formulate patterns of plausible reasoning and much of the text

Polya, G. (1962) *Mathematical Discovery:
On Understanding, Learning and Teaching Problem Solving, Combined Edition*.
New York. John Wiley & Sons.

Another of Polya's books on the art of problem
solving and a further elaboration on *How to Solve It*. Volume One contains
two parts. Part One is devoted to an examination of familarizing the reader
with useful patterns that arise in problem solving and contains four chapters
devoted to geometric loci, the reduction of problems to the solution of equations
(Polya refers to these as the Cartesian pattern), recursion, and suposition.
Part Two of volume I is entitled "Toward a General Method, contains two
chapters and is devoted to heuristics which are discussed in general in the
context of problems and patterns discussed earlier in the volume. The second
volume continues the discussion of heuristics. The first chapter lays out the
plan for the remaining chapters in this volume and develops and uses the context
of a geometric question to investigate the elements of the problem solving process.
These include: understanding the problem, ideas for getting started, formulating
a plan and program, the possibility of generating subsidiary problems and so
on. Also, within this context, Polya discusses issues relating to cognition
- how the mind is working and made to work during the problem solving process.
The subequent chapters of Volume Two explore these topics in much greater detail,
generally illustrated in the actual process of examining a particular problem.

Posamentier, Alfred and Salkind, Charles T.
(1996) *Challenging Problems in Geometry*. New York. Dover Publications,
Inc.

Posamentier, Alfred and Salkind, Charles T.
(1996) *Challenging Problems in Algebra*. New York. Dover Publications,
Inc.

Collections of problems, mostly at the high school level but some, particularly from the geometry book that can be used in a college course on problem solving.

Rubinstein, Moshe F. (1975) *Patterns of Problem
Solving*.Englewood Cliffs, NJ Prentice-Hall, Inc.

A book prepared for an interdisciplinary course of the same name. The book attempts a balance between problem solving techniques and cultivating the attitudes and attributes the author deems as essential for good problem solving. The first five chapters are devoted to the general foundation of problem solving and develops a framework for problem solving, including such mathematical tools as set theory. The succeeding five chapters are more specfic and emphasize specific classes of problems, models and methods, including probablistic models, decision models and optimatization models.

Schoenfeld, Alan H., Kaput, Jim and Dubinsky, Ed, Editors. (1998) Research in Collegiate Mathematics. III Rhode Island. American Mathematical Society.

The entire volume if worthwhile but the first three papers are most germane to my class and course portfolio and all deal with the same course in problem solving, taught by Schoenfeld at the University of California, Berkeley. The first article, "Teaching Mathematical Problem Solving: An Analysis of an Emergent Classroom Community" is authored collectively by Abraham Arcavi, Cathy Kessel, Luciano Meira and John P. Smith III, each of whom individually analyze a different aspect of Schoenfeld's class. Arcavi provides an overview of the course. This includes a background and statement of goals and objectives. He describes the kind of classroom environment fostered by Schoenfeld and the attributes he intends to bring out in students, the curriculum, the pedagogy and the effectiveness of the course. Meira analyzes the first problem introduced in Schoenfeld's course, on an infinite series to contrast the difference between presenting and doing mathematics and presenting on how to do mathematics. . Smith's paper is an account of a particular problem (inscribing a square in a triangle) that Schoenfeld introduces early in his class which illustrates both heurisic modeling and the way in which Schoenfeld asserts authority in the classroom. Finally, Kessel uses a class devoted to the magic square to discuss communication in Schoenfeld's classroom, in particular, she makes use of sociolinguistic analysis to contrast the types of discourse in Schoenfeld's classroom with that usually found in mathematical exposition (papers and texts) as well as the typical classroom. The article also addresses the rich heuristics which arise out of this simple problem.

The second paper, "On the Implementation of Mathematical Problem Solving Instruction: Qualities of Some Learning Activities," is written by Manuel Santos-Trigo. This paper focuses on the environment of Schoenfeld's course and consistent features that contributed to students' mathematical growth. Among these features are problems which stimulate discussion, offer mathematical lessons and heurisitic principles, that underscore the need to justify speculations drawn frominvestigations with mathematical arguments, and so on.

The final paper, "Reflections on a Course in Mathematical Problem Solving," is by Schoenfeld. In this article the author sketches a model of the teaching process developed by the Teacher Model Group at Berkeley. In subsequent sections the goals of the problem solving course are discussed and elaborated. The first several classes, which "make or break" the course, and set the tone for the entire semester are analyzed through the model.

Schoenfeld, Alan H. (1985) *Mathematical Problem
Solving*. New York. Academic Press, Inc.

An absolute must read for anyone interested in mathematical problem solving. More than anything else, this book influenced my own thinking and interest in teaching problem solving. The book is an outgrowth of the first decade of Schoenfeld's teaching of problem solving and his research, the purpose of which is to make sense of people's mathematical behavior, encompassing fresh students and experienced problem solvers. The heart of the book is a framework for an analysis of complex problem-sovling behavior which is developed in the first part, comprising chapters 1 through 5. The first chapter outlines and briefly describes Schoenfeld's framework which identifies four elements in expert problem solving: resources, heuristics, metacognition and control, and beliefs. These are then separately elaborated on in chapters 2-5. The second part of the book, chapters 6-10, present a series of empirical studies which document the way problem solvers make use of the resources that they bring to their mathematical activity.

Wickelgren, Wayne A. (1974) *How to Solve
Problems: Elements of a Theory of Problems and Problem Solving*. San Francisco.
W.H.Freeman and Company

Devoted entirely to a discussion of heuristics and mental strategies for solving mathematical problems. The book" teaches by example", with a new problem solving technique being discussed theoretically, then applied to a variety of problems. There is a good introductory chapter on problem theory followed by chapters on inference, state evaluation and hill climbing, subgoals, the use of contradiction, working backwards, relating problems to other problems and the use of such heuristics as simplifying a problem, specializing a problem and generalizing a problem and representation of problems.

Yaglom, A.M. and Yaglom, I.M. (1964) *Challenging
Mathematical Problems With Elementary Solutions. Volume I: Combinatorial Analysis
and Probability Theory. Volume II Problems From Various Branches of Mathematics*.
New York. Dover Publications, Inc.

Two collections of problems.

Yusof, Yudariah BT. Mohammad and Tall, David
(1999) "Changing Attitudes to University Mathematics Through Problem Solving,"
in* Educational Studies in Mathematics* **37**: 67-82

This paper describes the effect on students attitudes of a course which encouraged co-operative problem solving as measured by student responses to an attitudinal questionaire. I found the questionaire given to be useful in formulating my own questions to students and it will certainly have an influence on my design of instruments for obtaining data in subsequent offerings of the course.

Zeist, Paul (1999) *The Art and Craft of Problem
Solving*. New York. John Wiley & Sons, Inc.

A textbook for a college course in mathematical problem solving for novices. The book nearly contains it all: It discusses strategies for investigating problems, including pyschological strategies, getting started which includes useful heuristics such as search for a penultimate step, do some computation, apply wishful thinking and make the problem easier; types of mathematical arguments such as deduction, argument by contradiction and mathematical induction. The several sections of chapter three are devoted to tactics or what I have called techniques such as the exploitation of symmetry, the extreme principle, the pigeonhole principle and invariants. Chapter four develops three topics that are useful when employing an alternative representation: graph theory, complex numbers and generating functions and recurrence. The final chapters are devoted to problems and methods in different content areas including algebra, combinatorics, number theory and calculus. I drew quite a few problems from this text and would consider it for adoption in the next offering of my class.