E-books plus: role of interactive visuals in exploration of mathematical information and e-learning.

Sedig, Kamran

E-books promise to become a widespread delivery mechanism for educational resources. However, current e-books do not take full advantage of the power of computing tools. In particular, interaction with the content is often reduced to navigation through the information. This article investigates how adding interactive visuals to an e-book influences interaction with the information and learning. Two different versions of a mathematics e-book were designed containing information about Platonic and Archimedean solids: a hypertextual version and a version augmented with interactive visuals. A study of how learners explore mathematical information using these two different versions of the e-book is presented. The findings of this study show that the addition of interactive visuals to an e-book can increase learning by supporting diverse sense-making activities. Based on the findings of this study, we suggest five approaches for adding interactive visuals to e-books.

INTRODUCTION AND BACKGROUND

This article presents a study of how learners explore mathematical information in the context of two different versions of an e-book. Taking into consideration the learners' exploration and usage patterns, the article makes suggestions on how the addition of certain design features can improve the effectiveness of mathematics e-books, and e-learning tools in general.

Electronic books (e-books) are books in digital form (Rao, 2003). Exploiting the power of computing technology, e-books offer a number of advantages over traditional books. Some of these advantages include: embedded hyperlinks, connecting related information and paragraphs (Wilson, Landoni & Gibb, 2002); increased accessibility, allowing multiple readers, anytime and anywhere (Long, 2003; Rao, 2003); and extended utilities, including dictionaries, search, audio, and annotation tools (Hodas et al., 2001; Rao, 2003). Nevertheless, e-books are not yet widely used. The reasons why e-books are not universally adopted include: lack of a standard format for e-books, as the majority of existing e-books are hardware dependent (Bry & Kraus, 2002; Lee, Guttenberg & McCrary, 2002); need for a distribution and library model that provides wide-spread access while enforcing copyright laws (Connaway, 2001); and limited availability of e-book titles (Rao, 2003). In spite of the small number of current e-book readers, many researchers expect that the pervasiveness of the internet as a delivery mechanism and the evolution and improvement of e-books, will soon result in the wide-spread usage and popularization of e-books in personal, commercial and educational settings (Connaway, 2001; Long, 2003; Rao, 2003).

E-books have already generated some interest in educational settings. With the increasing popularity of e-learning and long distance education, it has become paramount for students to be able to remotely access learning material (Connaway, 2001). Publishers and instructors have increasingly been producing e-textbooks and other electronic course materials for various disciplines, such as chemistry, biology, physics, computer science, history and mathematics (e.g., see Crescenzi & Innocenti, 2003; Dyllick, 1997; Farabee, 1997-2002; National Science Teacher Association, 1998; Simon, 2001; Wachsmuth, 1994-2000; Weber & Specht 1997; Wilson, Landoni & Gibb, 2002). However, there has been some criticism of current educational e-books (Alessi & Trollip, 2001; Bottcher, 1998; Lison, Gunther, Ogurol, Pretschner & Wischnesky, 2004; Mari et al., 2002; Principe, Euliano & Lefebvre, 2000). Critics point out that current e-books adhere too strictly to the traditional book metaphor; as such, current e-books do not fully take advantage of the interactive possibilities afforded by computational technologies. Mari et al. (2002), for instance, state that, instead of using e-books in their electronic form, students often just print the books and use them offline, "thus losing the peculiar advantages that derive from interaction with the computer" (p. 1). Most current e-books and e-learning systems are rich in textual content, supplemented by alternative representations in the form of static visuals, sound, and video (Mari et al., 2002; Martinez-Unanue, Peredes-Velasco, Pareja-Flores, Urquiza-Fuentes & Velazquez-Iturbide, 2002; Vrasidas, 2004). Interacting with the information in these tools is often limited to either clicking on links to move through the information space, clicking on non-textual information to listen to or view explanations of concepts and events, or answering self-testing questions at the end of a chapter (Mari et al., 2002). Apart from navigating the information, often times what is lacking for learners is the possibility of rich interaction and active engagement with the information (Vrasidas, 2004).

A few researchers have started adding interactive features to e-books. Lison et al. (2004), for instance, introduce the e-book "Vision2003" for medical students. Vision2003 includes simulations of joint mechanics. Students, for example, can interact with graphical representations of bones to investigate the range of movement in healthy and fractured joints. Principle et al. (2000) have designed an e-book for teaching adaptive and neural systems to electrical and computer engineering students. In addition to textual information, the e-book contains simulation modules, allowing students to alter parameters and view the results. However, neither of the above systems has been empirically evaluated to find out how the addition of interaction and simulation enhances the e-books.

Currently, e-books can take full advantage of the power of computational tools to incorporate rich interaction with all kinds of visuo-textual information. The purpose of our research is to design more effective e-books. Specifically, we report a study in which a text-rich e-book has been augmented with highly interactive visual representations to investigate whether learners' understanding of the information improves. We also discuss how the addition of these interactive visual representations influences the way learners explore and reason about the information. Finally, we consider the implications that the learners' exploration and reasoning patterns would have for the design of more effective e-books.

Since the focus of this article is how learners explore interactive visuo-textual information, we will not discuss all possible design elements of e-books. Rather, in this section, we provide some conceptual and terminological background with regard to representing and interacting with information.

Representing Information

Information can be encoded and displayed using different representations. Generally, research distinguishes between depictive and descriptive representations (Schnotz, 2002). Depictive representations make use of icons or signs whose meanings are supposed to be conveyed through their visual form. Descriptive representations make use of notational symbols whose meanings are supposed to be conveyed through conventions and rules (Tversky & Lee, 1999). Examples of depictive representations include diagrams, maps, and graphs, and examples of descriptive representations include text and mathematical equations.

Depictions and descriptions are both external representations. Learners build internal, mental models of the information through interaction with external representations, for example, by reading a piece of text or studying a map (Scaife & Rogers, 1996). Good external representations act as external aids to support memory, thought, and reasoning (Norman, 1993). In the context of the design of e-books for education, an important question is: How should information be represented to facilitate the formation of internal, mental models by learners?

Depictive and descriptive representations are two different forms of representing information (Peterson, 1996). The same information can be encoded using both depictive and descriptive representations. Two representations are informationally equivalent if all the information that can be decoded and inferred from one can also be inferred from the other (Larkin & Simon, 1987; Peterson, 1996). Two informationally equivalent representations, however, may be psychologically or cognitively non-equivalent. This happens when learners may need to spend different amounts of cognitive effort to make sense of them or may have different feelings towards them. Two general advantages of depictive representations include: 1) they are more effective than descriptive ones for communicating complex content, and 2) they are cognitively less demanding for learners to process than descriptive representations (Robinson & Kiewra, 1995; Vekiri, 2002). For example, tally marks are depictive and Arabic numerals are descriptive representations of numeric information. The absolute value of a tally mark (e.g., III) is visual and can be processed perceptually, whereas the absolute value of an Arabic numeral (e.g., "3") is symbolic and needs to be retrieved from memory (Zhang & Norman, 1995). However, it is important to note that the suitability of representations is determined by the task for which they are used and the instructional objectives of that task. For instance, whereas graphs, because they are depictive, are more suitable for performing comparisons, numbers, because they are descriptive, are more suitable for performing computations and numeric operations (Norman, 1993). Hence, the choice of which representations to use depends on the task at hand.

Complementary, multiple, alternative representations can be used to allow for individual differences in learning as well as to support different reasoning and exploration strategies (Ainsworth, 1999; Ainsworth & Van Labeke, 2002). Depictive and descriptive representations can be used to complement one another and help direct learner attention to different aspects of the represented material (Myers & Konolige, 1995; Peterson, 1996). However, there does not exist sufficient research to prescribe how to integrate multiple representations into a single learning environment (Ainsworth, 1999; Brna, Cox & Good, 2001; Cheng, Lowe & Scaife, 2001). The research reported in this paper investigates how descriptive, textual information and depictive, visual information can be integrated in the context of a mathematics e-book to support learners' reasoning and exploration strategies.

Interacting With Information

As people think and reason about external representations, they form internal mental models of the structure, organization, and relationship of the elements in the representations (Jonassen, Beissner & Yacci, 1993; Markman, 1999; Sternberg, 1999). However, interpreting and making sense of information is not just through thinking and reasoning with static representations of the information on the screen. Interaction with external representation, via the human-computer interface, allows learners to act upon it, and get feedback from the representations, interpret and evaluate the feedback, and hence build their own mental models and understanding of the represented information. As such, interaction mediates between the information and the mind, allowing learners to become actively involved in the learning process. Interaction can enhance the communicative power of information by allowing learners to make different features of the information explicit and accessible when needed (Otero, Rogers & du Boulay, 2001; Sedig & Sumner, in press; Strothotte, 1998). Other general benefits of interaction include: supporting emergent interpretation and understanding of encoded ideas; providing opportunities for experimentation, discovery, and hypothetical reasoning; facilitating acquisition of qualitative insight into the nature of representations; and, serving as a coordinator between the internal mental models of learners and the external representations.

Just as alternative representations support alternative thinking styles, different interaction techniques allow learners to engage in different forms of reasoning (Sedig, Rowhani, Morey & Liang, 2003). In the context of e-books, this implies that learners need access to more interaction techniques than merely navigation tools. This is especially true if an e-book contains visual representations of information. Whereas text, static images, video and sound can be passively read, watched or heard, interaction with visual representations can challenge learners to become active in the learning process and increase motivation, attention, and involvement (Vrasidas, 2004).

The remainder of this paper comprises of three sections: Research Methodology, Findings, and Summary and Conclusions.

RESEARCH METHODOLOGY

The methodology used to address the goals of this research are as follows. First, two versions of a mathematics e-book were designed: a typical e-book and an augmented e-book. Second, an experimental study was designed and conducted in which two groups used the two versions of the e-book. Third, the two versions were compared in terms of 1) their educational effectiveness and 2) the learners' patterns of exploration of the information.

E-Book Used for the Study

Two different versions of a mathematics e-book, containing information about 3-dimensional geometric solids, were designed. The purpose of the e-book is to allow learners to explore information about Platonic and Archimedean solids. The first prototype mimics existing e-books that just convert a paper book to an electronic version by adding hyperlinks and minimal interaction. The second prototype augments the first prototype by adding interactive visuals to it. Before describing the two versions, a brief introduction is provided with regard to Platonic and Archimedean solids.

Platonic and Archimedean Solids

Platonic solids are 3-dimensional shapes that look identical from every vertex and are composed of only one type of regular polygon (1). Examples of Platonic solids are the cube and the tetrahedron. Archimedean solids also have identical vertices; however they are composed of two or more types of regular polygons. Platonic and Archimedean solids are closely related to each other. An Archimedean solid can be derived from a Platonic solid by simultaneous truncation of all the vertices of the Platonic solid. For example, truncating the vertices of a cube results in a truncated cube (see Figure 1 from left to right). The reverse process of truncation is called augmentation (read Figure 1 from right to left). Platonic and Archimedean solids provide a suitable testbed for this study, as the traditional way to represent these solids and their relationships is through textual descriptions and static images.

[FIGURE 1 OMITTED]

Version 1: PASE

The first version, Platonic and Archimedean Solids E-book (PASE), resembles a traditional print-based book with interactive, hypertextual features, as in typical e-books. It contains descriptive information that is entirely based on Wenninger's standard textbook, "Polyhedron Models" (1971). PASE is text-rich. But it also contains visual figures and sound, as is customary in other e-books. The presentation of the information regarding Platonic and Archimedean solids is mostly in terms of textual descriptions. In addition to the information in Wenningers' book, hypertextual links have been embedded in the information to enable learners to move through the information space.

Figure 2 shows a screen capture of the first page of PASE. The left panel contains an index of the Platonic and Archimedean solids in alphabetical order. Clicking on one of the names results in the selection of the corresponding solid. A textual description and visual representation of the selected solid are then displayed in the middle and right panels, respectively. The middle panel consists of a textual description of the selected solid and its composition. (2) The middle panel also contains a list of links to information about other solids that can be derived from the selected solid. Learners can interact with the visual representation in the right panel in a minimal form; that is, they can rotate the solid to view it from different angles and viewpoints. The button with the loudspeaker icon in the upper left hand corner allows learners to hear the name of the currently selected solid.

[FIGURE 2 OMITTED]

Version 2: PASE+

The second version, PASE+, is comprised of two modules: the PASE module (exactly the same as the PASE version of the e-book) plus an interactive visuals module (IVM). IVM contains interactive visual representations of the information. Upon starting PASE+, learners are presented with the first page of PASE, as seen in Figure 2. Learners can easily switch between the PASE module and IVM by clicking on tab buttons. IVM allows rich interaction with the visual information by means of providing learners with three toolkits: interactive diagrammatic maps, interactive solids, and query tools.

The first toolkit in IVM consists of 5 diagrammatic maps depicting relationships among related solids. The diagrammatic maps visually encode the transformational relationships of the solids. (3) Figure 3 shows an interactive diagrammatic map of the solids related to the cube. Rather than expressing the relationships textually as in PASE (e.g., "by truncating the vertices of the dodecahedron, one obtains a truncated dodecahedron"), these diagrammatic maps organize the relationships visually by laying out the images of the solids in maps. By clicking on a transitional link (i.e., arrow between two solids), learners can watch a simulation of how one solid gradually transforms into another.

[FIGURE 3 OMITTED]

The second toolkit in IVM consists of 16 interactive solids. Figure 4 shows a screen capture of this toolkit. Using this toolkit, learners can interact with a solid to morph it from one solid to another. Learners perform this type of transformation by dragging an interactive vertex (i.e., the dot in Figure 4) along a truncation path on a selected solid (see Figure 4). (4) This direct manipulation of the solids allows learners to control the speed of transformation and view details of the truncation and augmentation processes. Learners can also rotate each interactive solid to view it from different angles and viewpoints.

The third toolkit in IVM consists of two query tools: range sliders and composition checkboxes. Learners interact with the sliders and checkboxes to specify query values with respect to the number of vertices, edges, and faces of the solids, as well as their compositional makeup (see Figure 5). Only those solids that satisfy the specified queries are displayed.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Experimental Study

Design

A multi-method (quantitative and qualitative) research design was used, including a number of types of data-collection instruments (achievement results, video transcripts, interviews, and direct observations). The multi-method design was used to help triangulate and cross-validate the different types of findings.

Subjects

Twenty-four undergraduate students from computer science and engineering participated in the study. An e-mail message was sent to under-graduate students in these departments inviting them to participate in the study. The first twenty-four who responded were selected to participate in the study. None of the subjects had previously seen or used the e-book prototypes.

Procedure

Subjects were randomly combined into twelve pairs. These pairs were again randomly distributed into two groups, the treatment group and the comparison group. Similar to other experimental research in educational settings (Schumacher & McMillan, 1993), PASE+ was assigned to the treatment group ([G.sub.PASE+]) and PASE was assigned to the comparison or control group ([G.sub.PASE]). All subjects completed a multiple-choice geometry pretest. After the pretest, each group explored their version of the e-book. Subjects explored the e-book in pairs. (5) Each pair used its version for 150 minutes on three consecutive days: two 60-minute sessions held on two days and one 30-minute session held on the third day. During all three sessions, subjects in both groups were free to explore the information contained in their version in any order or manner they desired, without any interference or mediation from the study conductor.

As subjects used the e-book, their interaction with the system was videotaped and their overall usage patterns and verbal comments were recorded. After the third session, all subjects wrote a posttest, identical to the pretest. [G.sub.PASE+] also completed a design questionnaire, as will be explained below. After the posttest, subjects in both groups were interviewed individually, asking about their thoughts and impressions of the two prototypes.

Geometry Test

A paper-and-pencil geometry test (GT) about Platonic and Archimedean solids was constructed and used as the pre- and posttest instrument. It was intended to provide a comparative measure of the subjects' overall understanding of the geometric solids, thereby permitting comparison of the relative effectiveness of the two prototype versions of the e-book. The questions on the test were designed and presented in a format that would be understandable to anyone learning about Platonic and Archimedean solids, not just the subjects using this e-book. GT contained 33 questions, both descriptive and depictive, with varying degrees of difficulty. GT included questions about the solids themselves and transformations and transitions among the solids. Two examples of the types of question on the test are seen in Figure 6.

[FIGURE 6 OMITTED]

Design Questionnaire

The design questionnaire (DQ) was constructed to collect data about the underlying reasons for the subjects' usage patterns of PASE+. DQ was designed to collect both quantitative and qualitative data. The quantitative and qualitative data were intended to cross-validate each other. Many scaled questions terminated with an open-ended prompt ("explain why") inviting written comments. The written comments provided a good source of qualitative data, capturing subjects' opinions and feelings. They added depth to the subjects' quantitative responses. An example of a question on DQ is shown in Figure 7.

Sources of Data

Five sources of data were used to evaluate the prototypes: 1) achievement results, obtained from the statistical analysis of pre- and posttest scores on GT, 2) video transcripts, obtained from video tapes of the subjects' sessions using the prototype, 3) direct observations, obtained from the recordings the study coordinator made of the subjects' overall usage patterns and body language, not easily captured through video recordings of the interactions with the system, 4) design questionnaire results, obtained from the qualitative and quantitative analysis of the answers participants gave on the design questionnaire, and 5) interview transcripts, obtained from the video tapes of the post-hoc interviews with the subjects.

FINDINGS

This section reports the results of the experimental study. The results are presented in two sub-sections: quantitative and qualitative findings. All quotes from the subjects are reported verbatim.

Quantitative Findings

At a descriptive level, all (6) subjects achieved a higher score on the post-test than on the pretest. The median of the posttest is considerably higher than the median of the pretest (see Figure 8). All subjects improved their score of the pretest. The mean increase in percent for all subjects regardless of which e-book version they used is 40% with a standard deviation of 22%.

[FIGURE 8 OMITTED]

Table 1 presents the results of the overall statistical analysis. At an inferential level, a one-sided paired-sample t-test was performed. The hypothesis tested was that the mean difference in scores on the pre- and posttests is positive; that is, the subjects performed significantly better on the posttest. The results indicate a significant improvement in test scores ([t.sub.diff](22)=8.96; p < 0.001). Thus, overall, all subjects improved their performance after using either of the two e-book versions.

Quantitative Findings by Group

The test scores of both groups increased notably. Figure 9 displays the difference in percent between pre- and posttest scores for each group. The boxplot indicates that [G.sub.PASE+] achieved significantly better results than [G.sub.PASE].

[FIGURE 9 OMITTED]

Table 2 summarizes the results of the statistical analysis for the pre- and posttest results by group. The statistics in Table 2 indicate that both groups experienced improvements in their results from pre- to posttest. The increase was larger for [G.sub.PASE+] at 59%. The increase for [G.sub.PASE] was 24%. One sided paired-sample t-tests were performed for each treatment group. The hypothesis tested was that the subjects performed better on the posttest than on the pretest. The results indicate a significant improvement for both treatment groups ([t.sub.diff G(TP)](11)=9.41; p < 0.001; [t.sub.diff G(VTP)](10)=12.2; p < 0.001). Hence, the subjects in each group improved their test scores significantly after interacting with the software.

Moreover, an independent sample one-sided t-test was performed to test if [G.sub.PASE+] achieved statistically significant higher results than [G.sub.PASE]. The hypothesis tested was that the mean achievement results of [G.sub.PASE+] are greater than the mean achievement results of [G.sub.PASE]. Table 3 contains the results of the t-test. The results indicate that [G.sub.PASE+] performed significantly better than [G.sub.PASE]([t.sub.diff](21)=6.83; p < 0.001).

Qualitative Findings

The two groups explored the mathematical information differently. [G.sub.PASE] quickly fell into a rote-learning routine, memorizing the names of the solids and their relationships, and connecting the names with the shapes of the solids. At the beginning of their exploration, the [G.sub.PASE] subjects read the information about each solid. Then, they started with a certain Platonic solid (e.g., tetrahedron) and read about all the solids that could be derived from it. As evidenced by their comments and conversations, the [G.sub.PASE] subjects had considerable difficulty visualizing and making sense of the textual statements describing the relationships among the solids. For instance, while exploring relationships between two solids, one subject turned to her partner and in a frustrated tone asked: "I see the start solid and the end product when I click on the name, but I don't get how the truncation works. Do you get it? How did this come from that?" As a result, some would become frustrated and click systematically through the whole list of the geometric solids trying to memorize their properties and relationships, while some other subjects would simply focus on a subset of the solids.

A different pattern of exploration was observed for [G.sub.PASE+]. The subjects started in the PASE module, as it was the first thing they encountered. However, they soon ran into difficulty trying to understand and visualize the textual descriptions of the mathematical information, similar to difficulties experienced by the subjects in [G.sub.PASE]. In their conversations with each other, qualitative comments, and interviews, many [G.sub.PASE+] subjects expressed that they found it very difficult to visualize how one solid was transformed into another in the PASE module. For instance, one subject, conversing with his partner, said that "It [is] difficult to imagine how [to perform] the ... truncation ... in my brain." Another subject stressed that "Mental visualization can get quite confusing when the geometries become more complex; thus, seeing the transformations is extremely useful." Consequently, shortly after using the PASE module, subjects would move to IVM to find out if they could explore the information in a different way.

In IVM, subjects initially spent a good amount of time interacting with the dynamic maps and the solids. Subjects focused on a specific truncation or augmentation, watched the process a few times, or used the interactive vertex until they seemed to have made sense of the transition. For example, one subject, referring to simulating a particular transition, asked his partner: "Can you play that again, I don't quite get it yet." After the simulation, he commented: "Oh! Now I see it!" This process of experimentation with the interactive maps and solids would continue until the subjects felt more comfortable with the transformational transitions. Afterwards, the subjects started hypothesizing about the nature of the transitional relationships. In order to test their hypotheses, they gathered more detailed information about the solids (e.g., structural composition). They used the dynamic query tools to gather this type of information. For example, two subjects, trying to find out the number of faces which solids derived from the cube have, moved the vertex slider into the correct position and read the resulting numbers. They then tried to explain and justify these numbers to each other.

Sometimes, the subjects needed to understand the language of the information or they needed to communicate more accurately to perform a task together. To acquire the mathematical language and terminology and to review the descriptions of the solids and their relationships, the subjects would switch to the PASE module. The following is an example of a conversation between a pair of subjects discussing the nature of the cuboctahedron:

A: "Why is that one called cuboctahedron?"

B: "It's made of cubes and octahedrons."

A: "But it's made of triangles and squares?"

B: "Hmmm! I need more information about that solid."

At this point, they switched to the PASE module and read about the cuboctahedron. (7)

A: "Oh! I see; the squares come from the cube, and the triangles come from the octahedron."

 Then slowly pronouncing: "cube--octahedron," and
 immediately suggesting:

A: "Let's see how [it works]; go back to the map."

The two modules of PASE+ were used for different tasks and at different stages in the exploration process. IVM was primarily used for making sense of and visualizing the truncation and augmentation processes. PASE was mostly used for verbal, linguistic, and explanatory purposes. For instance, when asked about the reasons for using IVM, two typical responses were:

 "I learned how the objects [i.e., solids] are formed using
 truncating edges and vertices and what they looked like as the
 transformation progressed."

 "It allows you to see things that you can't imagine. I can
 imagine cutting a vertex, but imagining [truncating] edges is
 beyond what I can visualize."

When asked about the reasons for using the PASE module, two typical responses were:

 "It was more like a reference file than a main part of the
 program. The information contained within it was really good
 though. [It] brought together all of the visuals into a concise
 text format."

 "I used it just a little. I learned some names and descriptions
 of the solids [from it]."

The ability to engage with the visual representations through the rich interaction tools offered in IVM allowed the subjects not only to decode and understand the meaning of the information but also to recall and mentally simulate the latent processes in the information. For instance, referring to the interactive solids and the ability to morph them by interactively cutting their vertices, two comments, congruent with statements made by several other subjects, were:

 "When I wrote the geometry test the first time, I was trying to
 truncate and augment in my head, and I didn't know how to ... I
 tried to work from 1st principles but it was very difficult to
 keep everything in my head. The 2nd test, I now know how
 augmenting works ... It's much easier to visualize the result
 of the transformations. I would mentally grab the red dot
 [interactive vertex of the solid] and move it."

 "I could truncate at my own pace and get a feel for the
 relative sizes of the [solid in] focus."

The subjects, who had access to the interactive tools, also engaged in experimentation and posed questions to each other, an activity that was not observed among the [G.sub.PASE] subjects. Two common questions of an experimental nature included: "How many faces can a solid derived from the cube have?" or "Which types of faces can solids derived from the cube have?"

Finally, with regard to their attitude towards the two versions, [G.sub.PASE] and [G.sub.PASE+] differed. Whereas all subjects in [G.sub.PASE] experienced frustration with their version, all subjects in [G.sub.PASE+] commented that they liked the manner in which the information was presented. Ninety percent of the subjects stated that although the presented material was complex, "the design was clear and helped [them] make sense of the material."

SUMMARY AND CONCLUSIONS

This article has presented a study of how learners explore mathematical information in the context of an e-book. It investigated how adding interactive visuals to an e-book influences interaction with the information and learning. Two different versions of a mathematics e-book were designed containing information about Platonic and Archimedean solids. The first version, PASE, mimicked e-books that adhere to a hypertextual book-metaphor. PASE contained mainly textual descriptions of the mathematical content with embedded links. The second version, PASE+, augmented PASE by adding interactive visual representations to the textual information. PASE+ contained two informationally-equivalent modules: a textual module, which was identical to PASE, and an interactive visuals module (IVM), which allowed learners to have rich interaction with the visual representations of the mathematical information.

A multi-method study was conducted using the co-discovery learning method in an experimental setting. The purpose of this study was 1) to investigate whether learners' understanding of the information improved by augmenting the e-book with interactive visuals, and 2) to find out how the addition of these interactive visual representations influenced the way learners explored and reasoned about the information.

This study and its results have certain limitations. First, the study was limited to only one e-book. Second, the mathematical information was of a specific kind: characteristics of 3D geometric shapes and the relationships between these shapes. And, third, the subjects came from a limited sample space. Despite these limitations, this study can lead to a number of general conclusions and has implications for the design of effective e-books and e-learning tools.

The findings of the study showed that both prototypes of the e-book lead to performance improvements on the geometry test. The PASE+ group experienced a considerably larger gain than the PASE group. Hence, adding IVM to PASE supported and improved subjects' recall and learning. Moreover, the PASE group experienced frustration as they had difficulty making sense of the information. As the mathematical information required learners to mentally simulate transitional processes in their mind, we believe the PASE group's frustration stems from the lack of interaction techniques that support process-based reasoning with the mathematical information. While textual information can verbally describe the process, it does not give visual clues as how to mentally visualize and run through the process. As a result, it seems that textual information leaves gaps that are difficult to bridge by the unsupported mind of learners. The PASE+ group used the IVM to visually support their process-based sense-making and reasoning. Furthermore, the PASE+ group expressed enjoyment, which seems to be due to being able to more actively engage with the information.

The pattern of usage of the PASE and PASE+ groups differed considerably. While the PASE group fell into a memorization routine, the PASE+ group engaged in diverse exploration and sense-making activities. Glaser, Ferguson and Vosniadou (1996) state that learners who do not understand the symbolic expressions of textual representations resort to learning the subject matter through superficial memorization, as exhibited by the PASE group. The two modules within PASE+ complemented each other and were used for different purposes. Subjects used the interactive visual representations in IVM for visual imagery and dialectic, formative, and intuitive sense-making. The PASE module provided the language and vocabulary to explain the visual concepts encountered in the IVM module. Interaction with the depictive representations of the IVM module provided meaningful context in which the formalized, descriptive language of the PASE module was situated. Whereas the depictive representations allowed easier access to the apparent meaning of the information, the descriptive representations supported more formal and accurate communication about the information. Interaction with the visual representations added dynamism not only during the use of the e-book, but also during the recall of the information during the post-test, as evidenced by post-hoc interviews. PASE+ allowed learners to think both visually and verbally.

Generally, e-books should not be designed to strictly adhere to the metaphor of print-based static books. They should take more advantage of the representational and interactive power of new computational tools. This study suggests that text-rich educational e-books can be enhanced by adding to them dynamic, interactive visuals. Five possible approaches can be envisioned to enhance today's e-books with interactive visuals: parallel, sequential, simultaneous, text-on-demand, and visual-on-demand.

1) Parallel: This approach is identical to the way PASE+ was designed. That is, the descriptive and depictive representations of information are available in two parallel modules. Any typical text-rich e-book can have a supplemental module that allows learners to have rich interaction with visual representations of information in ways that enhance the exploration and understanding of the descriptive information. A drawback of this approach is that textual descriptions and interactive visual representations are spatially separated on different pages. This requires learners to switch contexts between the two modules and may cause mental disorientation, a problem of navigating virtual learning environments.

2) Sequential: This approach is based on the learners' patterns of usage observed during the study. That is, subjects interacted with the visual information first, and then moved to the textual information. In this approach, the e-book is designed in such a way that learners' interaction with the information moves from visual to textual. A drawback of this approach is that interactive visual representations and textual descriptions are temporally separated--that is, access to information is sequential and therefore transient. Another potential problem is that this approach does not easily fit into the structure of current e-books.

3) Simultaneous: This approach combines the parallel and sequential approaches. That is, interactive visuals and textual information are intermixed, just as static images are found on the same page as text in traditional books. A potential drawback of this approach is that learners may have to process the textual information and visual information simultaneously, find relationships between them, and make sense of the relationships--all of which may lead to cognitive overload for the learners (Betrancourt & Bisseret, 1998).

4) Visuals-on-demand: This approach provides interactive visuals on demand only. That is, the e-book is textual, and learners can click on embedded links to demand the display of interactive visuals. Any typical text-rich e-book can easily incorporate this feature without much change to its structure. A possible drawback of this approach, however, is that it violates the observed natural exploration pattern from visual to textual observed in this study.

5) Text-on-demand: This approach provides visuals as the primary source of information. Learners explore information by interacting with visual representations, demanding textual explanations when needed. That is, learners request textual descriptions via pop-up menus embedded in the visuals (for a system that uses this approach, see Sedig et al., 2003). The problem with this approach is that it does not adhere to the book metaphor anymore, requiring a drastic rethinking of what e-books are altogether.

Further research is needed to compare and contrast the advantages and disadvantages of the five approaches to find out how to best augment e-books for different subjects and learning contexts.

Truncating a cube by clicking and dragging the red dot helped me to
visualize the truncation process and the result in my mind better than
the truncation simulation
 a. strongly agree
 b. agree
 c. undecided
 d. disagree
 e. strongly disagree
Explain why?

Figure 7. Sample question from DQ

Table 1 T-test Analysis for Difference Between Pre- and Posttest Scores
(in %)

t-test for overall results

 Mean difference in %
 b/w pre- and posttest Std. t-statistic Df p-value
 results Deviation

All Subjects 40.40 21.62 8.96 22 < 0.001

Table 2 T-test Analysis for the Difference Between Pre- and Posttest
Scores (in %) by Group

Overall test scores: paired samples t-test for each version

 Mean difference in %
 b/w pre- & posttest Std. t-statistic Df p-value
 results Deviation

[G.sub.PASE] 23.80 8.76 9.41 11 < 0.001
[G.sub.PASE+] 58.50 15.9 12.2 10 < 0.001

Table 3 T-test to Probe for Difference Between the Means of the
Difference Between Pre- and Posttest Scores for Each Group

Overall test scores: independent samples t-test for difference between
versions

Mean[.sub.G(PASE+)]-Mean
G(PASE) [S.sub.pooled] t-statistic df p-value

34.7 12.18 6.83 21 < 0.001

Notes

(1) The sides of a regular polygon are all of equal length. A rectangle is not a regular polygon, but a square is a regular polygon.

(2) An example of the textual descriptions of a solid and its composition is: "The most commonly known and most widely used of all polyhedra is undoubtedly the cube or to give it a fancier name, the hexahedron. The hexahedron is a Platonic solid;" and "The hexa(hexa=six)hedron's six faces are all squares. Two squares or faces meet at each edge and three squares or faces meet at each vertex".

(3) A more detailed description of the diagrammatic maps of these solids appears in Sedig et al. (2003).

(4) A detailed description of the interactive nature of the morphing solids appears in Morey, Sedig & Mercer (2001).

(5) This method of software evaluation is called co-discovery learning in which two subjects explore an environment while collaborating and conversing with each other (Kennedy, 1989). The conversation is meant to help evaluators gain a better understanding of how the software is being used.

(6) One subject could not complete the study; therefore, the sample size for [G.sub.PASE+] was reduced to eleven.

(7) The text in the PASE module reads: "The name 'cuboctahedron' suggests a close relationship to the cube and the octahedron, and indeed this is so. It is a combinatorial solid. The six squares are on the facial plane of a cube and the eight triangles are the facial planes of an octahedron."

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SONJA ROWHANI AND KAMRAN SEDIG

University of Western Ontario Canada

[email protected]

[email protected]