AC 2011-691: OVERARCHING PROBLEMS IN SOPHOMORE MECHAN-ICS COURSES

Shawn P Gross, Villanova University

Shawn Gross is an Associate Professor of Civil and Environmental Engineering at Villanova University,where he teaches undergraduate and graduate courses in structural engineering and engineering mechan-ics.

David W Dinehart, Villanova University

Professor Assistant Chairman, Department of Civil and Environmental Engineering

Joseph Robert Yost, Villanova University

Joseph Robert Yost is an Associate Professor of Civil and Environmental Engineering at Villanova Uni-versity, where he teaches undergraduate and graduate courses in structural engineering mechanics anddesign

Aleksandra Radlinska, Villanova University

Aleksandra Radlinska is an Assistant Professor of Civil and Environmental Engineering at Villanova Uni-versity. She teaches introductory undergraduate courses on civil engineering materials as well as graduatecourses that relate fundamentals of materials science with applications to civil engineering materials.

c©American Society for Engineering Education, 2011

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Overarching Problems in Sophomore Mechanics Courses

Abstract In 2009, the Department of Civil and Environmental Engineering at Villanova University restructured its sophomore mechanics courses to present topics in a more integrated sequence. Courses in the classical areas of Statics, Mechanics of Solids, and Civil Engineering Materials were repackaged into a pair of four-credit mechanics courses which combine content from these areas. The first course (Mechanics I) integrates elements of Statics and Mechanics of Solids along with a few topics from Civil Engineering Materials. The second course (Mechanics II) integrates the remaining elements of Mechanics of Solids with the majority of Civil Engineering Materials. A key pedagogical component in this integrated curricular restructuring is a structured implementation of problem-based learning: the use of overarching problems. An overarching problem is a common design and/or analysis problem encountered in the discipline that involves numerous basic concepts brought together to compose a more complex problem. For example, students are able to use the Statics concepts of equilibrium and truss analysis, along with the traditional Mechanics of Solids concepts of stress, axial deformation, and factor of safety, and the Civil Engineering Materials concepts of steel material behavior, to analyze a decaying steel truss bridge in need of repair and retrofit. Other overarching problems from these courses include the analysis of a concrete gravity dam, the design of a water tower for a Third World country, analysis and material selection for a prestressed concrete highway bridge, the strengthening of wood I-beams using composite materials, and the 3-D analysis of a highway sign structure under combined loading. Overarching problems are used in two ways within these courses. First, they are used as an introductory context slide at the beginning of each lecture. This facilitates tying the lecture to real world applications and previous and future lectures. Each overarching problem is solved by students in a step-by-step fashion in a 2-1/2 hour recitation period, in which students work through individual mechanics steps in a structured fashion with assistance from instructors as necessary. Overarching problems have many potential pedagogical benefits, including presenting “real” engineering problems early in the curriculum, providing the context for simple “tool-like” mechanics concepts, and illustrating the interconnectivity of simpler mechanics calculations. Pre- and post-solution period surveys are used with each overarching problem to gauge the students’ perceptions on how these solution periods achieve these pedagogical outcomes and how they improve students’ ability to achieve certain specific learning outcomes in the course. Additional surveys at the end of the courses are used to understand the role that overarching

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problems play in student learning alongside more traditional instruments such as lectures, in-class examples, textbooks, quizzes, and exams. This paper describes a typical overarching problem and details how overarching problems are used in these two mechanics courses. Lessons learned from the use of overarching problems, including challenges encountered, are discussed. Quantitative assessment results from the tools described are presented. Structure of CEE Mechanics courses at Villanova University After two years of intense committee work, discussion, and course development, the Department of Civil and Environmental Engineering (CEE) at Villanova University began offering its required mechanics sequence in a new integrated format to sophomores beginning with the Fall 2009 semester. As shown in Table 1, the classical sequence of coursework in subjects of Statics, Dynamics, Mechanics of Solids, Fluid Mechanics, and Civil Engineering Materials was replaced with a series of three four credit courses. An overview of this curriculum restructuring process is provided by Glynn et al.1 and Wadzuk et al.2 As described by Wadzuk et al.3, the departmental mechanics committee used a Body of Knowledge (BOK) approach to identify the key concepts to be included in the new courses. The committee then packaged cohesive concepts into the three new courses. Although the primary intent of the BOK approach was to identify and pair key concepts, the process actually resulted in a net one credit hour reduction as concepts that were historically taught but not used in future courses were identified and eliminated.

Table 1. Villanova CEE old and new mechanics curriculum Old Mechanics Curriculum New Mechanics Curriculum

Course Credit Hours

Semester in Curriculum Course Credit

HoursSemester in Curriculum

Mechanics: Statics and Dynamics 4 Sophomore

Fall Mechanics

I 4 Sophomore Fall

Mechanics of Solids 3 Sophomore Fall

Mechanics II 4 Sophomore Spring

Civil Engineering Materials 2 Junior Fall Mechanics

III 4 Junior Fall

Fluid Mechanics 3 Junior Fall

Fluid Mechanics Lab 1 Junior Spring

The resulting three courses are summarized in Table 2. Mechanics I essentially focuses on traditional concepts of Statics and Mechanics of Solids with emphasis on axial loading. Basic material properties and linear elastic materials such as steels are introduced. Mechanics II consists of remaining concepts from Statics and Mechanics of Solids, and introduces more complex civil engineering materials such as concrete, composites, wood, and asphalt. Mechanics III consists of concepts from Fluid Mechanics, Fluids Mechanics Lab, and particle Dynamics.

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Table 2. Details of new mechanics curriculum (Mechanics I, II, and III)

Course Course Title Credit Hours

Semester in Curriculum Description

CEE 2105

Mechanics I: Fundamental

Behavior 4 Sophomore

Fall

Forces and moments; equilibrium of particles and rigid bodies; analysis of trusses; stress and strain; axial deformations; distributed force patterns; centroids and moments of inertia; dry friction; column buckling.

CEE 2106

Mechanics II: Material Behavior

4 Sophomore Spring

Shear and moment diagrams; bending and shear stresses; beam deflections; torsion; stress and strain transformations; combined loadings; characteristics of civil engineering materials including Portland cement concrete, masonry, wood, composites, and asphalt; experimental testing using recognized standards.

CEE 3107

Mechanics III: Fluid Behavior 4 Junior Fall

Fluid properties; kinematics of particles and flow; conservation of mass, energy and momentum; fluid resistance, boundary layer theory, flow in conduits; lift and drag; turbomachines.

The three courses are taken sequentially beginning with the first semester (Fall) of the sophomore year. Average section size is about 25 students, with a maximum of 35. All three courses are team-taught by a pair of faculty members and utilize a four meeting per week format, in which there are three 50-minute periods (Monday, Wednesday, and Friday) used primarily for lectures. The fourth period is a 165-minute “flex” period that meets on Thursdays, and can be used for lectures, laboratory exercises, exams, or for overarching problem solution periods. Aside from the integration of concepts described previously and the use of overarching problems as described herein, Mechanics I and II are taught in a fairly traditional manner. Most 50-minute lecture periods involve a set of PowerPoint lecture slides that run on average about 15 minutes, and then the instructor solves two or three example problems for the remainder of the period. Additionally, many classes use models and physical demonstrations to aid students in visualizing concepts. These demonstrations are usually five minutes or less in duration. Students are assigned simple homework problems that are similar to the in-class examples, and these problems are turned in by the students at the beginning of the next class period for credit. Quizzes are given weekly to gauge learning and reinforce the most important learning outcomes. Upon grading of the quizzes the instructors note and record the specific mistake(s) made by each student. This data is tabulated and used to identify potential areas of course improvement. Additionally, a list of “common mistakes” is presented to the students of the following year’s course offering in the class meeting immediately preceding the quiz. Three computation-based examinations are also given during the semester in addition to a comprehensive final exam.

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Overarching problems In the past fifteen years there has been a much needed shift in pedagogy towards problem based learning, the incorporation of concept oriented examples, and the use of interactive learning activities within undergraduate engineering, science, and medical school curricula4-7. Common sense dictates, students agree 8, and assessment bears out9-12, that students generally learn better when taught in these environments. Consequently, these initiatives have become more prevalent in engineering programs throughout the world.13-15 Many faculty members have implemented problem based learning within individual courses, while some departments and colleges have incorporated the philosophy systemically throughout entire programs.16-18 There exists a natural fit for this educational format within engineering mechanics as shown by Klosky et. al., 2007 and their work done at the United States Military Academy 19, among others20; however, the state of practice in mechanics education at most universities remains unchanged2. In most classical mechanics courses, students still learn a series of basic calculations, such as solving for support reactions in a beam, determining a column buckling load, computing a factor of safety, or locating a centroid. These concepts follow the chapters of the textbook. Students are encouraged to master these concepts, but are often not provided the real context of the calculations that they are making. Instead, these basic concepts become more like simple discrete tools that are not interconnected. Students are often given basic homework problems from the back of a chapter in a typical textbook that involve these calculations. As a result, the student may master, for example, determining the moment of inertia of an area, but may not understand why they are making the computation. The use of overarching problems is a specific structured implementation of Problem Based Learning. Simply defined, an overarching problem is a common design and/or analysis problem encountered in the discipline (in this case Civil Engineering) that involves numerous basic concepts brought together to compose a more complex problem. In this sense, it is a true engineering problem, much more complex than a textbook-type problem that focuses on a single concept. The complexity of the overarching problem is made simpler for use in a classroom environment, but not in a way that eliminates the interconnectivity of the many steps that make up the problem. Instead, this interconnectivity is emphasized as a teaching tool to provide context for the individual calculation steps. All three courses in the revised CEE mechanics curriculum at Villanova University utilize overarching problems in some manner to facilitate instruction and learning. However, only the first two courses (Mechanics I and II) utilize the specific format outlined herein. As a result, the use of overarching problems discussed relates only to experiences in these two sophomore-level courses. As of the writing of this paper, Mechanics I has been offered twice. The second offering of Mechanics II is currently ongoing. Three overarching problems are utilized in each course (Mechanics I and II). These problems, and the mechanics concepts that they address, are identified in Table 3. Note that each problem addresses topics from more than one of the classic mechanics areas (Statics, Mechanics of Solids, etc.). A typical overarching problem incorporates concepts from about ten to twenty regular lecture periods.

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Table 3. Description of overarching problems in Mechanics I and II courses

Course Overarching Problem Individual Concepts Addressed

Mechanics I

I-1 Steel Truss Retrofit Analysis and Design

Rigid body equilibrium, truss analysis, steel material properties, elastic axial deformations, normal and shear stress, allowable stress, factor of safety, design of simple connections

I-2 Gravity Wall (Dam) Analysis

Centroid/center of gravity, distributed loading, hydrostatic pressure, friction, rigid body equilibrium, impending motion analysis

I-3 Design of a Water Tower for a Third World Country

Centroid, moment of inertia of an area, parallel axis theorem, Euler column buckling (including end effects), basic cost analysis

Mechanics II

II-1 Prestressed Concrete Bridge Design

Concrete material properties, concrete mix proportioning, shear and bending moment diagrams, flexural stresses, beam deflection, combined stresses (axial load and bending), centroid, moment of inertia, composite beams

II-2 Strengthening of Wood I-Joists Using Composites

Centroid, moment of inertia, allowable stresses, flexural stresses, shear stress, shear flow, superposition, wood and composite material properties, composite beams, stress transformations, Mohr’s circle

II-3 Analysis of a 3-D Highway Sign Structure

Centroid, moment of inertia, 3-D rigid body equilibrium, flexural stresses, shear stress, torsional (shear) stress, biaxial bending, superposition, beam deflection, angle of twist

Overarching problems are formally used in the classroom in two ways. First, overarching problems are used in a PowerPoint slide at the beginning of most lectures, immediately after a title slide and a slide listing the lecture’s learning outcomes. The overarching problem is used to provide context for the lecture’s topics and learning outcomes. The instructor spends no more than one or two minutes using the problem to tie back to previous lectures and set up the upcoming lecture. An example of an overarching problem slide used in this manner is presented in Fig. 1. This particular lecture period was devoted to the design of simple bolted connections, a classical topic in most Mechanics of Solids courses. After all primary concepts that make up an overarching problem are presented in a regular lecture format, a Thursday flex period class meeting is then used for the overarching problem solution. Students are first given a brief PowerPoint overview of the problem with pertinent

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background information. Recall that students have already been introduced to the problem at the beginning of several previous lectures, so these overviews are kept quite brief. Students then begin solving the overarching problem either individually or in pairs, in a recitation-type format. The problem is broken down into a series of several “steps”, and students solve one step at a time. To minimize confusion on the part of the instructors and students, each step of the overarching problem is printed on a different color sheet. Examples of overarching problem solution sheets are provided in Fig. 2. Students must solve the question presented in each step correctly, and get their answer checked by one of the instructors before they are given the next step of the overarching problem to solve. Students assemble the problem as they progress through the steps, often using the results from a previous step as the input for a future calculation. Each overarching problem period typically utilzes most of the 165-minute class period and consists of about six to eight steps. During this time, both instructors circulate around the room answering students’ questions and checking answers. The overarching problem is collected at the beginning of the next class period and students are given credit. Solutions are posted on the course web site as with any typical homework problem. A detailed list of steps in a single (typical) overarching problem is given in Table 4.

CEE 2105 15‐3

Overarching Problem #1Steel Truss Bridge

How do we determine how large the pin in this connection needs to be?  Is the size of the truss member at the connection adequate to resist the force in that member?

Figure 1. Example of PowerPoint slide used in lecture introduction (Overarching Problem I-1)

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Figure 2. Example overarching problem solution sheets (Overarching Problem I-1, Pages 2 of 8

and 8 of 8 shown)

Table 4. Steps in solution of Overarching Problem I-1 Step Task Notes

1 Determine support reactions for truss 2 Determine forces in truss members using

method of joints Uses answer to Step 1 as input

3 Check member forces by method of sections Uses answer to Step 1 as input, answers checked using Step 2

4 Determine material properties (elastic modulus, yield stress, ultimate stress) from given tension test data for sample taken from bridge member

5 Determine actual axial stresses in truss members and determine factor of safety against yielding

Uses answer to Steps 2 and 4 as input

6 Determine elongations of all diagonal truss members

Uses answer to Steps 2 and 4 as input

7 Determine normal (net section) and bearing stresses on diagonal truss member with eyebar type end connection and check against allowable stresses

Uses answer to Steps 2 and 4 as input

8 Design (size) pin in connection at joint between main truss and deck hangers, given allowable shear stress in pin

Uses answer to Step 2 as input Page 22.1133.8

Pedagogical Outcomes Overarching problems in sophomore-level mechanics courses provide many obvious advantageous pedagogical benefits, including the presentation of real engineering problems early in the curriculum. More formally, the use of overarching problems within the mechanics sequence at Villanova University is intended to achieve the following specific outcomes:

1. Provide real-world context for individual simple mechanics concepts 2. Illustrate the interconnectivity of these individual concepts 3. Reinforce students’ understanding of specific learning outcomes related to these

individual concepts

The third pedagogical outcome above relates to students’ ability to satisfy specific learning outcomes corresponding to each lecture period. Examples of these learning outcomes include “Solve for unknown forces in a truss using the method of joints” and “Determine the centroid of a composite area”. In each course, there are roughly 80 to 100 specific learning outcomes of varying levels of complexity and importance. While it is logistically impossible to cover all of the learning outcomes with only three overarching problems in each course, the overarching problems that have been developed address a significant percentage of the most important learning outcomes identified by the instructors. In Mechanics I for example, the three overarching problems together address a total of 25 different learning outcomes, nearly all of which are addressed in some form on quizzes and exams. These learning outcomes are listed in the Appendix. Assessment There are numerous tools and formats for documenting student achievement of learning outcomes, including pre and post tests, pre and post surveys, exit interviews, and quizzes and examinations21-23. Three specific survey-based assessment methods have been implemented in the mechanics sequence at Villanova University in order to evaluate the effectiveness of overarching problems. These methods provide assessment data at different “scales”, addressing everything from the overall value of overarching problems to the impact of these problems on students’ achieving specific learning outcomes. These methods include:

(1) Macro-scale Assessment: Supplementary questions on end of the semester student course evaluation forms

(2) Meso-scale Assessment: Evaluation/feedback surveys given to students after each overarching problem-solving session

(3) Micro-scale Assessment: Detailed pre- and post-overarching problem surveys related to students’ understanding of specific learning outcomes

Data from each of these assessment methods are briefly discussed herein based on the first two offerings of Mechanics I and the first offering of Mechanics II. Since the meso- and micro-scale assessments were not introduced until the second offering of Mechanics I, these types of assessment data are only presented based a single semester. As noted earlier, detailed data is

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being collected on student performance on weekly quizzes and this data may be used in the future for additional assessment based on student work. Macro-scale Assessment – In their end of semester course evaluations, students were asked a series of supplemental questions in which they rated the overall value of several course instruments on their contribution to learning. As noted in Table 5, students gave overarching problems a high rating. It is interesting to note that overarching problems rated higher than some other instruments such as the textbook and lecture notes. Students’ perception that example problems in class (other than overarching problems) contribute most significantly to their learning is likely due to the fact that many exam problems are similar to these examples. It is also noteworthy that the mean rating for overarching problems was significantly lower for Mechanics II. This is likely attributed to the increased complexity of the overarching problems in the second course, as described later in the section of this paper entitled Challenges.

Table 5. Student ratings of course instruments in Mechanics I and II

Q: How well did the following contribute to your learning in this course?

Mean Rating Mechanics I

Mechanics II

Fall 2009

Fall 2010

Spring 2010

# Students 57 46 45 Course Instrument Textbook 3.4 3.2 3.1 Lecture Notes 4.2 4.5 4.3 Example Problems Solved in Class 4.9 4.9 4.9 Homework Problems 4.7 4.6 4.5 Quizzes 4.3 4.2 4.0 Practice Exams 4.6 4.7 4.8 Exams 4.7 4.7 4.4 In-class Demonstrations 4.7 4.4 4.4 Having the Course Co-Taught by Multiple Faculty 4.6 4.5 3.7 Overarching Problems 4.6 4.6 4.0 Rating scale: 1 = No contribution … 5 = Significant contribution

Meso-scale Assessment – Beginning with the second offering of Mechanics I (Fall 2010 semester), students were given a brief survey immediately after each overarching problem solution period. The anonymous survey was administered as a homework assignment and was due at the beginning of the subsequent class meeting. The survey asked for an evaluation of how each of the three pedagogical outcomes identified above was satisfied, and also asked two additional questions on effective use of class time and sense of accomplishment. Data from this survey is presented in Table 6. Ratings are overwhelmingly positive, and indicate that all three overarching problems successfully achieved the three pedagogical outcomes. The survey also allowed for comments and asked students for areas of potential improvement. Selected responses are shown in Table 7.

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Table 6. Overall overarching problem evaluation data for Mechanics I (Fall 2010)

Q: How well did the overarching problem do the following?

Mean Rating Overarching

ProblemI-1 I-2 I-3

Provided real-world context for the simple calculations performed in each individual step. 4.7 4.6 4.8

Demonstrated how the individual calculations interrelate to each other (such as output from a previous step becoming input for a later step).

4.8 4.8 4.7

Reinforced my understanding of individual learning outcomes from class. 4.6 4.7 4.6

Effectively used class time (i.e. the 165-minute “flex” period). 4.8 4.8 4.5 Made me feel a sense of accomplishment when I was finished. 4.7 4.7 4.6 Rating scale: 1 = Not well at all … 5 = Extremely well

Table 7. Selected student comments on overarching problems for Mechanics I (Fall 2010)

Examples of positive feedback The problem tied everything we have learned so far very nicely at the midpoint of

the semester. I think it really helped me recall everything from earlier in the semester. It was a

great way to use class time. I felt a huge sense of accomplishment when I was done. Didn’t realize I knew how to do all of that. Very cool. I felt awesome after completing this ☺. I was intimidated when I first thought of

the overarching problem, but I surprised myself with how much I knew. Overall it felt like a game or puzzle with different levels and difficulty which made

it entertaining. You guys did a great job with positive criticism on our mistakes and, like always, fostered a fun working environment. 

I actually felt like an engineer. Why? Data + specs = product. I know that if I tried to build that water tower, it would work reasonably well. 

The OA problems help me a lot. I'd use even more for future classes. Comments on logistics and potential improvements

In the beginning of the sheet, maybe list the steps that will be taken and the ultimate goals of the overarching problem. I feel like I am crunching numbers but don't know why. 

Give a brief description of what exactly we are solving for. In other words, tell why it is important to solve for each of the individual sections. I felt like I was just solving for numbers.

I think it’s more effective when everyone works individually, rather than with another person. I know that I feel a lot more confident coming out of the overarching problem knowing that I personally solved everything correctly and understood the concepts better than if I were working with someone else.

I felt mentally exhausted near the end and finishing felt more like a relief than an accomplishment. 

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Micro-scale Assessment – Beginning with the second offering of Mechanics I (Fall 2010 semester), students were also given a pre- and post- survey asking their perceived ability to achieve the specific learning outcomes covered by each of the three overarching problems. The pre-survey was administered at the beginning of each overarching problem solution period, and the post-survey was administered as a homework assignment jointly with the meso-scale survey described previously. Essentially, students rate their own abilities on each learning outcome just before and just after the overarching problem solution period. Ratings range from 1 for “No clue” to 5 for “Mastered”. A bar graph of the mean pre- and post-survey scores is presented in Fig. 3 for all 28 learning outcomes addressed on the three overarching problems in Mechanics I. A list of these learning outcomes may be found in the Appendix. Each graph is sorted with pre-survey scores in ascending order from left to right, such that those data to the left of each plot show the most potential room for improvement. Those outcomes which initially rate high on the pre-survey are typically those that have been introduced early in the course and been reinforced multiple times prior to the overarching problem, and those outcomes on which students have already had a major examination. Examples of this include Outcome 1-3 in Fig. 3(a) and Outcome 2-2 in Fig. 3(b), which respectively are “Solve for unknown forces and moments acting on a 2-D rigid body” and “Construct a free body diagram for a rigid body”. Both of these outcomes are introduced early in the course and covered on the first examination, prior to the first overarching problem solution period. It is noteworthy that an increase in ratings is shown for all 28 learning outcomes. As expected for the reason described above, the most significant increases tend to occur on the learning outcomes that have the lowest pre-survey scores. The “mean of mean” (i.e. average rating for all learning outcomes covered on a given overarching problem) rating for all students improved from 3.96 to 4.38 for Overarching Problem I-1, from 3.80 to 4.41 for Problem I-2, and from 3.68 to 4.38 for Problem I-3. Although the survey was administered anonymously and without names, each student was assigned a specific number so that individual student pre- and post-survey responses could be correlated and compared. Figure 4 shows the “mean of mean” score improvement by student for each overarching problem in Mechanics I (Fall 2010). The data presented in Fig. 4 is sorted similarly to the data in Fig. 3, with the lowest pre-survey scores to the left of the graph. It can also be generalized that the most significant improvements are for those students who do not understand a topic well going into the overarching problem. Still, the data shows that even those students with a solid understanding tend to show improvement, even if the improvement is limited by the upper bound of the 1 to 5 scale. In all but a few cases, students indicated overall improvement on a “mean of means” basis. In isolated cases where students did not improve, it is probable that the student felt more confused about a topic because they struggled to correctly solve a particular part (or parts) of the overarching problem.

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(a) Overarching problem I-1

(b) Overarching problem I-2

(c) Overarching problem I-3

Figure 3. Micro-scale assessment: pre- (red) and post- (blue) survey of student learning outcomes as sorted by individual outcome (Mechanics I, Fall 2010)

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(a) Overarching problem I-1

(b) Overarching problem I-2

(c) Overarching problem I-3

Figure 4. Micro-scale assessment: pre- (red) and post- (blue) survey of student learning outcomes as sorted by individual student (Mechanics I, Fall 2010)

Challenges The use of overarching problems has thus far worked well in the restructured mechanics courses that make up the Villanova University CEE required mechanics curriculum. The primary challenges, as with any type structured problem used in a classroom environment, are in developing the appropriate depth of overarching problems to foster student learning without it becoming too laborious to solve in the class time provided. In general, the overarching problems in Mechanics I have been of appropriate length and depth, and students have been able to solve the problems comfortably in the 165-minute sessions. Students appeared to leave these overarching problem sessions feeling confident and showing a sense of accomplishment, and the problems have covered a significant number of learning outcomes as discussed previously.

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In two of the three overarching problem solution periods in the first offering of Mechanics II, however, some students ran out of time to solve the problem in the 165-minute class period. In these problems, the level of computation was higher (as compared to problems in Mechanics I) given the more advanced concepts in the second course. These problems are being adjusted accordingly for the second offering of Mechanics II, in a manner that still retains the coverage of learning outcomes that the original problems provided. It should also be noted that the success of the overarching problem solution periods is directly linked to the ability of the professors to interact with students and provide real-time feedback on their computations for each step of the problem. In larger institutions where average course enrollments are well in excess of the 25 to 35 at Villanova University and in situations where team-teaching cannot be accommodated based on teaching loads, properly trained support personnel such as teaching assistants will be essential. Summary Overarching problems provide a structured opportunity for improved student learning in the recently restructured Villanova University CEE sophomore-level mechanics curriculum. Overarching problems are used to provide context for the basic computations and concepts that form the backbone of traditional sophomore-level mechanics courses, and to illustrate how these smaller calculations interrelate in a larger design or analysis problem. They also serve to reinforce student learning outcomes from standard lecture periods. Initial feedback on the value of the overarching problems, based on the first three semesters of the new courses, has been overwhelmingly positive. Macro- and meso-scale assessment surveys have indicated that overarching problems rate well above 4 on a 1 to 5 scale in their overall effectiveness with respect to the pedagogical outcomes described in this paper. Furthermore, more in-depth micro-scale assessment from pre- and post- overarching problem surveys clearly indicates that overarching problem solution periods increase students’ perception of their ability to achieve the individual learning outcomes that are core to any mechanics course. Acknowledgements The authors wish to thank the Villanova Institute for Teaching and Learning (VITAL) and the Department of Civil and Environmental Engineering for their support in establishing this innovative course sequence. Bibliography

1.  Glynn, E.F., Dinehart, D.W., Gross, S.P., Hampton, F.P., and Wadzuk, B.W. (2007), “Teaching Engineering Mechanics in a Problem-Structured Environment,” Proceedings of the Fall 2007 ASEE Middle Atlantic Section Conference, Philadelphia, PA.

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2. Wadzuk, B.M., Dinehart, D.W., Glynn, E.F., Gross, S.P., and Hampton, F.P. (2008), “'Survey of Engineering Mechanics in Civil Engineering Curricula',” Proceedings of the 2008 ASEE Annual Conference and Exposition, Pittsburgh, PA.

3. Wadzuk, B.M., Dinehart, D.W., Glynn, E.F., Gross, S.P., and Hampton, F.P. (2009), “A Methodology for Undergraduate Curriculum Modification,” Proceedings of the 2009 ASEE Annual Conference and Exposition, Austin, TX.

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Appendix: List of student learning outcomes covered in Mechanics I overarching problems 1-1 Identify reaction forces and moments associated with different support conditions. 1-2 Construct a free body diagram for a rigid body. 1-3 Solve for unknown forces and moments (such as support reactions) acting on a 2-D

rigid body. 1-4 Identify zero force members in a truss by inspection. 1-5 Solve for unknown forces in a truss by the method of joints. 1-6 Solve for unknown forces in a truss using the method of sections. 1-7 Compute the modulus of elasticity, yield stress, and ultimate stress from a stress-strain

curve. 1-8 Calculate the elastic displacement of an axially loaded member. 1-9 Calculate the average normal stress acting on a cross-section. 1-10 Calculate the average shear stress acting on a cross-section. 1-11 Compute actual factors of safety and/or stress ratios. 1-12 Determine the load capacity of a simple connection given a set of allowable normal

stresses, shear, and/or bearing stresses. 1-13 Design a simple connection for a given load and set of allowable normal, shear, and/or

bearing stresses. 2-1 Calculate the moment of a force or set of coplanar forces about a point by scalar

methods. 2-2 Construct a free body diagram for a rigid body. 2-3 Compute actual factors of safety and/or stress ratios. 2-4 Determine whether or not an object remains in static equilibrium under a given set of

loads. 2-5 Solve for unknown parameters in a problem involving the impending motion of a

body. 2-6 Determine the magnitude and location of an equivalent concentrated force to represent

a simple distributed force pattern. 2-7 Determine the magnitude(s) and location(s) of an equivalent concentrated force(s) to

represent a fluid pressure. 2-8 Determine the centroid of a composite area. 3-1 Compute actual factors of safety and/or stress ratios. 3-2 Determine the centroid of a composite area. 3-3 Determine the moment of inertia of area about a specified axis using the Parallel Axis

Theorem. 3-4 Determine the moment inertia of a composite area about a specified axis. 3-5 Explain the relationship between section properties and critical buckling load for a

compression member. 3-6 Explain the relationship between end conditions and critical buckling load for a

compression member. 3-7 Determine the effective length for a compression member with specified end

conditions. 3-8 Calculate the critical buckling load for a compression member using the Euler

buckling formula and appropriate effective slenderness ratio.

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