**Curriculum and 3D Geometry**

*To Karl Menger and Marshall H. Stone, in Memoriam*

## Introduction

Our chief concern in this paper is to contribute to a wider consideration of 3D Geometry in the compulsory mathematical curriculum. As a result of research results and practical experiences, space geometry in the curricular framework has experienced a positive growth. In the old days the special knowledge was restricted to a few figures (polyhedra-spheres-cylinders-cones) and transformations, and no attention was given to the essential aim of the subject: to develop visual thinking and mathematical reasoning. Nevertheless, beyond the curricular frame, the real time dedicated to geometry in the classrooms is still low and the level is weak. In general, plane geometry wins space geometry. But many research studies have shown the unacceptable low level of achievements in 2D geometry. The 3D case is still worse. Bearing this situation in mind we will defend the interest of space geometry, and how it can be developed by means of integrating appropriate resources (from hands-on materials to technological devices). We will make reference to recent research of A. Aubanell that we have been tutoring and, hopefully, we will clarify some key issue concerning this important topic.

## What is reality?

*³ is not our reality. The real world is our house and our land; it’s the objects we use and the transportation we ride, it’s our social and cultural life… reality is a complex issue and merits a mathematical approach.

The real world is not the recreational problems paradigm of crossing trains sheep in boats crossing rivers, castles with lakes, etc. Reality is not the Euclidean description of shapes. Many times we make the mistake of identifying reality with its mathematical models. Real space is not *³: it’s much more!

Moreover our reality is not only a historical heritage. Our reality is also the 21st Century, i.e., the problems and possibilities that we face today, from the use of the new technological tools to the environmental open problems.

We are three dimensional people in a complex three dimensional world,… and we have mathematics to discover and make actions in this 3D setting.

## Towards a spacial culture

The first aim for including 3D Geometry in the curricula is to give people opportunities for developing a spacial culture (Alsina, 2000a,b). We have identified eight principles that could guide our approach to 3D-Geometry in the classrooms:

- A first aim in learning 3D-geometry is to develop visual thinking which is the basic competency for a spacial culture;
- Common sense in 3D needs to be developed since it is not a genetic issue;
- There is no need to follow the artificial ordering 1D-2D-3D, since one can work at the same time in the different domains;
- We can base our approach to 3D-Geometry on interesting applied problems related to our reality;
- We can mix different representations, models and technologies to treat 3D problems, since the multi-modeling gives richer results;
- Spacial culture can facilitate connexions and interdisciplinary approaches (history, art, environment,…);
- Spacial culture can promote the research spirit of our students and the discovering of intriguing results;
- The final goal of a spacial culture is to develop human creativity.

## A theoretical framework

As a result of the research carried out in the last decades many theoretical frameworks of reference have been developed, from the topological approach of Piaget and Inhelder (1967) setting to the latest social constructivism. All these theories, which may orientate teaching and learning processes in mathematics, remark the importance of geometry.

Following Hershkowitz, Parzysz and Dormoleu we may consider that geometry is seen by many students like the “science of the physical space and its mathematization” and that *visualization* is the key tool to facilitate geometry’s learning. But visualization includes a rich combination of figures, diagrams, transformations, mental images, inter-active figures, etc. (see (Bishop, 1989), (Del Grande, 1990). The theories of Internal Representation have clarified as well the role of images in the learning progression.

My personal point of view is that Freudenthal’s ideas concerning “*mathematization*” are very relevant for space geometry. I follow in this work the realistic mathematics framework, which focuses in *modeling and applications* (Malkevitch, 1998) for working 3D-geometry in the classrooms. Results from PISA show the need of devoting more attention to this subject.

Under the direct influence of Hans Freudenthal, Pierre Marie and Dina Van Hiele started to develop in the 50’s a model concerning mathematical reasoning and its levels (Van Hiele, 1957). Since then a lot of research has proven that Van Hiele’s model may be useful in geometrical thinking (Clements and Battista, 1992) (Jaime, 1993) (Gutiérrez, 2007). Let us recall that Van Hiele distinguishes 5 levels of reasoning, from the very basic simple observations to descriptive understanding/ analytical thinking and more abstract settings including relations and proofs. Experimental studies (see e.g. (Clements and Battista, 1992)) have shown that all students need to develop in a sequential way these different levels.

This model had a direct influence in the NCTM principles and standards (NCTM, 2000) where the goal to attain at least the first three levels at grade K-12 guides the proposal of the curricular development.

## On teaching resources and technologies

Since the times of Maria Montessori where small cubes and colourful prisms became a key tool for the learning of arithmetic, a lot of resources have been produced for primary schools. The exhibition area at NCTM 2007 in Atlanta showed a very large collection of hands-on materials for school children.

The bad news is that there is a low production of hands-on materials for the higher grades where the commercial interest focuses on graphic calculators, software, polyhedra, etc. Nevertheless as noted in (Clements and Batista, 1992) the student’s representation of space is constructed from active manipulation.

But the good news is that for working 3D-geometry one has at hand (and free of charge!) our own 3D-world, i.e., instruments for measuring, geometrical objects with clever designs, all kinds of technical devices, etc. In addition one can easily produce materials which will facilitate teaching and learning activities (transparent polyhedra, soap films, mirrors, scale models, etc.).

The increasing power of technologies will have also a big impact on treating 3D-Geometry. In the initial approach to 3D-Geometry (at any level!) *virtual images can’t substitute the real observation and manipulation of hands-on materials.* To draw figures with ruler and compass by hand is not equivalent to manipulate an applet making such drawings; the direct measure of a building cant’be substituted by a virtual measure using Google-Earth; the effective construction of an icosahedron with card-board or plastic pieces can’t be substituted by clicking the mouse to move a virtual polyhedra on the screen.

But technology has its own interest, it is a motivating tool and opens great learning opportunities. So there is no doubt that technology needs to be integrated in the process of mastering 3D-Geometry. All drawing programs (Cabri-II, Geometry sketchpad, Geobra, etc.), the new 3D-software (Cabri 3D, CAD,…), the interactive web sides (see Internet resources at the end), etc. may be excellent tools to be used properly.

## Some new research results of A. Aubanell

For many years Anton Aubanell (Aubanell, 2006) has been working the subject of teaching with hand-on materials with 12-18 year old students. In 2005-2006 he had a one year leave to develop under my supervision a research program whose goal was to answer the question: can all parts of the curriculum be treated by appropriate hands-on materials? The answer is yes, results are available in the web side of our Department of Education (http://www.xtec.cat/~aaubanel/index.swf). Data accumulated after many years of case studies in the IES Sa Palomera (Blanes, Spain) showed that students become highly motivated when taking an active role in their our learning progress, when making cooperative work and developing a more visual approach.

In the case of 3D-Geometry (Aubanell’s study was covering all parts of the curriculum) we have identified twelve uses of hands-on materials which are of interest for teaching and learning purposes. The next diagram sums up our considerations.

At the same time we have seen that it’s necessary to combine all kinds of representations and resources if we want to face rich tasks, The idea is to break the usual linear development of the text-book strategy and combine problem solving, projects, technology, experimentation, etc.

## Implications for thinking, reasoning and proving

Let’s remark that beyond the concrete knowledge of shapes and transformations in the space the curricular work of 3D-Geometry needs to help students to develop visual thinking (with a rich variety of representation), to improve their ways of mathematical reasoning (induction, analogy, generalization,…) and to discover the need of proving. It’s a hot topic in today’s education (NCTM, 2000) to pay more attention to proofs and proving. Many researchers and national curricula recommend that the concept of *proof* and the activity of *proving* become an important goal of student’s mathematical experiences throughout the grades. We are convinced, following ideas extensively developed by Gila Hanna, that the *proofs* which are of interest for teaching are those that facilitate a better understanding of the results or concepts being faced. For making possible visual proofs we have developed in (Alsina and Nelsen, 2006) a full methodology on how to create what we call “images for understanding”, where 2D and 3D images are combined.

## Good curricula do not mean better teaching

While we are now more optimistic than in the past concerning the role of space geometry in many curricula, it is unclear to what extend this has been really developed in the classrooms. To make this possible much more training on space geometry must be offered in pre-service and in-service teacher’s courses and seminars.

Our own experiences in Spain and Argentina confirm also that the lack of geometrical knowledge among teachers is to be compensated with attractive in-service actions (e.g., with geometry workshops) and actions in professional development.

## To sum-up: 3D-Geometry wants an opportunity

We are convinced that the Geometry merits further attention in the curriculum. But we have also seen that there is a lot to do with teachers before some progress could be appreciated. In many occasions teachers are not confident with the mathematics of 3D-Geometry and the minimal-effort principle (text-book & blackboard) has a strong influence.

Our future citizens merit our efforts towards a better learning of 3D-Geometry.

## REFERENCES

ALSINA, CLAUDI. (2000a), *Sorpresas Geométricas, *Buenos Aires, OMA.

ALSINA, CLAUDI (2000b), Gaudi’s ideas for your classroom: Geometry for Three-Dimensional Citizens, *Selected Lectures ICME-9*, Makuhari-Tokyo (Japan) (CD-Rom Proceedings).

ALSINA, CLAUDI (2001), *Geometría y realidad en Aspectos didácticos de Matemáticas *8 ICE. Univ. Zaragoza, Zaragoza, 11-32.

ALSINA, CLAUDI (2007), Less Chalk, Less Words, Less Symbols,… More Objects, More Context, More Actions in W. Blum et al. (Eds). The 14^{th} ICMI Study: Modelling and Applications in Mathematics Education, Springer, Berlin, 35-55.

ALSINA, CLAUDI, Invitación a la tridimensionalidad en (Flores, 2007), pp. 119-139.

ALSINA, CLAUDI, BURGUÉS, CARMEN., FORTUNY, JOSEP M. (1990), *Materiales para construir la Geometría. *Síntesis, Madrid.

ALSINA, CLAUDI, FORTUNY, JOSEP M., PÉREZ, RAFAEL (1997), *¿Por qué Geometría? Propuestas Didácticas para la ESO, *Síntesis, Madrid.

ALSINA, CLAUDI, NELSEN, ROGER. (2006), Math Made Visual. Creating images for understanding mathematics. MAA, Washington.

AUBANELL, ANTON (2006), Recursos materials i activitats experimentals en l’educació matemàtica a adecundària. Dep. Educació, Generalitat de Catalunya, Barcelona.

BATTISTA, MICHAEL T. and CLEMENTS, DOUGLAS H. (1996), Student’s understanding for three-dimensional rectangular arrays of cubes. *Journal for Research in Mathematics Education *27, 258-292.

BISHOP, ALAN J. (1989), Review of research on visualization in mathematics education, *Focus on Learning Problems in Mathematics* 11, 1, pp. 7-16.

BOLT, BRIAN (1991), *Mathematics meets technology, *U.P., Cambridge

CLEMENTS, DOUGLAS H.; BATTISTA, MICHAEL T. (1992), Geometry and spatial reasoning, en Grouws, D.A. (ed.), *Handbook of research on mathematics teaching and learning*. (MacMillan: N. York, EEUU), p. 420-464.

CORBERÁN, ROSA, GUTIÉRREZ, ANGEL et al. (1994), *Diseño y evaluación de una propuesta curricular de aprendizaje de la Geometría en Enseñanza Secundaria basada en el Modelo de Razonamiento de Van Hiele* (colección “investigación” n. 95), CIDE, MEC, Madrid.

DEL GRANDE, JOHN (1990), Spatial sense, *Arithmetic Teacher* 37, 6, pp. 14-20.

FLORES, PABLO et al. (Eds.) (2007), Geometría para el siglo XXI, FESPM, Badajoz.

GUILLÉN, GREGORIA (1997), *El modelo de Van Hiele aplicado a la geometría de los sólidos. Observación de procesos de aprendizaje,* Univ. de Valencia, Valencia.

GUILLÉN, GREGORIA (2000), Sobre el aprendizaje de conceptos geométricos relativos a los sólidos. Ideas erróneas. *Enseñanza de las Ciencias,* 18-1, pp. 35-53.

GUTIÉRREZ, ANGEL (1996), Visualization In 3-dimensional geometry: In search of a framework, *Proceedings of the 20th PME Conference* 1, pp. 3-19.

GUTIÉRREZ, ANGEL (1998), Las representaciones planas de cuerpos 3-dimensionales en la enseñanza de la geometría espacial, *Revisa EMA* 3-3, pp. 193-220.

GUTIÉRREZ, ANGEL, JAIME, ADELA, FORTUNY, JOSEP M. (1991), An alternative paradigm to evaluate the acquisition of the Van Hiele levels, *Journal for Research in Mathematics Education *21, 3, pp.237-251.

GUZMÁN, MIGUEL de (1991), *Para pensar mejor, *Labor, Barcelona.

HANNA, GILA (1996), The ongoing value of proof. *Proceedings of the 20 ^{th} PME Conference* 1, pp. 21-34.

JAIME, ADELA (1993), *Aportaciones a la interpretación y aplicación del modelo de Van Hiele. La enseñanza de las isometrías del plano. La evaluación del nivel de razonamiento, *Univ. Valencia, Valencia.

LAKATOS, IMRE (1978), *Pruebas y refutaciones. La lógica del descubrimeinto matemático, *Alianza. Madrid.

MALKEVITCH, JOSEPH (1998), *Finding room in the curriculum for recent geometry en *(Mammana, 1998) 18-24.

MALKEVITCH, JOSEPH (1998), *Geometry and reality en *(Mammana, 1998) 85-99.

MALKEVITCH, JOSEPH (1998), *Geometry’s future, * COMAP, Lexington.

MAMMANA, C. et al. (1998), *Perspective on the Teaching of Geometry for the 21 ^{st} Century. *An ICMI Study, Kluwer, Dordrecht.

NCTM, (2000), Principles and standards for school mathematics, NCTM, Reston, EEUU.

PLUGH, G. (1976), *Polyhedra, a visual approach*. Univ. Calif. Press, Londres.

STEEN, LYNN A. (1994), *For all practical purposes, *COMAP Lexington. W.H. Freeman Co. New York, Versión española: *Matemáticas en la vida cotidiana, *Addison-Wesley, Madrid, 1999.

VAN HIELE, PIERRE M. (1957), El problema de la compresión en conexión con la comprensión de los escolares en el aprendizaje de la geometría, Univ. Utrecht, Utrecht, Holanda.

VAN HIELE, PIERRE M. (1986), *Structure and insight. A theory of mathematics education,* Academic Press, Londres.

VELOSO, EDUARD (1998), *Geometria: Temas actuais. *Min. Educ., Lisboa.

**Internet resources**