Bibliography

The bibliography which follows consists of an alphabetical listing of vol­umes (and a very few articles) which have been chosen because they areclassics, surveys, or show emerging trends in geometry and geometry edu­cation. When possible recent books are listed. Although no doubt everyonewill find some favorite omitted, it is hoped that this list will serve as a rep­resentative sample for people who want to probe further into geometry'smany aspects. This bibliography, thus, supplements the listings in the ref­erences of the individual contributions in this volume. Note that thesebooks are not meant to be a list we recommend that either be acquired byindividuals or libraries.

Although the bibliography lists items primarily in English, there area sprinkling of titles in other languages. The readers are encouraged toexplore books that are being written in languages other than their ownnative language so as to get a feel for what is going on in other countries.However, we recognize that every nation has its own equivalents of titlesthat are primarily in English in this list.

We hope you will find interesting and stimulating reading here!

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Exploring Mathematics, MIT Press, Cambridge Mass., 1981.

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Ed. Sintesis, Madrid, 1990.

[3] ALSINA, C., BURGUEs, C. & FORTUNY, J: Invitacion ala Didactica de la Geometria,

Ed. Sintesis, Madrid, 1987.

[4] ARCHIMEDES: The works, (Translations in many languages). A critical English edi­

tion is due to T. Heath, Dover, New York, 1956.

[5] BACHMANN, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer, Berlin

(Second Ed.), 1973.

[6] BAGLIVO, J. & GRAVER, J.: Incidence and Symmetry in Design and Architecture,

Cambridge Univ. Press, Cambridge, 1983.

[7] BARNETTE, D.: Map Coloring, Polyhedra, and the Four-Color Problem, Mathemat­

ical Association of America, Washington DC, 1983.

[8] BARNSLEY, M.: Fractals Everywhere, Academic Press, Boston (Second Ed.), 1993.

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Mathematik, Bd. 2. Geometrie, Vandenhoeck & Ruprecht, Gottingen, 1960. English

translation: Fundamentals of Mathematics, Vol. 2, Geometry, MIT Press, Cambridge

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[10] BERZOLARI, L., VIVANTI, G & GIGLI, D. (EDs): Encic10pedia delle Matematiche

Elementari, Vol II, Parts 1 & 2, Hoepli, Milano, 1937.

329

330 BIBLIOGRAPHY

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BIBLIOGRAPHY 331

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Geometry, CRC Press, New York, 1997.

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London, 1984.

[54] GRAENING, J.: Geometry: A Blended Approach, Merrill, Columbus, 1980.

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York, 1974. New Ed., 1993.

[57] GRUNBAUM, B. & SHEPHARD, G.: Tilings and Patterns, W.H. Freeman, New York,

1987.

[58] GUILLEN, G.: Poliedros, Ed. Sintesis, Madrid, 1990.

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[59J HADAMARD, J.: Lec;ons de geometrie, I geometrie plane, II geometrie dans l'espace,

Ed. Jacques Gabay, Paris, (reprint), 1988.

[60] HANSEN, V.: Geometry in Nature, A. K. Peters, Wellesley Mass, 1993.

[61J HENDERSON, D.: Experiencing Geometry on Plane and Sphere, Prentice-Hall, Upper

Saddle River, NJ., 1996.

[62] HENDERSON, K. (ED): Geometry in the Mathematics Curriculum, National Council

of Teachers of Mathematics, Reston, 1973.

[63] HENLE, M.: Modern Geometries, Prentice-Hall, Upper Saddle River, N.Y., 1997.

[64] HILBERT, D.: Grundlagen der Geometrie, Teubner, Leipzig, 1899. English transla­

tion: The Foundations of Geometry; Open Court, LaSalle, 1969.

[65] HILBERT, D. & COHN-VOSSEN, S.: Anschauliche Geometrie, Springer, Berlin, 1932.

English translation: Geometry and the Imagination, Chelsea, New York, 1952.

[66] HONSBERGER, R.: Episodes in Nineteenth and Twentieth Century Euclidean Geom­

etry, Mathematical Association of America, Washington DC, 1995.

[67] HORDERN, L.: Sliding Piece Puzzles, Oxford Univ. Press, New York, 1986.

[68] HOYLES, C. & Noss, R. (EDS): Learning Mathematics and Logo, MIT Press, Cam-

bridge Mass, 1992.

[69] JENNINGS, G.: Modern Geometry with Applications, Springer, New York, 1994.

[70] JOHNSON, R.: Advanced Euclidean Geometry, Dover, New York, 1929.

[71] KAPPRAFF, J.: Connections: The Geometric Bridge Between Art and Science, Mc­

Graw Hill, New York, 1991.

[72] KLEE, V. & WAGON, S.: Old and New Unsolved Problems in Plane Geometry and

Number Theory, Mathematical Association of America, Washington DC, 1991.

[73] KLEIN, F.: Geometry, Translated from the German. Dover, New York, 1939.

[74] KLEIN, F.: Vorlesungen tiber Nichteuklidische Geometrie, Giittingen, 1893, reprinted

by Springer, Berlin, 1968.

[75] KLINE, M.: Mathematical Thought from Ancient to Modern Times, Oxford Univ.

Press, New York, 1972.

[76] KOSTOVSKII, A.: Geometrical Constructions with Compass Only, Translated from

the Russian. Mir, Moscow, 1986.

[77] KRAUSE, E.: Taxicab Geometry, Addison-Wesley, Reading, 1975.

[78] LAKATOS, I.: Proofs and Refutations, Cambridge Univ. Press, Cambridge, 1976.

[79] LEBESGUE, H.: Les Coniques, Ed. Jacques Gabay, Paris, (reprint) 1987.

[80] LEBESGUE, H.: Lec;ons sur les Constructions Geometriques, Ed. Jacques Gabay,

Paris, (reprint), 1987.

[81] LEHMAN, D. & BKOUCHE, R.: Initiation ala geometrie, PUF, Paris, 1988.

[82] LINDGREN, H.: Recreational Problems in Geometric Dissections and How to Solve

Them, Dover, New York, 1972.

[83] LINDQUIST, M. & SHULTE, A.: Learning and Teaching Geometry, National Council

of Teachers of Mathematics, Reston, 1987.

[84] LOCKWOOD, E.: A Book of Curves, Cambridge Univ. Press, Cambridge, 1961.

BIBLIOGRAPHY 333

[85] LOCKWOOD, H. & MACMILLAN, R.: Geometric Symmetry, Cambridge Univ. Press,

New York, 1978.

[86] LOMBARD, P.: Geometrie eJementaire et ca1cul vectoriel, Topiques editions, Pont aMousson, France, 1994.

[87] LOOMIS SCOTT, E.: The Pythagorean Proposition, The Nat. Council of Teachers of

Mathematics (Second Edition), 1968.

[88] LYUSTERNIK, L.: The Shortest Lines: Variational Problems, Translated from the

Russian. Mir, Moscow, 1976.

[89] MADDUX, C.: Logo in the schools, Haworth Press, New York 1985.

[90] MANDELBROT, B.: The Fractal Geometry of Nature, W. H. Freeman, New York,

1983.

[91] MARTIN, G.: Transformation Geometry, Springer, New York, 1982.

[92] MARTIN, J. (ED.): Space and Geometry, EricjSmeac Center, The Ohio State Univ.,

Columbus Ohio, 1976.

[93] MAXWELL, E.: Geometry for Advanced Pupils, Oxford Univ. Press, Oxford, 1953.

[94] MELTER, R., ROSENFELD, A. & BHATTACHARYA, P. (EDS): Vision Geometry,

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[95J Mo·isE, E.: Elementary Geometry from an Advanced Standpoint, Addison-Wesley,

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[96] MORLEY, F. & MORLEY, F.V.: Inversive Geometry, 1933, reprinted by Chelsea,

New York, 1954.

[97] MORRIS, R. (ED): Teaching of Geometry, Studies in Mathematics Education, Vol­

ume 5, Unseco, Paris, 1986.

[98] NELSON, R.: Proofs Without Words, Mathematical Association of America, Wash­

ington DC, 1993.

[99] NIKULIN, V. & SHAFAREVICH, I.: Geometries and Groups, Translated from the

Russian. Springer, Berlin, 1987.

[100] NOEL, G. (ED): Colloque International sur l'Enseignement de la Geometrie, Mons

31 aout - 2 septembre 1982, Universite de l'Etat a Mons, Mons, 1982.

[101] OKABE, A., BOOTS, B. & SUGIHARA, K.: Spatial Tessellations: Concepts and Ap­

plications of Voronoi Diagrams, Wiley, New York, 1992.

[102] OLSON, D. & BIALYSTOCK, E.: Spatial Cognition: The Structure and Development

of Mental Representations of Spatial Relations, Lawrence Erlbaum, Hillsdale, 1983.

[103] ORE, 0.: Graphs and Their Uses, Random House, New York, 1963 (Now available

from Mathematical Association of America, in a revised edition edited by R. Wilson).

[104] PACH, J. & AGARWAL, P.: Combinatorial Geometry, Wiley, New York, 1995.

[105] PAPERT, S.: Mindstorms: Children, Computers and Powerful Ideas, Basic Books,New York, 1980.

[106] PEHKONEN, E. (ED): Geometry Teaching - Geometrieunterricht, Conference on the

Teaching of Geometry in Helsinki, Research Report 74, Department of Teacher Edu­

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[107] PElTGEN, H. & RICHTER, P.: The Beauty of Fractals, Springer, Berlin, 1986.

334 BIBLIOGRAPHY

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[110] PREPARATA, F. & SHAMOS, M.: Computational Geometry, Springer, New York,1985.

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[129] WELLS, D.: The Penguin Dictionary of Curious and Interesting Geometry, Penguin

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BIBLIOGRAPHY 335

[133] YAGLOM, I.: A Simple Non-Euclidean Geometry and its Physical Basis, Translated

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Holt, Rinehart and Winston, New York, 1961.

APPENDIX

PERSPECTIVES ON THE TEACHING OF GEOMETRYFOR THE 21st CENTURY

Discussion Document for an ICMI Study

(Reprinted from L'Enseignement MatMmatique, 40, 345-357, 1994)

1. Why a study on geometry?

Geometry, considered as a tool for understanding, describing and interacting withthe space in which we live, is perhaps the most intuitive, concrete and reality­linked part of mathematics. On the other hand geometry, as a discipline, restson an extensive formalization process, which has been carried out for over 2000years at increasing levels of rigour, abstraction and generality.

In recent years, research in geometry has been greatly stimulated by new ideasboth from inside mathematics and from other disciplines, including computerscience. At present, the enormous possibilities of computer graphics influencemany aspects of our lives; in order to use these possibilities, a suitable visualeducation is needed.

Among mathematicians and mathematics educators there is a widespreadagreement that, due to the manifold aspects of geometry, the teaching of geom­etry should start at an early age, and continue in appropriate forms throughoutthe whole mathematics curriculum. However, as soon as one tries to enter intodetails, opinions diverge on how to accomplish the task. There have been in thepast (and there persist even now) strong disagreements about the aims, contentsand methods for the teaching of geometry at various levels, from primary schoolto university.

Perhaps one of the main reasons for this situation is that geometry has somany aspects, and as a consequence there has not yet been found - and perhapsthere does not exist at all - a simple, clean, linear, "hierarchical" path from thefirst beginnings to the more advanced achievements of geometry. Unlike whathappens in arithmetic and algebra, even basic concepts in geometry, such as thenotions of angle and distance, have to be reconsidered at different stages fromdifferent viewpoints.

Another problematic point concerns the role of proofs in geometry: relationsbetween intuition, inductive and deductive proofs, age of students at which proofscan be introduced, and different levels of rigour and abstraction.

Thus the teaching of geometry is not at all an easy task. But instead of tryingto face and overcome the obstacles arising in the teaching of geometry, actualschool-practice in many countries has simply bypassed these obstacles, cuttingout the more demanding parts, often without any replacement. For instance,three-dimensional geometry has almost disappeared or has been confined to amarginal role in the curricula in most countries.

337C. Mammana and V. Villani (eds.).Perspectives on the Teaching ofGeometry for the 2IP Century. 337-346.© 1998 Kluwer Academic Publishers.

338 THE ICMI DISCUSSION DOCUMENT

Starting from this analysis, and specifically considering the gap between theincreasing importance of geometry for its own sake, as well as in research and insociety, and the decline of its role in school curricula, rCMr feels that there is anurgent need for an international study, whose main aims are:

- To discuss the goals of the teaching of geometry at different school levels andaccording to different cultural traditions and environments.

- To identify important challenges and emerging trends for the future and toanalyze their potential didactical impact.

- To exploit and implement new teaching methods.

2. Aspects of Geometry

The outstanding historical importance of geometry in the past, in particularas a prototype of an axiomatic theory, is so universally acknowledged that it de­serves no further comment. Moreover, in the last century and specifically duringthe last decades, as Jean Dieudonne asserted at rCME 4 (Berkeley, 1980), Geom­etry "bursting out of its traditional narrow confines [...] has revealed its hiddenpowers and its extraordinary versatility and adaptability, thus becoming one ofthe most universal and useful tools in all parts of mathematics" (J. Dieudonne:The Universal Domination of Geometry, ZDM 13 (1) , p. 5-7 (1981)).

Actually, geometry includes so many different aspects, that it is hopeless (andmaybe even useless) to write out a complete list of them. Here we mention onlythose aspects, which in our opinion are particularly relevant in view of theirdidactical implications:

- Geometry as the science of space. From its roots as a tool for describingand measuring figures, geometry has grown into a theory of ideas and methodsby which we can construct and study idealized models of the physical world aswell as of other real world phenomena. According to different points of view,we get euclidean, affine, descriptive, projective geometry, but also topology ornon-euclidean and combinatorial geometries.

- Geometry as a method for visual representations of concepts and pro­cesses from other areas in mathematics and in other sciences; e. g. graphs andgraph theory, diagrams of various kinds, histograms.

- Geometry as a meeting point between mathematics as a theory and math­ematics as a model resource.

- Geometry as a way of thinking and understanding and, at a higher level,as a formal theory.

- Geometry as a paradigmatic example for teaching deductive reasoning.- Geometry as a tool in applications, both traditional and innovative. The

latter ones include e. g. computer graphics, image processing and image manip­ulation, pattern recognition, robotics, operations research.

Another distinction should be made with respect to several different approachesaccording to which one may deal with geometry. Roughly speaking, possibleapproaches are:

• manipulative• intuitive• deductive• analytic.

APPENDIX 339

Also one may distinguish between a geometry which stresses "static" propertiesof geometric objects and a geometry where objects are considered in a "dynamic"setting, as they change under the effect of different types of space transformations.

3. Is there a crisis in the teaching of geometry?

During the second half of this century geometry seems to have progressivelylost its former central position in mathematics teaching in most countries. Thedecrease has been both qualitative and quantitative. Symptoms of this decreasemay be found for instance in recent national and international surveys on themathematical knowledge of students. Often geometry is totally ignored or only avery few items concerned with geometry are included. In the latter case questionstend to be confined to some elementary "facts" about simple figures and theirproperties, and performance is reported to be relatively poor.

What are the main causes of this situation?- From about 1960 to 1980 a general time pressure on traditional topics has

occurred, due to the introduction of new topics in mathematics curricula (e.g.probability, statistics, computer science, discrete mathematics). At the same timethe number of school hours devoted to mathematics has gone down. The "modernmathematics movement" has contributed - at least indirectly - to the decline ofthe role of euclidean geometry, favouring other aspects of mathematics and otherpoints of view for its teaching (e.g. set theory, logic, abstract structures). Thedecline has involved in particular the role of visual aspects of geometry, boththree-dimensional and two-dimensional, and all those parts which did not fit intoa theory of linear spaces as, for instance, the study of conic sections and of othernoteworthy curves.

- In more recent years there has been a shift back towards more traditionalcontents in mathematics, with a specific emphasis on problem posing and problemsolving activities. However, attempts to restore classical euclidean geometry ­which earlier in many parts of the world was the main subject in school geometry- have so far not been very successful. The point is that in traditional courses oneuclidean geometry the material is usually presented to students as a ready-madeend product of mathematical activity. Hence, in this form, it does not fit well intocurricula where pupils are expected to take an active part in the development oftheir mathematical knowledge.

- In most countries the percentage of young people attending secondary schoolhas increased very rapidly during the last decades. Thus the traditional way ofteaching abstract geometry to a selected minority has become both more difficultand more inappropriate for the expectations of the majority of students of thenew generations. At the same time, the need for more teachers has caused,on average, a decline in their university preparation, especially with respect tothe more demanding parts of mathematics, in particular geometry. Since youngerteachers have learned mathematics under curricula that neglected geometry, theylack a good background in this field, which in turn fosters in them the tendencyto neglect the teaching of geometry to their pupils.

The situation is even more dramatic in those countries which lack a priortradition in schooling. In some cases geometry is completely absent from theirmathematics curricula.

- The gap between the conception of geometry as a research area and as asubject to be taught in schools seems to be increasing; but so far no consensus

340 THE ICMI DISCUSSION DOCUMENT

has been found on how to bridge this gap, nor even whether it could (or should)be bridged through an introduction of more advanced topics in school curriculaat lower grades.

4. Geometry as reflected in education

In former sections, we have considered geometry mainly as a mathematicaltheory and have analyzed some aspects of its teaching. Since learning is unques­tionably the other essential pole of any educational project, it is now appropriateto pay due attention to the main variables which may affect a coherent teach­ing/learning process. Consequently, several different aspects or "dimensions"(considered in their broadest meaning) must be taken into account:

- The social dimension, with two poles:

• The cultural pole, i.e. the construction of a common background (knowl­edge and language) for all the people sharing a common civilization;

• The educational pole, Le. the development of criteria, internal to eachindividual, for self consistency and responsibility.

- The cognitive dimension, Le. the process which, starting from reality, leadsgradually to a refined perception of space.

- The epistemologic dimension, Le. the ability to exploit the interplay be­tween reality and theory through modelling (make previsions, evaluate their ef­fects, reconsider choices). Thereby axiomatization enables one to get free fromreality; this in turn may be seen as a side-step which allows further conceptual­ization.

- The didactic dimension, Le. the relation between teaching and learning.Within this dimension several aspects deserve consideration. As an example, welist three of them:

• To make various fields interact (both within mathematics and betweenmathematics and other sciences).

• To make sure that the viewpoints of the teacher and the pupils areconsistent in a given study. For instance, to be aware that different distancescales may involve different conceptions and processes adopted by the pupils,even though the mathematical situation is the same: in a "space of small ob­jects", visual perception may help to make conjectures and to identify geometricproperties; when dealing with the space where we are used to move around (theclassroom, for instance) it is still easy to get local information, but it may bedifficult to achieve an overall view; in a "large scale space" (as is the case in geog­raphy or in astronomy) symbolic representations are needed in order to analyzeits properties.

• To pay due consideration to the influence of tools available in teach­ing/learning situations (from straightedge and compass, as well as other concretematerials, to graphic calculators, computers and specific software).

It goes without saying that all these dimensions are interrelated with eachother and that they should also be related appropriately to different age levelsand school types: pre-primary level, primary level, lower secondary level, uppersecondary level (where differentiation into academic, technical, vocational tracksusually starts), tertiary (Le. university) level, including teacher preparation.

APPENDIX

5. New technology and teaching aids for geometry

341

There is a long tradition of mathematicians making use of technological tools,and conversely the use of these tools has given rise to many challenging math­ematical problems (e.g. straightedge and compass for geometric constructions,logarithms and mechanical instruments for numerical computations). In recentyears new technology, and in particular computers, has affected dramatically allaspects of our society. Many traditional activities have become obsolete, whilenew professions and new challenges arise. For instance, technical drawing is nolonger done by hand. Nowadays, instead, one uses commercial software, plottersand other technological devices. CAD/CAM and symbolic algebra software arebecoming widely available.

Computers have also made it possible to construct "virtual realities" and togenerate interactively animations or marvellous pictures (e.g. fractal images).Moreover, electronic devices can be used to achieve experiences that in everydaylife are either inaccessible, or accessible only as a result of time-consuming andoften tedious work.

Of course, in all these activities geometry is deeply involved, both in orderto enhance the ability to use technological tools appropriately, and in order tointerpret and understand the meaning of the images produced.

Computers can be used also to gain a deeper understanding of geometric struc­tures thanks to software specifically designed for didactical purposes. Examplesinclude the possibility of simulating traditional straightedge and compass con­structions, or the possibility of moving basic elements of a configuration on thescreen while keeping existing geometric relationships fixed, which may lead to adynamic presentation of geometric objects and may favour the identification oftheir invariants.

Until now, school practice has been only marginally influenced by these inno­vations. But in the near future it is likely that at least some of these new topicswill find their way into curricula. This will imply great challenges:

- How will the use of computers affect the teaching of geometry, its aims, itscontents and its methods?

- Will the cultural values of classical geometry thereby be preserved, or willthey evolve, and how?

- In countries where financial constraints will not allow a massive introductionof computers into schools in the near future, will it nevertheless be possible to re­structure geometry curricula in order to cope with the main challenges originatedby these technological devices?

6. Key issues and challenges for the future

In this section we list explicitly some of the most relevant questions whicharise from the considerations outlined in the preceding sections. We believe thata clarification of these issues would contribute to a significant improvement inthe teaching of geometry. Of course we do not claim that all the problems quotedbelow are solvable, nor that the solutions are unique and have universal valid­ity. On the contrary, the solutions may vary according to different school levels,different school types and different cultural environments.

6.1. AIMS

Why is it advisable and/or necessary to teach geometry?

342 THE ICMI DISCUSSION DOCUMENT

Which of the following may be considered to be the most relevant aims of theteaching of geometry?

- To describe, understand and interpret the real world and its phenomena.- To supply an example of an axiomatic theory.- To provide a rich and varied collection of problems and exercises for individual

student activity.- To train learners to make guesses, state conjectures, provide proofs, and find

out examples and counterexamples.- To serve as a tool for other areas of mathematics.- To enrich the public perception of mathematics.

6.2. CONTENTSWhat should be taught?Is it preferable to emphasize "breadth" or "depth" in the teaching of geometry?

And is it possible/advisable to identify a core curriculum?In the case of an affirmative answer to the second question above, which topics

should be included in syllabi at various school levels?In the case of a negative answer, why is it believed that teachers or local

authorities should be left free to choose the geometric contents according to theirpersonal tastes (is this point of view common to other mathematical subjects, oris it peculiar to geometry)?

Should geometry be taught as a specific, separate subject, or should it bemerged in general mathematical courses?

There seems to be widespread agreement that the teaching of geometry mustreflect the actual and potential needs of society. In particular, geometry of three­dimensional space should be stressed at all school levels, as well as the relation­ships between three-dimensional and two-dimensional geometry. How could andshould the present situation (where only two-dimensional geometry is favoured)therefore be modified and improved?

In which ways can the study of linear algebra enhance the understanding of ge­ometry? At what stage should "abstract" vector space structures be introduced?And what are the goals?

Would it be possible and advisable also to include some elements of non-euclidean geometries into curricula?

6. 3. METHODSHow should we teach geometry?Any topic taught in geometry can be located somewhere between the two

extremes of an "intuitive" approach and a "formalized" or "axiomatic" approach.Should only one of these two approaches be stressed at each school level, or shouldthere be a dialectic interplay between them, or else should there be a gradualshift from the former to the latter one, as the age of students and the school levelprogresses?

What is the role of axiomatics within the teaching of geometry? Should acomplete set of axioms be stated from the beginning (and, if so, at what age andschool level) or is it advisable to introduce axiomatics gradually, e.g. via a "localdeductions" method?

Traditionally, geometry is the subject where "one proves theorems". Should"theorem proving" be confined to geometry?

APPENDIX 343

Would we like to expose students to different levels of rigour in proofs (as ageand school level progress)? Should proofs be tools for personal understanding,for convincing others, or for explaining, enlightening, verifying?

Starting from a certain school level, should every statement be proved, orshould only a few theorems be selected for proof? In the latter case, should onechoose these theorems because of their importance within a specific theoreticalframework, or because of the degree of difficulty of their proof? And shouldintuitive or counterintuitive statements be privileged?

It seems that there is an international trend towards the teaching of analyticmethods in increasingly earlier grades, at the expense of other (synthetic) as­pects of geometry. Analytic geometry is supposed to present algebraic models forgeometric situations. But, as soon as students are introduced to these new meth­ods, they are suddenly projected into a new world of symbols and calculationsin which the link between geometric situations and their algebraic models breaksdown and geometric interpretations of numerical calculations are often neglected.Hence, at what age and school level should teaching of analytic geometry start?Which activities, methods and theoretical frameworks can be used in order torestore the link between the algebraic representation of space and the geometricsituation it symbolizes?

How can we best improve the ability of pupils to choose the appropriate toolsfor solving specific geometric problems (conceptual, manipulative, tecnological)?

6.4. BOOKS, COMPUTERS, AND OTHER TEACHING AIDS

Are traditional textbooks as appropriate for teaching and learning geometry,as we would like them to be?

How do teachers and pupils actually use geometry textbooks and other teachingaids? How would we like pupils to use them?

What changes could and should be made in teaching and learning geometryin the perspective of increased access to software, videos, concrete materials andother technological devices?

What are the advantages, from the educational and geometrical point of view,that can follow from the use of such tools?

Which problems and limitations may arise from the use of such tools, and howcan they be overcome?

To what extent is knowledge acquired in a computer environment transferableto other environments?

6.5. ASSESSMENTThe ways of assessment and evaluation of pupils strongly influence teaching and

learning strategies. How should we set out objectives and aims, and how shouldwe construct assessment techniques that are consistent with these objectives andaims? Are there issues of assessment which are peculiar to the teaching andlearning of geometry?

How does the use of calculators, computers and specific geometric softwareinfluence examinations as regards content, organization and criteria for the eval­uation of the answers of the students?

Should assessment procedures be based mainly upon written examination pa­pers (as it seems to be customary in many countries) or else what should be therole of oral communication, of technical drawing and of work with the computer?

344 THE ICMI DISCUSSION DOCUMENT

What is it exactly that should be evaluated and considered for assessment: Thesolution outcome? The solution process? The method of thinking? Geometricconstructions?

6.6. TEACHER PREPARATIONOne essential component of an efficient teaching/learning process, is good

teacher preparation, as regards both disciplinary competence and educational,epistemological, technological and social aspects. Hence, what specific prepa­ration in geometry is needed (and realistically achievable) for prospective andpracticing teachers?

It is well known that teachers tend to reproduce in their profession the samemodels they experienced when they were students, regardless of subsequent ex­posure to different points of view. How is it then possible to motivate the needfor changes in the perspective of teaching geometry (both from the content andfrom the methodological point of view)?

Which teaching supplies (books, videos, software, ... ) should be made availablefor in-service training of teachers, in order to favour a flexible and open-mindedapproach to the teaching of geometry?

6.7. EVALUATION OF LONG-TERM EFFECTSAll too often the success (or failure) of a curricular and/or methodological

reform or innovation for a certain school system is evaluated on the basis onlyof a short period of observation of its outcomes. Moreover usually there are nocomparative studies on the possible side effects of a change of content or methods.Conversely, it would be necessary to look also at what happens in the long term.For instance:

- Does a visual education from a very young age have an impact on geometricthinking at a later stage?

- How does an early introduction of analytic methods in the teaching of geom­etry influence the visual intuition of pupils? When these pupils become profes­sionals, do they rely more on the intuitive or on the rational parts of the geometryteaching to which they have been exposed?

- What is the impact of an extensive use of technological tools on geometrylearning?

6.8. IMPLEMENTATIONAt ICME 5 (Adelaide, 1984) J. Kilpatrick asked a provocative question: What

do we know about mathematics education in 1984 that we did not know in 1980?Recently the same question has been picked up again in the ongoing ICMI study:"What is research in mathematics education, and what are its results". As forgeometry, the possibility of relying on research results would be extremely usefulin order to avoid reproposing in the future paths already proved unsuccessful, andconversely in order to benefit from successful solutions. And, as for still unsettledand relevant questions, we would like research to give us useful information inorder to clarify the advantages and drawbacks of possible alternatives.

Hence, a key question might be:What do we already know from research about the teaching and learning

of geometry and what would we want future research to tell us?

7. Call for papers

APPENDIX 345

The ICMI study "Perspectives on the Teaching of Geometry forthe 21st Century" will consist of an invited Study Conference and a Pub­lication to appear in the ICMI study series, based on the contributions to, andthe outcomes of, the Conference.

The Conference is scheduled for September 1995 in Catania (Italy). The In­ternational Program Committee (IPC) for the study hereby invites individualsand groups to submit ideas, suggestions and contributions on major problems orissues related to this discussion document, not later than February 15, 1995.

Although participation in the conference requires an invitation from the IPC,"experts" and "newcomers" interested in contributing to and participating in theconference are encouraged to contact the chair of the IPC. Unfortunately, aninvitation to attend does not imply that financial support will be provided by theorganizers.

Papers, as well as suggestions concerning the content of the study conferenceprogram should be sent to

Prof. Vinicio VILLANIDipartimento di MatematicaUniversita. di PisaVia Buonarroti 21- 56127 PISA , ITALYe-mail: <villani@dm.unipi.it>

The IPC members are:Vinicio VILLANI (Chair of the IPC). Carmelo MAMMANA (Chair of the Lo­

cal Organizing Committee, Dipartimento di Matematica, Viale A. Doria 6, Citta.Universitaria, 1-95125 Catania, Italy, e-mail: <mammana@dipmat.unict.it»,Regine DOUADY (!REM, Univ. Paris VII, France), Vagn Lundsgaard HANSEN(Math. Inst., Technical Univ. of Denmark, Lyngby, Denmark), Rina HER­SHKOWITZ (Dept. of Science Teaching, the Weizmann Inst. of Science, Re­hovot, Israel), Joseph MALKEVITCH (Math., York College, CUNY, Jamaica,N.Y., USA), Iman OSTA (American Univ. of Beirut, Lebanon), Mogens NISS(Member ex officio, IMFUFA, Roskilde Univ., Denmark).

LIST OF PARTICIPANTS TO THE CATANIA CONFERENCE

347

N. Aguilera (Argentina)A. Ambrus (Hungary)M. Bartolini Bussi (Italy)R. Berthelot (France)R. Bkouche (France)P. Boero (Italy)G. Brown (USA)S. Casella (Uruguay)A. Chronaki (England)B. D'Amore (Italy)A. Douady (France)R. Douady (France)R. Duval (France)W. Ebeid (Kuwait)M. E. EI Tom (Qatar)E. K. Fainguelernt (Brazil)J. Fortuny (Spain)M. Galuzzi (Italy)C. Gaulin (Canada)G. Gholam (Egypt)N. Gorgori6 (Spain)K. Graf (Germany)H. B. Griffiths (England)P. Grushko (Russia)A. Gutierrez (Spain)M. de Guzman (Spain)V. L. Hansen (Denmark)D. Henderson (USA)R. Hershkowitz (Israel)B. Hodgson (Canada)C. Hoyles (England)M. Humbert (Switzerland)O. Iden (Norway)K. Jones (England)L. Jones (England)F. Kurina (Czech)

C. Laborde (France)P. Legisa (Slovenia)R. Lehrer (USA)A. J. Lopes (Brazil)Z. LuCie (Serbia)J. Madjarova (Sweden)G. Malaty (Finland)J. Malkevitch (USA)C. Mammana (Italy)M. Marchi (Italy)M. A. Mariotti (Italy)P. Maroscia (Italy)W. G. Martin (USA)M. Neubrand (Germany)M. Niss (Denmark)R. Olstorpe (Sweden)I. Osta (Lebanon)E. Partova (Slovak)B. Parszysz (France)J. Pegg (Australia)N. Pietrocola (Argentina)R. Porcaro (Italy)M. Rahim (Canada)T. Romberg (USA)N. Rouche (Belgie)J. Ruberu (Brunei)H. Schumann (Germany)F. Speranza (Italy)R. Straesser (Germany)D. Taimina (Lettland)S. C. Tang (China)D. Torri (Italy)S. Turnau (Poland)C. Vasco (Colombia)V. Villani (Italy)F. Zhang (China)

348

ADDRESSES OF THE CONTRIBUTORS

Maria G. Bartolini BussiDip. di Matematica Pura ed Appi.Universita, Via Campi 213fb41100 Modena, Italy

Rene BerthelotLADIST, Universite Bordeaux 140, rue Lamartine33400 Talence, France

Paolo BoeroDip. di Matematica, UniversitaVia Dodecaneso 3516146 Genova, Italy

Adrien DouadyDep. de MathematiquesUniversite de Paris-Sud91405 Orsay, France

Regine DouadyIREM - Universite Paris 7, C.P. 70182, Pi. Jussieu75251 Paris 05, France

Raymond DuvalUniversite du LittoralIUFM du Nord Pas-de-Calais59820 Gravelines, France

Massimo GaluzziDip. di Matematica, UniversitaVia Saldini, 5020133 Milano, Italy

Ghada K. GholamDar Al HandasahP.O.Box 895Cairo, Egypt

Klaus-Dieter GrafFreie Universitat BerlinInstitut fUr InformatikD-14195 Berlin, Germany

H. Brian GriffithsUniversity of SouthamptonFaculty of Mathematical StudiesHighfield, SouthamptonS017 lBJ, England&Mathematics CentreChichester Institute ofHigher Education,Bognor Regis, West Sussex,P021 lHR, England

Angel GutierrezUniversitat de Valencia,Dep. de Did. de la Matematica46071 Valencia, Spain

Vagn Lundsgaard HansenDept. of Mathematics,Technical University of Denmark,Building 303, DK-2800,Lyngby, Denmark

Rina HershkowitzDept. of Science TeachingWeizmann Institute of Science76100 Rehovot, Israel

Bernard R. HodgsonDep. de Mathern. et de Statist.Universite Laval,Quebec GIK 7P4 Canada

Celia HoylesInstitute of Education,University of London,London, WCIH OAL, England

Pedro HuertaUniversitat de Valencia,Dep. de Did. de la Matematica,46071 Valencia, Spain

Keith JonesUniversity of SouthamptonSchool of EducationHighfield, Southampton,S017 IBJ, England

Colette LabordeDidaTech LSD, BP 53 University38041 GrenobleFrance

Peter LegisaDept. of Mathematics, UniversityJadranska 19, 1000 LjubljanaSlovenia

Richard LehrerNCRMSE - University of WisconsinMadison WI, 53706 USA

Joseph MalkevitchMathematics and Computing Dept.York College (CUNY)Jamaica, New York 11451-0001USA

Carmelo MammanaDip. di Matematica, Citta Univ.Viale A. Doria 695125 Catania, Italy

Walter MeyerAdelphi UniversityGarden City, New York 11530USA

Michael NeubrandUniversitat FlensburgInst. fur Mathematik und ihre DidaktikMurwiker Str. 77D-24943 Flensburg, Germany

Mogens NissIMFUFA, Roskilde UniversityP. O. Box 260DK-4000 Roskilde, Denmark

Iman OstaAmerican University of BeirutDept. of EducationP. O. Box 11-0236Beirut, Lebanon

Bernard ParzyszIUFM de Lorraine,UniversiteMetz, France

349

John PeggDept. of Curriculum Studies,University of New EnglandArmidale NSW 2351, Australia

Thomas RombergNCRMSE - University of WisconsinMadison, WI, 53706 USA

M. H. Salin,LADIST, Universite Bordeaux 140, rue Lamartine33400 Talence, France

Tang Sheng ChangShanghai High School400 Shang Zhong Road200231 Shanghai, P. R. China

Jerzy TockiInstytut MatematykiWyzszej Szkoly PedagogicznejUl. Rejtana 16 A35-310 Rzeszow, Poland

Stefan TurnauInstytut MatematykiWyzszej Szkoly PedagogicznejUl. Rejtana 16 A35-310 Rzeszow, Poland

Carlos E. VascoUniversidad Nacional de ColombiaAvenida 32, n. 15-31Santafe de Bogota, 1, DC, Colombia

Vinicio VillaniDip. di Matematica, Universita,Via Buonarroti, 256127 Pisa, Italy

Zhang FushengEducational Commission of Shanghai200041 Shanghai, P. R. China

351

INDEX

921510

1232,53-57

4239, 124

24102

2, 13, 1622

16, 24, 90, 95

F

Hamming codeshelicoidHerodotus

EudoxusEulerexplanations

GaussGeometric supposergraph theory

G

EEinstein 3, 87empirical reasoning 123Erlangen program 2, 154, 245error-correcting codes 24Euclid 2, 18, 146, 195, 237Euclidean geometry 11, 26, 29, 86

121, 249, 253, 25410

15, 16, 94, 23731, 126

Fermatfields of experiencefigural changefigurefrieze patternsfullerenes

D

data compression 97Dedekind 3deductive process 47deductive reasoning 110, 123, 241delivery problem 19Descartes 1, 12, 231didactic engineering 159, 173didactic variables 181Dieudonne 3dimensional change 40discursive process 45drawings 115, 124, 164, 217dynamic geometry 121, 243dynamic geometry software 109, 123

H

1182,13

1616

102

2173

1312, 256, 258

402, 11, 87

852, 87, 146

14, 22, 186, 189258, 322

1219

263-268, 275, 326268151

21, 23, 216, 26022, 26, 47, 218

22, 27, 123, 259, 324110

128, 133, 3243

5419615114745

14424

27, 109, 113, 155227, 260, 324

109187451126

11, 256, 322124

30, 38134, 164, 166

11, 22, 204, 235, 321

Bblack-box tasksBolyaiBorsukBrouwerbuckyball

C

CabriCabri-geometreCADCantorcartographyCayleycentral reflectioncircular inversioncognitive processesCollatzcompression codescomputer

Aristotleart museum problemassessmentassessment modesaxial reflectionaxiomaticsaxioms

computer technologyconcrete/abstractconfigural processcongruenceconicsconic sectionsconjectureconstructioncubecurriculum

A

Albertialgebra (and geometry)algebraic geometryanalytic geometryanchorage changeApolloniusapplicationsArchimedesarea

Iimage manipulation and processing 93intuition 80, 82, llOisometry 65, 253, 322isoperimetric problem 14

352

HeronHilberthistoryhyperbolic geometryhyperbolic plane

152, 18, 238, 319

2392813

N

national British curriculum 122, 199nautilus 17New Math 205, 209, 220, 236

257, 258, 297, 298323, 325

Newton ll, 87non-Euclidean geometry 13, 14, 21

27, 203, 237

0

J operations research 94operative apprehension 41, 43

justification 29,242 orientation 75

K P

kaleidoscope 96, 149 parallelogram 217Kant 2 parallel postulate 13, 21, 27Kepler 87 parallel projection 130, 322Klein 2, 154, 209 parameter 175knots 24 Pasch 2knowing/seeing 164 patterns 24

perimeter 22, 186Perry 196

Lperspective drawing 58

Leibniz 16 Piero della Francesca 2linear algebra 258 plane representations 239, 322Lobachevsky 2, 13,87 Plato 12local deductions 79,323 Playfair 13logarithmic spiral 17 Poincare 13LOGO 23 polyhedra 24,96

proof 11, 26, 30, 31, 38, 48M 49, 59, 70, llO, III

121,127, 170,172macrospace 72 186, 190, 250magnitudes 239 256, 322maps 74 proving 29, 31, 323mathematical machines 59 Ptolemy 2measuring 54, 258, 322 Pythagoras 10, 87, 237medical imaging 91mesospace 25,72 QMeusnier 15 quaternions 26microspace 25,72 quilt design 33,63-64minimal surface 15Minkovski 3modelling 85,89,226 RMoebius band 96Monge 2 reasoning 29-33, 38, 39, 45motions 65 241, 258, 322

353

reference systemsrelativityrepresentationsRiemannrigidityrigourroboticsrotationsruled surface

S

14114

164, 1652

101242

91, 10422, 247

15

v

van Hiele levelsvectors

vector spacesVeronesevisionvisualizationvisual reasoningvolume

79, 202, 278, 288142, 143, 202, 214

216, 238, 253256, 260

3, 230, 241, 260, 3122

3938, 48, 82

3322,258

145

3,14

W

yin-yang

y

Weierstrass

106100

33, 54, 166-17111, 253, 255

258, 32222, 324163, 171281-293

20, 25, 71, 110170, 236, 241

302, 32228

103216

11, 22, 103, 32212

Sketchpadsocio-cognitive interactionsSOLO taxonomyspace

sphere (geometry on the -)strip patternsstructural approachsymmetrysynthetic geometry

scalingscan conversionshadowssimilarity

T

160297132

9110,87,237

32024,96

53, 165, 19022, 59, 116, 206

210, 217, 253, 258translations 22, 143, 247trends in geometry curricula 260trigonometry 196, 249, 253, 258turtles 153two-column proof 30

teacherjpupils relationshipteacher qualificationstechnical drawingtelecommunicationsThalesThorntilingstool-object dialectictransformations