Table of Contents

Cover

Title

Preface

Introduction

I.1. Observations and motivations
I.2. Contributions
I.3. Book outline

1 Construction of Spatial Representation and Perspective in Students

1.1. Spatial representation in children according to Piaget
1.2. The representation of geometric objects: the status of drawings
1.3. From the physical shape to its planar representation
1.4. Benefits of new technologies and dynamic 3D geometry

2 Mobile Devices and 3D Interactions

2.1. Why mobile devices?
2.2. Mobile devices
2.3. Interactions on mobile devices and physiology
2.4. 3D interaction techniques
2.5. “Language” of interactions and classifications

3 Elaboration and Classification of Interactions

3.1. Human-centered design
3.2. Study of the needs and behaviors of users
3.3. Our grammar and interaction language
3.4. Evaluation of the acceptance of interactions (selection, translation and rotation)
3.5. Conclusion and perspectives

4 Evaluation of the Educational Benefits for 3D Geometry

4.1. Partnerships
4.2. Limits
4.3. Evaluation of problem solving aids
4.4. Evaluation of the benefits in learning 3D geometry
4.5. Partial conclusions

Conclusion and Perspectives

C.1. Contributions
C.2. Perspectives

Bibliography

Index

End User License Agreement

List of Tables

1 Construction of Spatial Representation and Perspective in Students

Table 1.1. Synopsis of the development of geometrical representations in children [PIA 48]
Table 1.2. Summary of the development of perspectives in children [PIA 48]
Table 1.3. Summary of the development of surface representations in children. There are two types of representation: a) profile view; b) lateral surface development [PIA 48]
Table 1.4. Summary of the results obtained from the “Nuremburg’s scissors” experiment [PIA 48]
Table 1.5. Summary of the primary curriculums for 3D learning

3 Elaboration and Classification of Interactions

Table 3.1. Analysis and comparison of three 3D geometry programs
Table 3.2. Classification of tactile interactions depending on the number of contacts, their directivity, mobility and duration
Table 3.3. Test protocol for the Year 7s
Table 3.4. Summary of the technology required by each interaction in the context of learning 3D geometry
Table 3.5. Classification of interactions based on their entry system

4 Evaluation of the Educational Benefits for 3D Geometry

Table 4.1. Summary of the experimental protocol
Table 4.2. User feedback on the adaptation of interactions to the exercises and situations suggested on the Likert eight-point scale (median and interquartile)
Table 4.3. Test results: success rate and duration (in seconds) for each condition
Table 4.4. Measurement of the probability of work charge where the highest probability shows a high cognitive load. For a color version of this table, see www.iste.co.uk/bertolo/geometry.zip
Table 4.5. Summary of the experimental protocol put into place with the two classes
Table 4.6. Comparison of the success rates at the pre-test and post-test for each week (Khi², significant values in bold)

List of Illustrations

1 Construction of Spatial Representation and Perspective in Students

Figure 1.1. Solids used by children for playing or in day-to-day life: a) wooden solids; b) Lego; c) daily life
Figure 1.2. For a student, what is represented? Is it a cube or a square and two parallelograms?
Figure 1.3. a) Tadpole people, a classic example of synthetic incapacity; b) a house with an inverted roof, representing the difficulty of the neighboring relationship [LUQ 27]
Figure 1.4. Examples of drawings representing intellectual realism: a) drawing of a duck in its egg; b) drawing without coordination of different points of view [LUQ 27]
Figure 1.5. Piaget’s three mountains. One has a house at its summit, another a cross and the last snow. They are provided in three different colors [PIA 48]. See www.fondationjeanpiaget.ch
Figure 1.6. Affine transformations of lozenges: “Nuremberg’s scissors”[PIA 48]. See www.fondationjeanpiaget.ch
Figure 1.7. Representation of the relationships between physical objects, geometrical objects and sketches [CHA 97]
Figure 1.8. Representation of the relationships between physical and geometrical shapes and sketches in the context of 3D geometry [CHA 97]
Figure 1.9. Diagram showing the geometrical workspace [HOU 06]
Figure 1.10. On the left, a Rubik’s cube in the shape of a cube and on the right a cube in isometric perspective that can be seen as an assembly of three lozenges
Figure 1.11. Two instrumental deconstructions of a square, as suggested by Mithalal [MIT 10]
Figure 1.12. Decomposition into figural units by dimensional deconstruction of a parallelogram: a) from its side (b); its diagonals (c); from a group of outlines (d); [DUV 05]
Figure 1.13. Dimensional hiatus [DUV 05]
Figure 1.14. Representations in perspective used by primary education: a) La Tribu des maths CP (year 1); b) Maths+ CP (year 1); c) Cap Maths CE1 (year 2); d) Cap Maths CE2 (year 2)
Figure 1.15. Different types of models found in primary and secondary school: a) paperboard; b) rigid rods; c) Polydron^©
Figure 1.16. Different types of representations in perspective: a) central perspective; b) cavalier projection; c) isometric perspective; d) spherical perspective
Figure 1.17. Two student productions of the front view from a view in perspective [AUD 87]
Figure 1.18. Diagram showing the organization of the categories relative to KNOWING [COL 93]
Figure 1.19. Procedures carried out from: a) a FACE; b) a BASE [COL 93]
Figure 1.20. Three different dynamic 3D geometry programs: a) Cabri 3D; b) Calque 3D; c) Geospace
Figure 1.21. Interfaces of the two most widely used programs: a) Geospace; b) Cabri-3D
Figure 1.22. Button on the tool bar allowing the choice of certain regular polyhedrons after clicking

2 Mobile Devices and 3D Interactions

Figure 2.1. The first calculation tools in mathematics: a) abacus (photograph by Mike Colshaw, CC BY SA 3.0, source: Wikipedia); b) Chinese abacus; c) classroom abacus
Figure 2.2. a) A pascaline, signed by Pascal in 1652, found in the “arts et métiers” museum in Paris (photograph by David Monniaux CC BY SA 3.0, source Wikipedia); b) an educational version of the pascaline called “Zéro+1”
Figure 2.3. Evolution of calculators from the ICC 804D made by Sanyo in 1971 to the TI-Nspire of nowadays: a) Sanyo, b) HP-35, c) Casio fx92, d) TI-89, shows that calculators, such as the TI-NSpire CX CAS, have truly become small laptops. (Teclasorg – Flickr. CC BY SA 2.0, source: Wikipedia)
Figure 2.4. Two iPhone programs of the “graphing-tracing” type for use in studying functions [TRO 10]. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 2.5. Different uses of tablets and smartphones of 50 Year 6-7 students in 2014
Figure 2.6. Evolution of touchscreen tablets between 1989 and nowadays: a) Newton MessagePad (photograph by Rama, CC BY SA 2.0, source: Wikipedia); b) Lenovo Yoga 3 (photograph by Maurizio Pesce, CC BY 2.0, source: Wikipedia); c) iPad Air
Figure 2.7. Different types of mobile devices¹
Figure 2.8. Different MD keyboards: a) a BlackBerry’s mechanical keyboard; b) keyboard program of the iPad; c) separated keyboard in the iPad
Figure 2.9. Different types of joysticks on mobile phones and tablets: a) joystick on a mobile phone; b) virtual joystick; c) physical and adaptable joystick for tablet or smartphone
Figure 2.10. First tactile system, the saqueboute from 1948: a) left hand commands; b) device controlled by the index finger with the pad represented by the dotted line
Figure 2.11. Different tactile systems since 1991: a) DiamondTouch (photograph by MERL, CC BY SA 3.0, source: Wikipedia); b) Microsoft Surface (photograph by Ergonomidesign, CC BY SA 3.0, source: Wikipedia); c) Apple iPhone
Figure 2.12. Description of capacitive technology [MES 01]
Figure 2.13. Description of resistive technology [MES 01]
Figure 2.14. Description of infrared technology [MES 01]
Figure 2.15. Description of surface wave technology [MES 01]
Figure 2.16. Our choice: iPad tablets due to their interactive possibilities
Figure 2.17. Evolution of PC and tablet sales in France between 2010 and 2014 (Source GFK – via ZDNet.fr/chiffres-cles). For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 2.18. Two techniques to mitigate the finger’s lack of precision: a) Shift; b) TapTap
Figure 2.19. Asphalt 6 on iPhone using the accelerometer to steer the car [DEC 09]
Figure 2.20. Illustration of the limits to hand movement, application limits: a) in flexion/extension; b) pronation/supination; c) ulnar/radial deviations [RAH 09]
Figure 2.21. Different references used in 3D programs (source image, [DEC 09])
Figure 2.22. Euler’s angles: the xyz system (fixed) is shown in blue, the XYZ system (rotation) in red. The knotted line, marked N, appears in green (source Wikipedia). For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 2.23. Illustration of different mathematical 3D transformations: a) translation, b) rotation; c) homothety
Figure 2.24. Widgets used in Blender© to apply elementary transformations to objects: a) translation; b) rotation; c) scaling. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 2.25. Skitters and Jacks [BIE 86]: a) movement of a skitter along an axis and jack placement; b) positioning of copies of the solids on the jacks
Figure 2.26. Nielson and Olsen et al. [NIE 86] manipulation technique: a) partition of the representation plane including the widget into six zones; b) use of characteristic elements of the solid to restrict or define transformations. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 2.27. Transitory widget [SCH 08]: a) invocation of the widget by moving a finger on the screen; b) apparition of the widget with an orientation corresponding to that of the finger movement on the screen; c) orientation of the crown in order to have a larger contact zone; d) disappearance of the translation arrow when the operation is impossible. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 2.28. Cohé et al’s tBox. [COH 11]: a) presentation of the tBox and the Maya widget; b) translation technique; c) rotation technique; d) scaling technique. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 2.29. Shallow Depth 3D interaction [HAN 07], translation and rotation around the Z-axis using two fingers. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 2.30. 3D Screen-Space RST technique from Reisman et al. [REI 09]: a) ambiguity on the axis of rotation when the three fingers form an equilateral triangle; b) switching perspectives using four fingers
Figure 2.31. 3D positioning techniques from [MAR 10]: a) Z-technique; b) extension of the four view system. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 2.32. Different uses of two fingers on the same hand from Kin et al.’s [KIN 11] language of movements: a) one contact; b) two contacts with two fingers; c) a double joint contact using two fingers
Figure 2.33. All of Kin et al.’s [KIN 11] movements
Figure 2.34. Four movements managing the 6 DOF in Liu et al’s technique: a) translation in X and Y; b) translation in Z; c) rotation around the Z-axis; d) rotation around other axes in the XY plane. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 2.35. a) Accelerometer measures acceleration along the X-, Y- and Z-axes; b) gyroscope measures the angular speed around the X-, Y- and Z-axes
Figure 2.36. i3D [FRA 11]: a) face-tracking: b) example use of the i3D software
Figure 2.37. Group of movements suggested by Wobbrock et al. after having studied 1080 propositions from 20 people for 27 commands [WOB 09]
Figure 2.38. Examples of atomic gestures in the grammar developed by [KAM 10]
Figure 2.39. Example of a complex gesture as described by the GeFormMT syntax (right) and the resulting movement (left). (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 2.40. Classification of Roudaut and [LEC 07] of entry and exit interaction techniques on mobile devices
Figure 2.41. Classification of gestual interactions using sensors from [BAG 09]
Figure 2.42. Taxonomy from [MAR10] extended by [LIU 12] based on the number and type of contact (direct: d, indirect: i) then on the fact that the contact is in movement (m) or fixed (f)

3 Elaboration and Classification of Interactions

Figure 3.1. Interfaces of the two programs: a) Cabrl-3D; b) Calque 3D
Figure 3.2. Calque 3D with the two boxes: on the left, the 3D rendering box and on the right the creation window
Figure 3.3. Representation of the nets: a) with Geospace; b) with Cabri-3D.
Figure 3.4. Distribution of the use of 3D geometry software by teachers
Figure 3.5. Initial classification based on functionalities suggested by the users and our state of the art
Figure 3.6. Second classification based on interactions and the number of contacts and directivity
Figure 3.7. Different solids in FINGERS
Figure 3.8. Cyclical selection of a solid: a) state 1, solid represented with edges and vertices; b) state 2, solid as before with the object’s reference; c) state 3, wire representation with only edges and vertices shown. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.9. Axis selection: a) state 2, solid with edges, vertices and reference shown, with one selected axis; b) definition of an axis of rotation by selection of two vertices. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.10. Selection/deselection: a) selection of a solid; b) selection of an axis; c) deselection of a solid. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.11. Translations with 2 DoF: a) direct translation; b) indirect translation. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.12. Principle of a spiral: a) screwing; b) unscrewing
Figure 3.13. The spring: a) movement along X and –Z; b) movement along –X and Z, c) movement along 3 DoF, –X, –Y and –Z; d) movement along Y and Z
Figure 3.14. Implementation and recognition of the spiral using two different techniques
Figure 3.15. Translation with 3 DoF: one-handed, two-fingered gesture. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.16. Different types of rotations possible with FINGERS: a) rotation with 3 DoF in the reference of the screen; b) rotation in the reference around an axis defined by two vertices of the solid; c) rotation around an axis in the object’s reference. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.17. Rotations: a) rotation around the Z-axis in the screen’s reference; b) rotation around the red axes of the screen; c) rotation around an axis of the object’s reference. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.18. Exchange between translation and rotation in the case of an un-eliminated ambiguity. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.19. Nets: a) opening; b) decomposition of the solid into its faces to recreate a net. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.20. Modification of the net of a solid: a) basic net generated by the software; b) net obtained through modification of the placement of the faces; c) verification of the validity of a net, with the bicolored face indicating two superimposed faces, indicating its invalidity. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.21. Illustration of the validation algorithm after face movement. In this example, three iterations are needed to validate. The arrows of the same color indicated one same iteration. The dotted lines indicate paths that were not taken into account due to the fact that they lead to faces that were already taken into account. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.22. Illustration of the folding edge determination algorithm. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.23. Solid duplication: a) continuous duplication; b) duplication by teleportation. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.24. Assembly of two solids. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.25. The use of Euler’s angles to align the orientation of two solids (source Wikipedia). For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.26. Managing the observer around the scene as well as point of view using the gyroscope and metaphor of the video camera. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 3.27. Gesture that activates the gyroscope to change the point of view of the observer. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.28. Creation of a solid: a) contacts on a tangible solid taken into account through pattern detection; b) followed by the creation of the solid
Figure 3.29. Deletion of a solid: a) by removing it from the screen; b) by picking it up. (CC BY SA 2.0, Gestureworks. See http://gestureworks.com)
Figure 3.30. Plan of experiments in the acceptability and usability of the interactions
Figure 3.31. Task no. 2: finding cubes with a red face and their position
Figure 3.32. Average and standard deviation of the time (in sec.) required by the participants to carry out the task
Figure 3.33. Percentage of successful attempts at the “finding the red faces” task for all participants
Figure 3.34. Feedback on the interactions by users (FR: rotation with 3 DoF, OR: rotation around an axis of the object’s reference, WR: rotation around an axis defined by two vertices, CPV: change in the point of view using the gyroscope): a) ease of use; b) intuitiveness; c) register of interactions
Figure 3.35. Experimental programs: a) training program; b) test program
Figure 3.36. Median number of cubes with a single red face found by the students depending on the type of interaction used
Figure 3.37. Evolution of the average time (in minutes and seconds) taken to carry out each test
Figure 3.38. Student’s strategies for using interactions: a) face-tracking; b) gyroscope
Figure 3.39. Classification of the interactions carried out by the students. In dark gray: the number of students who ranked an interaction in first place. In light gray: the weighted score obtained by taking into account the interactions that were ranked second by students
Figure 3.40. Comparison of acquisition of the different interactions (FINGERS) by students between the first learning and the last sessions as percentages

4 Evaluation of the Educational Benefits for 3D Geometry

Figure 4.1. Educational teaching plan
Figure 4.2. Example of a series of four exercises given to students during their evaluation
Figure 4.3. Material used in the experiments: a) exercise 1; b) exercise 2; c) exercise 3; d) exercise 4
Figure 4.4. Success rate in percentages for each group and session
Figure 4.5. Evaluation by the students of the use of their tools on a scale of 0 to 7 (median)
Figure 4.6. Different types of cognitive load according to [SWE 05] and the factors linked to them
Figure 4.7. Rey-Osterrieth complex figure
Figure 4.8. Teaching materials used: polydrons. (Photograph by Nevit Dilmen, CC BY SA 3.0)
Figure 4.9. Example of the obtained data using the ABM system: a) the test system as installed; b) monitoring screen; c) EEG recordings. For a color version of this figure, see www.iste.co.uk/bertolo/geometry.zip
Figure 4.10. Problem solving under three different conditions
Figure 4.11. Materials used during the experiments with both classes: a) and c) are screenshots from FINGERS; b) wooden blocks; d) polydrons. For a color version of this table, see www.iste.co.uk/bertolo/geometry.zip
Figure 4.12. Results of both pre and post-tests based on the tool used as well as the percentage between the two during week 1
Figure 4.13. Results of both pre- and post-tests based on the tool used as well as the percentage between the two during week 2
Figure 4.14. Results of both pre- and post-tests based on the tool used as well as the percentage between the two during week 3
Figure 4.15. Median of the validated items (out of 22) of the final test for each group
Figure 4.16. User feedback on FINGERS after a week’s use for each group

First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of David Bertolo to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016941694

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A CIP record for this book is available from the British Library

ISBN 978-1-84821-926-7

Preface

Multipoint digital terminals have grown largely in popularity over the past few years. An increasing number of schools are experimenting with the introduction of digital tablets in their classrooms in a hope that the educational experience will benefit. However, stores dedicated to these new devices have developed almost no programs for the learning of 3D geometry throughout primary and secondary schools. The main obstacle for any application of this type is the ability to manipulate three dimensions from a two-dimensional device, and hence, young students learning about spatial structures are often unable to do so with classic desktop programs. In this book, we will focus on use of the new technologies supported by digital tablets. We have several goals: allowing 9- to 15-year-old students to manipulate, observe and modify 3D spaces, as well as measuring the educational contributions of an approach that is not technology based but rather anthropo-centered.

By taking a user-centered approach, we will first suggest an interactional grammar, adapted to young learners. We will then evaluate the accessibility and the ease of both use and learning of our interactions. Finally, we will study the in situ educational benefits of the use of digital tablets equipped with a program based on the previously developed grammar.

We will note that by using a collection of adapted interactions, students will manipulate, observe and modify 3D spaces intuitively. Furthermore, the use of such programs during spatial geometry learning has shown to be of significant benefit to Year 5s, particularly in terms of linking perspectives and the investigation of patterns.

This book will first and foremost take a two-pronged approach, of both human–machine interactions and educational science, and then suggest a grammar and an implementable language for interactions on multi-touch digital tablets for all 3D geometry applications. All these are aimed at 9- to 15-year-old students.

I will also take an advantage of this foreword to acknowledge all the people thanks to whom the writing and the publication of this book have been made possible.

David BERTOLO

May 2016

Introduction

I.1. Observations and motivations

Although multipoint touchscreens have existed since the 1980s, they have only become popular over the last few years. The rapid development of smartphones and among others, in 2007 the launch of the iPhone, contributed to this rise in popularity. Following this, the recent trend for touchscreen tablets has increased, with several reports confirming the high penetration coefficient of these devices and their use in households [ARC 13, DEL 13]. These new mobile devices offer further opportunities for interaction through the integration increasing complimentary technology. Furthermore, it is now usual for smartphones and tablets, already equipped with multi-touch screens, to also be fitted with cameras and sensors of all types (accelerometer, gyroscope, compass, etc.). With the view of increasing the efficiency of their teaching, educational institutions have quickly begun integrating these technologies into their classrooms. Numerous experiments are taking place in many countries where tablets have been introduced into schools: in France, for example, in the department of Correze, all children entering secondary school have been equipped with an iPad. On a more general level, in 2013, Apple had already sold over 8 million iPads directly to institutions dealing with education [ETH 13]. Experience, however, shows that the advancement of learning has never been solely techno-centered. Nonetheless, the new interactive possibilities made possible by these tablets make possible new learning opportunities for certain concepts that may be able to make the most of such technologies. Among these, the one which seems to show the most promise in linking to these new interactions is without a doubt that of 3D geometry. For example, it is difficult for young students to establish the link between 3D solids and their planar representations. In parallel, many studies are focused on the manipulation of 3D spaces from 2D devices, particularly those that are also multi-touch. This observation guides us to fully investigate this lead.

I.2. Contributions

In this context, we will show that the existing interactive 3D geometry programs are subject to limitations, particularly for 9- to 15-year-old young students. Throughout this book, our main contribution will be to show that by using a group of adapted interplays, we are able to overcome these limitations and create an ongoing link between real objects and their 3D representations for students learning about spatial structure. For this we will suggest, from a human-centered design/approach, a formal grammar as well as an interactive language adapted to the investigated theme and our target public.

Next, we will present those interplays that have been developed and show the complementarity of different installed technologies on current tablets throughout the development of an app based on our developed grammar and language.

Finally, we will show through in situ evaluations that when this technology is introduced into a classroom, it has educational benefits in the learning of 3D geometry.

I.3. Book outline

This book is divided into 4 chapters and finishes with conclusions:

– Chapter 1 will describe the state of the art in learning 3D geometry. In order to conceptualize the interactions that facilitate the learning of 3D geometry, it is essential to know and understand the main elements of its teaching. This chapter will focus on the different stages of structuring the 3D space as well as the difficulties faced by the students during the learning process.
– Chapter 2 will cover the state of the art in interactions on digital terminals. After describing the use of mobile devices in our context, we will cover a brief material history before describing the group of interplays now made possible on these devices.
– Chapter 3 will first present the principle of the user-centered approach that has been the guiding thread throughout the studies described in this book. It will then describe the formal grammar that we have developed as well as the following interactive language. Finally, we will hear some user experiences to evaluate the acceptation and the ease of assimilation of our program.
– Chapter 4 will present evaluations relative to the learning of 3D geometry and show the benefits of the prototype developed based on our grammar and interactive language. As such, we will link to the first chapter by showing that in situations of discovery and investigation, we facilitate the link between real objects and their planar representations.
– Finally, the Conclusion will present our conclusions as well as the upcoming prospects for the future of our research.