Mathematical Problem Solving Research in New Learning Environments

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Series F: Computer and Systems Sciences Vol. 89

The ASI Series Books Published as a Result of
Activities of the Special Programme on
ADVANCED EDUCATIONAL TECHNOLOGY
This book contains the proceedings of a NATO Advanced Research Workshop held
within the activities of the NATO Special Programme on Advanced Educational
Technology, running from 1988 to 1993 under the auspices of the NATO Science
Committee.
The books published so far as a result of the activities of the Special Programme are:
Vol. F 67: Designing Hypermedia for Learning. Edited by D. H. Jonassen and H.
Mandl. 1990.
Vol. F76: Multimedia Interface Design in Education. Edited by A. D. N. Edwards and
S. Holland. 1992.
Vol. F 78: Integrating Advanced Technology into Technology Education. Edited by
M. Hacker, A. Gordon, and M. de Vries. 1991.
Vol. F80: Intelligent Tutoring Systems for Foreign Language Learning. The Bridge to
International Communication. Edited by M. L Swartz and M. Yazdani. 1992.
Vol. F81: Cognitive Tools for Learning. Edited by PAM. Kommers, D.H. Jonassen,
and J.T. Mayes. 1992.
Vol. F84: Computer-Based Learning Environments and Problem Solving. Edited by
E. De Corte, M. C. Linn, H. Mandl, and L. Verschaffel. 1992.
Vol. F85: Adaptive Learning Environments. Foundations and Frontiers. Edited by M.
Jones and P. H. Winne. 1992.
Vol. F86: Intelligent Learning Environments and Knowledge Acquisition in Physics.
Edited by A. Tiberghien and H. Mandl. 1992.
Vol. F87: Cognitive Modelling and Interactive Environments in Language Learning.
Edited by F. L. Engel, D. G. Bouwhuis, T. Basser, and G. d'Ydewaile. 1992.
Vol. F89: Mathematical Problem Solving and New Information Technologies. Edited
by J. P. Ponte, J. F. Matos, J. M. Matos, and D. Fernandes. 1992.
Vol. F90: Collaborative Learning Through Computer Conferencing. Edited by A. R.
Kaye. 1992.

Mathematical Problem Solving and
New Information Technologies
Research in Contexts of Practice
Edited by
Joao Pedro Ponte
Joao Filipe Matos
Departamento de Educac;ao, Faculdade de Ciâlcias
Universidade de Lisboa, Av. 24 de Julho, 134-4°
P-1300 Lisboa, Portugal
Jose Manuel Matos
Sec'1ao de Ciâlcias da Educac;ao, Faculdade de Ciâlcias e Tecnologia
Universidade Nova de Lisboa
P-2825 Monte da Caparica, Portugal
Domingos Fernandes
Instituto de Inova'1ao Educacional, Travessa Terras de Sant'Ana-15
P-1200 Lisboa, Portugal
Springer-Verlag Berlin Heidelberg GmbH

Proceedings of the NATO Advanced Research Wor1<.shop on Advances in
Mathematical Problem SoIving Research, held in Viana do Castelo, Portugal,
27-30 ApOI, 1991
CR Subiect Classification (1991): K.3.1
ISBN 978-3-642-83483-3 ISBN 978-3-642-58142-7 (eBook)
DOI 10.1007/978-3-642-58142-7
This work is subject to copyright. M rights are reservoo.wh!!ther the whole Of" part 01 tha material is concemed.
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undar the German COpyright Law.
C Springar-Variag Berlin He1de1barg 1992
Originally published by Springer-Verfag Berlin Heidelberg NfI'N Yor( in 1992
Soflcover reprint of !ha har<loover 1si edition 1992
Typasatting: C!mefa f88.dy by aultlors
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Plinted on acid-rrea papar

Preface
A strong and fluent competency in mathematics is a necessary condition for scientific,
technological and economic progress. However, it is widely recognized that problem solving,
reasoning, and thinking processes are critical areas in which students' performance lags far
behind what should be expected and desired. Mathematics is indeed an important subject, but is
also important to be able to use it in extra-mathematical contexts. Thinking strictly in terms of
mathematics or thinking in terms of its relations with the real world involve quite different
processes and issues. This book includes the revised papers presented at the NATO ARW
"Information Technology and Mathematical Problem Solving Research", held in April 1991, in
Viana do Castelo, Portugal, which focused on the implications of computerized learning
environments and cognitive psychology research for these mathematical activities.
In recent years, several committees, professional associations, and distinguished
individuals throughout the world have put forward proposals to renew mathematics curricula,
all emphasizing the importance of problem solving. In order to be successful, these reforming
intentions require a theory-driven research base. But mathematics problem solving may be
considered a "chaotic field" in which progress has been quite slow. There are many questions
still to be resolved, including:
- The purpose of problem solving: what is the problem solving activity? What are suitable
problems for teaching purposes? How are they best explored in class? How can teachers
develop their awareness about it?
- Its curricular status: how are problems integrated with the remaining classroom activities?
What instructional strategies can be used to improve students' competency? How should they
be assessed?
- The nature of the necessary research: what is the kind of knowledge that should be striven
for? What are the most appropriate research methodologies? Are there critical variables to
control, manipulate, and measure, and what are they?
There are four main questions running through the papers presented in this book: (a) What
is a problem-solving activity and how can it be assessed? (b) What psychological theories can
be used to explain and improve students' problem-solving ability? (c) What are the implications
of new information technologies for mathematical problem solving? (d) How can these issues

VI
be carried out to practice, namely to teacher education and to the instructional environments?
The papers are sequenced according to their focus on one of those themes; however, most of
these aspects pervade back and forth through the book. A simple glance at the abstracts and
keywords will tell the reader the main concerns ofeach author.
Indeed, the nature of problem-solving as a mathematical activity is a critical issue.
Problem solving has become a very popular expression, meaning quite different things for
different people. It can be seen as an add-on to the existing practices, or more radically as a
replacement. Alternatively, it may be seen as a new perspective which enables us to see new
and old things in a different way. In some views, problem solving mostly consists of well-
defined activities, which may have a peripheral or central role in mathematics learning. For
others, problem solving is just an aspect of the more general concept of mathematical
experience. Looking beyond well-defined and clearly formulated problems, one can devise
other highly significant kinds of activity such as open-ended investigations, making and testing
conjectures, and problem formulation, which may extend the flavour of creative mathematical
thinking into the mathematics classroom. A problem formulated as such by a researcher or by a
teacher may not have the same meaning (in fact any meaning at all) for a student. But the
purpose of education is not that the students should just be expected to be able to solve the
problems and tasks proposed by the teachers, but also to be able to raise, solve, and evaluate
their own significant questions.
A closely related issue concerns students' assessment. Without proper assessment the
teacher does not have the possibility of ascertaining if the pupils are making any progress at all.
Without assessment instruments and procedures there is no way of conducting research.
However, assessing problem solving has been a major difficulty both for research and practice.
If one is not just concerned with the number of exact solutions but also with the strategies and
thought processes (successful or unsuccessful), and these are not overtly shown, then how
does one assess them? Analytic scoring schemes, drawing upon Polya's four-stage model of
problem solving, have been proposed as a frame for getting a composite picture of students'
work. Alternatively, problem solving can be assessed in a holistic way, given the complex
nature of the mental processes involved and the fact that this activity is always both socially and
culturally situated. The tasks proposed may range from well-defined mathematical questions to
open-ended real-life situations. However, the difficulties do not lie just in the researchers'
models and in the teachers' capability to use new approaches. They have to do also with the
students who are used to being tested on content knowledge rather on thinking processes,
reasoning, or strategies arising in complex problems, and simply do not understand the new
demands that are required from them. In a word, significant problem-solving experiences and
new assessment tasks may imply not just new classroom procedures but also a significant
cultural change.

VII
Psychological research has been a major theoretical influence on mathematical problem
solving. In the past, most research efforts in this field followed an established tradition in
psychology: students are presented with ready-made, well-defmed, tasks in laboratory settings.
Much valuable information can be gathered from this kind of research, but current educational
thinking stresses the importance of activities of a very different kind, in which (a) students
participate in the process of defining the nature and goal of the task, (b) things are not so well-
defined from the very beginning but follow processes which may bear successive
reformulations, and (c) the setting is much more complex in terms of roles and interactions than
the usually straightforward researcher-subject relationship. Problems presented to the students
in previous research include, among others, puzzles, word problems, simple and complex real-
world applications, concept recognition tasks, strategy games, and questions relying on the
knowledge of established formal mathematics. It is quite unlikely that general theories can be
drawn up to encompass all these activities. And it is certainly questionable whether all of them
have equivalent educational usefulness and value. However, one can recognize important steps
that have been accomplished more recently. A general agreement has been reached around the
need to take into consideration students' thought processes (especially metacognitive processes)
and not just their problem solving competency measured by success rates in given kinds of
problems. The need for theoretical orientations guiding research efforts and the value of the
contributions of cognitive psychology to study those processes have been widely recognized.
At this meeting the notions of conceptual field, didactic contract, cognitive conflict, sense
making, and noticing were proposed to further extend the theoretical frameworks to study
mathematical problem solving. Further progress is also dependent on refining the definition of
key concepts and methodologies for research in this area.
The availability of modern information technologies with its possibility to empower
mathematical thinking and extend the range and scope of applications of mathematics poses new
challenges for mathematical problem-solving research and practice. The computer can be
viewed as a powerful intellectual tool, providing the means to automate routine processes and
concentrate strategic thinking. Alternatively, it can be seen as a base to construct microworlds
deliberately designed to foster specific kinds of knowledge. Attempts are being made to apply
artificial intelligence techniques to software design in mathematics. But several contributions at
the meeting made it clear that the software never works just by itself, and a critical role is played
by the teacher in setting up the learning situation. With the computer, mathematics can become
a much more experimental activity. However, the possibility of conducting easily a large
number ofexperiences may prevent the most appropriate thinking from taking place - especially
if students are not properly encouraged to develop critical and metacognitive processes.
Problem solving is related to skills learning, concept and principle development, and
reasoning processes. The way it is integrated into the classroom practices bears close
connections with the teachers' confidence, feeling, educational agenda and espoused

VIII
philosophy of mathematics. In this meeting, there was a special concern with the issues of
classroom practice. In the countries in which problem solving is part of the curriculum, the
activities that are performed in schools, if they exist, tend to not go much further than simple
applications of content knowledge. How will teachers acquire confidence as problem solvers?
How will they acquire the competence to conduct problem solving activities? What sort of
support will be required? It is necessary to study the role of the teacher and the processes of
teacher development, both through formal training programmes and through their participation
in innovation processes. Formal pre-service and in-service courses are needed but they cannot
replace the initiative of the teachers themselves giving rise to grass-roots innovative
experiences. These, in tum, to be fruitful, consistent, and lasting should have support from
research and teacher education institutions.
The perspective of the computer as an intellectual tool, enabling explorations, and
empowering the students, dominates current thinking concerning the use of this instrument in
mathematical problem solving activities. However, the generalization of its use imposes stricter
demands both on software development and teacher education. The growing interest of
researchers in what happens in the classrooms may hopefully represent a turning point in this
field. The concern with investigations, conjecturing, problem posing, and seeing problem
solving in a wider context of mathematical activities may also represent a very important new
direction of thinking. If that is the case, the computer with its strong invitation to
experimentation may well have been a decisive factor in this evolution.
Lisbon, May 1992 Jo3:o Pedro Ponte
JOOo Filipe Matos
Jose Manuel Matos
Domingos Fernandes

Contributors and Participants
Canada
Joel Hillel
Department of Mathematics and Statistics. Conconlia University
Loyola Campus. 7141 Sherbrooke Street West. Montreal. Quebec H4B lR6. CANADA
E_mail: jhillel@vax2.concordia.ca
France
Colette Laborde
Laboratoire LSD2 - IMAG. Univ. Joseph Fourier-CNRS
BP 53 X. 38041 Grenoble Cedex. FRANCE
Jean Marie Laborde
Laboratoire LSD2 - IMAG. Univ. Joseph Fourier-CNRS
BP 53 X. 38041 Grenoble Cedex. FRANCE
E_mail: laborde@imag.fr
Jean ｆｾｩｳ＠ Nicaud
CNRS University ofParis-Sud. Laboratoire de Recherche en Informatique
Batiment 490. 91405 Orsay Cedex. FRANCE
E_mail: jfn@lri.lri.fr
Greece
Chronis Kynigos
University of Athens
19 Kleomenous St.• 10675 Athens. GREECE
E_mail: sip6O@grathunl.bitnet
Israel
Tommy Dreyfus
Center for Technological Education
P.O. Box 305. Holon 58102. ISRAEL
Fax: 972-1-5028967
Italy
Ferdinanda Anarello
Dipartimento di Matematica. University of Torino
Via Carlo Alberto. 10. 10123 Torino. ITALY
Fax: 39-11-534497

PaoloBoero
Dipartimento di Matematica,University ofGenova
Via L.B. Alberti,4, 16132 Genova, ITALY
Fax: 39-10-3538769
Pier Luigi Ferrari
University of Genova, Dipartimento di Matematica
E_mail: grandis@igecuniv
Fax:39 10 3538763
Fulvia Furinghetti
Dipartimento di Matematica, University ofGenova
Fax: 39-10-3538769
The Netherlands
Henk van der Kooij
ow&OC, Utrecht University
Tiberdreef4, 3561 GG Utrecht, THE NETHERLANDS
E_mail: jan@owoc.ruu.nl
Fax: 31-30-660403
Portugal
Domingos Fernandes
Instituto de ｉｮｯｶｾｮｯ＠ Educacional
x
Travessa Terras de Sant'Ana - IS, 1200 LISBOA, PORTUGAL
Fax: 351-1- 690731
Henrique Manuel Guirnaries
Departamento de ｅ､ｾＬ＠ Faculdade de Cimcias, Universidade de Lisboa
Av. 24 Julho - 134 - 4°, 1300 LISBOA, PORTUGAL
E_mail: ejfm@fcvaxl.fcl.rccn.pt
Fax: 351-1-604546
Maria Cristina Loureiro
Escola Superior de ｅ､ｾ＠ de Lisboa
Av. Carolina Micaelis, 1700 LISBOA, PORTUGAL
Joao Filipe Matos
Departamento de ｅ､ｾｮｯＬ＠ Faculdade de Cimcias, Universidade de Lisboa
Av. 24 Julho - 134 - 4°,1300 LISBOA, PORTUGAL
E_mail: ejfm@fcvaxl.fcl.rccn.pt
Fax: 351-1-604546
Jose Manuel Matos
ｓｾ＠ de Cimcias da ｅ､ｾＬ＠ Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa
2825 MONTE DA CAPARICA, PORTUGAL

XI
Jolo Pedro Ponte
Departamento de ｅ､ｾＬ＠ Faculdade de Ci&1cias, Universidade de Lisboa
Av. 24 Julho - 134 - 4°, 1300 LlSBOA, PORTUGAL
E_mail: ejp@fcvaxl.fcl.rccn.pt
ｆ｡ｸＺＳＵＱＭＱｾＵＴＶ＠
Jaime Carvalho e Silva
Departamentode Matem4Iica. Universidade de Coimbra
Apartado 3008 COIMBRA, PORnJGAL
E_mail: Jaimecs@ciuc2.uc.rccn.pt
Fax: 351-039-32568
MariaGraciosa Ve1oso
Departamento ､･ｾＬ＠ Faculdade de Ci&1cias, Universidade de Lisboa
Av. 24 Juiho - 134 - 4°, 1300 LISBOA, PORTUGAL
E_mail: ejfm@fcvaxl.fclrccn.pt
Fax: ＳＵＱＭＱｾＵＴＶ＠
Spain
Juan Dfaz Godino
Dep. Didactica de la Matematica, Escuela Universitaria del Professorado
Campus de Cartuja. 18071 GRANADA, SPAIN
E_mail: Jdgodino@ugr.es
Fax: 58-203561
United Kingdom
Alan Bell
Shell Centre for Mathematical Education
University Park, Nottingham, NG7 2RD, UK
Fax: 0602-420825
PaulEmest
School of Education, University ofExeter
St Luke's, Heavitree Road, Exeter, EXI 2LU, UK
E_mail: (Janet)emest.p@uk.ac.exeter
Fax: 0392-264857
John Mason
Centre for Mathematics Education,The Open University
Walton Hall, Milton Keynes, MK76AA, UK
E_mail: jh_mason@vax.acs.open.ac.uk
USA
Patricia A. Alexander
Department ofEDCI/EPSY, College of Education, Texas A&M University
College Station, TX 77843, USA
E_mail: ellOpa@tamvml.tamu.edu
Fax:409-845-6129

XII
Randall I. Charles
Dep. Mathematics and Computer Science, San Jose State University
San Jose, CA 95192-0103, USA
Fax: 408-924-4815
Robert Kansky
Mathematical Sciences Education Board
818 Connecticut Avenue, NW, Suite 500, Washington, DC 20006, USA
E_mail: RKansky@nas.bitnet
Fax:202-334-1453
Jeremy Kilpatrick
Department of Mathematics Education, University of Georgia
105 Aderhold Hall Athens, Georgia 30602, USA
E_mail: jkilpat@uga.bitnet
Fax:I-404-542-SOIO or 4551
Frank Lester, Jr.
Mathematics Education Development Center, School of Education, Indiana University
W.W. Wright Education Building, Suite 309, Bloomington, IN 47405, USA
E_mail: lester@iubacs.bitnet
Fax: 812-855-3044
Judah Schwartz
Educational Technology Center, Harvard Graduate School of Education
Nichols House, Appian Way, Cambridge MA 02138, USA
E_mail: jUdah@hugsel.bitnet
Fax: 617-495-0540
Lora J. Shapiro
Learning Research and Development Center, University of Pittsburgh
3939 O'Hara Street, Pittsburgh, PA 15260, USA
E_mail:Shapiro@Pittvms
Fax: 412-6249149

Table of Contents
A Framework for Research on Problem-Solving Instruction. . . . . . . . . . . . . . . . . . .. 1
Frank K. Lester, Jr., Randalll. Charles
Researching Problem Solving from the Inside . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . 17
John Mason
Some Issues in the Assessment of Mathematical Problem Solving. . . . . . . . . . . . . .. 37
Jeremy Kilpatrick
Assessment of Mathematical Modelling and Applications ......................45
Henkvan tier Kooij
A Cognitive Perspective on Mathematics: Issues of Perception, Instruction,
and Assessment ................................................. 61
Patricia A. Alexander
The Crucial Role of Semantic Fields in the Development of Problem Solving Skills
in the School Environment ......................................... 77
Paolo Boero
Cognitive Models in Geometry Learning ................................ 93
Jose Manuel Matos
Examinations of Situation-Based Reasoning and Sense-Making in Students'
Interpretations of Solutions to a Mathematics Story Problem .................. 113
Edward A. Silver, Lora J. Shapiro
Aspects of Hypothetical Reasoning in Problem Solving ..................... 125
PierLuigi Fe"ari

XIV
Problem Solving, Mathematical Activity and Learning: The Place of Reflection
and Cognitive Conflict. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137
Alan Bell
Pre-Algebraic Problem Solving ................................... " 155
Ferdinando Anarello
Can We Solve the Problem Solving Problem Without Posing the
Problem Posing Problem? ......................................... 167
JudahL. Schwartz
Problem Solving in Geometry: From Microworlds to Intelligent
Computer Environments .......................................... 177
Colette Laborde, Jean-Marie Laborde
Task Variables in Statistical Problem Solving Using Computers ................ 193
J. Dfaz Godino, M. C. Batanero Bernabeu, A. Estepa Castro
The Computer as a Problem Solving Tool; It Gets a Job Done, but Is It Always
Appropriate? .................................................. 205
Joel Hillel
Insights into Pupils' and Teachers' Activities in Pupil-Controlled Problem-Solving
Situations: A Longitudinally Developing Use for Programming by All in a Primary
School ...................................................... 219
Chronis Kynigos
Cognitive Processes and Social Interactions in Mathematical Investigations. . . . . . .. 239
Jotio Pedro Ponte, Jotio Filipe Matos
Aspects of Computerized Learning Environments Which Support Problem Solving ... 255
Tommy Dreyfus
A General Model of Algebraic Problem Solving for the Design of
Interactive Learning Environments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267
Jean-Franfois Nicaud

xv
Problem Solving: Its Assimilation to the Teacher's Perspective. . . . . . . . . . . . . . .. 287
Paul Ernest
Computer Spreadsheet and Investigative Activities: A Case Study of an
Innovative Experience ............................................ 301
Jodo Pedro Ponte, Susana Carreira
Examining Effects of Heuristic Processes on the Problem-Solving Education of
Preservice Mathematics Teachers .................................... 313
Domingos Fernandes
Mathematics Problem Solving: Some Issues Related to Teacher Education, School
Curriculum, and Instruction ........................................ 329
RandallI. Charles
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343

A Framework for Research on Problem-Solving Instruction
Frank K. Lester, Jr.1, Randall!. Charles2
ISchool ofEducation, Indiana University, Bloomington, Indiana 47405, USA
2San Jose State University, San Jose, California 95192-0103, USA
Abstract: Research on mathematical problem solving has provided little specific infonnation
about problem-solving instruction. There appear to be four reasons for this unfortunate state of
affairs: (1) relatively little attention has been given to the role of the teacher in instruction; (2)
there has been little concern for what happens in real classrooms; (3) there has been a focus on
individuals rather than small groups or whole classes; and (4) much of the research has been
largely atheoretical in nature. This paper discusses each of these reasons and presents a
framework for designing research on mathematics problem-solving instruction. Four major
components of the framework are discussed: extra-instruction considerations, teacher planning,
classroom processes, and instructional outcomes. Special attention is given to factors that may
be particularly fruitful as the focal points of future research.
Keywords: problem-solving instruction, teacher education, teaching, classroom processes,
affects, instructional outcomes
Despite the popUlarity of problem solving as a topic of research for mathematics educators
during the past several years, there has been growing dissatisfaction with the slow pace at
which our knowledge has increased about problem-solving instruction. Research has
demonstrated that an individual must solve many problems over a prolonged period of time in
order to become a better problem solver. Unfortunately, beyond this there is little specific
advice for teachers that can be gleaned from research. In this paper we provide a brief
discussion of some of the reasons for this unfortunate condition, we propose a step toward a
solution, and we discuss some important considerations for future research concerned with
mathematical problem-solving instruction.
Why Has Progress Been Slow?
Among the reasons why the research conducted thus far has provided so little direction for
problem-solving instruction, four stand out as especially prominent: (a) lack of attention to the
role of the teacher in instruction, (b) the absence of concern for what actually happens in real

2
classrooms, (c) a focus on individuals rather than groups or whole classes, and (d) the
atheoretical nature of much of the research.
The Role of the Teacher
Silver [35] has pointed out that the typical research report might describe in a general way the
instructional method employed, but rarely is any mention made of the teacher's specific role. A
step toward structuring descriptions of what teachers do during instruction was taken by the
Mathematical Problem Solving Project (MPSP) at Indiana University with the identification of
several teaching actions for problem solving [38]. Subsequent to the work of the MPSP, we
identified ten teaching actions for problem solving [5, 6]. These teaching actions were selected
by listing the thinking processes and behaviors that the research literature and other sources
suggested as desirable outcomes of problem-solving instruction. We then identified teaching
behaviors that seemed likely to promote these behaviors. Our task was made especially difficult
by the fact that none of the literature on mathematical problem-solving instruction discussed the
specifics of the teacher's role and very little of the research literature on teaching dealt with
problem solving. As reasonable as our effort seemed at the time, we now view it as
incomplete. What is needed is to consider teaching behavior, not simply as an agent to effect
certain student outcomes, but rather as one dimension of a dynamic interaction among several
dimensions.
Observations of Real Classrooms
A few years ago we conducted a large-scale study of the effectiveness of an approach to
problem-solving instruction based on the ten teaching actions mentioned above. The research
involved several hundred fifth and seventh grade students in more than 40 classrooms [6]. The
results were gratifying: students receiving the instruction over the course of a year benefited
tremendously with respect to several key components of the problem-solving process.
However, despite the promise of our instructional approach, the conditions under which the
study was conducted did not allow us to make extensive systematic observations of
classrooms. Ours is not an isolated instance. Good and Biddle [13], Grouws [14], and Silver
[35] have noticed an absence of adequate descriptions of what actually happens in the
classroom. In particular, there has been a lack of descriptions of teachers' behaviors, teacher-
student and student-student interactions, and the type of classroom atmosphere that exists. It is
vital that such descriptions be compiled if there is to be any hope of deriving sound
prescriptions for teaching problem solving.

3
Focus on Individuals Rather than Groups or Whole Classes
Much of the research in mathematical problem solving has focused on the thinking processes
used by individuals as they solve problems or as they reflect back on their work solving
problems. When the goal of research is to characterize the thinking involved in a process like
problem solving, a microanalysis of individual performance seems appropriate. However,
when our concerns are with classroom instruction. we should give attention to groups and
whole classes. We agree with the argument of Shavelson and his colleagues [32] that small
groups can serve as an appropriate environment for research on teaching problem solving. But.
the research on problem-solving instruction cannot be limited to the study of small groups.
Lester [25] suggests that. in order for the field to move forward. research on teaching ｰｲｯ｢ｬ･ｾ＠
solving needs to examine teaching and learning processes for individuals, small groups, and
whole classes.
Atheoretical Nature of the Research
The absence of any widely-accepted theories to guide the conduct of research is a serious
problem [1, 17,20]. The adoption of a theory orientation toward research in this area is crucial
if progress is to be made toward establishing a stable body of knowledge about problem-
solving instruction. In particular, we would like to see the development of theories of problem-
solving instruction become a top priority for mathematics education research.
Toward a Solution
An important ingredient in making research on problem-solving instruction more fruitful is the
clear description of the factorsl involved. In this section we provide an analysis of the factors
relevant to the study of problem-solving instruction. This analysis is intended as a structure
within which to design research.
Categories of Factors Related to Problem-solving Instruction
Our first attempt to provide an analysis of the factors that might be related to problem-solving
instruction was made by Lester [24] in his general discussion of methodological issues
lWe are aware that the word "factors" is commonly associated with experimental and quasi-experimental research
methods. Our use of this word should not be interpreted as indicating that we have a preference for this sort of
research to study problem-solving instruction. We have no such preference. Indeed, we believe that progress in
this area will bemade primarily by means ofcarefully conducted ethnographic studies, longitudinal case studies
of classrooms, observations of individuals and small groups, and clinical interviews. The word "factors" is used
here exclusively to indicate what we consider to be key ingredients in the success or failure of problem-solving
instruction.

4
associated with research in this area. Lester's analysis was based on the earlier efforts of
Dunkin and Biddle [9] and Kilpatrick [16]. Dunkin and Biddle's work was a broad view of the
important categories of factors associated with research on teaching and was not specifically
concerned with either mathematics or problem solving. Kilpatrick's analysis was restricted to
research on teaching problem-solving heuristics in mathematics. After considering research on
teaching behavior and teacher-student interaction (see e.g., [22, 23, 30, 31, 41, 42]), Charles
[5] developed a refinement of Lester's original analysis. The primary improvement was that
Charles' analysis incorporated two factors which were not present in Lester's work: a teacher
planning category and interactions among categories of classroom processes. The structure
presented below is a further refinement of our previous conceptualizations and other more
recent conceptualizations of teaching. As is true of Charles's categorization, we have chosen to
classify factors on the basis of three broad categories: Extra-classroom Considerations,
Classroom Processes, and Instructional Outcomes. We have also identified a category that cuts
across each of these categories: Teacher Planning. The four categories are discussed in the
following sections.
Category 1: Extra·classroom Considerations. What goes on in a classroom is
influenced by many things. For example, the teacher's and students' knowledge, beliefs,
attitudes, emotions, and dispositions all play a part in determining what happens during
instruction. Furthermore, the nature of the tasks (i.e., the activities in which students and
teachers engage during instruction) included during instruction, as well as the contextual
conditions present also affect instruction. We have identified six types of considerations:
teacher presage characteristics, student presage characteristics, teacher knowledge and affects,
student knowledge and affects, tasks features, contextual (situational) conditions.
Teacher and Student Presa&<; Characteristics. These are characteristics of the teacher and
students that are not amenable to change but which may be examined for their effects on
classroom processes. In addition, presage characteristics serve to describe the individuals
involved. Typically, in experimental research these characteristics have potential for control by
the researcher. But, awareness of these characteristics can be useful in non-experimental
research as well help researchers make sense of what they are observing. Among the more
prominent presage characteristics are age, sex, and previous experience (e.g., teaching
experience, previous experience with the topic of instruction). Factors such as previous
experience may indeed be of great importance as we learn more about the ways knowledge
teachers glean from experience influences practice (see [33]).
Teacher and Student Affects. Cognitions. and Metacognitions. The teacher's and students'
knowledge (both cognitive and metacognitive) and affects (including beliefs) can strongly
influence both the nature and effectiveness of instruction. As a category, these teacher and
student traits are similar to, but quite different from, presage characteristics. The similarity lies
in the potential for providing clear descriptions of the teacher and students. The difference
between the two is that affects, cognitions, and metacognitions may change, in particular as a
result of instruction, whereas presage characteristics cannot.

5
Much recent attention has been given to the role of the teacher's knowledge as it relates to
planning and classroom instruction (see e.g.,[33]). Doyle [8] suggests that teaching can be
viewed as a problem-solving activity where the knowledge a teacher brings to the teaching
situation is craft knowledge [34], that is, knowledge gleaned from experience. Doyle suggests
that "teaching is, in other words, fundamentally a cognitive activity based on knowledge of the
probable trajectory of events in classrooms and the way specific actions affect situations" [8,
p. 355]. Thus, there is growing evidence that the knowledge a teacher brings to the classroom
has a significant impact on the events that follow, and that there are a variety ofkinds of teacher
knowledge that need to be considered in research on teaching (e.g., content knowledge,
classroom knowledge).
Task Features. Task features are the characteristics of the problems used in instruction and
for assessment. At least five types of features serve to describe tasks: syntax, content, context,
structure, and process (see [12]). Syntax features refer to the "arrangement of and relationships
among words and symbols in a problem" [18, p. 16]. Content features deal with the
mathematical meanings in the problem. Two important categories of content features are the
mathematical content area (e.g., geometry, probability) and linguistic content features (e.g.,
terms having special mathematical meanings such as "less than," "function," "squared").
Context features are the non-mathematical meanings in the problem statement. Furthermore,
context features describe the problem embodiment (representation), verbal setting, and the
format of the information given in the problem statement. Structurefeatures can be described as
the logical-mathematical properties of a problem representation. It is important to note that
structure features are determined by the particular representation that is chosen for a problem.
For example, one problem solver may choose to represent a problem in terms of a system of
equations, while another problem solver may represent the same problem in terms of some sort
of guessing process (see for example, Harik's discussion of "guessing moves" [15]). Finally,
process features represent something of an interaction between task and problem solver. That
is, although problem-solving processes (e.g., heuristic reasoning) typically are considered
characteristics of the problem solver, it is reasonable to suggest that a problem may lend itself to
solution via particular processes. A consideration of task process features can be very
informative to the researcher in selecting tasks for both instruction and assessment.
Contextual Conditions. These factors concern the conditions external to the teacher and
students that may affect the nature of instruction. For example, class size is a condition that may
directly influence the instructional process and with which both teacher and students must
contend. Other obvious contextual conditions include textbooks used, community ethnicity,
type of administrative support, economic and political forces, and assessment programs. Also,
since instructional method provides a context within which teacher and student behaviors and
interactions take place, it too can at times be considered a factor within this category.
The six types of extra-classroom considerations are displayed in Table 1. It is important to
add that we do not regard these six areas of consideration as comprehensive. It is likely that
there are other influences that may be at least as important as the ones we have discussed.

Table 1: Key Extra-classroom Considerations
Teacber ｐｲ･ｳ｡ｾ＠ Characteristics
Examples:
6
Task Features
In particular:
• age, sex, ... • syntax
• teacher education experiences • content
• teaching experience • context
• teacher traits (IQ, personality, teaching • mathematical and logical structure
skills) • processes (e.g., "inherent" heuristics)
Teacher Affects. Co&nitions & MetaCOKDitions Contextual Conditions
In particular: Examples:
• knowledge of content, curriculum, & • classroom contexts (e.g., class size,
pedagogy textbook used)
• affects about self, students, mathematics, • schooVcommunity contexts (e.g., ethnicity,
teaching administrative support)
• beliefs about self, students, mathematics, • sociaVeconomic/political forces
problem solving, and teaching • mathematics content to be leamed
Student PreS" Characteristics
Examples:
• age, sex, ...
• instructional history
• student traits (IQ, personality, ...)
Student Affects. Cognitions & MetacoKDitions
In particular:
• mathematical knowledge, "world"
knowledge
• affects about self, teacher, mathematics, &
problem solving
• beliefs about self, teacher, mathematics, &
problem solving
Rather, our intent is to point out the importance ofpaying heed to the wide range of factors that
can have an impact on what takes place during instruction.
Category 2: Teacher Planning. Teacher planning is not clearly distinct from the other
categories; in fact, it overlaps each of them in various ways. Of particular interest for research
are the various decisions made before, during and as a result of instruction about student
presage characteristics, instructional materials, teaching methods, classroom management
procedures, evaluation of student performance, and amount of time to devote to particular
activities and topics. Unfortunately, teacher planning has been largely ignored as a factor of
importance in problem-solving instruction research. Indeed, in most studies teacher planning
has not even been considered because the teachers in these studies have simply implemented a
plan that had been predetermined by the researchers, not the teachers. Furthermore, it is no
longer warranted to assume that the planning decisions teachers make are driven totally by the

7
content and organization of the textbooks used and, therefore, need not be considered as an
object ofresearch [10].
Category 3: Classroom Processes. Classroom processes include the host of teacher
and student actions and interactions that take place during instruction. We have identified four
dimensions of classroom processes: teacher affects, cognitions, and metacognitions; teacher
behaviors; student affects, cognitions, and metacognitions; and student behaviors. Table 2
provides additional details on each of these dimensions.
Table 2: Oassroom Processes
Teacher Affects. Cognitions & Metacognitions Student Affects. Cognitions & Metacognitions
• with respect to problem-solving phases • with respect to problem-solving
(understanding, planning, carrying out plans, (understanding, planning, carrying out plans,
& looking back) & looking back)
• attitudes about self, students, mathematics, • attitudes about self, teachers, mathematics, &
problem solving, & teaching problem solving
• beliefs about self, students, mathematics, • beliefs about self, teachers, mathematics, &
problem solving, & teaching problem solving
Teacher Behaviors Student Behaviors
Examples: questioning, clarifying, guiding, Examples: identifying information needed to
monitoring, modeling, evaluating, solve a problem, selecting strategies for
diagnosing, aiding generalizing, ... solving problems, assessing extent of
progress made during a solution attempt,
implementing a chosen strategy, determining
the reasonableness ofresults
Both the teacher's and the students' thinking processes and behaviors during instruction are
almost always directed toward achieving a number of different goals, sometimes
simultaneously. For example, during a lesson the teacher may be assessing the appropriateness
of the small-group arrangement that was established prior to the lesson, while at the same time
trying to guide the students' thinking toward the solution to a problem. Similarly, a student may
be thinking about what her classmates will think if she never contributes to discussions and at
the same time be trying to understand what the activity confronting her is all about. For
convenience, we have restricted our discussion of classroom processes to the thinking
processes and behaviors of the teacher and students that are directed toward activities (both
mental and physical) associated with P6lya's [29] four phases of solving problems:
understanding, planning, carrying out the plan, and looking back. That is, we have restricted
our consideration to what the teacher thinks about and does to facilitate the student's thinking

8
and what the student thinks about and does to solve a problem. We have not attempted to
include a complete menu ofobjects or goals a teacher might think about during instruction.
Teacher Affects. CoWitions. and MetacQWitions. These processes include those attitudes,
beliefs, emotions, cognitions and metacognitions that influence, and are influenced by, the
multitude of teacher and student behaviors that occur in the classroom during instruction. In
particular, this dimension is concerned with the teacher's thinking and affects while facilitating
a student's attempt to understand a problem, develop a plan for solving the problem, carry out
the plan to obtain an answer, and look back over the solution effort.
Teacher Bebayiors. A teacher's affects, cognitions, and metacognitions that operate during
instruction give rise to the teacher's behaviors, the overt actions taken by the teacher during
problem-solving instruction. Specific teacher behaviors such as those shown in Table 2 can be
studied with regard to use (or non-use) as well as quality. The quality of a teacher behavior can
include, among other things, the correctness of the behavior (e.g., correct mathematically or
correct given the conditions of the problem), the clarity of the action (e.g., a clear question or
hint), and the manner in which the behavior was delivered (e.g., the verbal and nonverbal
communication style ofthe teacher).
Student Affects. Coe;nitions. and Metacoe;nitions. Similar to the teacher, this subcategory
refers to the affects and the cognitive and metacognitive processes that interact with teacher and
student behaviors. The concern here is with how students interpret the behavior of the teacher
and how the students' thinking about a problem, their affects, and their work on the problem
affects their own behavior. Also of concern here is how instructional influences such as task
features or contextual conditions directly affect a student's affects, cognitions, metacognitions,
and behaviors.
Student Behaviors. These behaviors include the overt actions of the student during a
problem-solving episode. By restricting our attention to the problem-solving phases mentioned
earlier, we can identify several behaviors students might exhibit as they solve problems.
Sample behaviors are shown in Table 2.
Category 4: Instructional Outcomes. The fourth category of factors consists of three
types ofoutcomes ofinstruction: student outcomes, teacher outcomes, and incidental outcomes.
Most instruction-related research has been concerned with short-term effects only.
Furthermore, transfer effects, effects on attitudes, beliefs, and emotions, and changes in teacher
behavior have been considered only rarely. Table 3 provides a list of the key outcomes
associated with each of the three types.
Student Outcomes. Both immediate and long-term effects on student learning are included,
as are transfer effects (both near and far tranSfer). Illustrative of a student outcome, either
immediate or long-term, is a change in a student's skill in implementing a particular problem-
solving strategy (e.g., guess and check, working backwards). An example of a transfer effect
is a change in students' performance in solving non-mathematics problems as a result of
solving only mathematics problems. Also, of special importance is the consideration ofchanges
in students' affects about problem solving or about themselves as problem solvers and the

9
Table 3: Instructional Outcomes
Student Outcomes
1. Immediate effects on student leaming with respect to:
a. problem-solving skills b. perfonnance c. general problem-solving ability
d. affects e. beliefs f. mathematical content knowledge
2. Long-tenn effects on student learning with respect to:
a. problem-solving skills b. perfonnance c. general problem-solving ability
d. affects e. beliefs f. mathematical content knowledge
3. Near and far transfer effects
Teacher Outcomes
1. Effects on the nature of teacher planning
2. Effects on teacher behavior during subsequent instruction
3. Effects on teacher affects and beliefs about:
a. effectiveness of instruction b. "worthwhileness" of instructional methods
c. "ease" of use of instructional methods
Incidental Outcomes
Examples: student perfonnance on achievement tests, influence of instruction on student/teacher
behavior in non-mathematics areas, affective changes with respect to mathematics in general,
schooling, ...
effect of problem-solving instruction on mathematical skill and concept leaming. For example,
how is computational skill affected by increased emphasis on the thinking processes involved in
solving problems?
Teacher Outcomes. Teachers, of course, also change as a result of their instructional
efforts. In particular, their attitudes and beliefs, the nature and extent of their planning, as well
as their classroom behavior during subsequent instruction are all subject to change. Each
problem-solving episode a teacher participates in changes the craft knowledge of the teacher
[34]. Thus, it's reasonable to expect that experience affects the teacher's planning, thinking,
affects, and actions in future situations.
Incidental Outcomes. Increased perfonnance in science (or some other subject area) and
heightened parental interest in their children's school work are two examples of possible
incidental outcomes. Although it is not possible to predetermine the relevant incidental effects of
instruction, it is important to be mindful of the potential for unexpected side effects.

10
Discussion
There are several factors in the structure described in this paper that deserve particular attention.
In the following paragraphs we discuss these factors.
Extra-clagroom Comidemdom: Teacher's Affects, Cogniliom and Metacogniliom
Research on teaching in general points to the important role a teacher's knowledge and affects
play in instruction. However, they have received relatively little attention in research on the
teaching of problem solving [3, 7, 39, 40]. Future research should consider all of these factors
when real teachers are involved. Questions such as the following need to be investigated: What
knowledge (in particular, content, pedagogical, and curriculum knowledge) do teachers need to
be effective as teachers of problem solving? How is that knowledge best structured to be useful
to teachers? How do teachers' beliefs about themselves, their students, teaching mathematics,
and problem solving influence the decisions they make prior to and during instruction?
Extra-classroom Considerations: Task Features
Although there has been considerable research on task variables (see [12]), it may be time to
consider the very nature of the tasks used for instruction relative to problem solving. In most
research, tasks have been relatively brief problem statements presented to students in a printed
format Thus, research on task variables has considered variables like problem statement length
and grammatical complexity. But, very little attention has been given to the identification of task
variables that should be considered when the problem-solving task is presented, for example,
through a videotape episode from an Indiana Jones movie (see [2]). Or, suppose the real-world
tasks used for instruction were selected from those used in an instructional approach modeled
after the concept of an apprenticeship [21]. Would the task variables of importance be the same
as those examined and discussed in the existing research literature?
Teacher Planning
A teacher's behavior while teaching problem solving is certainly influenced by the teacher's
affects, cognitions and metacognitions during instruction. However, some of this behavior
during instruction is likely to be determined by the kinds of decisions the teacher makes prior to
entering the classroom. For example, a teacher may have planned to follow a specific sequence
of teaching actions for delivering a particular problem-solving lesson knowing that the exact
ways in which these teaching actions are implemented evolve situationally during the lesson.
Or, if the knowledge teachers use to plan instruction is case knowledge, that is knowledge of
previous instructional episodes [8], then we would search for those cases that significantly

11
shape the craft knowledge teachers use as a basis for planning and action. Future research
should consider how teachers go about planning for problem-solving instruction and how the
decisions made during planning influence actions during instruction.
Classroom Processes: Teacher Affects, Cognitions and Metacognitions
Students' cognitive and metacognitive behavior during mathematical problem solving have
received a great deal of attention in recent years (see, for example, [7, 11, 36]). And, there is an
indication that students' attitudes, beliefs, and emotions also are beginning to be the object of
study [27, 28]. Research is needed that describes the thinking a teacher does during the
teaching of problem solving. What does the teacher attend to during instruction? How do
teachers interact with students during instruction? What drives the teacher's decisions at those
times? Are teachers aware of their actions during instruction? Do they consciously assess their
behaviors during instruction? Rich descriptive studies are seriously needed in this much
neglected area. The work ofLampert [19] provides a promising model for consideration.
Classroom Processes: Teacher Behaviors
It is especially important that future research pay closer heed to what teachers actually do during
instruction (i.e., teacher actions). In particular, there is a need to document teacher behaviors as
they relate to the process of solving problems. It would be a significant contribution to our
understanding of how to teach mathematical problem solving if careful, rich descriptions were
prepared of a teacher's actions during each of P6lya's four phases (refer to Table 2). Such
descriptions would help develop a clearer picture of what teachers should do to aid students to
understand problem statements, to plan methods of solution, to carry out their plans, and to
look back (evaluate, reflect, generalize) at their efforts.
Classroom Processes: Student Affects, Cognitions, and Metacognitions
As mentioned earlier, these factors have received substantial attention in the mathematical
problem-solving literature. It would be valuable to know more about the conditions under
which students' cognitions and metacognitions change and when these changes begin to occur.
In a study we conducted [6] continual improvement in students' understanding, planning, and
implementing behaviors (cognitions) occurred during the course of 23 weeks of problem-
solving instruction. However, the design of the study did not allow us to make conjectures
about the aspects of the program that were responsible for the improvement. The effects of a
student's attitudes, beliefs, and emotions during problem solving on problem-solving
performance is another area in need of serious attention. Recent work by Lester, Garofalo, and

12
Kroll [26, 27] illustrates the often strong interaction among affects, cognitions, and
metacognitions. McLeod and Adams [28] provide numerous questions for additional research
in this area.
Classroom Processes: Student Behaviors
As important as it is to begin to study carefully the behavior of teachers during instruction, it is
just as important to observe the behavior of students during problem-solving sessions. In
particular, it would be valuable to know more about the way in which students' interactions
affect their thinking processes and actions. For example, to what extent are students' choices
and use of a particular problem-solving heuristic or strategy influenced by their peers'
decisions? What group processes promote or inhibit successful problem solving? How are
individual differences exhibited during problem solving? What kinds of individual differences
influence success in solving problems and how can these differences be handled by the teacher?
How are students' classroom behaviors influenced not only by their cognitions and
metacognitions during instruction, but how are their behaviors influenced by the attitudes and
beliefs they bring to the lessons? What kinds of student behaviors are attended to by teachers?
How do student-student and teacher-student communications shape affects, beliefs, and
cognitions demonstrated while solving problems and the knowledge students extract from the
experience? Which instructional influences have the greatest effect on a student's actions?
Instructional Outcomes: Students
Both immediate and long-term effects on student learning should be considered with respect to
cognitions and affects. In addition, possible transfer effects (both near and far) should be
investigated. For example, does instruction involving geometry problems have any effect on
students' solutions to non-geometry problems? Does instruction in solving routine story
problems influence students' solutions to non-routine problems? Furthermore, it is not enough
to be content with some vague, general improvement in the ability to get more correct answers
(cf. [36, 37]). Instead, it is essential that researchers look to see if their instructional
interventions actually develop the kinds of student behaviors they were designed to promote.
Finally, how does problem-solving instruction influence the amount and nature of students'
mathematics content knowledge? As an illustration of this last question, consider the situation in
which a third grade teacher has emphasized during the course of an entire school year the
development of various problem-solving strategies. To maintain this emphasis she has found it
necessary to devote less attention than in the past to basic fact drill activities. Has the students'
mastery of these basic facts been affected by this change in emphasis?

13
Instructional Outcomes: Teachers
Lampert [19] illustrates how reflecting on instructional practice influences subsequent planning
and classroom behavior. It would be a useful contribution to document how systematic
problem-solving instruction affects teachers' planning, affects and beliefs. Furthermore it
would be valuable to know how such instruction influences teacher behavior during subsequent
instruction.
A Final Comment
Our analysis of factors to be considered for research on problem-solving instruction is intended
as a general framework for designing investigations of what actually happens in the classroom
during instruction. As mentioned earlier, we recognize that there may be other important factors
to be included in this framework and that certain of the factors may prove to be relatively
unimportant. Notwithstanding these possible shortcomings, the framework we have presented
can serve as a step in the direction of making research in the area more fruitful and relevant.
Finally, at the beginning of this paper we recommended that the development of theories of
problem-solving instruction should become a top priority for mathematics educators. We also
believe that the type of theory development that is likely to have the greatest relevance should be
"...grounded in data gathered from extensive observations of 'real' teachers, teaching 'real'
students, 'real' mathematics, in 'real' classrooms" [24, p. 56]. It is only by adopting such a
perspective that we are likely to make any significant progress in the foreseeable future.
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NJ: LEA 1985
37. Silver, E., & Kilpatrick, J.: Testing mathematical problem solving. In: The teaching and assessing of
mathematical problem solving (R. Charles & E. Silver, eds.), pp. 178-186. Reston, VA: LEA & NCTM
1988
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Bloomington, IN: Mathematics Education Development Center 1977
39. Thompson, A.: Changes in teachers' conceptions of mathematical problem solving. Paper presented at the
annual meeting of the American Educational Research Association, Chicago, April 1985
40. Thompson, A.: Learning to teach mathematical problem solving: Changes in teachers' conceptions and
beliefs. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp.
232-243. Reston, VA: LEA & NCTM 1988
41.Winnie, P. H., & Marx, R. W.: Reconceptualizing research on teaching. Journal of Educational Psychology
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Educational Psychology 72, 257-264 (1980)

Researching Problem Solving! from the Inside
10hnMason
Centre for Mathematics Education, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK
Abstract: A methodology is offered for studying problem solving from the inside. This
approach is first set in the context ofmillenia of educational thinkers, and then in the context of
providing support for the professional development of teachers of mathematics. The approach
is based on the development and strengthening of each teacher's own awareness of their own
mathematical thinking so that they can more readily enter and appreciate the thinking of their
pupils. It is described and illustrated, through offering experience of working on mathematics,
through offering challenging assenions, and through suggesting ways of working on
mathematics teaching. Observations are made about validity and validation through this
methodology. The heart of the methodology is based on the observation that what really matters
in teaching is to be awake to possibilities in the moment, as a lesson unfolds; moments when
the teacher experiences the true freedom of a conscious choice. Three extracts from different
styles ofpresentation of mathematics for students at a distance are offered for consideration.
Keywords: awareness, metacognition, frameworks, noticing, conjectures, investigations,
insights, generalizing, specializing, teaching
Introduction
What are we doing when we engage in research on problem solving? What is the enterprise?
Is it to understand and appreciate the processes involved?
Is it to locate a framework, a programme for "teaching" problem solving?
The first aim already begs many questions in referring to the processes, implying as it does that
processes have independent existence and that the researcher's job is to locate them in some
Platonic world of forms. Modern epistemological views suggest that such processes are
constructed by the observer as part of the process of distinguishing features and making
distinctions on the way to understanding. Though they may "exist" for one researcher, they
may not for others. Ifdistinctions are to be useful to others, then it is necessary to assist others
to construct and enter the same reality. The second aim assumes that direct instruction is
possible and even desirable. The trouble with direct approaches is that they may not leave room
for contingent or collateral learning.
1 I take the word problem to refer to a person's state of being in question, and problem solving to refer to
seeking to resolve or reformulate unstructured questions for which no specific technique comes readily to mind

18
What would useful research conclusions look like? Who would try to use them?
There is a long history of thoughtful advice about problem-solving, from such eminent names
as Plato, Aristotle, Pappus, Bacon, Descartes, Fermat, Euler, Bolzano, Boole, Hadamard, and
P6lya [29]2. They sought to organize thinking, and some of them even sought to mechanize it,
as well as to assist novices in learning mathematics. The impulse to mechanize thought has
appeared in every generation. The availability of sophisticated computers has stimulated a
burgeoning of activity by numerous workers in the fields of Artificial Intelligence and Cognitive
Psychology, who have likewise contributed to the mechanization of problem solving in
mathematics3 in the idiom of their times.
When research findings are translated into practice, they turn from observation into rules,
from heuristics into content. Attempts to pass on insights become attempts to teach patterns of
thought. Once the "patterns of thought", the heuristics, become content to be learned,
instruction in problem-solving takes over and thinking tends to come to a halt. Dewey put it
nicely, when he said
Perhaps the greatest of all pedagogical fallacies is the notion that a person learns
only the particular thing he is studying at the time. Collateral learning ... may be
and often is much more important than the actual lesson.
Approaching the "content" too directly may not always be wise. In an age noted for its
pragmatism which begins to rival that of Dewey's times, collateral learning may need to be re-
emphasized. To counteract the functionalism of modern educational curriculum specification
and assessment, Bill Brookes, in a seminar, suggested that
We are wise to create systems for spin-offs rather than for pay-offs.
Mathematicians too have been very slow to acknowledge and accept distinctions made by
researchers, fearful perhaps that they will lose touch with their own creativity. People do not,
on the whole, relish being told how they think or how they might think differently.
In order to influence teaching, researchers have traditionally moved to programme
development, almost always text based, with or without training for teachers in the use of the
materials. Even where teachers are supported, it is usually to help them to "deliver the
programme", like some mail-order firm.
My approach is to work from the inside, from and on my own experience. This is not as
solipsistic and idiosyncratic as it sounds, for it draws on results of outer research which speaks
2 See particularly George POlya [29] for a study of the use and teaching of heuristics. For recent surveys, see
also Alan Schoenfeld [32] and H. Burkhardt, S. Groves, A. Schoenfeld, and K. Stacey [2].
3 Particularly notable is the work of Edmund Furse [7]. Furse has managed to build a program that will read
standard mathematical texts (translated into a suitable formal language), and then solve the end-of-chapter
exercises. Its solutions look like those of an expert. Furthermore, there is no mathematics embedded in the
program, only a syntactic pattern recognition engine.

19
to my experience. Furthennore, there is a long history of people working effectively in this
way, particularly in the phenomenological and hermeneutic traditions, building as they do on
psychological insight of ancient peoples in India, China and the Middle East. Recently there has
been a notable increase in interest from academics, with attention being drawn to metacognition
and reflection4• Humboldt observed that the essence of thinking lies in abstraction, Dewey
spoke of turning a subject over in the mind, Piaget stressed the role of reflective abstraction,
Vygotsky drew attention to the internalization ofhigherpsychologicalprocesses through being
in the presence of more expert thinkers, Skemp lays great stress on the development of
reflective as well as intuitive intelligence, and Kilpatrick [19] made an explicit call for the
development of self-awareness in mathematics education.
Apart from practical considerations such as that teacher-proof materials have never succeeded,
and that there is no evidence for a single way to teach mathematics, there are three principle
justifications for an approach from the inside:
to appreciate the struggles of another, it is necessary to struggle oneself, and to be
able to re-enter that struggle;
to learn to struggle and to make best use of available resources, it is useful to be in
the presence of someone who themselves is in question, who is struggling to
know;
teaching mathematics is ultimately about being mathematical oneself, in front of and
with pupils, and so in order to develop one's teaching, to work at being
mathematical, it is necessary to develop one's mathematical being.
I do not mean to imply that it is best for teachers to be struggling with the same mathematics as
their pupils, although often it is the case that a teacher who does not know answers can be of
more help than a teacher who does. More precisely, knowing the/an answer can make it harder
to appreciate pupils' thinking, and more likely that you will direct them to your own solution.
Instead of dwelling on mathematical answers, teachers can be more helpful by dwelling in their
own questions about what the pupils are thinking, how their powers can be evoked, what
advice might be helpful, and leave the actual mathematical thinking to the pupils. To do this
requires awareness of their own thinking, which comes from inner research.
To influence pupils in how to come into question, to be in question, and to resolve
questions it is helpful if the teachers are themselves in question, either about some other
mathematics, or about their teaching of mathematics, or both. I concur with Brown, Collins,
and Duguid [1] who advocate a cognitive-apprenticeship vocabulary for describing one form of
effective teaching, and this is consistent with current social-constructivist thinking about the
role and significance of peers and experts in learning. The medieval Guild system, when
functioning effectively, had much to recommend it.
4 Some would attribute this to the dawning of the age of Aquarius. Others. such as Julian Jaynes. [17] would
claim that we are experiencing an evolutionary change in the structure ofconsciousness.

20
The difficulty as always is to translate description into action. I do not advocate trying to
describe effective teaching in behavioural tenns so that others can then (supposedly) "do it". All
attempts at this of which I am aware, have failed. What I advocate is establishing a collegial
atmosphere in which teachers work on and develop their awareness of their own mathematical
thinking, and of themselves as teachers.
I stress the collegial, because it is not enough to be a hennit, operating alone behind the
closed door of the classroom. It is essential to test out insights and conjectures with others, to
participate in a wider community of fellow seekers. Such peer-checking is an essential part of
the larger methodology ofthe Discipline of Noticing (see Mason and Davis [28] and Davis [3]).
In this paper I want to illustrate rather than derme what I mean by researching/rom the inside,
and to make some remarks about what I have learned over the past twenty-five years. I shall
offer some mathematical questions5 to illustrate a little bit of what I mean by working from the
inside, and to indicate that it is fundamental to me to be consistent in my approach. To speak
about working fonn the inside is to monger words. To appreciate what I have learned, you
must experience it from the inside. If my words speak to your experience, then all well and
good. Ifnot, then we have both wasted time!
When working with teachers I often suggest at the beginning that I shall be working on
three levels:
on Mathematics,
on Teaching and Learning Mathematics,
and on Ways of Working:
- on Mathematics,
- on Teaching and Learning Mathematics.
Having come to mathematics education from mathematics research, I find much that is similar
between working on mathematics and working on teaching and learning of mathematics,
particularly in the need for particular examples, and for attention to be devoted to how
generalizations are drawn from or seen in the particular.
I draw attention to these three levels because I neither claim to know, nor wish to show or
tell people, how to teach. Rather, I wish to draw out from their experience for conscious
attention the powers that they already possess, and through that awareness, try to help them be
able to do the same for their pupils. Socrates' image of midwife for ideas in Plato's dialogue
the Thaetetus has much to recommend it, though care is needed in isolating the useful features
of the much invoked but frequently misunderstood Socratic Dialogue (see for example [12,
13]).
Instruction, that term so beloved of American educators, is much more subtle than directing
pupils through the stages of some institutional regime. It involves more than being sensitive to
pupils' needs, more than actualizing a few slogans. It is about being yourself, being aware of
5 Mindful that a question is ink on paper, and a problem is a state which arises in a person in the presence of a
question, in a context.

21
yourself and your powers, and evoking those same powers in others. Thus teaching and
learning are for me almost synonymous. They are about entering the potential space
(D. Winnicot, quoted in [18]) between you and another, and inviting others to join you in that
space.
Sometimes direct instruction, that is, giving pupils specific instructions as to what to do, is
appropriate. Telling pupils things can indeed be effective (despite the current bad press which
exposition receives at present), especially if pupils are prepared to hear what is being said
because of recent experience which has raised doubts or questions. Sometimes a less direct
prompt to recall prior advice or instruction is enough, and at other times being verbally silent
but physically and mathematically present, or even being physically absent is sufficient. Pupil
dependency on teacher intervention can easily and unintentionally be fostered. Tasks are
sometimes directive, sometimes indicatively prompting investigation, sometimes spontaneously
arising from pupil or group. Samples of attempts to manifest these in print, purely as support
materials, are given at the end of the paper.
Examples of Approaching from the Inside
In this section I offer some mathematical questions, together with a few remarks about what
often emerges when using them. If the questions are too easy, then nothing will be learned
from them. If you get stuck, then there is a chance something can be learned. If you simply
scan or skim through them, then nothing will be gained.
One Sum
Take any two numbers that sum to one. Which will be the larger sum: the square of
the larger added to the smaller, or the square of the smaller added to the larger?
Everyone naturally specializes. Some use simple fractions, some use decimals, and some use
algebraic symbols. I refer to all of these as specializing, since people naturally tum to
something specific which they find confidence inspiring. Pointing this out can help to ease a
transition from number to algebra.
Square Sums
32 + 42 = 52
102 + 112 + 122 = 132 + 142
212 + 222+ 23 2 + 242 = 252 + 262 + 272
Most people find the first statement a well known and unexceptional fact. Juxtaposed with the
second, which is somewhat surprising, interest mounts. When the third is offered, there is

22
conviction that there must be a pattern which continues. Locating that pattern evokes powers of
pattern spotting and expression which lie at the heart of algebraic generalization and thinking.
The assumption of pattern is endemic in mathematics, but not always born out:
32 + 42 = 52
33 + 43 + 53 = 63
... ?
Map Scaling
Imagine that you are sitting at home with a map of your country, and you wish to
scale it down by a factor of a half. To do this, you put a transparent piece of paper
over the map, and using your own home-town as centre, scale every other feature
of the map halfway along the line joining it to your centre. Now imagine that a
friend uses the same map, but using a different centre. Will your two scaled maps
look the same or different, and why?
Many people have incomplete intuitions about scaling. There is something paradoxical about
being able to use any point as centre, and still get the same map. Even experienced
mathematicians can be caught out in a conflict between their educated intuition and their gut
response. A computer programme such as Cabri-geometre [20] enables a simultaneous scaling
of the same line or circle from two different points, and watching the two images develop is
often a surprise even when you know what to expect. The following has a similar effect,
particularly on astronomers:
The Half Moon Inn
In England you often see pub signs showing a vertical half-moon (the straight
diameter is vertical) in a black sky with a few stars shining nearby. When, if ever,
can you see a vertical half-moon?
These questions are useful for opening up discussion about the extent to which intuition can be
educated, and the extent to which early intuitions are always present, if suppressed. Fischbein
[6] takes the view that early intuitions are robust against teaching, and Di Sessa [4] develops the
notion ofpsychologicalprimitives as basic building blocks in constructing stories to account for
things that happen in the world. The development of much of mathematics can be seen as the
gradual refinement of intuitions: for example from discontinuity as a jump in the graph, to a
more complex phenomenon; from a function having a derivative of 0 at a point being flat
nearby, to the possibility that it could have arbitrary slope arbitrarily close to such a point.

23
Planets [33]
Consider a collection of identical spherical planets in space. From some places on
the the surface of some of the planets, it is possible to see one or more other planets
in the collection. What is the total·area on all the planets of these places?
This question has proved fruitful because, after successfully specializing to two-dimensions,
most people develop an argument which generalizes only with great difficulty to three
dimensions. After some initial success, it is usually necessary to prompt people to go back to
the two-dimensional case and seek an argument which will generalize.
Remarks about my Approach
Inner research begins with self-observation. This is a constant and on-going process, guided by
the concerns of the people I am trying to assist, and by my own propensities. Teachers often
remark to me that it is very rare that they have the time to work on mathematics themselves,
either alone or with colleagues. I take this as resonance with the notion that it would be better if
they did engage in mathematical thinking, but that conditions are not supportive. Many are the
programmes of in-service support for mathematics teachers; the most significant benefit to
participants of any programme is rarely the content, and more often the opportunity and
stimulus to "indulge" in mathematics, and to compare reflective notes with colleagues. Where
teachers are transfixed by "picking up some idea they can use tomorrow" the benefit is short-
lived, as evidenced by their return next time for another "fix".
Give someone a fish, and you feed them for a day;
Teach them to fish, and you feed them for a lifetime.6
After many years of inviting people to engage in mathematical thinking, it became clear that on
the whole it was much more effective to begin with simply stated tasks. As with any useful
observation about teaching, this is not meant to be an unbroken rule; rather it is meant to inform
choices made in the moment. I am always tempted to start with something "interesting", which
means "interesting to me", and because of the complexity, people find it difficult to separate off
a little bit of their attention to observe themselves. Starting with apparently very simple
questions which invoke the type of thinking which is needed when you get stuck - specializing
and generalizing, conjecturing and convincing, animating the static and freezing the dynamic,
working backwards and working forwards - enables participants to attend to their experience as
well as to the task, and thereby to find discussion and elaboration fruitful. More important than
the completion of tasks is awareness of what you are doing. But attention is usually drawn into
the mathematics. For example, when teachers watch videotaped episodes from classrooms,
6 Purportedly a Chinese proverb, source unknown.

24
their attention is drawn initially to the task posed to the pupils, and whether they, the viewers,
can do it. Only when they feel confident about and familiar with the mathematics can they attend
to the acts of teaching.
I struggled for a long time with Gattegno's memorable observation that
Only Awareness is Educable [9].
He has a quite specific meaning which I only partially appreciate, connected with his Science of
Education. This is an approach which has much in common with mine, in that it seeks the roots
of mathematics and of thinking in actions and experience, although Gattegno emphasizes
particularly the actions and experience of the neonate in its cot. I do not pretend to have
mastered Gattegno's ideas, nor to be carrying them further. I simply note similarities. I found
increased meaning in his observation when I augmented it by twoother assertions, informed by
that most ancient of metaphors for the psychological make-up of human beings, the image of
the Chariot? The chariot is drawn by a horse, and has a driver as well as an owner. One way to
read it is as follows:
the chariot represents the body, which needs maintenance;
the horse represents the emotions (the source of motion) which need grooming and
feeding and sometimes require blinkering;
the driver represents the intellect which needs challenging tasks;
the reins represent the power of mental imagery to direct, which need to be kept
subtle and loosely-taut;
the shafts represent the psychosomatic interplay between emotion and the
musculature (see [31]);
the owner represents a different form of awareness that is possible, but only when
the chariot is functioning correctly will the owner deign to use it.
The two extra observations suggested by this image are that
Only behaviour is trainable, and
Only emotion is hamessable.
I capitalize the only in each case, because it is often the only which causes the most strife when
people encounter and contemplate these assertions. The definitiveness often generates interest
and thought, whether in favour or in opposition. The three onlys also act to inform an approach
to the construction of mathematical tasks for pupils, outlined in Griffith and Gates [10].
The positive role of frameworks, and for slogans too [14], is that they can serve to heighten
awareness, to remind you in the midst of preparation or lesson, of some alternatives to your
standard, automatic reaction. The negative side is that instead of in-forming, they come to pro-
form, even de-form awareness. The education of awareness, and the integration of
7 Katha Upanishad 1.3.3-5, see for example S. Rhadakrishnar [30]. A more modem version of the image can
be found in Gurdjieff [111.

25
observations through subordination8 are part of the development of being. but the metaphor of
growth suggests a monotonic progression. and this does not conform with my experience.
Often what once seemed sorted out, needs to be re-evaluated and re-questioned. and stimulus to
do this comes through working with colleagues in order to seek resonance with their
experience.
Mathematical questions. and the recall of teaching or learning incidents through anecdotal
re-telling or through video-tape are devices for focusing attention. In recalling an event in
retrospect. there is a temptation to judge. to berate oneself for failing to do something. This is a
waste of energy. To notice something about yourself is like suddenly taking a breath of clean
air. like suddenly coming upon an unexpected vista. The energy which accompanies the insight
has to go somewhere. and it is all too natural to throw it away on judgement and negativity. To
make use of what is observed. it is better simply to note. to "in-spire" the in-sight into oneself.
to accept. By re-calling and re-entering specific salient moments. moments that come readily
and vividly to mind soon after an event. it is possible to take a three-centred approach to
developing one's being. The horse. carriage. reins. shafts and driver can be employed in a
balanced and intentional storing of experience which can serve to inform practice in the future.
This is the theory behind the presentation and recommendations in Mason. Burton and Stacey
[24]. By vividly re-entering a significant moment through the power of mental imagery. you
contribute to a rich store ofexperience to be resonated at a later date. To stimulate the growth of
an inner monitor. it is necessary to direct some of the energy released from noticing. for that
purpose.
What tends to happen is that you become aware of missed opportunity. After the lesson it
suddenly comes to you what you could have done; a few moments after a pupil asks a question.
you find yourself answering in your usual automatic fashion; despite intending to listen to what
pupils have to say. you find yourself telling them what you think; while working on a question
you gradually become aware that you are not getting anywhere. These moments of awareness
come at fll'St in retrospect. The challenge is to move them into the moment. so that you become
aware of opportunities when they are relevant. Strategies and techniques for doing this are part
of the Discipline of Noticing. and have been described elsewhere.
Noticing Action in the Moment
To become aware of your own thinking. and to be able to make use of personal insights when
teaching. it is necessary to be able to catch action in the moment that it happens. But action is
very hard to catch in yourself when you are being successful. It is likely. for example. that you
found at least one of the tasks presented earlier so easy that you simply "did it". The trouble
8 A reference to Gattegno's memorable title. What we owe children: The subordination ofteaching to learning.
Gattegno [8].

26
with success is that although it breeds psychological confidence, it offers no support if and
when things start to get sticky.
It is relatively easy to catch yourself when you are stuck, because attention naturally drifts
from the task at hand to awareness of being stuck. Unfortunately, being stuck does not involve
much action, and again, this state is not terribly helpful in itself. It is an honourable state, full of
potential, and it can be turned to positive advantage, if you recognize and acknowledge it, and
cast around intentionally for advice and assistance. A passing teacher can at this point say many
things to you, some of which you can hear, while others will go right past you. If the teacher
always provides the same sort of help, then you may come to recognize that constant help, or
you may simply come to depend on it
Let me take one class of examples: teacher interventions when pupils get stuck. There is
considerable interest at present in scaffolded instruction in mathematics, in which the teacher,
with Vygotsky's zone ofproximal development in mind, asks questions to locate just where the
pupil's current "zone" lies, and then tries to "operate in that zone". But the sorts of questions
that teachers ask in these circumstances have been described over and over again over the years.
The new metaphors make little difference to what happens, just different ways of reading them.
Edwards and Mercer [5] concluded after extensive study of transcripts that
Despite the fact that the lessons were organized in terms of practical actions and
small-group joint activity between pupils, the sort of learning that took place was
not essentially a matter of experiential learning and communication between
pupils.... While maintaining a tight control over activity and discourse, the
teacher nevertheless espoused and attempted to act upon the educational principle of
pupil-centred experiential learning ... (p.156)
John Holt [15] has a most succinct version in an interaction with Ruth:
We had been doing math, and I was pleased with myself because, instead of telling
her answers and showing her how to do problems, I was "making her think" by
asking her questions. It was slow work. Question after question met only silence.
She said nothing, did nothing, just sat and looked at me through those glasses, and
waited. Each time, I had to think of a question easier and more pointed than the last,
until I found one so easy that she would feel safe in answering it. So we inched our
way along until suddenly, looking at her as I waited for an answer to a question, I
saw with a start that she was not at all puzzled by what I had asked her. In fact, she
was not thinking about it. She was coolly appraising me, weighing my patience,
waiting for that next, sure-to-be-easier question. I thought, "I've been had!" The
girl had learned to make all her previous teachers do the same thing. If I wouldn't
tell her the answers, very well, she would just let me question her right up to them.
Bauersfeld calls this thefunnel effect, as teacher and pupil are together drawn down a funnel of
increasing detail and special cases. The outcome is usually a question so trivial that the pupil is

27
in wonder that the teacher could ask it, but their attention is absorbed by each successive
question and is drawn away from the larger task.
There are alternatives. When you recognize that you have something specific that you are
trying to get pupils to "see" or "say", you can, for example, stop your current questioning,
acknowledge what you are doing, and simply tell them the idea that you have in mind. This
requires more presence and strength than might appear, and that presence and strength are built
up over a period of time through noticing missed opportunities. You can then suggest that they
reconstruct it in their own language, perhaps by repeating it to a colleague. Then they can
explore variations of the idea, and pose themselves more general questions which succumb to
the same approach. That works if they see the relevance of what is said to what they are trying
to do. Unfortunately, when a teacher leaves a group of pupils after an intervention it is often the
case that the pupils act as if the teacher had never been. They carry on as before. I conjecture
that one reason is that teachers remain caught in giving direct advice, much of which does not
relate to the pupils' experience.
More fruitful is a gradual movement from direct advice, though indirect prompts which
evoke recent memories of similar situations, with the overall aim of withdrawing prompts
altogether and detecting pupils using the suggestions spontaneously themselves. Brown,
Collins and Duguid [1] referred to this asfading. I have found it useful to keep in mind a
transition in intervention from directed, to prompted, to spontaneous use of technical terms and
heuristic advice by pupils. When you are stuck on a problem, you tend to be deeply involved in
that problem. When you seek advice, you immediately try to apply what you are told to the
problem, and so tend not to be aware of the shape or form of that advice. In order to assist
pupils to become aware of the existence of useful advice, metacognitive shifts can be effective.
Instead of asking Can you give me an example?, the question What question am I going to ask
you? can draw attention away from the particular to the general, prompting a shift in the
structure of attention [23, 26]. The pupil may not know the first few times, so you can resort to
your original question. But at least you are signalling that the advice itself is of some, indeed
more importance than success on the particular task. In this way, pupils can move from
requiring direct advice, to prompted advice, and eventually, to giving themselves that same
advice. In the process they will have internalized their own mathematical monitor, and
intentionally contributed to the growth of their inner teacher ([16, 27]).
In terms of evaluating the effectiveness of some framework such as directed, prompted,
spontaneous, advice on what to do when you are stuck, or any other proposed to pupils and/or
teachers, I want to know whether people find the ideas informing their practice. But I distrust
asking them directly. Even observing lessons to see if there is some manifestation of the
framework is a form of prompting probe which is likely to distort, and assumes that what is of
value is observable. In seeking to evaluate it is all too tempting to be caught in a cause-and-

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