introduction
In Houghton’s (1989) review of educational paradigms, he highlights the gaining importance of chaos theory. Chaos theory is often characterized by the term non-linear. Chaos theory can be found in many disciplines; in structural engineering, the behaviour of a structure under earthquake loads is often seen in terms of non-linear behaviour. Another characteristic of chaos theory is unpredictability. The implications for educational theory, as Houghton suggests, is that we have a realistic model for what happens in highly interactive systems.
If the process of teaching and learning is seen as a highly interactive environment, then the parallels to chaos theory can be easily seen. The nature of a lecture can change when a student asks a question. This results in a non-linear learning environment. Students affect how something is taught by their own unique ways of understanding. Houghton (1989) suggests that the use of computers in education is supported by chaos theory. He suggests that computers should play a significant and active role with learning. Chaos theory not only supports the concept of using computers in education, it suggests that with non-linear programming (e.g., hypertext), education can change from the traditional linear format to a non-linear methodology that is alive and vibrant.
background
Many theorists have been focusing on the importance of introducing non-linear methodology into education. Often, computer software is seen as the ideal method for implementing such a methodology. Saint-Germain (1997) identified teaching styles as an important factor in the use of multimedia and non-linear capable teaching software. She notes that previous conceptions of non-linear education were centered on developing software that complemented teaching styles rather than the other way around.
Gardner (1993), in his theory of multiple intelligences, theorized that people learn in different ways and, traditionally, education is not equally effective for all people, as it tends to favour certain types of learning. Lampropoulou (2001) recognizes the potential of computer-based or computer-assisted learning to address this concern. Well thought-out multimedia instruction could focus on more of Gardner’s learning styles than traditional lectures and improve access to education. Computer use could not only help to improve student understanding, but also allow traditionally marginalized groups to access education in a manner that might be better suited to their learning style. Lampropoulou describes learning as often being of a non-linear nature and sees multimedia as one of the possible applications of computer technology, with a great potential to learning.
Boger-Mehall (1997), drawing on the work of Spiro, Feltovich, Jacobson and Coulson (1992), supports the use of a non-linear teaching methodology for teaching complex, ill-structured subject matter. The basis of this approach is rooted in the theory of cognitive flexibility. This theory is based on constructivism and the idea that students construct their own learning based on their own experiences. Spiro et al. suggest that when the subject matter is complex, traditional linear instruction may be ineffective (Boger-Mehall, 1997). The traditional linear instruction model will cause an oversimplification of key factors, resulting in a student’s difficulty in transferring this knowledge into a new situation. She suggests that the goal of many professional educational programs is to help students transfer their knowledge into new and unique circumstances.
The use of computers to facilitate non-linear instruction is seen to be a natural progression. This approach is wellsuited to the subject of mechanics of materials, where the application ofprinciples is needed for success. The various components of mechanics of materials are not organized in a linear manner. Often, components from one theory are needed to explain another theory, and vice-versa. Cognitive flexibility theory provides a potential solution to the problem. Again, the use of computers is seen as the ideal mechanism to accomplish this. This process can be explained with the following statement:
Ill-structured aspects of knowledge pose problems for advanced knowledge acquisition that are remedied by the principles of Cognitive Flexibility Theory. This cognitive theory of learning is systematically applied to an instructional theory, Random Access Instruction, which in turn guides the design of non-linear computer learning environments we refer to as Cognitive Flexibility Hypertexts. (Spiro et al., 1992, p. 59)
Ashmann (2000) also supports the use of non-linear computer-based methods for displaying information. He sees the main advantage as the ability of the reader to choose which links to follow in whatever order seems logical. Ashmann focuses on the American science standards, where he notes that displaying information in a list leads to the implicit assumption that the first listed item is the most important. The use of a Web-based system with appropriate graphics and hyperlinks can eliminate this assumption and allow readers to view the material in the manner that seems most appropriate for them. Ashmann continues by indicating that the use of colour on the graphics can lead to important connections.
Unfortunately, the computer-based options for supplementing the teaching and learning of mechanics topics are primarily linear in nature, and most are geared towards a very basic level. Existing computer-based resources can be broken into three distinct types: Web-based information, computer algebra systems and specific computer programs. Web-based information consists of Web sites on the teaching of mechanics (e.g., Fanous & Billingsley, 2003) Web sites to accompany textbooks (e.g., Prentice Hall, 2003). Hillsman and Tomovic (1995) support the use of computer algebra systems to teach beam deflections. They suggest that generic computer algebra systems such as Mathematica, MathCad and Maple can be used to speed up the calculation process and allow students to examine more beam configurations within the same amount of time. Blackmon and Fenves (1997) have developed the STructural Analysis Resource (STAR) tool. This tool allows students to construct the objects needed for any given mechanics of materials problem, and then the STAR tool will create a spreadsheet or the input file for a computer algebra system. This allows students to concentrate on the concepts rather than the mathematics. Other specific computer programs for teaching and learning mechanics include Dr. Beam (Miller, 2001), MDSolids (Philpot, 2001), Statics Tutor (DeVore, 2000) and a concrete beam design program (Rostom, 2003).
Perhaps the most significant research in this area has been conducted by Shepherdson (2001). Her research has focused on the fundamental concern that students lack adequate understanding of the basic concepts. Her focus was in the area of structural mechanics. Shepherdson developed a new software program to help combat this problem, but her resulting software is still linear in nature and aimed at a very introductory level. While Shepherdson’s software is intended to emulate a tutor, it provides little true interaction and behaves in a manner that leads the student through a predetermined path.
Other research efforts available within engineering education not specific to mechanics ofmaterials provide an insight into current research into computer-based engineering education. One of the current trends is towards the development of “tutor” systems that attempt to emulate a live tutor. Zaitseva and Zakis (1997) have been focusing on creating tutoring systems. They also recognize the importance of aesthetics in the design of the software. They specify the need to develop both a model of the subject matter as well as a model of a specialist. They stress that it is important to identify what knowledge is needed for someone to be a considered a specialist in a particular subject and to include all of the relevant material in the design of the software.
Scott and Stone (1998) have been experimenting with computer-based tutorial systems that ask students to solve carefully selected problems and attempt to determine the type of error that occurred in wrong answers by codifying typical errors. Their research has been in the area of engineering dynamics. They recognize that there are many cognitive steps leading from the problem to the solution. They look to provide an explanation to students of why something is wrong, not just that it is wrong. Questions are designed to “trap” common misconceptions. The main problem with this approach is that it does not catch all possible errors, and only focuses on the statistically most common errors.
Zywno, Brimley and White (2000) have been conducting research on the effectiveness of multimedia courseware. Their research indicates students are more motivated and have superior achievement to that previously obtained using traditional methods of instruction for the same course content. In addition, they note that these results are not unique to the specific program that they studied; a review of student work from other similar programs indicates similar results. They also note that there is a relationship between the non-linear nature of student access to online material and the improvements in comprehension of theory.
main thrust of article
The Beam Analysis Tool (BAT), developed by Burrage (2004), was designed to address a new approach to teaching the mathematically based courses that make up the fundamentals for anyone working within the field of civil engineering. The approach focuses on increasing the level of students’ understanding of the concepts at all levels of a problem, from introductory concepts to more advanced.
The teaching method centers on the use of computer programs to supplement the traditional lecture. The online program allows students to work at their own pace in a non-linear manner. As the software is used individually, students are able to focus on the areas they find the most challenging. Additionally, the software is intended to allow students to improve their understanding of the relationship between the description of the problem and the mathematical formulation. Secondary concepts can also be reinforced, particularly as there is extensive use of graphical representations of the problem.
The primary goal in the design of the software was to allow students to explore the various stages in the beam deflection process. This recognizes that the traditional process of learning the analysis of beam deflection relies on the students obtaining knowledge of the various pieces of the process in a methodical, linear manner. The biggest drawback with this approach is that students must obtain a good understanding of each section before they can perform the calculations on the next stage. The software was designed to separate each of the “steps” found in a typical analysis to allow students to focus on learning any section in any order. However, the complete solution to a beam deflection problem will still require all of the components to be carried out in sequence.
As the moment-area method for beam deflections relies on a semi-graphical approach, a programming platform that stressed graphics was essential. Macromedia’s Flash development software enabled both cross-platform development and a strong graphic capability. The development of the Flash programming tools included addition of a scripting language that worked within the Flash environment to allow developers to add more complex interactivity to their movies. This scripting language, called ActionScript, is very similar to the JavaScript programming language common in many Web pages. As the development of Flash and ActionScript continued, ActionScript has evolved into a full object-oriented programming language capable of handling the complex mathematical formulations normally associated with more conventional programming languages.
During the design of the software, careful consideration was given to the user interface. The user interface is the key to creating a program that is both intuitive and visually appealing. Both of these factors are important for creating a program that students actually use. The program was designed to be as easy to use as possible. A computer program that required extensive training to use would not be as effective a teaching tool, since classroom time would be needed to teach the software use in addition to the technical content.
The first screen displayed once the program is launched is a menu screen. This initial screen shows a number of example beam configurations (see Figure 1). The examples shown are all dynamically created so they can be easily changed. This allows the examples to be tailored to any specific course. Students are not limited to the examples shown on the initial screen. They can choose a blank beam to enter directly into the beam configuration editor, or they can select one of the pre-configured examples and then modify it by selecting the configuration button. As the user makes a change to the beam, the program instantly updates the graphical representation of the beam.
When an example is selected from the main screen or the solve button is selected from the configuration screen, the software solves for all of the reactions, internal shear forces and bending moments, and the slopes and deflections at all points on the beam. This information is then shown on a generalized “solved” screen, which shows the loading diagram at the top of the screen with the shear diagram directly below it. The moment diagram lies directly beneath the shear diagram, with the slope diagram and the deflected shape shown below them. No numbers or labels are shown on these diagrams, so students can start to appreciate the relationships between the diagrams (see Figure 2). A complete description of the design process, program specifications, all tools and examples is included in the project report by Burrage (2004).
Figure 1. Software menu screen
Figure 2. Shear equation screen
Evaluation of the software
Two separate groups evaluated the software and answered questions about their use of it: three instructors who actively teach mechanics of materials courses, and 18 second-year civil engineering technology students who had recently completed the course. Instructors were interviewed guided by a set of core questions intended to determine how they used the software and whether they would make use of it in their teaching. They were also asked to assess whether the software addressed areas of typical student difficulty, their awareness of other computer programs for mechanics of materials, and strengths and weaknesses of this software and other programs. Student participants answered questions about their course mark, personal evaluation and how they used the software. Other questions related to difficulties students had learning the materials and recommendations for improvement.
All three instructors used a non-linear thinking approach to review the software and indicated the program would be useful in bridging knowledge gaps for students, allowing the instructor to defer questions relating to prerequisite material to the software, and said they plan to use it as a teaching resource. All but one student felt that the software would have been very useful for helping them learn when they took the mechanics courses. This student’s primary concern is that the software does all of the work for the student. Comments on the difficulty of certain concepts suggest one non-linear approach to use the software—focusing only on the parts of the course giving difficulty. Other comments included the value of being able to quickly construct different models and see the effects immediately and the ability to create complex examples. Many helpful comments related to the interface, display and examples were also given.
conclusion
Results from a review of use of the software indicate that the BAT met its intended goals. The software allows students to view questions related to beam deflection problems in a non-linear manner and effectively links the diagrams to the equations. The software also emphasizes key concepts. However, this software program looks at only a very limited area within mechanics of materials. Using software programs to supplement traditional lectures effectively will require adequate non-linear computer programs for many more areas of mechanics of materials.
However, using this software as a guide, there are three important areas where online software can become a useful addition to the traditional lecture format: increasing the number of examples covered, reducing emphasis on pre-requisite knowledge and as an out-of-class resource for students.
key terms
Actionscript: A full object-oriented scripting language developed by Macromedia to work within the Flash environment that allows developers to add more complex interactivity to their movies. It is capable of handling the complex mathematical formulations normally associated with more conventional programming languages.
Beam Analysis Tool (BAT): A non-linear capable, online software developed using the Macromedia Flash programming environment and which focuses on beam deflection problems using the moment-area method.
Chaos Theory: A theory that describes systems that are apparently disordered or uncertain, but which may have an underlying order. An underlying tenet is that a small change in the initial conditions can drastically change the long-term behaviour of a system.
Cognitive Flexibility: The ability to spontaneously restructure one’s knowledge, in many ways, in adaptive response to radically changing situational demands.
Computer Algebra Systems: Generic mathematical tools that allow users to perform complex calculations and algebraic manipulations of equations.
Constructivism: A learning theory based on the premise that students construct their own learning based on their own experiences.
Flash: A tool that is a registered trademark of Macromedia and originally created to allow developers to create simple animations or movies that could be inserted into a Web page and displayed on any computer that had the appropriate browser plug-in.
Multiple Intelligences: A theory that suggests there are a number of distinct forms of intelligence that each individual possesses in varying degrees. The seven primary forms are: linguistic, musical, logical-mathematical, spatial, body-kinaesthetic, intrapersonal (e.g., insight, metacognition) and interpersonal (e.g., social skills).
Non-Linear Learning: A system in which learners are provided with a variety of options, they choose their own path, different learners can follow different paths, and the outcomes are emergent and cannot be foretold.
Random Access Instruction: The principled use of flexible features inherent in computers to produce non-linear learning environments.
