Interactions on Digital Tablets in the Context of 3D Geometry Learning

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.

1

Construction of Spatial Representation and Perspective in Students

From a young age, children play with and manipulate solid objects such as cubes and blocks. Whether these are the first wooden blocks used by babies or later the bricks of all shapes used in construction games such as Lego (see Figures 1.1(a) and (b)), children observe, manipulate and use solids. They come across these objects in their day-to-day life (see Figure 1.1(c)). We could, therefore, expect that these students would be entirely familiar with the main characteristics of solids, given that they are also studied in primary and secondary education. However, in national French evaluations of Year 6 in 2011, 40% of students were unable to correctly describe a cuboid (number of faces and edges) from its representation in oblique perspective (see Figure 1.2). Similarly, nearly 50% of students were unable to complete this same task for a prism (source: French National Education).

Figure 1.1. Solids used by children for playing or in day-to-day life: a) wooden solids; b) Lego; c) daily life

Numerous avenues have been studied with the aim of facilitating the learning of 3D geometry - for example, the manipulation of solids in real life, as well as the use of interactive geometry programs, or more recently, the use of digital touchscreen tablets equipped with multi-touch interfaces and various sensors. An increasing number of academies in partnership with political institutions (regional, departmental, etc.) are putting experiments in place within schools, with the aim of evaluating any potential educational benefit of these new tools. These new devices have now made new interplays possible and bring new possibilities of visualization and manipulation. However, for these avenues to benefit the teaching of this complex concept, it is necessary to understand the reasons behind these difficulties in order to find ways within them to facilitate teaching. Human-computer interactions (HCI) are centered on humans, and it would be unrealistic to think it is possible to design devices to help students without understanding their obstacles beforehand and the reasons behind the obstacles.

Figure 1.2. For a student, what is represented? Is it a cube or a square and two parallelograms?

In order to cover difficulties in teaching 3D geometry, we must quote Mithalal [MIT 10]:

3D geometry is one of the more delicate subjects within the teaching of mathematics; such is the difficulty of seeing 3D": both for students, who can no longer use drawings to underpin their reasoning, as well as for teachers who lose the illustrative purpose of drawings. Whichever the perspective, it becomes a question of visualization."

In this chapter, we will see the causes of these problems from both a didactic and an educational viewpoint, in order to suggest relevant interactions for our audience – students aged 9 to 15 years. The didactic elements will allow us to justify the chosen age range, among other things. First, we describe spatial representation to the child according to Piaget [PIA 48]. Then, we study the representation of geometrical objects, particularly the status of drawing. From these inputs, we articulate a theory on the transition of the physical 3D object to its planar representation and conclude by describing new technologies in the learning of 3D geometry.

1.1. Spatial representation in children according to Piaget

1.1.1. From perception to representation

According to Piaget [PIA 47, PIA 48]:

The biggest difficulty in psychogenetic analysis of space concerns the fact that the progressive construction of spatial markers takes place on two distinct planes: the perceptive or sensorial-motor plane, and the representative or intellectual plane.

From birth, children construct their own sensorial-motor space that evolves alongside the development of their perception and motility. Piaget sets this period between birth and the age of two. After this, from the development of language, imagery and intuitive thinking, children progressively enter the representative space between two and seven years of age.

Piaget and Inhelder’s studies have put forth three stages of development of spatial representation in children, which are based on the stages of infant drawing put forward by Luquet [LUQ 27]:

– synthetic incapacity between 3 and 4 years of age, at which time drawings do not correspond to perception;

– intellectual realism between 4 and 8 years of age, at which time children are capable of spatial analysis through observation. Spatial relationships become coordinated and projection relationships emerge;

– visual realism from 8-9 years onward, when children start to use perspective.

1.1.1.1. Stage I: synthetic incapacity

At the synthetic incapacity stage, spatial representation in children is characterized by the fact that Euclidean and projective markers are neglected. Children also do not take distances or perspectives into consideration. They begin constructing topographical reports, without necessarily mastering cases such as the drawing of a person. Among these, the following are some significant relationships:

– of proximity (against, close, far, etc.): this is respected in general but not in the detail of the drawings. A good example of this is the tadpole man studied by Luquet (see Figure 1.3);

– of separation: children have difficulties in separating elements from one another, such as the edges of a quadrangle;

– of order: this only begins at this stage, allowing the better determination of the relative position in a couple. Reversals such as mouth, nose and eye types are noted (see Figure 1.3(a));

– circling or neighboring (in, inside, on, under, etc.): at this stage, children draw things such as eyes outside of the head or a roof pointing inside the house (see Figure 1.3(b));

– of continuity and discontinuity: at this stage, children are content juxtaposing elements without taking into account continuous relations, such as a hat drawn above the head of a person.

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]

Therefore, during the stage of synthetic incapacity, topographical relationships appear without necessarily becoming generalized when it comes to complex shapes such as those children prefer to draw: people, animals, houses, etc. At this stage, the graphical space is lacking the relationships of distance and proportion; however, above those is missing directional relationships in three dimensions, and therefore, all perspective logic is missing.

1.1.1.2. Stage II: intellectual realism

Following from the synthetic incapacity, children enter a new stage named intellectual realism. Here, children do not draw what they see of the object but rather everything ‘in it’ [LUQ 27]. Intellectual realism is a method of representation that is marked by the acquisition of topographical relationships seen during the previous stage. We may note the importance taken by neighboring relationships that are often used to mark transparence, for example, in the drawing of a duck in its egg (see Figure 1.4(a)).

Projection and Euclidean relationships are only beginning here and are used incoherently and without the coordination of points of view. When these enter into opposition with the topographical relationships, the latter will win out in the representation (see Figure 1.4(b)). Intellectual realism is also marked by the appearance of simple geometrical shapes, even if the lengths and distances are not always accurate. There is, however, still no Euclidean structuring of spaces.

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]

1.1.1.3. Stage III: visual realism

Visual realism appears, on average, around the age of 8-9 years. This stage is marked not only by the respect of topographical relationships already developed during the intellectual realism stage, but also by the care of respecting perspectives, proportions and lengths in the drawings.

The stage of visual realism highlights three main points:

– that the representation of projection and Euclidean relationships appear after their perception by the child;

– projective relationships do not precede Euclidean ones, nor vice versa, but they are developed simultaneously and nourishing each other;

– children go from step by step constructions induced by the principal use of topographical relationships to constructions of whole bodies, linked to the very nature of projective and Euclidean relationships that conserve positions and distances between figures.

In order to verify their hypotheses and to define the different stages in spatial representation, Piaget and Inhelder developed an experiment that consisted of reproducing the whole or part of a group of 21 models. Table 1.1 is a synopsis of the results and some examples obtained from it.

Table 1.1. Synopsis of the development of geometrical representations in children [PIA 48]

1.1.2. Projective space

According to Piaget, the discovery of the straight line is the simplest manifestation of research into an organization of relationships between objects given projection and coordinate relationships. Indeed, representations of straight lines require the introduction of the understanding that points of the line are hidden from each other (and therefore a certain perspective), or the introduction to lengths and movements along straight lines. Furthermore, even if perceptive recognition of lines can develop early, its representation will not develop until later on. Let us also note that although a line remains a line even when the point of view is changed (the perspective system), for other geometrical shapes such as the circle this is not the case. The experience of constructing a straight line represented by the method of Piaget’s aims has shown this discrepancy between recognition of perspective and its representation, linked to the transition from perceptive to representative