One of the first considerations in the production of lecture videos is the value of live lecture recording vs. that of an in-office lecture recording. A live lecture environment will be most familiar to the student and will likely be the most engaging when considering the body language, facial expressions and personality of the lecturer. The best live lecture production includes classroom video along with audio and screen capture. The classroom video adds considerable cost and complexity to the production in terms of manpower required for video recording and the post-production process of superimposing the classroom and screen capture videos onto a single frame. Without classroom video, unintended audio irregularities (such as extraneous noises and periods of silence) become distracting to the viewer. The in-office lecture recording model allows for a more polished presentation. Audio quality is more easily controlled. Rehearsing the lecture prior to recording and re-recording segments of the lecture both lead to a better final product. Most screen capture recording software allows for "talking head" video insets of the instructor to be included in the lecture recording. The value of these insets throughout the lecture should be considered [Sorden (2005), Nielsen (2005)]. Do they add communicational value? Is the space on the screen better used by written lecture notes? An alternative is to include a short, full-screen video of the instructor at the beginning and/or end of the lecture when the lecture topics are introduced and/or summarized.

#### **4.1.2 Video solutions**

The HigherEd 2.0 program has shown the strength of the worked-example on learning in foundational engineering courses. A solution video and a textbook worked-example share the same pedagogical deconstruction of a problem into its relevant parts and the delineation of problem-solving strategies. The power of the video solution lies in the modality of an audio description with text and graphics. The design and production of solution videos should always remained focused on optimizing the positive impact of this modality. Simply reading what is being written on the screen adds little value. Use the audio component to amplify the problem-solving strategy. Explain nuances and provide context of a problem-solving step to others. Provide considerable detail in the handwritten portion of the video. Although some

1. *Synchronize all temporal input*: Spoken narration should be synchronized with

HigherEd 2.0: Web 2.0 in Higher Education 79

2. *Eliminate extraneous information*: Do not include extra information that is not

3. *Keep it short*: Longer videos can contain too much information for students to

4. *Use consistent color schemes, formatting, and overall aesthetics*: Take care to use color and other visual elements consistently both *within and across* videos, so that students can easily recognize the meaning of specific formatting conventions. written descriptions using color ➎; e.g., the red equation accompanies the red annotations

5. *Keep it current*: Deploy videos when students need them, and make sure students

6. *Don't write a script, but do rehearse*: Use Mayer's personalization principle; use

7. *Compress*: Large videos (>100 MB in size) may present problems for students with slow connections at home. Compress your videos using a modern standard, such as H.264. Good-quality videos can be produced at a size of 1 MB/minute for

8. *Spend the time and money to polish your hardware and software setup*: Creating your videos on a substandard setup increases the cognitive load on the instructor! Make sure your setup makes it easy for you to produce and distribute your videos. 9. *Distribute video in a format and manner that encourages use*: Use a standard, platform-independent video format, such as Quicktime, so that students can access

10. *Critique your productions on a regular basis*: Revise (or eliminate) material that has been determined to be ineffective for learning. There is no need to keep media that

blog serves as an organizing framework for course material, particularly for the multimedia material for the course. The function and form of each dictates the design and implementation of a successful course blog. This video ➏ describes a typical HigherEd 2.0 course blog, its

The discussion threads of the blog should facilitate an uninhibited exchange of ideas among the students leading to cooperative learning. This exchange could mimic that of a small group of students sitting around a table discussing a homework problem for the course. In this small group, the students offer up their opinions without concern of judgement from their peers. Not all opinions are correct; however, after a period of time the group settles on a consensus opinion. The blog discussion can follow this model; however, the blog group includes the entire class, and the participants can remain anonymous via careful choice of

relevant to the problem, because this increases the ECL for the learner.

assimilate. Our experience is that 15 minutes is about right.

are aware when useful new videos become available.

conversational language to describe your approach.

and play videos on a wide range of devices.

Table 1. Best practices in video production and distribution.

on-screen action.

and text on a figure.

tablet-based videos.

students do not use.

features, and how it is used and organized.

students may only listen and watch, many will be taking notes; complete written thoughts are important for this group of students. Write legibly. Use color creatively for graphical emphasis, but do so selectively. Rehearsal prior to recording is highly recommended; however, working without a script is good practice in that in leads to a fresher presentation.

#### **4.1.3 Animations and simulations**

Animations and simulations allow students to observe the dynamic and visual consequences of mathematical concepts. Animations are motion recordings whose parameters and initial conditions are set by the designer, whereas in simulations the user has control over selected input parameters and initial conditions. Learning objectives totally dictate the design of these components. If the goal is to demonstrate global motion of a system, the animation/simulation should allow visualization of the entire system. Alternately, the learning objective might be focused on a qualitative assessment of analytical results, in which case the visualization needs to focus on graphical representation of those results. Unlike lecture videos and video solutions, animations and simulations are not standalone media components; they require contextual connection to analysis. The best practice is to embed them within a focused learning module. In this module, the problem is concisely described with text and images/videos followed by a summary of analytical results (possibly along with a solution video). The critical component of the module is discussion directed at one aspect of the solution alongside the animation/simulation. Multiple animations (or a single simulation module allowing student interactivity) are required for "what if" discussions. Spatial contiguity principles should be followed in the page layout for the module allowing for students to simultaneously observe related components on the page without scrolling or links to other pages.

#### **4.1.4 Case studies**

The instructional method for case studies is based on the illustration of a concept through an example. The case study leads from a specific situation back to the general principle, generally the reverse of the other tools discussed here. The case study is more difficult to design from a pedagogical standpoint. From a media standpoint, its design employs the components described above. The layout of a case study module shares similarities with those used with animations and simulations. The problem is introduced using text and graphics/videos and is connected to relevant lecture and solution videos by textual discussion that lays out the thought process in arriving at the general underlying concept. At the end, the student is allowed to study the relationship of the original example problem to the underlying mathematics through animation and/or simulation components. Spatial contiguity principles dictate the layout the module in the same way as for the animation and simulation modules described earlier.

Table 1 summarizes a set of best practices for multimedia production as learned from the HigherEd 2.0 program.

#### **4.2 Best practices in course blog design**

The course blog serves two primary functions. First, the discussion thread of the blog is a social platform that encourages interaction and sharing between students. Second, the 12 Will-be-set-by-IN-TECH

students may only listen and watch, many will be taking notes; complete written thoughts are important for this group of students. Write legibly. Use color creatively for graphical emphasis, but do so selectively. Rehearsal prior to recording is highly recommended; however, working without a script is good practice in that in leads to a fresher presentation.

Animations and simulations allow students to observe the dynamic and visual consequences of mathematical concepts. Animations are motion recordings whose parameters and initial conditions are set by the designer, whereas in simulations the user has control over selected input parameters and initial conditions. Learning objectives totally dictate the design of these components. If the goal is to demonstrate global motion of a system, the animation/simulation should allow visualization of the entire system. Alternately, the learning objective might be focused on a qualitative assessment of analytical results, in which case the visualization needs to focus on graphical representation of those results. Unlike lecture videos and video solutions, animations and simulations are not standalone media components; they require contextual connection to analysis. The best practice is to embed them within a focused learning module. In this module, the problem is concisely described with text and images/videos followed by a summary of analytical results (possibly along with a solution video). The critical component of the module is discussion directed at one aspect of the solution alongside the animation/simulation. Multiple animations (or a single simulation module allowing student interactivity) are required for "what if" discussions. Spatial contiguity principles should be followed in the page layout for the module allowing for students to simultaneously observe related components on the page without scrolling or

The instructional method for case studies is based on the illustration of a concept through an example. The case study leads from a specific situation back to the general principle, generally the reverse of the other tools discussed here. The case study is more difficult to design from a pedagogical standpoint. From a media standpoint, its design employs the components described above. The layout of a case study module shares similarities with those used with animations and simulations. The problem is introduced using text and graphics/videos and is connected to relevant lecture and solution videos by textual discussion that lays out the thought process in arriving at the general underlying concept. At the end, the student is allowed to study the relationship of the original example problem to the underlying mathematics through animation and/or simulation components. Spatial contiguity principles dictate the layout the module in the same way as for the animation and simulation modules

Table 1 summarizes a set of best practices for multimedia production as learned from the

The course blog serves two primary functions. First, the discussion thread of the blog is a social platform that encourages interaction and sharing between students. Second, the

**4.1.3 Animations and simulations**

links to other pages.

**4.1.4 Case studies**

described earlier.

HigherEd 2.0 program.

**4.2 Best practices in course blog design**

Table 1. Best practices in video production and distribution.

blog serves as an organizing framework for course material, particularly for the multimedia material for the course. The function and form of each dictates the design and implementation of a successful course blog. This video ➏ describes a typical HigherEd 2.0 course blog, its features, and how it is used and organized.

The discussion threads of the blog should facilitate an uninhibited exchange of ideas among the students leading to cooperative learning. This exchange could mimic that of a small group of students sitting around a table discussing a homework problem for the course. In this small group, the students offer up their opinions without concern of judgement from their peers. Not all opinions are correct; however, after a period of time the group settles on a consensus opinion. The blog discussion can follow this model; however, the blog group includes the entire class, and the participants can remain anonymous via careful choice of

1. *Don't assume anything*: On the first day of class, (and several times thereafter) discuss the role of the course blog. Students will better understand your expectation

HigherEd 2.0: Web 2.0 in Higher Education 81

2. *Limit your involvement in discussion threads*: Let students discuss the issues on their

3. *Reward blog participation*: Giving minimal credit for participation lets the students

4. *Control who can add to the discussion thread*: Make it easy for students to comments; however, require a login and act as moderator to prevent outside contributors. 5. *Create leads on discussion threads*: You want to keep the students focused on relevant

6. *Learn from discussion threads*: Watch the discussion threads to identify student difficulties or misconceptions that can be addressed in the next class meeting. 7. *Use good organization, and keep it all in one place*: Use tags and categories to help students locate material. Know your 'pages' from your 'posts'. Have all course content on the same blog. A student who leaves the blog to an external link might

8. *Allow user anonymity*: Allow students to choose a user name that protects their identity, if they wish. The instructor should be the blog administrator, and will therefore know users' identities. But allowing anonymity can reduce anxiety

9. *Keep it current*: An out-of-date blog signals to the students that you are not

10. *No funny stuff*: Do not post irrelevant material on the blog. Discourage students

The HigherEd 2.0 program leverages student creativity and ambition to amplify the voice of the learners. While we have employed a wide range of SGC formats and strategies [including wikis and podcasting assignments (Berger, 2007), (Berger, 2009)], we focus here on the two

The course blog (the technical details of which were presented in Sec. 4.2) presents a simple, direct avenue for student authoring and sharing. Student participation in blog-based, asynchronous discussion–in a public forum–supports the collective construction of understanding celebrated in *social constructivist* theories of learning. Social constructivism emphasizes that learning is: (i) a process that requires social interactions and is time-evolving; (ii) contextual and reliant on the culture of the learning community; and (iii) personalized, with students influencing the direction and format of the learning. The blog is an inherently

most productive forms of SGC: blogging and construction of worked examples.

social learning tool that encourages discussion, collaboration, and sharing.

own. Your involvement will likely curtail participation.

issues. Start out the discussion with a hint.

associated with participation in a public forum.

interested in them using the blog.

**5.1 The course blog and social constructivism**

know that you consider blog participation to be important.

throughout the course.

not return.

from doing so.

Table 2. Best practices in blogging.

their blog username3. With the larger group on the blog, the accuracy of the group consensus is improved greatly. The course instructor is able to monitor the discussion but should intervene only when necessary to keep the student bloggers on track. By silently observing the blog discussion, the instructor has gained considerable insight on the depth of the students' understanding. Difficult concepts, as learned from the blog, can become part of a lecture discussion in the next class period.

Issues of learner-control vs. instructor-control need to be considered when maintaining the blog during the course term. We have observed that if the initiation of discussion threads is left up to the students, multiple posts related to the same issue (such as the solution of a homework problem) can appear on the blog. This produces many disconnected threads with few comments per thread. The continuity and interaction of discussion is lost in this way. We recommend that the course instructor add blog posts to initiate discussion threads on anticipated issues of interest to the students (homework problems, exam reviews, etc.). These posts can be used to set the tone and focus of discussion by the blogging students. Students should also be allowed blog permissions to author posts, permitting them to start discussion threads on topics not anticipated by the instructor.

Students are more likely to be engaged in course discussion if the blog is spatially connected with the course multimedia content. To accomplish this, course material is accessed through links to content pages on the blog. With this layout, students do not need to leave the blog as they review course material. Blog comments and posts can be added directly from the course content pages.

Note that blog discussion threads are temporally organized, generally in reverse chronological order. Course material, on the other hand, is best organized topically. This difference in form for the two blog functions should be kept in mind when setting the organizational standards for the blog. One should not insert topical material within the temporally-organized discussion thread. Furthermore, the blog designer should make extensive usage of tags on both discussion threads and content pages to assist the students in locating relevant material.

Table 2 summarizes a set of best practices for blogging as learned from the HigherEd 2.0 program.

#### **5. Student Generated Content (SGC)**

The web 2.0 era not only empowers instructors to author multimedia learning materials for their students; it also offer unprecedented empowerment to students to create learning materials for each other. The pedagogical rationale for integrating students into the production and sharing of learning material is simple: in order for students to effectively "teach" their peers, they must develop a high level of expertise in the subject. This peer element of instruction has existed in a variety of forms and with many names essentially since the dawn of time, and it has consistently been shown to add value to learning (Hsiao et al., 2010). But the advent of powerful new tools always injects new excitement and pedagogical opportunities. Web 2.0 tools specialize in authorship and collaboration, two critical elements that empower participants to easily create materials and share them with a large audience.

3 Of course, the instructor–as blog administrator–will know the user's identity, but other participants on the blog may not.

14 Will-be-set-by-IN-TECH

their blog username3. With the larger group on the blog, the accuracy of the group consensus is improved greatly. The course instructor is able to monitor the discussion but should intervene only when necessary to keep the student bloggers on track. By silently observing the blog discussion, the instructor has gained considerable insight on the depth of the students' understanding. Difficult concepts, as learned from the blog, can become part of a lecture

Issues of learner-control vs. instructor-control need to be considered when maintaining the blog during the course term. We have observed that if the initiation of discussion threads is left up to the students, multiple posts related to the same issue (such as the solution of a homework problem) can appear on the blog. This produces many disconnected threads with few comments per thread. The continuity and interaction of discussion is lost in this way. We recommend that the course instructor add blog posts to initiate discussion threads on anticipated issues of interest to the students (homework problems, exam reviews, etc.). These posts can be used to set the tone and focus of discussion by the blogging students. Students should also be allowed blog permissions to author posts, permitting them to start discussion

Students are more likely to be engaged in course discussion if the blog is spatially connected with the course multimedia content. To accomplish this, course material is accessed through links to content pages on the blog. With this layout, students do not need to leave the blog as they review course material. Blog comments and posts can be added directly from the course

Note that blog discussion threads are temporally organized, generally in reverse chronological order. Course material, on the other hand, is best organized topically. This difference in form for the two blog functions should be kept in mind when setting the organizational standards for the blog. One should not insert topical material within the temporally-organized discussion thread. Furthermore, the blog designer should make extensive usage of tags on both discussion threads and content pages to assist the students in locating relevant material. Table 2 summarizes a set of best practices for blogging as learned from the HigherEd 2.0

The web 2.0 era not only empowers instructors to author multimedia learning materials for their students; it also offer unprecedented empowerment to students to create learning materials for each other. The pedagogical rationale for integrating students into the production and sharing of learning material is simple: in order for students to effectively "teach" their peers, they must develop a high level of expertise in the subject. This peer element of instruction has existed in a variety of forms and with many names essentially since the dawn of time, and it has consistently been shown to add value to learning (Hsiao et al., 2010). But the advent of powerful new tools always injects new excitement and pedagogical opportunities. Web 2.0 tools specialize in authorship and collaboration, two critical elements that empower participants to easily create materials and share them with a large audience.

3 Of course, the instructor–as blog administrator–will know the user's identity, but other participants on

discussion in the next class period.

content pages.

program.

the blog may not.

threads on topics not anticipated by the instructor.

**5. Student Generated Content (SGC)**

Table 2. Best practices in blogging.

The HigherEd 2.0 program leverages student creativity and ambition to amplify the voice of the learners. While we have employed a wide range of SGC formats and strategies [including wikis and podcasting assignments (Berger, 2007), (Berger, 2009)], we focus here on the two most productive forms of SGC: blogging and construction of worked examples.

#### **5.1 The course blog and social constructivism**

The course blog (the technical details of which were presented in Sec. 4.2) presents a simple, direct avenue for student authoring and sharing. Student participation in blog-based, asynchronous discussion–in a public forum–supports the collective construction of understanding celebrated in *social constructivist* theories of learning. Social constructivism emphasizes that learning is: (i) a process that requires social interactions and is time-evolving; (ii) contextual and reliant on the culture of the learning community; and (iii) personalized, with students influencing the direction and format of the learning. The blog is an inherently social learning tool that encourages discussion, collaboration, and sharing.

progressed along the novice-to-experienced transition to a sufficient degree, they are in a position to create (video) worked examples for their peers. Video creation is only possible now, with the rise of web 2.0, because the intrinsic cognitive load related to authoring in the video production environment is almost negligible. Students create videos seamlessly, because the authoring tools are so powerful and user friendly that they essentially fade into the background4; students are free to focus entirely on the germane cognitive load of constructing worked-out examples for their peers. Throughout their experience in a HigherEd 2.0 course, student have encountered a large number of instructor-generated worked examples, each one a specific example of how to effectively construct multimedia learning materials (Mayer, 2009). Despite their lack of formal training in multimedia pedagogy, students generally create quality solutions consistent with multimedia learning principles, and challenge themselves to truly master the problem solving approach. This example ➐ of a hand-written, student-generated video solution uses a white board, multiple

HigherEd 2.0: Web 2.0 in Higher Education 83

It might seem counterintuitive that the HigherEd 2.0 approach promotes construction and sharing of worked-out examples by students–students who have already substantially moved beyond the novice phase of problem solving. The worked-example effect is known to be most powerful with novice learners, so why would student-generated content be valuable in this form? The answer lies in the pedagogy; the worked-out solution assignment is not designed to explicitly benefit the learning community (although it certainly can do that by providing solutions to problems with which students might struggle). Rather, students construct these learning materials as an exercise of *schema automation*, therefore accelerating their transition to experts. Indeed, as instructors already know, teaching any material to novices requires a high level of expertise in the subject matter (cf. Section 3.3). Using the familiar Bloom's Taxonomy (Krathwohl, 2002) language, when students create authoritative solutions for problems, they focus on *creation* and *evaluation*, the two highest cognitive domains in the taxonomy. Students build a solution that they perceive to be the "best" solution to a given problem, explain the solution clearly using available tools and techniques, and present it using digital tools so that

The HigherEd 2.0 program uses a mixed-methods evaluation that emphasizes both quantitative and qualitative data. Because these strategies are deployed in real higher education courses across an entire semester, the study cannot be conducted in the well-controlled environments characteristic of much of the educational psychology or cognitive science literature. Instead, we have constructed the evaluation with a combination of usage statistics, survey data, gradebook information, and student and faculty interviews. The results presented here are a subset of the broader evaluation data, more of which is included in Orange et al. (2011). Because the HigherEd 2.0 paradigm rests upon such a

4 It is worth noting that when we started developing the HigherEd 2.0 paradigm in 2006, we had to provide training to students on video production using tools like GarageBand. Now, in 2011, no such training is required, and students often simply use their mobile phone cameras to capture video

colors, live narration, and a hand-held video.

others can learn from their example.

**6. Evaluation methods and results**

solutions written on paper or a white board.

**6.1 Evaluation overview**

In the parlance of CLT, the blog promotes schema acquisition via social means. Students share ideas about how to approach problems, discuss the details and meanings of specific facets of problems, and collectively construct an understanding of how to approach the problems. A typical example of this process is shown in Figure 1, in which students use the nested commenting features of the course blog to asynchronously collaborate on homework solutions. Each individual student brings his/her prior knowledge and schema (under construction) to the discussion, and the outcome can be considered as a transcript of schema construction by these individuals.

Fig. 1. A sample threaded discussion on the course blog.

#### **5.2 Student worked examples**

Another component of student-generated content with HigherEd 2.0 is student creation of worked examples. Throughout the course, students consume instructor-created worked examples as they learn key concepts and problem-solving strategies. Once students have 16 Will-be-set-by-IN-TECH

In the parlance of CLT, the blog promotes schema acquisition via social means. Students share ideas about how to approach problems, discuss the details and meanings of specific facets of problems, and collectively construct an understanding of how to approach the problems. A typical example of this process is shown in Figure 1, in which students use the nested commenting features of the course blog to asynchronously collaborate on homework solutions. Each individual student brings his/her prior knowledge and schema (under construction) to the discussion, and the outcome can be considered as a transcript of schema

construction by these individuals.

Fig. 1. A sample threaded discussion on the course blog.

Another component of student-generated content with HigherEd 2.0 is student creation of worked examples. Throughout the course, students consume instructor-created worked examples as they learn key concepts and problem-solving strategies. Once students have

**5.2 Student worked examples**

progressed along the novice-to-experienced transition to a sufficient degree, they are in a position to create (video) worked examples for their peers. Video creation is only possible now, with the rise of web 2.0, because the intrinsic cognitive load related to authoring in the video production environment is almost negligible. Students create videos seamlessly, because the authoring tools are so powerful and user friendly that they essentially fade into the background4; students are free to focus entirely on the germane cognitive load of constructing worked-out examples for their peers. Throughout their experience in a HigherEd 2.0 course, student have encountered a large number of instructor-generated worked examples, each one a specific example of how to effectively construct multimedia learning materials (Mayer, 2009). Despite their lack of formal training in multimedia pedagogy, students generally create quality solutions consistent with multimedia learning principles, and challenge themselves to truly master the problem solving approach. This example ➐ of a hand-written, student-generated video solution uses a white board, multiple colors, live narration, and a hand-held video.

It might seem counterintuitive that the HigherEd 2.0 approach promotes construction and sharing of worked-out examples by students–students who have already substantially moved beyond the novice phase of problem solving. The worked-example effect is known to be most powerful with novice learners, so why would student-generated content be valuable in this form? The answer lies in the pedagogy; the worked-out solution assignment is not designed to explicitly benefit the learning community (although it certainly can do that by providing solutions to problems with which students might struggle). Rather, students construct these learning materials as an exercise of *schema automation*, therefore accelerating their transition to experts. Indeed, as instructors already know, teaching any material to novices requires a high level of expertise in the subject matter (cf. Section 3.3). Using the familiar Bloom's Taxonomy (Krathwohl, 2002) language, when students create authoritative solutions for problems, they focus on *creation* and *evaluation*, the two highest cognitive domains in the taxonomy. Students build a solution that they perceive to be the "best" solution to a given problem, explain the solution clearly using available tools and techniques, and present it using digital tools so that others can learn from their example.

#### **6. Evaluation methods and results**

#### **6.1 Evaluation overview**

The HigherEd 2.0 program uses a mixed-methods evaluation that emphasizes both quantitative and qualitative data. Because these strategies are deployed in real higher education courses across an entire semester, the study cannot be conducted in the well-controlled environments characteristic of much of the educational psychology or cognitive science literature. Instead, we have constructed the evaluation with a combination of usage statistics, survey data, gradebook information, and student and faculty interviews. The results presented here are a subset of the broader evaluation data, more of which is included in Orange et al. (2011). Because the HigherEd 2.0 paradigm rests upon such a

4 It is worth noting that when we started developing the HigherEd 2.0 paradigm in 2006, we had to provide training to students on video production using tools like GarageBand. Now, in 2011, no such training is required, and students often simply use their mobile phone cameras to capture video solutions written on paper or a white board.

(a) Spring 2011 blog usage for Purdue (8.3 visits per week per student).

HigherEd 2.0: Web 2.0 in Higher Education 85

(b) Spring 2008 blog usage for UVa (5.3 visits per week per student).

valuable tools for a better conceptual understanding of the material. However, they also report that these components should be more closely integrated into course assignments in

On the other hand, both the *course blog* and the *video solutions* are perceived to be of high value to students. Student report strong usage of the blog and typically find the anytime, anywhere collaboration through threaded discussions on the blog to be useful. Moreover, they report strong satisfaction with the learning environment shaped by the course blog. Students indicate that they appreciate the easy navigation, excellent organization, and convenient access. Video solutions are viewed similarly, with students generally appreciating their ease of

Fig. 2. Blog usage during two semesters at two different sites.

order for the learning impact to be maximized.

strong pedagogical foundation (Section 3), for the rest of this section we focus on high-level conclusions about student participation, satisfaction with the learning environment, and overall rates of adoption of the technologies for learning.

#### **6.1.1 Usage data**

Student usage of the course blog and multimedia learning materials has remained consistently high throughout implementation of the HigherEd 2.0 program. Student usage of the blog (Figure 2) is typically cyclic but strong, averaging about 5-8 visits per week per student for the entire semester; students are clearly avid blog users. While students routinely view the blog, their comment rate appears to be quite sensitive to the local student environment and the set of incentives in place to promote active participation. The *student environment* concerning collaboration is important; in settings with a very small student population, it is not uncommon for students to collaborate almost exclusively face-to-face. However, with a larger student population, asynchronous collaboration via blogging, text messaging, etc. is much more common.

Perhaps more importantly, *incentives* provide an important impetus for students to actively use certain technologies, and our experience with blogging clearly shows the impact of incentives. For both Purdue courses and the Spring 2011 UVa course in Table 3, a small portion of the final course grade (3%) was tied to blog participation and asynchronous collaboration via commenting in threaded discussions on the course blog. The Spring 2009 UVa course used no incentives for blog participation. We discuss the role of incentives more in Section 6.2.2, but both our usage data, as well as student interview data detailed in Section 6.1.3, suggest that a course-grade-related incentive is an important motivator of early and active blog participation by students. When students are sufficiently motivated, fruitful discussions take place on the blog (Figure 1).

Table 3. Comparison of blog comments and per capita comment rate over two sites and two semesters.

#### **6.1.2 Survey data**

Survey data captures student attitudes about the HigherEd 2.0 paradigm and its perceived usefulness in helping them achieve course learning outcomes. Here we present a *very high-level summary* of the trends we have observed in our survey data, which has been collected over at least 6 semesters of HigherEd 2.0 courses at multiple institutions. Students across all semesters perceive the *lecture videos* to be of low-to-medium value, and this is largely because the lecture videos capture conceptual information and derivations. Students perceive this information to be less germane to their problem solving efforts in homework and exams, and therefore typically only consult lecture videos when they miss class due to illness or travel. Students report that they generally appreciate the *animations, simulations, and case studies* as 18 Will-be-set-by-IN-TECH

strong pedagogical foundation (Section 3), for the rest of this section we focus on high-level conclusions about student participation, satisfaction with the learning environment, and

Student usage of the course blog and multimedia learning materials has remained consistently high throughout implementation of the HigherEd 2.0 program. Student usage of the blog (Figure 2) is typically cyclic but strong, averaging about 5-8 visits per week per student for the entire semester; students are clearly avid blog users. While students routinely view the blog, their comment rate appears to be quite sensitive to the local student environment and the set of incentives in place to promote active participation. The *student environment* concerning collaboration is important; in settings with a very small student population, it is not uncommon for students to collaborate almost exclusively face-to-face. However, with a larger student population, asynchronous collaboration via blogging, text messaging, etc. is

Perhaps more importantly, *incentives* provide an important impetus for students to actively use certain technologies, and our experience with blogging clearly shows the impact of incentives. For both Purdue courses and the Spring 2011 UVa course in Table 3, a small portion of the final course grade (3%) was tied to blog participation and asynchronous collaboration via commenting in threaded discussions on the course blog. The Spring 2009 UVa course used no incentives for blog participation. We discuss the role of incentives more in Section 6.2.2, but both our usage data, as well as student interview data detailed in Section 6.1.3, suggest that a course-grade-related incentive is an important motivator of early and active blog participation by students. When students are sufficiently motivated, fruitful discussions take place on the

*Site/Semester #comments #comments/student*

Table 3. Comparison of blog comments and per capita comment rate over two sites and two

Survey data captures student attitudes about the HigherEd 2.0 paradigm and its perceived usefulness in helping them achieve course learning outcomes. Here we present a *very high-level summary* of the trends we have observed in our survey data, which has been collected over at least 6 semesters of HigherEd 2.0 courses at multiple institutions. Students across all semesters perceive the *lecture videos* to be of low-to-medium value, and this is largely because the lecture videos capture conceptual information and derivations. Students perceive this information to be less germane to their problem solving efforts in homework and exams, and therefore typically only consult lecture videos when they miss class due to illness or travel. Students report that they generally appreciate the *animations, simulations, and case studies* as

UVa, Spring 2011 1012 9.45 Purdue, Spring 2011 2904 9.50 UVa, Spring 2009 26 0.40 Purdue, Spring 2009 1764 7.41

overall rates of adoption of the technologies for learning.

**6.1.1 Usage data**

much more common.

blog (Figure 1).

semesters.

**6.1.2 Survey data**

(a) Spring 2011 blog usage for Purdue (8.3 visits per week per student).

(b) Spring 2008 blog usage for UVa (5.3 visits per week per student).

Fig. 2. Blog usage during two semesters at two different sites.

valuable tools for a better conceptual understanding of the material. However, they also report that these components should be more closely integrated into course assignments in order for the learning impact to be maximized.

On the other hand, both the *course blog* and the *video solutions* are perceived to be of high value to students. Student report strong usage of the blog and typically find the anytime, anywhere collaboration through threaded discussions on the blog to be useful. Moreover, they report strong satisfaction with the learning environment shaped by the course blog. Students indicate that they appreciate the easy navigation, excellent organization, and convenient access. Video solutions are viewed similarly, with students generally appreciating their ease of

These characteristics, the goal of any innovation seeking widespread adoption, and the ways the HigherEd 2.0 paradigm expresses them, are shown in Table 4 and discussed in more depth

HigherEd 2.0: Web 2.0 in Higher Education 87

*HigherEd 2.0 feature RA Cp Cx T O* goal of innovation ↑ ↑ ↓ ↑↑

Table 4. Summary of HigherEd 2.0 approaches within the diffusion of innovation "5 characteristics" framework (Rogers, 2003). [↑=high value, ↓=low value, − = neutral]

The shared, collaborative nature of HigherEd 2.0 strategies and resources essentially guarantee that the *trialability* and *observability* goals are met. Students perceive the course materials to be available for usage and experimentation anytime, anywhere, and (because of the public nature of the course blog) the impacts of this educational approach have high visibility among the student population. As described in the Introduction, the *complexity* associated with web 2.0 tools is generally low, but students do initially reserve some skepticism for the SGC components of their work. Because creating multimedia content for their peers typically presents a new approach to learning, students are not immediately confident that they will be able to successfully navigate the technology and produce a quality product. They quickly–and with little training–overcome this perception, but it nonetheless requires careful and frequent encouragement from the instructor to ensure that students

In two technology areas (communication via blogging and video solutions), both the *relative advantage* and *compatibility* of the technology are essentially self-evident to students. Video solutions allow students to access expertise in problem solving when they need it, providing both a relative advantage and compatibility. The video solutions are a favorable alternative to time-constrained office hours, offer expertise that a student's peers might not possess, and fit smoothly into both the workflow and digital lifestyle of today's students. Students can simply download the videos to their portable device and use them anytime, anywhere. The course blog, with its RSS technology, platform-independence and robust navigation features, fits easily within students' daily workflow, and these features typically provide a performance advantage over a traditional course management system (CMS). However, for collaboration, students do not always perceive a relative advantage to the threaded discussions on the blog, nor does it always fit with their conception of what collaboration in course work looks like. Our experience has shown that students sometimes prefer synchronous, in-person collaborations with their peers on problem sets and exam preparation. Here, we have

in Sections 6.2.1 and 6.2.2.

blogging

**6.2.1 Complexity, trialability and observability**

**6.2.2 Relative advantage and compatibility**

understand and exploit the simplicity of current authoring tools.

use, expert perspective, and constant availability. Students report that the value of the video solutions increases as the course material become more conceptually complex (Orange et al., 2011) and the germane cognitive load therefore becomes higher.

#### **6.1.3 Interview data**

Student interview data reveals that the course blog and video solutions are considered extremely valuable by the majority of students. Purdue students offered comments like this about the course blog: "The discussions on the blog were very beneficial to view...I was promptly helped by both Prof. Krousgrill and other students." The virtues of video solutions were also recognized: "Helpful, good explanations, [I] would not get grade I am going to have without [the] videos", or "Very useful, well put together, easy to follow", or "They're super effective". On balance, student qualitative data is overwhelmingly positive about these and other features of the HigherEd 2.0 program. Nonetheless, students also recognize that social tools (such as the course blog) require a critical mass of users in order to deliver maximum value: "It's [the blog] only helpful if everyone uses it." Comments like these have stimulated our use of incentives to promote participation, yielding comments like this one from a Purdue student: "Assigning a portion of the course grade to blog posting made me look at the blog. Once I was there I wanted to post on the blog."

Students also describe their experience with technology adoption, specifically their gradual adaptation to using the technology for learning. A UVa student: "I was really resistant to it [using the technology] last semester, but I now see the value and efficiency of it. There's definitely an adjustment period to get comfortable though." Another UVa student: "I didn't have Dr. Berger for Statics last semester, so the blog and related tools forced me to change the way I use technology in support of learning. There was a slight learning curve/adjustment period at the beginning, but now I'm a pretty big fan of the blog...". The data from the Spring 2009 UVa course in Table 3, from a semester when no blog participation incentives were in place, bear out this general feeling from students. Incentives certainly promote technology adoption, and they therefore help to create the user base necessary for social tools like the blog to be truly beneficial to students as a collaboration tool.

#### **6.2 A conceptual framework for technology adoption**

The diffusion of innovation framework provides a useful lens through which to view the HigherEd 2.0 program and our evaluation data. Rogers' theory states (Rogers, 2003) that diffusion is "the process by which an innovation is communicated through certain channels over time among members of a social system" (p.5). Rogers further explains the five key characteristics of innovations, and managing these characteristics has a profound effect on the rate of diffusion: (i) *relative advantage* (RA) is the value of an innovation compared to available alternatives; (ii) *compatibility* (Cp) captures the consistency of an innovation with the prevailing values of the user community; (iii) *complexity* (Cx) represents the perceived difficulty in using an innovation; (iv) *trialability* (T) refers to an innovation's ability to be used experimentally by the user base; and (v) *observability* (O) denotes the visibility of the innovation's impact on the community. The goal for any innovation is obviously to present high RA, Cp, T, and O, while introducing low Cx.

20 Will-be-set-by-IN-TECH

use, expert perspective, and constant availability. Students report that the value of the video solutions increases as the course material become more conceptually complex (Orange et al.,

Student interview data reveals that the course blog and video solutions are considered extremely valuable by the majority of students. Purdue students offered comments like this about the course blog: "The discussions on the blog were very beneficial to view...I was promptly helped by both Prof. Krousgrill and other students." The virtues of video solutions were also recognized: "Helpful, good explanations, [I] would not get grade I am going to have without [the] videos", or "Very useful, well put together, easy to follow", or "They're super effective". On balance, student qualitative data is overwhelmingly positive about these and other features of the HigherEd 2.0 program. Nonetheless, students also recognize that social tools (such as the course blog) require a critical mass of users in order to deliver maximum value: "It's [the blog] only helpful if everyone uses it." Comments like these have stimulated our use of incentives to promote participation, yielding comments like this one from a Purdue student: "Assigning a portion of the course grade to blog posting made me look at the blog.

Students also describe their experience with technology adoption, specifically their gradual adaptation to using the technology for learning. A UVa student: "I was really resistant to it [using the technology] last semester, but I now see the value and efficiency of it. There's definitely an adjustment period to get comfortable though." Another UVa student: "I didn't have Dr. Berger for Statics last semester, so the blog and related tools forced me to change the way I use technology in support of learning. There was a slight learning curve/adjustment period at the beginning, but now I'm a pretty big fan of the blog...". The data from the Spring 2009 UVa course in Table 3, from a semester when no blog participation incentives were in place, bear out this general feeling from students. Incentives certainly promote technology adoption, and they therefore help to create the user base necessary for social tools like the

The diffusion of innovation framework provides a useful lens through which to view the HigherEd 2.0 program and our evaluation data. Rogers' theory states (Rogers, 2003) that diffusion is "the process by which an innovation is communicated through certain channels over time among members of a social system" (p.5). Rogers further explains the five key characteristics of innovations, and managing these characteristics has a profound effect on the rate of diffusion: (i) *relative advantage* (RA) is the value of an innovation compared to available alternatives; (ii) *compatibility* (Cp) captures the consistency of an innovation with the prevailing values of the user community; (iii) *complexity* (Cx) represents the perceived difficulty in using an innovation; (iv) *trialability* (T) refers to an innovation's ability to be used experimentally by the user base; and (v) *observability* (O) denotes the visibility of the innovation's impact on the community. The goal for any innovation is obviously to present

2011) and the germane cognitive load therefore becomes higher.

Once I was there I wanted to post on the blog."

blog to be truly beneficial to students as a collaboration tool.

**6.2 A conceptual framework for technology adoption**

high RA, Cp, T, and O, while introducing low Cx.

**6.1.3 Interview data**

These characteristics, the goal of any innovation seeking widespread adoption, and the ways the HigherEd 2.0 paradigm expresses them, are shown in Table 4 and discussed in more depth in Sections 6.2.1 and 6.2.2.

Table 4. Summary of HigherEd 2.0 approaches within the diffusion of innovation "5 characteristics" framework (Rogers, 2003). [↑=high value, ↓=low value, − = neutral]

#### **6.2.1 Complexity, trialability and observability**

The shared, collaborative nature of HigherEd 2.0 strategies and resources essentially guarantee that the *trialability* and *observability* goals are met. Students perceive the course materials to be available for usage and experimentation anytime, anywhere, and (because of the public nature of the course blog) the impacts of this educational approach have high visibility among the student population. As described in the Introduction, the *complexity* associated with web 2.0 tools is generally low, but students do initially reserve some skepticism for the SGC components of their work. Because creating multimedia content for their peers typically presents a new approach to learning, students are not immediately confident that they will be able to successfully navigate the technology and produce a quality product. They quickly–and with little training–overcome this perception, but it nonetheless requires careful and frequent encouragement from the instructor to ensure that students understand and exploit the simplicity of current authoring tools.

#### **6.2.2 Relative advantage and compatibility**

In two technology areas (communication via blogging and video solutions), both the *relative advantage* and *compatibility* of the technology are essentially self-evident to students. Video solutions allow students to access expertise in problem solving when they need it, providing both a relative advantage and compatibility. The video solutions are a favorable alternative to time-constrained office hours, offer expertise that a student's peers might not possess, and fit smoothly into both the workflow and digital lifestyle of today's students. Students can simply download the videos to their portable device and use them anytime, anywhere. The course blog, with its RSS technology, platform-independence and robust navigation features, fits easily within students' daily workflow, and these features typically provide a performance advantage over a traditional course management system (CMS). However, for collaboration, students do not always perceive a relative advantage to the threaded discussions on the blog, nor does it always fit with their conception of what collaboration in course work looks like. Our experience has shown that students sometimes prefer synchronous, in-person collaborations with their peers on problem sets and exam preparation. Here, we have

to transform the way they teach, engage students, share information, and collaborate. We suggest further, detailed evaluation of our strategies with diverse student populations, in multiple settings, and a variety of subject areas. Careful studies in real higher education environments, such as semester-long courses, will inform the on-going conversations about thoughtful technology deployment, and will surely suggest lively areas for future research.

HigherEd 2.0: Web 2.0 in Higher Education 89

• ➊ sample lecture video: http://people.virginia.edu/~ejb9z/Media/Sample\_

• ➋ sample video solution: http://people.virginia.edu/~ejb9z/Media/Sample\_

• ➌ sample learning module: http://people.virginia.edu/~ejb9z/Media/

• ➍ sample case study: http://people.virginia.edu/~ejb9z/Media/Sample\_

• ➎ sample use of color in video: http://people.virginia.edu/~ejb9z/Media/

• ➏ video description of HigherEd 2.0 course blog: http://people.virginia.edu/

• ➐ sample student-generated content: http://people.virginia.edu/~ejb9z/

We gratefully acknowledge the financial support of the National Science Foundation under grant DUE-0717820. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We also reserve our greatest thanks for our students, who have engaged deeply in the HigherEd 2.0 program, provided essential feedback, and demonstrated that the

Annis, L. F. (1983). The processes and effects of peer tutoring, *Human Learning: Journal of*

Benware, C. A. & Deci, E. L. (1984). Quality of learning with an active versus passive motivational set, *American Educational Research Journal* 21(4): 755–765. Berger, E. J. (2007). Podcasting in engineering education: A preliminary study of content, student attitudes, and impact, *Innovate–the Journal of Online Education* 4(1). Berger, E. J. (2009). Highered 2.0: Enhanced and video podcasts in engineering mechanics,

Cole, M. (2009). Using wiki technology to support student engagement: lessons from the

DeAngelo, L., Hurtado, S., Pryor, J. H., Kelly, K. R. & Santos, J. L. (2009). *The American*

*college teacher: national norms for the 2007-2008 HERI faculty survey*, Higher Education

**8. Index of embedded hyperlinks**

media/vibrations\_homework

media/merry\_go\_round

Media/DynProb16\_134.mov

**9. Acknowledgment**

**10. References**

Sample\_media/impact\_problems

Sample\_media/p5\_23\_snippet.mov

web 2.0 era empowers everyone to author great content.

*Practical Research & Applications* 2(1): 39–47.

trenches, *Computers & Education* 52: 141–146.

Research Institute, UCLA, Los Angeles.

*Academic Intersections* 1(3).

~ejb9z/Media/Sample\_media/blog\_description.mov

media/lecture\_video\_example.mov

found that *incentives* are usually needed to jump-start student excitement about asynchronous collaboration via the course blog. A small amount of course credit can be awarded to promote participation in blog discussions. Remember also that although students frequently use social technologies, their attitudes about such technology for purely social purposes versus academic purposes can be quite different (Cole, 2009).

The lecture videos, according to student perceptions, possess little relative advantage compared to attending class meetings in person. Moreover, the length of the lecture videos (up to about 40 minutes) is somewhat incompatible with students desires; students would rather navigate directly to the section of the video with which they need help, instead of watching the entire video. While the lecture videos serve students well in specific circumstances (e.g., when they miss class due to illness), they generally are not compatible with student needs and therefore do not enjoy significant usage.

Student-generated content presents perhaps the most challenges to widespread adoption, and the mechanisms are largely about context. Students taking a HigherEd 2.0 course are likely also enrolled in non-HigherEd 2.0 courses as well. The dearth of usage of active learning techniques in higher education is well documented (DeAngelo et al., 2009), so it should be no surprise that the general learning environment often does not challenge students to create and share educational materials for their peers. In addition to students' perceptions about the complexity of producing multimedia materials, they are typically unconvinced of its relative advantage versus other approaches to learning, and often find it incompatible with the prevailing educational environment of which they are a part. Our work has shown that instructors can convince students of the benefits of SGC by explaining SGC pedagogy, modeling clear examples of good multimedia materials (such as video solutions), and providing user-friendly authoring options.

#### **7. Summary and conclusion**

The HigherEd 2.0 program has been deployed since 2006 in engineering education classrooms with strong success. The course blog provides the critical course backbone, from which instructors can serve multimedia content, and on which students can have productive asynchronous discussions. The HigherEd 2.0 paradigm is built upon the scholarship in engineering education, educational psychology, and cognitive science. In particular, it represents a real-world deployment of such powerful learning approaches as the worked-example effect. When placed in a cognitive load theory framework, the HigherEd 2.0 program clearly seeks to optimize cognitive load on the learner by adhering to best practices in multimedia production (Table 1) and blog usage (Table 2), both of which are built upon our extensive experience and evaluation data as well as the scholarship of multimedia pedagogy. When viewed through a diffusion of innovations framework, the HigherEd 2.0 program largely follows Rogers' "5 characteristics" framework (Table 4), with incentives and consistent coaching/reinforcement from the instructor motivating students to begin active use of the technology resources.

As new tools and technologies become available, their use in higher education will continue to evolve. The prevailing trend of increased "socialization" of technology, as well as its commoditization, will make the HigherEd 2.0 paradigm even more relevant in the future. The coming ubiquity of mobile tablet devices presents an essential opportunity for educators 22 Will-be-set-by-IN-TECH

found that *incentives* are usually needed to jump-start student excitement about asynchronous collaboration via the course blog. A small amount of course credit can be awarded to promote participation in blog discussions. Remember also that although students frequently use social technologies, their attitudes about such technology for purely social purposes versus

The lecture videos, according to student perceptions, possess little relative advantage compared to attending class meetings in person. Moreover, the length of the lecture videos (up to about 40 minutes) is somewhat incompatible with students desires; students would rather navigate directly to the section of the video with which they need help, instead of watching the entire video. While the lecture videos serve students well in specific circumstances (e.g., when they miss class due to illness), they generally are not compatible with student needs and

Student-generated content presents perhaps the most challenges to widespread adoption, and the mechanisms are largely about context. Students taking a HigherEd 2.0 course are likely also enrolled in non-HigherEd 2.0 courses as well. The dearth of usage of active learning techniques in higher education is well documented (DeAngelo et al., 2009), so it should be no surprise that the general learning environment often does not challenge students to create and share educational materials for their peers. In addition to students' perceptions about the complexity of producing multimedia materials, they are typically unconvinced of its relative advantage versus other approaches to learning, and often find it incompatible with the prevailing educational environment of which they are a part. Our work has shown that instructors can convince students of the benefits of SGC by explaining SGC pedagogy, modeling clear examples of good multimedia materials (such as video solutions),

The HigherEd 2.0 program has been deployed since 2006 in engineering education classrooms with strong success. The course blog provides the critical course backbone, from which instructors can serve multimedia content, and on which students can have productive asynchronous discussions. The HigherEd 2.0 paradigm is built upon the scholarship in engineering education, educational psychology, and cognitive science. In particular, it represents a real-world deployment of such powerful learning approaches as the worked-example effect. When placed in a cognitive load theory framework, the HigherEd 2.0 program clearly seeks to optimize cognitive load on the learner by adhering to best practices in multimedia production (Table 1) and blog usage (Table 2), both of which are built upon our extensive experience and evaluation data as well as the scholarship of multimedia pedagogy. When viewed through a diffusion of innovations framework, the HigherEd 2.0 program largely follows Rogers' "5 characteristics" framework (Table 4), with incentives and consistent coaching/reinforcement from the instructor motivating students to begin active use

As new tools and technologies become available, their use in higher education will continue to evolve. The prevailing trend of increased "socialization" of technology, as well as its commoditization, will make the HigherEd 2.0 paradigm even more relevant in the future. The coming ubiquity of mobile tablet devices presents an essential opportunity for educators

academic purposes can be quite different (Cole, 2009).

therefore do not enjoy significant usage.

and providing user-friendly authoring options.

**7. Summary and conclusion**

of the technology resources.

to transform the way they teach, engage students, share information, and collaborate. We suggest further, detailed evaluation of our strategies with diverse student populations, in multiple settings, and a variety of subject areas. Careful studies in real higher education environments, such as semester-long courses, will inform the on-going conversations about thoughtful technology deployment, and will surely suggest lively areas for future research.

#### **8. Index of embedded hyperlinks**

#### **9. Acknowledgment**

We gratefully acknowledge the financial support of the National Science Foundation under grant DUE-0717820. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We also reserve our greatest thanks for our students, who have engaged deeply in the HigherEd 2.0 program, provided essential feedback, and demonstrated that the web 2.0 era empowers everyone to author great content.

#### **10. References**

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**6**

*Serbia*

**Multimedia Approach in Teaching Mathematics**

Research on multimedia approach efficiency was carried out using lessons about the definite integral and isometric transformations (line and point reflection, translation and rotation), since these areas are among the basic ones in the fields of mathematical analysis and geometry. These topics are also important because of their presence in the mathematics programmes in the great majority of high schools and faculties, both directly and through their multipurpose character. Consequently, proper approach in presenting these topics is one of the most important segments of teaching mathematics on all educational levels.

In teaching mathematics, it is remarkably important to avoid so-called 'knowledge/ information adoption' as the only way of work. Students often solve problems mechanically, by following the algorithm steps without real awareness of their actual meaning. For example, in case of the definite integral, one of the common problems is that students calculate its value by following steps of the algorithm, without real understanding of its definition, the meaning of upper and lower limit or relation between the definite integral and the size of an area or a volume etc. In case of isometric transformations, it was noticed that students learned what are line and point reflection, translation and rotation, but the problem appeared when they have been faced to the realistic task, where they were supposed to use the adopted knowledge. One of the most important aims of the modern approach in teaching mathematics is to combine information adoption with so-called 'knowledge transfer', i.e. learning through defining the

Mathematics teachers show great interest in visualisation of the mathematical terms and emphasize that visualised lectures are of the great help in abstract thinking in mathematics (Bishop, 1989; Tall, 1986) believes that it is of the major importance to connect the existing pictures that students have on certain terms in order to develop them further and to enable students to accept the further knowledge. Therefore, in teaching mathematics it is necessary to combine the picture method and the definition method in order to improve the existing knowledge and to enlarge it with the new facts, which is one of the points of the cognitive

**1. Introduction**

facts and explaining their interrelations.

theory of multimedia learning (Mayer, 2001, 2005).

**– Examples of Interactive Lessons from**

Marina Milovanović1, Đurđica Takači2 and Aleksandar Milajić3

**Mathematical Analysis and Geometry**

*1Faculty of Real Estate Management, Union University, Belgrade, 2Faculty of Natural Sciences, University of Novi Sad, Novi Sad, 3Faculty of Management in Civil Engeneering, Union University,*

### **Multimedia Approach in Teaching Mathematics – Examples of Interactive Lessons from Mathematical Analysis and Geometry**

Marina Milovanović1, Đurđica Takači2 and Aleksandar Milajić3 *1Faculty of Real Estate Management, Union University, Belgrade, 2Faculty of Natural Sciences, University of Novi Sad, Novi Sad, 3Faculty of Management in Civil Engeneering, Union University, Serbia*

#### **1. Introduction**

26 Will-be-set-by-IN-TECH

92 Interactive Multimedia

Sweller, J. (2010). Element interactivity and intrinsic, extraneous, and germane cognitive load,

Sweller, R. (1988). Cognitive load during problem solving: Effects on learning, *Cognitive*

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Wagner, M. & Gansemer-Topf, A. (2005). Learning by teaching others: a qualitative study exploring the benefits of peer teaching, *Landscape Journal* 24(2): 198–208. Wheeler, S., Yeomans, P. & Wheeler, D. (2008). The good, the bad, and the wiki: evaluating

Yale, M. S., Bennett, D., Brown, C., Zhu, G. & Lu, Y.-H. (2009). Hybrid content delivery

*International Conference on Frontiers in Education*, IEEE Press, Piscataway, NJ.

student-generated content for collaborative learning, *British Journal of Educational*

and learning styles in a computer programming course, *Proceedings of the 39th IEEE*

Sweller, J., Ayres, P. & Kalyuga, S. (2011). *Cognitive Load Theory*, Springer, New York.

*Educational Psychology Review* 22: 123–138. URL: *DOI 10.1007/s10648-010-9128-5*

Harvard University Press, Cambridge.

*Science* 12(2): 257–285.

*Technology* 39(6): 987–995.

Research on multimedia approach efficiency was carried out using lessons about the definite integral and isometric transformations (line and point reflection, translation and rotation), since these areas are among the basic ones in the fields of mathematical analysis and geometry. These topics are also important because of their presence in the mathematics programmes in the great majority of high schools and faculties, both directly and through their multipurpose character. Consequently, proper approach in presenting these topics is one of the most important segments of teaching mathematics on all educational levels.

In teaching mathematics, it is remarkably important to avoid so-called 'knowledge/ information adoption' as the only way of work. Students often solve problems mechanically, by following the algorithm steps without real awareness of their actual meaning. For example, in case of the definite integral, one of the common problems is that students calculate its value by following steps of the algorithm, without real understanding of its definition, the meaning of upper and lower limit or relation between the definite integral and the size of an area or a volume etc. In case of isometric transformations, it was noticed that students learned what are line and point reflection, translation and rotation, but the problem appeared when they have been faced to the realistic task, where they were supposed to use the adopted knowledge. One of the most important aims of the modern approach in teaching mathematics is to combine information adoption with so-called 'knowledge transfer', i.e. learning through defining the facts and explaining their interrelations.

Mathematics teachers show great interest in visualisation of the mathematical terms and emphasize that visualised lectures are of the great help in abstract thinking in mathematics (Bishop, 1989; Tall, 1986) believes that it is of the major importance to connect the existing pictures that students have on certain terms in order to develop them further and to enable students to accept the further knowledge. Therefore, in teaching mathematics it is necessary to combine the picture method and the definition method in order to improve the existing knowledge and to enlarge it with the new facts, which is one of the points of the cognitive theory of multimedia learning (Mayer, 2001, 2005).

Multimedia Approach in Teaching Mathematics – Examples

equation:

integrals

Given problem is show on Figure 12. The area of the rectangle is:

of Interactive Lessons from Mathematical Analysis and Geometry 95

The quadrature of the parabola problem is formulated as follows: *For any given parabola y = x2 and rectangle with nodes A(0,0), B(a, 0), C(a, a2) and D(0, a2), (a > 0 )the parabola divides the*

If we mark the zone under the parabola – limited by sides AB and BC of the rectangle and the arc AC – with S, and the other one with P, our next task is to prove the following

*SPS* :( ) 1:2 , i.e. 1 1 3

Fig. 1. Illustration of given problem taken from the multimedia lesson about the definite

2 Figure 1 is taken from the multimedia lesson about the definite integrals.

The next step is to divide the interval [0, a] of the *Ox* axis in *n* equal parts of length *a/n*, where *n* is a natural number, which should be shown to the students by animation (step-bystep), as shown in Figure 2a. Within each of these intervals, we construct two rectangles:

3 3

2 3 *P aa a* (1)

*SPa* (2)

*rectangle in two zones of which the area of one is twice bigger than the area of the other one.*

Nowadays, use of different kinds of multimedia is largely included in the education because it allows the wider spectrum of possibilities in teaching and learning. Visualisation is very useful in the process of explaining mathematical ideas, abstract terms, theorems, problems etc.

Experience in work with students showed that they are highly interested in modern methods in learning which include all kinds of multimedia, such as educational software, internet, etc.

Modern methods in multimedia approach to learning include the whole range of different possibilities applicable in mathematics lectures for different levels of education and with different levels of interactivity (Deliyiannis et al. 2008a, 2008b; D. Đ. Herceg, 2009; Milovanovic, 2005, 2009; Milovanovic et al., 2011; Takači, et al., 2003, 2004, 2006, 2008). The authors usually work on suggestions on using different kinds of software in education, especially in the field of mathematics: geometry, algebra, numerical analyses etc. (Deliyiannis et al., 2008a, 2008b; D. Đ. Herceg, 2009; Milovanovic, 2005, 2009; Milovanovic et al., 2011), as well as the definite integrals and isometric transformations.

All the above-mentioned resulted in an idea of making applicative software which would be helpful in modern and more interesting approach to the field of teaching mathematics and rising the students' knowledge from the scope of definite integrals and isometric transformation to a higher level. The aim of our research was to recognize the importance of multimedia in the teaching process as well as to examine the students' reaction to this way of learning and teaching. Therefore, we have developed experimental software with multimedia lessons about the definite integral and isometric transformations and tested them in class in order to see how they would affect teaching process and results.

#### **2. Multimedia presentation of given problems from the fields of integrals and isometric transformations**

We would like to emphasize the importance of using computers, i.e. multimedia software in teaching and learning in the both areas, because visual presentation offers much more possibilities. Beside that, using multimedia lessons on isometric transformations enables students to actually see not only the final solution, but also the "movements" that have led to it.

#### **2.1 Multimedia presentation of the area calculation**

The problem of area calculation in the filed of integration calculations lead us directly to the definition of the definite integral. The basis for the defining and calculation of the definite integrals was made by Archimedes1 and his quadrature of the parabola. Because of that, we decided to use modern, multimedia approach in explaining Archimedes' quadrature of the parabola to the students. In order to use PC as a teaching aid, we were led by suggestions of Tall (Tall, 1991), who emphasized the importance of PC in the teaching because of its great possibilities in the scope of visual presentations.

 1 Archimedes of Syracuse (Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

Nowadays, use of different kinds of multimedia is largely included in the education because it allows the wider spectrum of possibilities in teaching and learning. Visualisation is very useful in the process of explaining mathematical ideas, abstract terms, theorems, problems etc.

Experience in work with students showed that they are highly interested in modern methods in learning which include all kinds of multimedia, such as educational software,

Modern methods in multimedia approach to learning include the whole range of different possibilities applicable in mathematics lectures for different levels of education and with different levels of interactivity (Deliyiannis et al. 2008a, 2008b; D. Đ. Herceg, 2009; Milovanovic, 2005, 2009; Milovanovic et al., 2011; Takači, et al., 2003, 2004, 2006, 2008). The authors usually work on suggestions on using different kinds of software in education, especially in the field of mathematics: geometry, algebra, numerical analyses etc. (Deliyiannis et al., 2008a, 2008b; D. Đ. Herceg, 2009; Milovanovic, 2005, 2009; Milovanovic

All the above-mentioned resulted in an idea of making applicative software which would be helpful in modern and more interesting approach to the field of teaching mathematics and rising the students' knowledge from the scope of definite integrals and isometric transformation to a higher level. The aim of our research was to recognize the importance of multimedia in the teaching process as well as to examine the students' reaction to this way of learning and teaching. Therefore, we have developed experimental software with multimedia lessons about the definite integral and isometric transformations and tested

**2. Multimedia presentation of given problems from the fields of integrals and**

We would like to emphasize the importance of using computers, i.e. multimedia software in teaching and learning in the both areas, because visual presentation offers much more possibilities. Beside that, using multimedia lessons on isometric transformations enables students to actually see not only the final solution, but also the "movements" that have led

The problem of area calculation in the filed of integration calculations lead us directly to the definition of the definite integral. The basis for the defining and calculation of the definite integrals was made by Archimedes1 and his quadrature of the parabola. Because of that, we decided to use modern, multimedia approach in explaining Archimedes' quadrature of the parabola to the students. In order to use PC as a teaching aid, we were led by suggestions of Tall (Tall, 1991), who emphasized the importance of PC in the teaching because of its great

1 Archimedes of Syracuse (Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC) was a Greek mathematician,

et al., 2011), as well as the definite integrals and isometric transformations.

them in class in order to see how they would affect teaching process and results.

internet, etc.

**isometric transformations**

**2.1 Multimedia presentation of the area calculation**

possibilities in the scope of visual presentations.

physicist, engineer, inventor, and astronomer.

to it.

The quadrature of the parabola problem is formulated as follows: *For any given parabola y = x2 and rectangle with nodes A(0,0), B(a, 0), C(a, a2) and D(0, a2), (a > 0 )the parabola divides the rectangle in two zones of which the area of one is twice bigger than the area of the other one.*

Given problem is show on Figure 12. The area of the rectangle is:

$$P = a \cdot a^2 = a^3 \tag{1}$$

If we mark the zone under the parabola – limited by sides AB and BC of the rectangle and the arc AC – with S, and the other one with P, our next task is to prove the following equation:

$$S: \{P - S\} = 1: 2 \text{ , i.e. } S = \frac{1}{3}P = \frac{1}{3}a^3 \tag{2}$$

Fig. 1. Illustration of given problem taken from the multimedia lesson about the definite integrals

The next step is to divide the interval [0, a] of the *Ox* axis in *n* equal parts of length *a/n*, where *n* is a natural number, which should be shown to the students by animation (step-bystep), as shown in Figure 2a. Within each of these intervals, we construct two rectangles:

2 Figure 1 is taken from the multimedia lesson about the definite integrals.

Multimedia Approach in Teaching Mathematics – Examples

of Interactive Lessons from Mathematical Analysis and Geometry 97

Fig. 2. Illustrations of the quadrature of the parabola problem taken from the multimedia

lesson about the definite integrals

'circumscribed' one, with upper right vertex on the parabola (the animation of this step is shown on Figure 2b), and 'inscribed' one, with upper left vertex on the parabola (shown on Figure 2d). It is obvious that the first part have no inscribed rectangle. The heights of these rectangles are shown on Figure 2b, and their areas are as follows:

$$\frac{a}{n} \cdot \left(\frac{a}{n}\right)^2 \cdot \frac{a}{n} \cdot \left(2\frac{a}{n}\right)^2 \cdot \dots \cdot \frac{a}{n} \cdot \left((n-1)\frac{a}{n}\right)^2 \cdot \frac{a}{n} \cdot \left(n\frac{a}{n}\right)^2 \dots$$

(Part of this animation is shown on Figure 2c.). Area of each inscribed rectangle is difference between the area of adjoining circumscribed rectangle and the area of 'added' rectangle (part of this animation is shown on Figures 2d and 2e).

It is obvious that:

*P SP U O* (3) Therefore: 33 3 33 3 2 2 32 32 6 6 *aa a aa a S n n n n* , that is: 3 3 33 3 2 2 2 32 6 6 *a a aa a S n n n n* . These inequalities are correct for any given natural number *n*. Since 33 33 2 2 lim( ) lim( ) 0 *n n* 2 2 6 6 *aa aa n n n n* , we can conclude that: 1 1 3 *SPa* (4)

This was the solution of the given problem via numerical method, but we can offer much more by using the multimedia lessons. In animation shown on Figure 3, students can clearly see that with increasing of *n*, i.e. the number of circumscribed and inscribed rectangles, these areas will get closer and closer, until they, according to our intuition and visual perception, both become equal to the area S.

3 3

Led by the similar idea as in previous example, we will try to calculate the area of the curvilinear trapezium (students will see it in animation, as shown in Figure 4a and b, etc.) formed by the graph of the function *y f* ( ), , *x x ab* , the abscissa's segment *a b*, and the two segments of the lines *x = a* and *x = b* making the figure closed (Figure 4a).

If the values 01 1 , ,..., , *n n xx x x* define points on *x* axis as follows:

$$a = \mathbf{x}\_0 < \mathbf{x}\_1 < \dots < \mathbf{x}\_{n-1} < \mathbf{x}\_n = b$$

these points divide interval [*a, b*] into *n* sub-intervals:

$$[\mathfrak{x}\_0, \mathfrak{x}\_1] / [\mathfrak{x}\_1, \mathfrak{x}\_2] , \dots [\mathfrak{x}\_{n-1}, \mathfrak{x}\_n] \ . $$

Therefore, we can name the *n* 1 -plet 01 1 ( , ,..., , ) *n n xx x x* as *division of interval a b*, . For simplification, we will mark it as Π = 01 1 ( , ,..., , ) *n n xx x x* .

If we choose any of these sub-intervals (Figure 4b), for example 1 ( ,) *i i x x* , and if *i* is arbitrary value within that sub-interval, the area of the rectangle whose basis is sub-interval 1 [ ,] *i i x x* and height is ( ). *i f* can be calculated as:

'circumscribed' one, with upper right vertex on the parabola (the animation of this step is shown on Figure 2b), and 'inscribed' one, with upper left vertex on the parabola (shown on Figure 2d). It is obvious that the first part have no inscribed rectangle. The heights of these

> 22 22 , 2 ,..., ( 1) , *aa a a a a a a*

*nn n n n n n n* .

(Part of this animation is shown on Figure 2c.). Area of each inscribed rectangle is difference between the area of adjoining circumscribed rectangle and the area of 'added' rectangle

> 1 1 3 3 3

This was the solution of the given problem via numerical method, but we can offer much more by using the multimedia lessons. In animation shown on Figure 3, students can clearly see that with increasing of *n*, i.e. the number of circumscribed and inscribed rectangles, these areas will get closer and closer, until they, according to our intuition and visual perception,

Led by the similar idea as in previous example, we will try to calculate the area of the curvilinear trapezium (students will see it in animation, as shown in Figure 4a and b, etc.) formed by the graph of the function *y f* ( ), , *x x ab* , the abscissa's segment *a b*, and the

01 1 ... *n n ax x x x b*

*xx xx x x* 01 12 1 , , , ,..., , *n n* .

Therefore, we can name the *n* 1 -plet 01 1 ( , ,..., , ) *n n xx x x* as *division of interval a b*, . For

arbitrary value within that sub-interval, the area of the rectangle whose basis is sub-interval

If we choose any of these sub-intervals (Figure 4b), for example 1 ( ,) *i i x x* , and if *i*

can be calculated as:

*n n*

*P SP U O* (3)

*SPa* (4)

is

3 3 33 3 2 2 2 32 6 6 *a a aa a S n n n n* .

rectangles are shown on Figure 2b, and their areas are as follows:

(part of this animation is shown on Figures 2d and 2e).

33 3 33 3 2 2 32 32 6 6 *aa a aa a S n n n n* , that is:

33 33 2 2 lim( ) lim( ) 0 *n n* 2 2 6 6

*aa aa n n n n*

both become equal to the area S.

1 [ ,] *i i x x* and height is ( ). *i f*

These inequalities are correct for any given natural number *n*.

, we can conclude that:

two segments of the lines *x = a* and *x = b* making the figure closed (Figure 4a).

If the values 01 1 , ,..., , *n n xx x x* define points on *x* axis as follows:

these points divide interval [*a, b*] into *n* sub-intervals:

simplification, we will mark it as Π = 01 1 ( , ,..., , ) *n n xx x x* .

It is obvious that:

Therefore:

Since

Fig. 2. Illustrations of the quadrature of the parabola problem taken from the multimedia lesson about the definite integrals

Multimedia Approach in Teaching Mathematics – Examples

rectangles – figure S – with total area:

1 [ ,] *i ii*

exists 0

values 1 2 ( , ..., ) *n*

(a) (b)

multimedia lesson (step-by-step)

of Interactive Lessons from Mathematical Analysis and Geometry 99

1 ( )( ) *Pf xx i ii i*

If we do the same with every sub-interval 1 [ ,] *i i x x* , *i n* 1,2,..., , we will get the series of

( ) ( )( )

For a given curvilinear trapezium – i.e. for given interval *a b*, and given function *f x*( ) – the shape of figure S depends on division Π = 01 1 ( , ,..., , ) *n n xx x x* and on choice of values

sub-intervals 1 [ ,] *i i x x* , 1,2,..., *i n* , are 'small', shape of figure *S* will be 'very' similar to the

If we mark value of 1 , 1,2,... *iii xxx i n* , than set 1 2 { , ,..., } *n xx x* is finite set of positive numbers, and consequently has the largest element, which we will mark as *d*:

1 2 ( ) max{ , ,..., } *n dd x x x*

If the value of *d* is small enough natural number, it means that sub-intervals are smaller and the division Π is 'fine'. If we introduce new breaking points, *d* gets smaller and smaller so the division gets finer. Consequently – and according to our intuition and multimedia presentation – figure *S* will get more and more similar to the curvilinear trapezium, so we can conclude that following definition of the area of the curvilinear trapezium *F* is valid:

*Definition 1:* Real number S is the area of the curvilinear trapezium F if for every 0

Fig. 5. The second part of solution of the quadrature of the parabola problem taken from the

, such that for every division Π for which *d*( )

in correspondent sub-intervals:

*PS P f x x*

*i ii i*

1

*x x* , 1,2,..., *i n* . Let us mark this n-plet of choices as 1 2 ( , ..., ) *n*

curvilinear trapezium *F* (which are shown on Figures 5 and 6a).

*i*

*n*

(5)

(6)

and for any chosen set of

, there

. If all the

Fig. 3. Illustration of the quadrature of the parabola problem taken from the multimedia lesson (step-by-step)

Fig. 4. The first part of solution of the quadrature of the parabola problem taken from the multimedia lesson (step-by-step)

(b)

Fig. 3. Illustration of the quadrature of the parabola problem taken from the multimedia

(c)

Fig. 4. The first part of solution of the quadrature of the parabola problem taken from the

lesson (step-by-step)

(a)

multimedia lesson (step-by-step)

(a) (b)

$$P\_i = f(\xi\_i)(\mathbf{x}\_i - \mathbf{x}\_{i-1}) \tag{5}$$

If we do the same with every sub-interval 1 [ ,] *i i x x* , *i n* 1,2,..., , we will get the series of rectangles – figure S – with total area:

$$P(S) = \sum\_{i=1}^{n} P\_i = f(\xi\_i)(\mathbf{x}\_i - \mathbf{x}\_{i-1}) \tag{6}$$

For a given curvilinear trapezium – i.e. for given interval *a b*, and given function *f x*( ) – the shape of figure S depends on division Π = 01 1 ( , ,..., , ) *n n xx x x* and on choice of values 1 [ ,] *i ii x x* , 1,2,..., *i n* . Let us mark this n-plet of choices as 1 2 ( , ..., ) *n* . If all the sub-intervals 1 [ ,] *i i x x* , 1,2,..., *i n* , are 'small', shape of figure *S* will be 'very' similar to the curvilinear trapezium *F* (which are shown on Figures 5 and 6a).

If we mark value of 1 , 1,2,... *iii xxx i n* , than set 1 2 { , ,..., } *n xx x* is finite set of positive numbers, and consequently has the largest element, which we will mark as *d*:

$$d = d(\Pi) = \max\{\Delta\mathbf{x}\_1, \Delta\mathbf{x}\_2, \dots, \Delta\mathbf{x}\_n\}$$

If the value of *d* is small enough natural number, it means that sub-intervals are smaller and the division Π is 'fine'. If we introduce new breaking points, *d* gets smaller and smaller so the division gets finer. Consequently – and according to our intuition and multimedia presentation – figure *S* will get more and more similar to the curvilinear trapezium, so we can conclude that following definition of the area of the curvilinear trapezium *F* is valid:

*Definition 1:* Real number S is the area of the curvilinear trapezium F if for every 0 , there exists 0 , such that for every division Π for which *d*( ) and for any chosen set of values 1 2 ( , ..., ) *n* in correspondent sub-intervals:

Fig. 5. The second part of solution of the quadrature of the parabola problem taken from the multimedia lesson (step-by-step)

Multimedia Approach in Teaching Mathematics – Examples

(a) (b)

determining the area of a plane figure

Slant height of the cone is defined as line:

**Example:** Determining the volume of body by revolving.

Therefore, according to the formula for calculus of volume:

0

*V x dx*

around *Ox* or *Oy* axis.

given problem step-by-step. The final result is illustrated on Figure 8b.

of Interactive Lessons from Mathematical Analysis and Geometry 101

**Solution:** Animation shows the graphs of given curves (Figure 8a) and their intersection points *A(1, 1), O(0, 0)* and *B(-1, 1)* (with numerical and graphic presentation). We can see that given figure consists of two identical parts. The next step in the animation is solving the

Fig. 8. Animation parts which represents the graphs of the given task and solution in

*Task*: Determine the volume of a right circular cone with altitude *h* and base radius *r*.

Numerical solution of given problem is also shown step-by-step, by using animation.

Animation parts which represents the given task and the triangle revolution.

2

*h h*

In a similar way as shown for determining the area of a given figure, we used multimedia animations to explain application of the determined integrals for calculus of volumes of solids, as well as volumes of solids of revolution obtained by revolving a plane figure

**Solution:** The cone is generated by revolving the right-angled triangle *OAB* around the *Ox*axis (Figure 9a), which can be clearly shown by using animation (Figures 9b, and 9c).

> *y x tg x h*

> > 2

( ) 3 3 0 *h r r x hr h*

.

*r*

2 3 2

Fig. 6. *Definition 1*: Area of the curvilinear trapezium F – visualization method

$$\left| \sum\_{i=1}^{n} f(\xi\_i)(\mathbf{x}\_i - \mathbf{x}\_{i-1}) - \mathbf{S} \right| < \varepsilon \tag{7}$$

Or it is simplified as:

$$P(F) = S = \lim\_{d \to 0} \sum\_{i=1}^{n} f(\xi\_i)(\mathbf{x}\_i - \mathbf{x}\_{i-1}) \text{ i.e.} \tag{8}$$

*Definition 2:* Let the real-valued function *f* be defined on interval *a b*, . The real number *I* is definite integral of the function *f* on the interval *a b*, if for every 0 *,* there exists 0 , such that for every division П = 01 1 ( , ,..., , ), *n n xx x x* 01 1 ... *n n ax x x x b* for which 1 2 ( ) max{ , ,..., } *n dd x x x* and for any chosen set of values 1 2 ( , ..., ) *n ,*  1 [ ,] *i ii x x* , *i n* 1,2,..., (animation, Figure 7 shows further development step-by-step).

Fig. 7. *Definition 2:* Area of the curvilinear trapezium F – visualization method

With numerous visual presentations, animations, illustrations and examples we can also introduce and explain *integrability*, *integral sum*, *integrand*, *limits of integration*, *Newton-Leibniz formula*, *applications of integrals,* etc.

**Example:** Determining the area of plane figure.

*Task:* Determine the area of the figure in the *xOy* plane bounded by the curves *x – y = 0* and *x – y3 = 0*.

Fig. 6. *Definition 1*: Area of the curvilinear trapezium F – visualization method

1

definite integral of the function *f* on the interval *a b*, if for every 0

Fig. 7. *Definition 2:* Area of the curvilinear trapezium F – visualization method

*i*

Or it is simplified as:

1 [ ,] *i ii*

*x – y3 = 0*.

1 2 ( ) max{ , ,..., } *n dd x x x*

(a) (b)

*formula*, *applications of integrals,* etc.

**Example:** Determining the area of plane figure.

*n*

1

1 0 1

*ii i d i*

*Definition 2:* Let the real-valued function *f* be defined on interval *a b*, . The real number *I* is

such that for every division П = 01 1 ( , ,..., , ), *n n xx x x* 01 1 ... *n n ax x x x b* for which

With numerous visual presentations, animations, illustrations and examples we can also introduce and explain *integrability*, *integral sum*, *integrand*, *limits of integration*, *Newton-Leibniz*

*Task:* Determine the area of the figure in the *xOy* plane bounded by the curves *x – y = 0* and

*x x* , *i n* 1,2,..., (animation, Figure 7 shows further development step-by-step).

(7)

, i.e. (8)

and for any chosen set of values 1 2 ( , ..., ) *n*

*,* there exists 0

 ,

> *,*

( )( )

( ) lim ( )( ) *n*

*PF S f x x*

*f xx S*

*ii i*

**Solution:** Animation shows the graphs of given curves (Figure 8a) and their intersection points *A(1, 1), O(0, 0)* and *B(-1, 1)* (with numerical and graphic presentation). We can see that given figure consists of two identical parts. The next step in the animation is solving the given problem step-by-step. The final result is illustrated on Figure 8b.

Fig. 8. Animation parts which represents the graphs of the given task and solution in determining the area of a plane figure

In a similar way as shown for determining the area of a given figure, we used multimedia animations to explain application of the determined integrals for calculus of volumes of solids, as well as volumes of solids of revolution obtained by revolving a plane figure around *Ox* or *Oy* axis.

**Example:** Determining the volume of body by revolving.

*Task*: Determine the volume of a right circular cone with altitude *h* and base radius *r*. **Solution:** The cone is generated by revolving the right-angled triangle *OAB* around the *Ox*axis (Figure 9a), which can be clearly shown by using animation (Figures 9b, and 9c). Animation parts which represents the given task and the triangle revolution.

Numerical solution of given problem is also shown step-by-step, by using animation.

Slant height of the cone is defined as line:

$$y = \mathbf{x} \cdot t \mathbf{g} \,\alpha = \frac{r}{h} \cdot \mathbf{x}$$

Therefore, according to the formula for calculus of volume:

$$V = \pi \int\_0^h (\frac{r}{h} \cdot \mathbf{x})^2 d\mathbf{x} = \frac{\pi \cdot r^2}{h^2} \cdot \frac{\mathbf{x}^3}{3} \bigg|\_0^h \qquad = \frac{\pi \cdot h \cdot r^2}{3} \dots$$

Multimedia Approach in Teaching Mathematics – Examples

Fig. 10. Homepage of animation about isometric transformations

**Example:** Basic idea of this example is to help students to see, comprehend and implement the line reflection in different cases before giving them the exact definition. In the first task (Figure 11a), students were asked to recognize the common characteristic of given figures and to find which two of them do not belong in the group. After that, the solution was

of Interactive Lessons from Mathematical Analysis and Geometry 103

Fig. 9. Animation parts which represents the graphs of the given task and solution in determine the volume of a right circular cone with altitude *h* and base radius *r*

#### **2.2 Assorted examples and problems from multimedia lectures on isometric transformations**

Our lectures on isometric transformations (homepage shown on Figure 10), consist of four units: line and point reflection, translation and rotation. Every transformation is presented by the following chapters:

Since the field of isometric transformations is very broad, we have conducted the research in only one area – line reflection. Multimedia lessons in line reflection are presented here by characteristic examples which have enabled us to use different, multimedia approach than in classical lectures.

Great advantage of multimedia lessons is particularly evident in chapter *Basics*, because the definitions of line reflection, axis of symmetry etc. are not only 'given' (written and drawn), but also illustrated by numerous animations which show 'movements', i.e. isometric transformation. We have also paid special attention to enabling students to find out the solutions by themselves.

(a) (b)

(c)

Fig. 9. Animation parts which represents the graphs of the given task and solution in determine the volume of a right circular cone with altitude *h* and base radius *r*

**2.2 Assorted examples and problems from multimedia lectures on isometric**

Our lectures on isometric transformations (homepage shown on Figure 10), consist of four units: line and point reflection, translation and rotation. Every transformation is presented

Since the field of isometric transformations is very broad, we have conducted the research in only one area – line reflection. Multimedia lessons in line reflection are presented here by characteristic examples which have enabled us to use different, multimedia approach than

Great advantage of multimedia lessons is particularly evident in chapter *Basics*, because the definitions of line reflection, axis of symmetry etc. are not only 'given' (written and drawn), but also illustrated by numerous animations which show 'movements', i.e. isometric transformation. We have also paid special attention to enabling students to find out the

**transformations**

1. Basics 2. Examples

4. Exercises 5. Problems

in classical lectures.

solutions by themselves.

by the following chapters:

3. Some characteristics

6. Examples from everyday life

Fig. 10. Homepage of animation about isometric transformations

**Example:** Basic idea of this example is to help students to see, comprehend and implement the line reflection in different cases before giving them the exact definition. In the first task (Figure 11a), students were asked to recognize the common characteristic of given figures and to find which two of them do not belong in the group. After that, the solution was

Multimedia Approach in Teaching Mathematics – Examples

(a) (b)

transformations. (Figure 13)

of Interactive Lessons from Mathematical Analysis and Geometry 105

In chapter 'Examples', we have used series of animations to ask different questions, such as which of the given figures are symmetrical, how many axis of symmetry they have etc. All the answers were illustrated by complete multimedia presentations of isometric

Fig. 13. Example of symmetrical figure (b), which is obtained by animated isometric

Chapter 'Exercises' offers variety of multimedia Q/A, quizzes and tests with purpose of

One of the examples is presented in Figure 14. Its purpose was to use an interesting example

Chapter 'Problems' include variety of different interesting tasks, ranged from easier to more complex ones. All tasks are solved, majority with complete solutions and some with instruction how to solve them. The main idea was to enable student to get to the right solution individually, before he or she see it on the screen. Animations do not offer complete solution instantly, but gradually, step-by-step. Some of tasks are typical ones, as in traditional classes, but there are also non-standard tasks taken from the mathematics

transformation of picture (a), point by point, by pressing the button (left)

from the everyday life in order to enable students to resume their knowledge.

Fig. 14. Task: How many axes of symmetry have these flags?

competitions.

resuming and exercising adopted knowledge.

offered (for all the figures except the third and the last one) in which it was shown that there is at least one line along which we can fold the paper and every point from one side would fall on corresponding point on the other side (Figure 11b).

Fig. 11.

In next step, students were asked to look at the figures shown on Figure 12a and to find out if there is an axis of symmetry for any given pair of figures. Afther that, multimedia animation led them to the correct answer (Figure 12b)

offered (for all the figures except the third and the last one) in which it was shown that there is at least one line along which we can fold the paper and every point from one side would

In next step, students were asked to look at the figures shown on Figure 12a and to find out if there is an axis of symmetry for any given pair of figures. Afther that, multimedia

fall on corresponding point on the other side (Figure 11b).

(a)

(b)

animation led them to the correct answer (Figure 12b)

(a)

(b)

Fig. 11.

Fig. 12.

In chapter 'Examples', we have used series of animations to ask different questions, such as which of the given figures are symmetrical, how many axis of symmetry they have etc. All the answers were illustrated by complete multimedia presentations of isometric transformations. (Figure 13)

Fig. 13. Example of symmetrical figure (b), which is obtained by animated isometric transformation of picture (a), point by point, by pressing the button (left)

Chapter 'Exercises' offers variety of multimedia Q/A, quizzes and tests with purpose of resuming and exercising adopted knowledge.

One of the examples is presented in Figure 14. Its purpose was to use an interesting example from the everyday life in order to enable students to resume their knowledge.

Fig. 14. Task: How many axes of symmetry have these flags?

Chapter 'Problems' include variety of different interesting tasks, ranged from easier to more complex ones. All tasks are solved, majority with complete solutions and some with instruction how to solve them. The main idea was to enable student to get to the right solution individually, before he or she see it on the screen. Animations do not offer complete solution instantly, but gradually, step-by-step. Some of tasks are typical ones, as in traditional classes, but there are also non-standard tasks taken from the mathematics competitions.

Multimedia Approach in Teaching Mathematics – Examples

of Interactive Lessons from Mathematical Analysis and Geometry 107

In teaching mathematics, we are sometimes supposed to explain abstract terms that rarely or hardly can be seen in reality, but students often ask for proof that theory can be seen and implemented in everyday life. With help of multimedia aids, we can show numerous

(b)

Fig. 16. Examples of symmetry in everyday life: (a) This photograph of the Taj Mahal has two axes of symmetry; Beside the vertical one, there is also a horizontal one, along the water line; (b) Famous Leonardo da Vinci's drawing The Vitruvian Man is also called Canon of

Thanks to the experiences of some previous researches and results, some of the questions

1. Are there any differences between results of the first group of students, who had traditional lectures (control group – *traditional group*) and the second group, who had multimedia lectures (experimental group – *multimedia group*)? Where were these

2. What do students from the experimental group think about multimedia lectures? Do

3. In students' opinion, where did multimedia learning help them more, in geometry or analysis (based on lessons on the definite integrals and the isometric transformations)? 4. Do students think it is easier to understand and learn the matter individually and

The research was conducted on two groups of 50 students, divided on subgroups of 25, at two faculties: the Faculty of Architecture and the Faculty of Civil Construction Management of the UNION University, Belgrade, Serbia. In both cases, one subgroup, consisting of 25 students,

Proportions or Proportions of Man. It shows the symmetry of the human body

**3. Research methodology**

(a)

**3.1 Aim and questions of the research**

during this research were as follows:

differences the most obvious?

**3.2 Participants of the research**

they prefer this or traditional way and why?

during the classes by multimedia lectures?

examples of symmetry in architecture, art, nature, psychology, religion, etc.

**Example:** Two billiard balls, A and B, are on the rectangular table, as shown on figure. How should we hit the ball A if we want it to strike all four rails before hitting the ball B?

Let us mark the rectangle (billiard table) as XYUV, and A'=IXV(A), A''=IXY(A'), B'=IUV(B), B''=IUY(B').

(Multimedia presentation shows transformation step by step.)

If we mark the intersection of lines A''B'' and XY as M, the intersection of lines A''B'' and UY as N, the intersection of lines A'M and XV as P, and the intersection of lines B'N and UV as Q, it can be noticed that the following angles are equal: APV=A'PV=XPM, PMX=XMA''=NMY, MNY=B''NU=UNQ, and NQU=B'QV=VQB.

(Multimedia presentation shows drawing of every line and their intersections, i.e. above mentioned points)

Therefore, ball in point A should be hit in such a way that would send it through points P, M, N and Q, and it will finally hit the ball in point B.

Fig. 15. Solution of a given task given by the multimedia animation

**Example:** Two billiard balls, A and B, are on the rectangular table, as shown on figure. How

Let us mark the rectangle (billiard table) as XYUV, and A'=IXV(A), A''=IXY(A'), B'=IUV(B),

If we mark the intersection of lines A''B'' and XY as M, the intersection of lines A''B'' and UY as N, the intersection of lines A'M and XV as P, and the intersection of lines B'N and UV as Q, it can be noticed that the following angles are equal: APV=A'PV=XPM,

(Multimedia presentation shows drawing of every line and their intersections, i.e. above

Therefore, ball in point A should be hit in such a way that would send it through points P,

should we hit the ball A if we want it to strike all four rails before hitting the ball B?

(Multimedia presentation shows transformation step by step.)

(a)

(b)

PMX=XMA''=NMY, MNY=B''NU=UNQ, and NQU=B'QV=VQB.

M, N and Q, and it will finally hit the ball in point B.

(c)

Fig. 15. Solution of a given task given by the multimedia animation

B''=IUY(B').

mentioned points)

In teaching mathematics, we are sometimes supposed to explain abstract terms that rarely or hardly can be seen in reality, but students often ask for proof that theory can be seen and implemented in everyday life. With help of multimedia aids, we can show numerous examples of symmetry in architecture, art, nature, psychology, religion, etc.

Fig. 16. Examples of symmetry in everyday life: (a) This photograph of the Taj Mahal has two axes of symmetry; Beside the vertical one, there is also a horizontal one, along the water line; (b) Famous Leonardo da Vinci's drawing The Vitruvian Man is also called Canon of Proportions or Proportions of Man. It shows the symmetry of the human body

### **3. Research methodology**

#### **3.1 Aim and questions of the research**

Thanks to the experiences of some previous researches and results, some of the questions during this research were as follows:

#### **3.2 Participants of the research**

(c

The research was conducted on two groups of 50 students, divided on subgroups of 25, at two faculties: the Faculty of Architecture and the Faculty of Civil Construction Management of the UNION University, Belgrade, Serbia. In both cases, one subgroup, consisting of 25 students,

Multimedia Approach in Teaching Mathematics – Examples

How many axes of symmetry are there in the circle?

3. How many axes of symmetry are there in the following alphabet letters?

4. For given sharp angle *aOb* and point *C*, find the points *A* and *B*, such that *A* belongs to *Oa*, *B* belongs to *Ob*, and the triangle *ABC* has the minimal possible circumference.

5. For given line *p* and points *A* and *B* on the same side of *p*, find the point *P* on the line *p*, so the ray of light which starts in point *A* hits the point *P* and passes through the point

*B*. (Use the fact that entering ray is symmetrical with reflected ray.)

Immovable lines in the line reflection are:

 Circle Line

 Kite 2. Line reflection:

> a. 2 b. 4

Fig. 17.

Fig. 18.

 Parallelogram Isosceles triangle Isosceles trapezium

a. is a plane isometry b. is not a plane isometry

c. Infinite number

a. parallel with axis b. intersect axis

c. rectangular with axis

of Interactive Lessons from Mathematical Analysis and Geometry 109

had traditional lectures while the other one had multimedia lectures. Groups were formed randomly, so the previous knowledge needed for the lectures about limited integrals and isometric was practically the same, which was confirmed by pre-test. The pre-test included tasks from the area of the continuous functions (analysis and graph-drawing), as well as the tasks about the basic figures in analytic geometry (circle, ellipse, parabola etc.). Average score of this pre-test was practically equal in these groups (I: 72.35, II: 71.25 out of 100).

#### **3.3 Methods, techniques and apparatus**

Lectures in both groups included exactly the same information on given topics, i.e. axioms, theorems, examples and tasks. It is important to emphasize that the lecturer and the number of classes were the same, too. The main information source for the multimedia group was software created in Macromedia Flash 10.0, which is proven to be very successful and illustrative for creating multimedia applications in mathematics lectures (Bakhoum, 2008). Our multimedia lecturing material was created in accordance with methodical approach, i.e. cognitive theory of multimedia learning (Mayer, 2001, 2005), as well as with principles of multimedia teaching and design based on researches in the field of teaching mathematics (Atkinson, 2005). This material includes a large number of dynamic and graphic presentations of definitions, theorems, characteristics, examples and tests from the area of the definite integrals based on step-by-step method with accent on visualisation. An important quality of making one's own multimedia lectures is the possibility of creating combination of traditional lecture and multimedia support in those areas we have mentioned as the 'weak links' (definite integral definition, area, volume, etc.)

After the lectures were finished, all students had the same tests consisting of tasks on definite integrals and isometric transformations. Besides that, students were interviewed after the classes and transcripts of the most characteristic opinions are also included here. In order to get as objective results as possible, participation in the interviews was voluntary and anonymous, and the interviewer was not a member of the teaching staff at any of the faculties.

#### **Test 1 – Definite integral:**

#### **Test 2 – Isometric transformations:**

	- Ray

had traditional lectures while the other one had multimedia lectures. Groups were formed randomly, so the previous knowledge needed for the lectures about limited integrals and isometric was practically the same, which was confirmed by pre-test. The pre-test included tasks from the area of the continuous functions (analysis and graph-drawing), as well as the tasks about the basic figures in analytic geometry (circle, ellipse, parabola etc.). Average score

Lectures in both groups included exactly the same information on given topics, i.e. axioms, theorems, examples and tasks. It is important to emphasize that the lecturer and the number of classes were the same, too. The main information source for the multimedia group was software created in Macromedia Flash 10.0, which is proven to be very successful and illustrative for creating multimedia applications in mathematics lectures (Bakhoum, 2008). Our multimedia lecturing material was created in accordance with methodical approach, i.e. cognitive theory of multimedia learning (Mayer, 2001, 2005), as well as with principles of multimedia teaching and design based on researches in the field of teaching mathematics (Atkinson, 2005). This material includes a large number of dynamic and graphic presentations of definitions, theorems, characteristics, examples and tests from the area of the definite integrals based on step-by-step method with accent on visualisation. An important quality of making one's own multimedia lectures is the possibility of creating combination of traditional lecture and multimedia support in those areas we have

After the lectures were finished, all students had the same tests consisting of tasks on definite integrals and isometric transformations. Besides that, students were interviewed after the classes and transcripts of the most characteristic opinions are also included here. In order to get as objective results as possible, participation in the interviews was voluntary and anonymous, and the interviewer was not a member of the teaching staff at any of the faculties.

1. Use Archimedes' method to determine the area of plane figure bounded by the *Ox*-axis,

2

sin cos *x xdx*

4. Determine the area of the plane figure F bounded by *Oy* axis, graph of the function

5. Determine the volume of the body of revolution obtained by rotating figure F bounded

.

of this pre-test was practically equal in these groups (I: 72.35, II: 71.25 out of 100).

mentioned as the 'weak links' (definite integral definition, area, volume, etc.)

/4

0

by parabolas 2 *y* 8*x* and 2 *x y* in the *xOy* plane around the *Oy* axis.

line *x a* , and part of the curve 3 *y x* for 0 *x a* .

2 *y x* and the tangent on this graph in point (1, 1).

1. Which of the following figures are symmetrical:

**3.3 Methods, techniques and apparatus**

**Test 1 – Definite integral:**

2. Write the definite integral definition.

3. Determine the definite integral

**Test 2 – Isometric transformations:**

Ray

	- a. is a plane isometry
	- b. is not a plane isometry
	- How many axes of symmetry are there in the circle?
	- a. 2
	- b. 4
	- c. Infinite number

Immovable lines in the line reflection are:

Fig. 17.

4. For given sharp angle *aOb* and point *C*, find the points *A* and *B*, such that *A* belongs to *Oa*, *B* belongs to *Ob*, and the triangle *ABC* has the minimal possible circumference.

Fig. 18.

5. For given line *p* and points *A* and *B* on the same side of *p*, find the point *P* on the line *p*, so the ray of light which starts in point *A* hits the point *P* and passes through the point *B*. (Use the fact that entering ray is symmetrical with reflected ray.)

Multimedia Approach in Teaching Mathematics – Examples

of Interactive Lessons from Mathematical Analysis and Geometry 111

Fig. 21. Total average test scores for (a) Test 1 (definite integral) and (b) Test 2 (line reflection) for traditional and multimedia groups at the Faculty of Civil Construction Management

Fig. 22. Average test scores by single tasks for (a) Test 1 (definite integral) and (b) Test 2 (line

When asked whether they prefer classical or multimedia way of learning, 12% (3 students) answered classical and 82% (22 students) answered multimedia at the Faculty of Architecture, while at the Faculty of Civil Construction Management 20% (5 students) answered classical and 80% (20 students) answered multimedia, explaining it with the

reflection) for traditional and multimedia groups at the faculty of Architecture

following reasons:

Test scores by tasks for all groups are given in Figures 22 and 23.

Fig. 19.

Both tests were scored within the interval from 0 to 100 (20 points per task) and the average scores in both tests separately were calculated for the traditional and the multimedia group.

Results were analysed with *Student's t-test for independent samples* using SPSS (version 10.0) software. The result was considered significant if the probability *p* was less than 0.05.

#### **3.4 Results**

At the Faculty of Architecture, average score on Test 1 (definite integrals) in the traditional group was 67.75 with standard deviation 14.51, and in the multimedia group, average score was 83.21 with standard deviation 15.01. Average score on Test 2 (line reflection) in the traditional group was 76.04 with standard deviation 15.25, and in the multimedia group, average score was 87.92 with standard deviation 12.5. (Figure 20)

Fig. 20. Total average test scores for (a) Test 1 (definite integral) and (b) Test 2 (line reflection) for traditional and multimedia groups at the faculty of Architecture

At the Faculty of Civil Construction Management, average score on Test 1 (definite integrals) in the traditional group was 60.04 with standard deviation 16.2, and in the multimedia group, average score was 76.37 with standard deviation 19.13. Average score on Test 2 (line reflection) in the traditional group was 72.21 with standard deviation 17.32, and in the multimedia group, average score was 84.37 with standard deviation 15.27. (Figure 21)

In all groups, statistical comparison with t-test for two independent samples showed that multimedia groups had remarkably higher score in comparison with the traditional groups, with statistical significance p < 0.05.

Both tests were scored within the interval from 0 to 100 (20 points per task) and the average scores in both tests separately were calculated for the traditional and the multimedia group. Results were analysed with *Student's t-test for independent samples* using SPSS (version 10.0)

At the Faculty of Architecture, average score on Test 1 (definite integrals) in the traditional group was 67.75 with standard deviation 14.51, and in the multimedia group, average score was 83.21 with standard deviation 15.01. Average score on Test 2 (line reflection) in the traditional group was 76.04 with standard deviation 15.25, and in the multimedia group,

software. The result was considered significant if the probability *p* was less than 0.05.