This paper reports on the design of a dynamic environment for the learning of stereometry (DALEST) and the teaching of spatial geometry and visual thinking. The development of the software was in the framework of DALEST project which aimed at developing a dynamic three-dimensional geometry microworld that enables students to construct, observe and manipulate geometrical figures in space, and to focus on modelling geometric situations. The environment will also, support teachers in helping their students to construct a suitable understanding of stereometry. During the developmental process of software applications the key elements of spatial ability and visualization were carefully taken into consideration with emphasis on enhancing dynamic visualization as an act of construction of transformations between external media and student’s mind

Boytchev, Pavel

Christou, Constantinos

Jones, Keith

Mousoulides, Nicholas

Pitta, Demetra

Pittalis, Marios

Sendova, Evgenia

English

Southampton (e-Prints Soton)

Christou, C., Pittalis, M., Mousoulides, N., Pitta, D., Jones, K., Sendova, E. & Boytchev, P. (2007), Developing an Active Learning Environment for the Learning of Stereometry. Paper presented at the 8th International Conference on Technology in Mathematics Teaching (ICTMT8), Hradec Králové, Czech Republic, July 1-4, 2007. 1 Developing an Active Learning Environment for the Learning of Stereometry  Christou, C.*, Pittalis, M.*, Mousoulides, N.*, Pitta, D.*, Jones, K.**, Sendova, E.*** & Boytchev, P.***  * Department of Education, University of Cyprus ** School of Education, University of Southampton, UK *** Department of Information Technologies, Sofia University, Bulgaria edchrist@ucy.ac.cy, m.pittalis@ucy.ac.cy, n.mousoulides@ucy.ac.cy, dpitta@ucy.ac.cy, d.k.jones@soton.ac.uk, jsendova@mit.edu, pavel@elica.net   Abstract This  paper  reports  on  the  design  of  a  dynamic  environment  for  the  learning  of  stereometry (DALEST)  and  the  teaching  of  spatial  geometry  and  visual  thinking.  The  development  of  the software was in the framework of DALEST project which aimed at developing a dynamic three-dimensional  geometry  microworld  that  enables  students  to  construct,  observe  and  manipulate geometrical figures in space, and to focus on modelling geometric situations. The environment will also, support teachers in helping their students to construct a suitable understanding of stereometry. During the developmental process of software applications the key elements of spatial ability and visualization were carefully taken into consideration with emphasis on enhancing dynamic visualization as an act of construction of transformations between external media and student’s mind. Keywords: stereometry, spatial sense and reasoning, visualization, computer applications, software design Introduction and Theoretical Background Spatial  visualization  is  an  important  skill  that  deserves  instructional  attention.  Space  is  a fundamental category of thought, one that plays a deep role in many aspects of human cognition. Because  space  is  a  fundamental  feature  of  our  environment,  spatial  knowledge  and  spatial perception play crucial roles in even the most ordinary human problem solving (National Science Foundation, 1997). Strong evidence supports the claim that “measures of mathematical ability tend to be strongly correlated with spatial ability”. The Principles and Standards of the National Council of Teachers of Mathematics (2000) recommend that 2D and 3D spatial visualization and reasoning are core skills that all students should develop. There is every reason to believe that time spent helping students to develop their spatial visualization skills will have additional benefits to their mathematical growth (Chavez, Reys & Jones, 2005). A possible way to improve students’ 2D and 3D spatial visualization and reasoning abilities is to provide  students  with  possibilities  to  explore  the  properties  of  3D  objects  in  appropriately developed dynamic and interactive computer applications. Computer-based learning environments continue to be a seductive notion in mathematics education and the promise is that through using particular software in carefully-designed ways, it is possible for learners simultaneously to use and come to understand important aspects of mathematics, something that in other circumstances can be particularly elusive (Jones, 2001). This holds true especially in the field of space and 3D geometry where students are required to inspect, encode, transform, and construct information from visual displays. Since children’s initial representations of space are based on action an important feature of such applications is that they should be action-based, i.e. the students should be able to slide and rotate 3D objects creating dynamic images, combine 3D objects in more complex compositions, measure the basic 3D objects and study their properties and relationships and represent and solve problems using geometric models (Boytchev, Chehlarova & Sendova, 2006). When provided with carefully  designed  sequences  of  educational  activities  within  such  computer  applications  the   2 students are expected to develop an appreciation of geometry as a means of describing the physical world. In recent years, much disquiet has been expressed about the limitations of traditional approaches to the  teaching  and  learning  of  geometry  and  specifically  of  spatial  abilities  that  constrain  the visualisation  and  advancement  of  teaching  and  learning  in  geometry.  Most  current  curriculum materials  and  computer-based  systems  address  the  investigation  of  3D  geometry  by  requiring students to operate within two dimensional representations of 3D geometry which is not a natural way of investigating human 3D spatial ability. To overcome the limitations of these representations the  DALEST  project  aimed  at  developing  a  dynamic  three-dimensional  geometry  microworld  that enables students to construct, observe and manipulate geometrical figures in space, and to focus on modelling geometric situations, and supports teachers in helping their students to construct a suitable understanding  of  stereometry.  In  this  paper  we  focus  on  presenting  applications’  design  and capabilities  and  possible  teaching  potential  with  emphasis  on  dynamic  visualization.  In  the theoretical background below, we delineate fundamental theoretical spatial ability considerations upon  which  the  design  of  the  software  applications  were  based  and  in  the  following  sections provide a description of the capabilities of the software for the teaching of spatial geometry.  The aim of developing the Dalest software was to enhance the abilities and processes in students that are closely associated with the idea of visual imagery as a mental scheme depicting spatial information (Presmeg, 1986). It is  generally  accepted that learning 3D  geometry is strongly associated with spatial and visual ability. From this perspective, spatial abilities are important cognitive factors in learning geometry and incorporating spatial visualization and manipulation into learning activity could improve geometric learning. As a result, mathematics curricula in a range of countries (see, as an example, the National Council of Teachers of Mathematics, 2000) emphasize the importance of spatial abilities in mathematics education and recommend that 2D and 3D spatial visualization and reasoning are core skills that all students should develop. A 3D dynamic geometry software should provide to the learner a variety and richness of spatial images. The richer and more diverse the store of spatial representations, the easier is to use images in solving  problems.  Dynamic  visualization  can  be  a  very  powerful  tool  to  gain  a  greater understanding  of  many  mathematical  concepts  or  it  can  be  a  resource  to  solve  mathematical problems  (Harel  &  Sowder,  1998;  Presmeg,  1986).  Taking  into  consideration  the  gap-bridging power of computers which has been extensively reported on in the research literature, we claim that dynamic representations of mathematical processes may enable “the mind to manipulate them in a far more fruitful way than could ever be achieved starting from static text and pictures in a book”. Dynamic spatial abilities are those that are required to reason about moving stimuli. The advent of computer testing has led to renewed interest in the ability to reason about motion and the ability to integrate spatial information over time (Hegarty & Waller, 2005). Dynamic visualization is an act of construction of transformations between external media and individual’s mind. Individuals who possess this ability can reason about the essential properties of moving, shrinking, and rotating figures, which appear on the screen or, in their mind, and thus they can solve the mathematical problems  (Boz,  2005).   There  are  research  evidences  too,  which  show  that  some  students spontaneously develop this ability to move figures in their heads, to stretch and shrink them, to rotate them, to see them interact, and hence solve problems by using this kind of technique (Harel & Sowder, 1998).  Dalest Software Design In the framework of DALEST project several software applications have been developed aiming at enhancing middle school students’ 3D geometry understanding and spatial visualization skills by utilizing dynamic visualization images. In this study we present the general characteristics of the stereometry learning environment and describe the application called Cubix Editor.   3 General  Characteristics  of  Stereometry  Learning  Environment:  The  distinguished  feature implemented in the software is the ability to construct geometrical objects in successive steps that correspond  to  their  mathematical  properties.  For  example,  a  cuboid  is  constructed  is  three successive movements, defining in each movement the length of one of the bone edges of the solid (see figure 1). The manual construction of solids gives to the user and intuitive conception of the 3D nature of the solid. The geometrical objects created on the screen can be manipulated, moved and reshaped interactively by means of the mouse. The realisation of the software allows students to see a  geometric  solid  represented  in  several  possible  ways  on  the  screen  and  to  transform  it,  the intention here being to help students to acquire and develop abilities of visualization in the context of 3D geometry. As Gutiérrez (1996) argues, when a person handles a real 3-dimensional solid and rotates it, the rotations made with the hands are so fast and unconscious that the person can hardly reflect  on  such  actions.  The  design  of  the  software  facilitates  the  informed  manipulation  of directions of rotation.          Figure 1: The construction of a cuboid in three successive steps                        Figure 2: Different representations of a cube The design of the software focuses on the processes of observation and exploration in the following ways: Observation allows students to see and understand the third dimension by changing the spatial system of reference (axes), choosing perspectives and displaying visual feedback on objects. Students can rotate a geometric object in reference to the three axes and thus gain a holistic view of the object. The software is designed such that the drawing style of any object can be in a ‘solid colour’ view or in a ‘transparent line’ view, as illustrated in Figure 2, and students can select, label and colour the edges and faces of the objects. Exploration allows students to explore and discover the geometrical properties of a figure. This is the main procedure adopted in most of the teaching scenarios that are available to accompany the software. For example, users can take advantage of the grouping wizard of the software to explore how different solids can be grouped to form another solid. As illustrated in Figure 3a the user can  group the two cubes to construct the cuboid shown in Figure 3b.  In the  grouping procedure, the user can select the faces of the cubes that will coincide. In the case that the faces do not have the same dimensions the user has the following options: (a) to resize the first object, (b) to resize the second object, and (c) to group the objects as is, as illustrated in Figure 3c (the two solids in Figure 3b were grouped as is).   4              (a)                   (b)        (c) Figure 3: Grouping of cubes and cuboids The dragging capability of the software enables students to rotate, move and resize 3D objects in much the same way as the commonly available 2D dynamic geometry software environments. The design approach focused on enabling rotation to be executed in all directions through the provision of an on-screen rotation cursor that could also be used to determine the speed of the rotation. In addition,  the  design  was  made  such  that  students  are  able  to  resize,  proportionally,  all  the dimensions of the object or resize it only in one dimension, according to the requirements of the problem with the use of hot points. For example, students can resize a cone, as illustrated in Figure 4,  by  shrinking  or  enlarging  the  radius  BC  (by  dragging  the  hot  point  B),  or  by  shrinking  or enlarging  the  height  AC  (by  dragging  the  hot  point  A).  The  dynamic  characteristic  of  the measurement  facility  allows  learners  dynamically  to  explore  the  properties  within  and  amongst figures; for example, students can obtain the measure of the volume of the cone and then double its height and see how its volume is altered.                            Figure 4: Resizing the dimensions of a cone Cubix Editor: An editor which can be used by students to create 3D structures built of unit-sized cubes. The Cubix Editor allows the students to save their structures and reload them, thus making a library of various figures (see Figure 5). The user can also recolor the cubes and get measures for the volume and surface area of the solids constructed. A very useful characteristic of the application is the rotation of the whole platform of the construction, procedure which creates dynamic images and gives to the user the capability to develop dynamic visualizations of the front, side and top view of the object. The dragging capability of the application is a valuable tool that can be used to develop didactical situations enhancing visual abilities, such as “Perceptual constancy” and “Mental rotation”  by  producing  dynamic  mental  images  and  visualizing  a  solid  in  movement,  and investigating mathematical properties. As illustrated in Figure 5, student’s visual abilities can be enhanced when are challenged to change the place of one of the unit cubes in Figure 5(a) so that the new construction could take the position of Figure 5(b) in the space.   5  (a)  (b) Figure 5: Two compositions of unit-size cubes as part of a logical 3D problem  Conclusion The design of the DALEST applications is in the process of development. As presented earlier, the main purpose of these applications is to enhance students’ understanding 3-D geometrical reasoning and visualization  and  spatial  thinking.  Thus,  during  the  developmental  process  the  key  elements  of visualization as defined by several researchers are carefully taken into consideration. We expect that the final library of DALEST applications will constitute a powerful tool in the teaching of stereometry and spatial geometry. We also expect that applications will enhance students’ dynamic visualization ability and enable them to gain a greater understanding of 3D mathematical and spatial concepts.  Acknowledgements This project was part-funded by the European Union (grant 224269-CP-1-2005-1-CY-MINERVA-M). The views expressed are those of the authors and do not necessarily reflect the views of the EU or any of its sub-agencies. The authors acknowledge the contribution to the project of all members of the project consortium. References Boytchev, P.,  Chehlarova, T.,  & Sendova, E. (2007).  Enhancing spatial imagination of young students by acticities in 3D Elica applications. Paper presented at the Spring 2007 conference of the Bulgarian Union of Mathematicians. Boz, N. (2005) Visualization and software environments. Turkish Online Journal of Educational Technology, 4(1), article 3, http://www.tojet.net/articles/413.htm Chávez, Ó., Reys, R. & Jones, D. (2005). Spatial visualization: What happens when you turn it? Mathematics Teaching in the Middle School, 11(4), 190-196. Harel,  G.,  &  Sowder,  L.  (1998).  Student  proof  schemes  result  from  exploratory  studies.  In  A. Schoenfeld, J. Kaput and E. Dubinsky (Eds.), Research in Collegiate Mathematics III (pp. 234-282). American Mathematical Society. Hegarty, M., & Waller, D. (2005). Individual differences in spatial abilities. In P. Shah & A.Miyake (Eds.), The Cambridge Handbook of Visuo-spatial Thinking (pp. 121-169). Cambridge University Press. Jones, K. (2001), Learning Geometrical Concepts using Dynamic Geometry Software. In  K.Irwin (Ed),  Mathematics  Education  Research:  A  catalyst  for  change.  Auckland,  New  Zealand: University of Auckland. National  Council  of  Teachers  of  Mathematics  (2000).  Principles  and  Standards  for  School Mathematics. Reston, Va: NCTM. National  Science  Foundation  (1997).  Report  on  The  National  Science  Foundation  Research Planning Workshop, http://www.princeton.edu/~jjg/nsf_report.html Presmeg, N. (1986) Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42-46. Main theme: Designing and Using Dynamic Mathematics environments 

Enhancing spatial imagination of young students by acticities in 3D Elica applications. Paper presented at the Spring

Individual differences in spatial abilities.

Learning Geometrical Concepts using Dynamic Geometry Software. In K.Irwin (Ed), Mathematics Education Research: A catalyst for change.

Report on The National Science Foundation Research Planning Workshop, http://www.princeton.edu/~jjg/nsf_report.html Presmeg, N.

Student proof schemes result from exploratory studies. In

Visualization and software environments.

Developing an active learning environment for the learning of stereometry

Developing an active learning environment for the learning of stereometry

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