1 BACKGROUND AND RESEARCH RATIONALE
In our efforts to investigate the problem-solving skills of first-year students enrolled in the engineering-mechanics course at the University of Auckland's Faculty of Engineering, we seek to address the following research questions:
1. Designing a problem-solving heuristic framework --based on how both experts and proficient first-year students solve problems, what are the key elements of engineering-mechanics that might not be visible to beginners?
2. Measure the effect of a structured problem-solving strategy--(a) how can we measure the strategy's affect on an engineering-mechanics' problem-solving skills? and (b) what improvements can we observe as a result?
3. Designing a more effective problem-solving pedagogy--based on the measured results, how could the activities and instructions be made more effective?
Engineers are problem solvers that use their intellectual, analytical and mathematical abilities. This often involves finding effective solutions for problems that occur in our everyday lives. In the 21st century, these skills have become increasingly essential.
The objective of engineering education is to develop conceptual knowledge and problem-solving skills with all students. From a national survey in higher education learning communities within New Zealand, many first-year engineering students commented how they expect to produce a high standard of work, and to learn problem-solving and many facts at university (Cronjie & Coll, 2008). This is consistent with expectations from the University of Auckland, Faculty of Engineering, first-year students (Paton, 2009). Indeed, problem-solving is one of the most significant types of cognitive processing that occurs during teaching and learning in higher education (Frederiksen, 1984; Reif et al, 1976). This view is also supported by Christiansen & Rump (2007), who argued that it is vital that engineering programs address not only problem-solving skills but also the conceptual development from novice beginner to experienced engineer. Consequential, it is important for any in-course interventions to address both the development of conceptual knowledge and problem-solving skills. This will be explained further in subsequent sections.
At the University of Auckland, one of the main goals of the first-year engineering courses is to help students learn how to solve problems effectively. Often lecturers in the first-year engineering courses communicate to their students that they will be tested on their ability to solve problems (Paton, 2009), and about how they are expected to produce clear, effective and structured solutions. In particular, this is communicated further through course materials, online learning activities and assignments. The key goals of first-year engineering courses could be summarised as follows:
1. Students will acquire problem-solving experience and gain familiarity with expert problem-solving strategies.
2. Students will develop both a conceptual and procedural understanding of engineering, physics and mathematics.
3. Students will observe and employ the connections between multiple external representations of physical systems.
4. Beyond repeating theory and formulas, students will understand how to apply their engineering, physics and mathematics knowledge to different situations and challenges.
In an engineering context, a problem is a situation where the steps required are initially unclear or fuzzy, and sometimes remain so throughout the process of finding a solution. The problem might be to answer a question, design a structure to meet specifications or reduce the number of electronic components in a circuit. Often, in order to find the way forward in the process, one needs to make assumptions and approximations that enable the problem to be solved in parts. Clearly, these assumptions need to be relevant to the described situation. The surface features of the problem also need to be avoided, as this can lead to distractions (Ogilvie, 2009). Overall, problem-solving refers to one's efforts, when he or she does not have an obvious way forward to plan, and computes a solution that meets the goal of the problem.
Based on this description, not all learning activities involve problem-solving in this sense. For example, when learning activities have logical steps of rules to follow, they are not strictly referred to as problem-solving. Furthermore, when solving problems, students typically follow procedure-based rules or algorithmic steps through direct instruction from teachers. This mostly occurs during instruction in high-school physics (Roth & Roychoudhury, 1994) and in first-year physics courses at university, whereby, students typical rely on rote and surface learning to learn how to solve physics problems. As suggested by Ogilvie (2009), students simply find and apply the correct equation(s) or strategy to answer well structured algorithmic problems, and this is supported by other research on students' attitudes towards learning physics. Where many believe that when solving physics problems they should apply set procedures or algorithms (Redish et al, 1998), this is different from how experts solve problems in engineering-mechanics. This paper will explore these elements in detail.
What does physics and engineering education research reveal about problem-solving instruction in higher education? Specifically, investigators of problem-solving in physics have found that explicit instruction helps improve a student's performance; for example, the instruction and application of structured problem-solving approaches (Heller & Heller, 1995). Often this is achieved through a cognitive apprenticeship instructional approach (Collins et al, 1991), whereby instructors "think aloud" and explain their interpretation of events and results in real time. In particular, one study (Gaigher et al, 2006), demonstrated how this could be achieved through following seven algorithmic steps. However, other studies have also found that many students never adequately acquire the key problem-solving skills through the usual lectures (Lasry & Aulls, 2007; Redish et al, 1998). Also, it is important to note that Physics Education Research (PER) has found that, if the instructions about problem-solving are explicit (McMillan & Swandener, 1991), then qualitative understanding of quantitative problem situations is improved.
Furthermore, PER has determined that, in mechanics, successful problem-solvers initial translate the problem by drawing diagrams that represent the problem. They visualise the physical situation (Johnson, 2001; Lasry & Aulls, 2007; Larkin & Reif, 1979; Kook & Novak, 1991), but, as described in these papers, this is not the same for novices. Novice problem-solvers appear to search and apply a known algorithm or search for any useful equations based on the given and unknown values (Larkin & Reif, 1979; Ogilvie, 2009; Redish et al, 1998). In addition, as discussed in subsequent sections, novices do not plan for their solutions and use a surface approach to learning problem-solving (Johnson, 2001; Lasry & Aulls, 2007). By trying to master an ever-increasing list of algorithms, this could be characterised by the G.U.E.S.S. method--Givens, Unknowns, Equations, Search and Solve (this is discussed further in subsequent sections). Two possible reasons for this could be the mode of direct instruction typically promotes rote memorisation, rather than meaningful understanding, and/or the students' inexperience at solving problems.
Quality in higher education could be considered as a matter of having clear learning objectives and enabling students to achieve them (Sparkes, 1995). As educators, clearly it is vital that we develop students' skills at solving complex problems. Hence, in order to improve performance in problem-solving skills, teaching and learning should reflect what experts do. For example, many studies suggest that the qualitative aspects of problem-solving through explicit instruction can help develop students' understanding of not only problem-solving, but conceptual knowledge as well (Huffman, 1997). Hence students should be working on the skills they need to learn, in particular their qualitative knowledge.
Given the above perspective, a structured problem-solving heuristic framework was designed and its effects measured. That is, a general-purpose problem-solving strategy was introduced through the tutorials in the dynamics section of the engineering-mechanics first-year engineering course (ENGGEN 121) at the University of Auckland, Faculty of Engineering. However, to investigate such issues explicitly, it is necessary to obtain further information and theoretical models about how experts approach problem-solving in mechanics. In particular, given that much research has been carried out within the PER community, it is important to consider findings from the introductory mechanics curricula. Next, we describe these relevant theories.
2 MODELS OF PROBLEM SOLVING
It is clear that effective and successful problem-solving typically requires both qualitative knowledge and meta-cognitive skills, as listed in table 1.
These cognitive skills are clearly essential for effective problem-solving. As described by Reif (1994), one of the most basic and important abilities is the ability to interpret and connect relevant engineering-physics concepts and principles to problem situations. In addition, other skills are needed, such as the ability to analyse problems, construct multiple external representations of the problem situation, plan a solution and evaluate the final solution. Experts also have strong meta-cognitive skills (Ogilvie, 2009). The following section will explore these elements in more detail with reference to effective pedagogies.
What differentiates experts from problem-solving novices as described in the introduction? Huffman (1997) suggested that experts tend to represent a problem qualitatively …