AC 2011-847: IDENTIFYING AND ADDRESSING STUDENT DIFFICUL-TIES IN ENGINEERING STATICS

Andrea Brose, Hamburg University of Technology

Andrea Brose earned her Ph.D. in mathematics from the University of Colorado at Boulder. From 1999to 2008 she was in the Department of Mathematics at UCLA where she taught undergraduate math, ledand developed the mathematics teaching assistant and faculty training program, and contributed to otheraspects of academic administration. Since 2009, she is involved in a project on ”Active Learning inEngineering Education” at Hamburg University of Technology.

Christian H. Kautz, Hamburg University of Technology

Christian H. Kautz received his doctorate degree from the University of Washington for research on stu-dent understanding of hydrostatics and thermal physics and was involved in the curriculum developmentprojects of the Physics Education Group. After teaching physics at Syracuse University for three years,he moved to Hamburg University of Technology where he has since carried out research on student un-derstanding in various introductory engineering subjects, such as Engineering Mechanics, Electrical En-gineering, and Engineering Thermodynamics.

c©American Society for Engineering Education, 2011

Identifying and Addressing Student Difficulties in Engineering Statics

1 Introduction

A thorough understanding of basic science and engineering concepts is one of the core objectivesof engineering education. It is frequently observed that for many students, typical introductorycourses are not successful in generating such an understanding. Studies show that this contributesto high drop-out rates in technical (STEM) fields.1,2,3 Changes in the instructional settings, suchas incorporating more active learning formats, can help enhance learning outcomes, but effectiveinstruction in introductory science and engineering may require more than the mere adoption of aproper learning format. Educational research based in the science disciplines, especially inphysics, has identified specific conceptual and reasoning difficulties that often prevent studentsfrom developing a functional understanding of many of the topics taught in these courses.4 Thereis evidence that instructional materials that take into account such difficulties and foster activelearning are more likely to improve student learning.5

About five years ago we began a program at Hamburg University of Technology (TUHH) to adaptthe approach to engineering education in Germany. The program follows a three-step process thatconsists of (1) identifying student difficulties, (2) designing instructional materials to addressthese difficulties, and (3) using assessment data to test the effectiveness of the materials. Ourfocus has been on the three introductory courses in mechanics, electric circuits andthermodynamics. The present article is concerned with identifying and addressing studentdifficulties with statics concepts in mechanics.

In the present case we have used results from assessment tests to identify student difficulties.Subsequent to this analysis we invited students for an interview in which we asked them toanalyze or solve problems involving concepts that our analysis flagged as potentially problematic.In this article we will show how we identified four student misconceptions, one of which – to ourknowledge – has not been identified by previous research. We will then use one of thesemisconceptions to illustrate the development of instructional materials.

The remainder of this article is as follows: In section 2 we give a review of concepts that are dealtwith in this paper, describe our findings about misconceptions regarding static equivalence, detailhow they came to be identified and give some possible causes for their existence. Section 3outlines the instructional strategy we use and describes the development, implementation andassessment of worksheets. We conclude with a general discussion in section 4.

2 Identifying Student Difficulties

In this section we give a brief review of the mechanics concepts involved in this research,introduce our methodology and present evidence for our argument that a common difficulty ofstudents is their failure to recognize the reference point invariance of force couples.

2.1 Concept Review

In a first-year engineering mechanics course, statics is traditionally covered as one of the firsttopics. For instance, at our institution, the basic elements of statics, i.e., forces and moments, areintroduced in the first few lectures. These fundamental concepts form the building blocks formost of what follows in the remaining three to four semesters of mechanics. Misconceptions thatarise at this early stage, and for basic concepts such as these, tend to persist and thereby influencethe ability of students to master more advanced concepts.

Figure 1: Moment of a Force

Forces are usually introduced as vectors. Moments are introduced as arising from forces. Whenintroducing moments the point where a force is applied (or the line of action) matters. Studentslearn, as they probably already know from high school, that a force F applied to a body maycontribute to a translation as well as rotation of the body, the latter being measured by a momentM about a point A with perpendicular distance d from the line of action of F , according to theformula M (A) = d · F or vectorially, ~M (A) = ~r× ~F , where ~r is any vector from the point A to theline of action of ~F (Figure 1). Implicitly one should realize that a moment due to a force dependson the reference point chosen.

Force couples, with which this paper concerns itself, are introduced following the introduction ofmoments, and are identified as moments due to a set of several forces, such that the resultant forceis zero while there is a (non-vanishing) resultant moment. Most often one considers two forces ofequal magnitude and opposite direction having different lines of action. The resultant force iszero, but a non-vanishing moment is induced (which can be calculated by adding the momentsdue to each single force with respect to a given point). The non-vanishing moment that arises inthis example is called a couple, or force couple, likely reflecting the fact that it can be thought of

as the result of two opposing forces whose net moment does not vanish. An essential, butnon-intuitive property of a force couple is that it is invariant with respect to the point of referencefor its constituent moments. The German phrase for a force couple is Freies Moment, whichliterally means free moment, and emphasizes precisely this property of reference point invariance,which distinguishes it from a moment due to a single force (Moment in German).

Consider the example of a couple by two forces of equal magnitude, oppositely directed withdifferent lines of action. Then it does not matter where the two forces are applied as long as thedistance between their lines of actions remains the same. One can take this a step further, bynoticing that the same couple or free moment is induced if one chooses the magnitude F of theforces larger while reducing the distance between their lines of actions so that the couple M is thesame. If one takes the limit F ↑ ∞ and d ↓ 0 such that M remains the same, one can relate to thenotation used for couples or free moments in figures. Instead of drawing two forces of the abovekind, which can be ‘moved’ around, as long as the relative distances of their points of applicationremains the same, without changing the effect of the forces on the body, one simply draws onemoment with the symbola .

We shall argue below that students typically misunderstand the defining property of a forcecouple, and that the misunderstanding may originate from the way in which the concept isintroduced, and can be compounded by the way it is described and notated.

2.2 Background and Methodology

At TUHH we have led a program to investigate student understanding in engineering courses toguide the development of instructional material. This project is rooted in the methods of PhysicsEducation Research, and involves a sequence of steps comprised of, among other steps,standardized testing and interventions through interviews. These are summarized below; for moredetails on the methodology the reader is referred to Brose, Kautz (2010).6

Since 2005 the Force Concept Inventory, developed by Hestenes et al. has been given as a pre-testprior to the start of instruction in Mechanics at TUHH.7 At the end of the first semester the StaticsConcept Inventory (SCI) by Steif et al. has been given as a post-test.8 Based on an analysis ofresults from the SCI we identified topics that students struggle to master. We consistently find thatthe most difficult concept covered by the SCI is that of static equivalence. To further explore thisapparent conceptual shortcoming we conducted interviews with students from two mechanicscourses in the summer semesters of 2009 and 2010. The interview questions were designed basedon assessment questions and the content of course lectures. Interviewees were self selecting;during the course of the semester students were asked if they were willing to be interviewed. Overthis two year period nine interviews were conducted, always at the end of the semester, but beforethe final exam was given. The students were generally among the more highly achieving studentsof the class. All but one were ranked in the upper half of their class, and four in or above the 90thpercentile as measured by the SCI. The interviews benefited from the fact that we (the

aOther symbols, e.g., a double-headed arrow in the direction of the moment, are also used.

interviewers) were not involved in assigning course grades, and could be free (i.e., unbiased) inhow we posed our questions. Likewise, the students could also answer more freely asmisconceptions would not be held against them. All interviews were recorded with a digital videorecorder, and subsequently reviewed. Because the interviews are critical to our insight intostudent understanding, or in this case misunderstanding, the details of the interviews and theevidence derived from them are presented in more detail below.

2.3 Interviews

The interviews are generally designed to go into more depth on topics that, based on the SCI,prove difficult for students to master, in the present case this being static equivalence. Althoughthe total number of interviews, relative to students taking the SCI, is quite small, our hypothesis isthat a misconception that emerges consistently in interviews with even a few students is likely tobe more widely shared.

Each interview consisted of two parts. In both parts of the interview we posed questions to thestudentsb and asked them to think ‘out loud’ as much as possible so that we could gain insightinto the roots of any misunderstandings. Already more than thirty years ago it was documentedby Brohn and Cowan, who have investigated student understanding of structural behavior, howvaluable and surprising it can be to listen to how students think when they solve problems.9

2.3.1 First Part of Interview: Gedankenexperiment

The first part of the interview involved having the students participate in a thought experiment.The students were shown a physical object consisting of a moveable Γ-shaped object (slider)mounted on a beam that was assumed to be weightless (see Figure 2). The slider could bemounted in one of two ways: with the overhanging branch directed along the beam away from, ortoward, the person. We will refer to this object as slider-and-beam. The students beinginterviewed were asked to imagine holding the end of the beam in their hand and to predictwhether they would feel a difference in their hand while holding the beam as a function of theorientation of the slider. The students were then asked to explain their prediction using conceptsdeveloped in the mechanics course.

2.3.2 Second Part of Interview: SCI Revisited

In the second part of the interview the students were asked to answer one question from the SCIand another question modeled after an SCI question and with one distractor added (Figure 3).This additional distractor was intended to explore the anticipated student misunderstanding thatforces and moments are interchangeable.

bAlthough we refer to the intervieews collectively, and hence the students in plural, the interviews were conductedin serial, i.e., the procedure was repeated for each student in turn.

Figure 2: At the conclusion of the interview students were given the opportunity to holdthe imagined slider-and-beam to test their understanding of the concepts explored in thethought-experiment part of the interview.

A 60 Nm couple acting clockwise keeps the member in equilibrium while it issubjected to other forces acting in the plane shown schematically at the top. Thefour dots denote vertically aligned, equally spaced points along the member.

Assuming the other forces stay the same, what load(s) could replace the 60 Nmcouple and maintain equilibrium?

Mark all possible answers.

otherforces

2 m

60 Nm

(a) (b) (c) (d) (e) (f)

otherforces

otherforces

otherforces

15 Nm

otherforces

60 Nm

otherforces

otherforces

15 N 10 N

10 N

30 N

30 N

60 N

Figure 3: Example of an SCI-like interview question designed to explore the student misun-derstanding that forces and moments are interchangeable.

2.4 Student Misconceptions

Through our analysis of student responses in both parts of the interview we identified thefollowing misconceptions:

• belief that forces and moments are interchangeable;

• failure to recognize that forces and moments are interactions between two bodies;

• belief that couples are not independent of the reference point chosen;

• difficulties in understanding the choice of system and that independent thereof the result ofthe analysis should be the same.

The first point has been discussed by Newcomer and Steif, and in a somewhat different context byOrtiz et al. , the second by McDermott et al.10,11,12 To our knowledge the third point has notpreviously been identified, most likely since it is not specifically targeted as a concept in the SCI.Although this was not a concept we had been intending to explore in the interviews, gettingstudents to think aloud revealed persistent misconceptions that we would not have otherwisenoticed. The fourth point is probably unrelated to student understanding of couples and thereforeonly briefly discussed below.

Below we review material taken from the interviews as evidence for some of the studentmisconceptions identified above.

2.4.1 Misconception Concerning Choice of System

The answers and explanations received from the students in relation to the slider-and-beamGedankenexperiment revealed various difficulties associated with identifying a specific system tobe analyzed and recognizing the implications of a specific choice of system. Students generallystated somewhat vaguely that the beam with outward-directed slider would ”feel heavier” than theinward-directed case. They also could explain this in terms of the change in the center of mass ofthe system consisting of the beam and the slider. However, when we asked the students toconsider just the beam as a system, as this was the object that communicated the change in theorientation of the slider to their hand, they became confused and in most cases actually ended uparguing a different answer. From this point of view, or choice of system, the students haddifficulty in identifying a reason as to why the orientation of the slider should be noticeable.Students who realized that their answer should not depend on the choice of the system oftenchanged their initially correct answer and said that the orientation of the extension would not befelt by the person holding the beam, i.e., that they would not feel a difference holding theslider-and-beam with an inward-directed slider in comparison to an outward-directed slider.

2.4.2 Misconception Concerning Couples

While the question in Figure 3 was primarily intended to explore our suspicion that studentswould think of forces and moments as interchangeable (distractors (a) and (f)), we found through

the course of some interviews, that students fail to interpret a couple’s action on a body properly.More precisely, students dismissed one of the two correct answers (d) by claiming that the coupleof 60 Nm in the given member would (if no other forces or moments are acting) cause it to rotateabout the point where the couple is drawn, that is around the second point from top, while in (d)the member would rotate about the bottom point. We asked students to show how they wouldimagine the effect of a couple alone on such a member by giving them a piece of paper shapedlike the object in the problem. In every case the student would pin down the paper with their penor finger where the moment was drawn and rotate the paper about that point. Some examples oftheir transcribed (and translated) explanations are given below as examples, but the flavor of theresponse is typical of those students we specifically asked. Those arere only from the latter half ofthe interviewees after we had begun to suspect this misconception from listening to explanationsin the earlier interviews.

Student 1: ”One would induce a couple at a different point in (c) from that in the given situation... i.e., we do not have a moment with respect to the point above student points at the second pointfrom top. Thus (c) is false.” Student starts debating whether one could move this moment, later in(d) he revisits the idea of moving the couple but is not sure whether that is allowed, i.e., whether itwould produce a statically equivalent system.

Student 2: Student pointing at answer (c) answers: ”This couple is applied at the wrong point,which would induce a rotation about that point and not about the point about which it would rotate[in the original member]. Pointing at answer (d) That is also false because the couple is applied atthe wrong point.”

Interviewer: How can I imagine the effect of a couple on a member?

Student 2: ”I would try to rotate about this point, but this point Student points to the point at whichthe couple is drawn would be continually fixed. When I try to rotate about the other point pointingat answer (d), the original point would move.”

Interviewer (to Student 3): How can I imagine the effect of a couple on a member?

Student 3: I picture the couple’s center of gravity student 3 holds his pen as the axis of rotationorthogonal to the paper at the point where the couple is drawn, which is where the couple isapplied, through which I put an orthogonal axis, which is the axis of rotation about which thesystem would revolve.

Below we attempt to find some causes as to when and why the misconception concerning couplesexists.

2.5 Possible Causes

We believe a number of factors contribute to this misconception. One being the simple form ofthe symbol used to denote a moment, another being the examples that are used to introduce theconcept. The symbol used for a couple is , as used in Figure 3 and mentioned in section 2.1. It

suggests that there is a rotation about the center of the symbol, as it invokes the sense of a swirlabout a point. Secondly, often textbooks introduce couples with examples in which a force coupleis acting on a symmetric body and is applied symmetrically about its center of mass, which indeedis the fixed point under the ensuing rotation caused by the couple. Examples include handwheelsbeing turned by two hands holding on the left and right side diametrically opposed (Figure 4); ascrew being driven by a screw driver perfectly centered at the center of the screw, and so on.

13

14 15

Figure 4: Three examples taken from published instructional material to illustrate a forcecouple.

As discussed above, the German term for (force) couple is literally translated ‘free moment’,which suggests to the student that this moment is ‘free’ with respect to something. While duringthe introduction of ‘free moments’ the terminology is often explained (in textbooks and lectureslides used at TUHH), students struggle with the idea of a force couple being independent of itspoint of application. A lack of intuition related to objects free from a natural reference point, suchas a hinge or contact point to another object, and because dynamics related concepts are nottreated until much later, probably contributes to student difficulties with this concept.

3 Addressing Student Difficulties

In this section we illustrate our general approach in addressing student difficulties and describethe implementation and assessment of our materials.

3.1 Design of Instructional Strategies

The underlying pedagogical paradigm of our work is constructivism, i.e., the idea that studentsneed to construct knowledge for themselves and give meaning to the technical terms that they use

in their discourse about the subject matter. This idea goes back to Jean Piaget’s research onchildren in the first half of the previous century.16 Often in technical subjects, the construction ofknowledge is impeded by incorrect or partly correct preconceptions held by the students. Thisobservation has led various researchers and educators in the physical sciences to adopt strategiesto help students overcome (partly) incorrect ideas, and amend or replace them by a more correctunderstanding. One such instructional strategy, developed for this purpose by McDermott, can becharacterized by the terms elicit, confront, resolve: Student ideas need to be elicited at thebeginning of the learning process, students then need to be confronted with the possibility thattheir ideas may be (partly) incorrect through observations or logical arguments drawing onprevious results, and finally the conflict of contradictory ideas has to be resolved.17 This strategyhas successfully been employed in a set of instructional materials for introductory physics, theTutorials in Introductory Physics by McDermott et al. , which are now widely used in physicsinstruction at colleges and universities in the United States and some other countries.18

In the present example, typical incorrect ideas included the belief that forces and moments areinterchangeable (as illustrated in our companion paper). An application of the strategy outlinedabove must therefore include an opportunity for the student to face the question whether this is infact the case, a question that often does not arise explicitly in traditional instruction in engineeringmechanics. Students who hold this idea will then need to recognize that their belief is incontradiction with prior results, namely that the net force on an object (whether zero or not)determines the object’s translational motion and must therefore be kept the same when sets offorces or moments are replaced by equivalent sets. Finally, their newly gained understanding thata couple cannot be replaced by a single force can then be strengthened by the recognition thatsuch an incorrect substitution in general also leads to an incorrect net moment.

3.2 Development of Worksheets

Starting in the Fall of 2009 we developed weekly worksheets modeled after the Tutorials inIntroductory Physics. These worksheets included problems and exercises, some of which weredesigned to address specific difficulties that our research had previously uncovered. The othermajor difference between these worksheets and traditional exercises given in engineering coursesis the emphasis on qualitative understanding rather than quantitative problems often masteredthrough memorized algorithms.

An example of this type of question is provided by the situation shown in Figure 5. Studentsconsider a fictional student’s answer to a qualitative problem on static equivalence. They are thenasked to answer the following three questions

a. Do you agree with Peter? Justify your answer.

b. Compare the resulting forces in systems I and II.

c. Compare the resulting moments relative to point Q for both systems.

which roughly correspond to the three steps in the instructional strategy outlined above.

Peter: "System II is equivalent to sys-tem I. Remember, ~M = ~d× ~F . Hence,a moment of 12 Nm with respect toP can be replaced by a 3 N force, 4 mto the right of P ."

I

1m

12Nm 6N

P Q

II

1m

3N

6N

P Q

Figure 5: An example of material for a qualitative question designed to address misunder-standings of interchangeability of forces and moments

3.3 Implementation

Courses in engineering mechanics at our institution traditionally consist of lectures and recitationsections. While the latter have for some years involved active student work (in addition to apresentation of solutions by teaching assistants), their focus has previously been only on standardquantitative end-of-chapter problems. In the project described here, we were able to use half ofeach weekly 90-minute recitation section for collaborative group work focusing on ourworksheets. As our teaching assistants had not been exposed to these materials during their ownstudies of mechanics, this change required a more elaborate training program for the TAs. In turn,the weekly preparatory meetings held for them contributed to the development of the worksheets.Often, the teaching assistants recommended changes to the materials that they felt would makethem more easily understood by the students. At times, some of the teaching assistants still hadsimilar difficulties and thereby allowed us some deeper insight into students’ ideas.

In one of the subsequent courses, the engineering dynamics course taught to third-semestermechanical engineering students, the use of recitation sections for student work on our materialsdid not seem feasible. Based on our experience with the first-year course, we felt that theseactivities would not leave enough time for quantitative problems which still form the majority ofthe tasks that students encounter in their exams. We therefore decided to include the worksheetsin one of the weekly lecture hours that traditionally had been set aside for the presentation ofpractical examples of the subject matter. In this large-lecture setting, students would form groupsof two to four individuals while working on the materials. At various times throughout the lecturehour, the instructor would discuss some results, point out possible pitfalls and motivate studentsfor the next section of the worksheet. As this mode of implementation has only been startedrecently, the assessment data in the subsequent section only concerns the use of our worksheets inthe small-group (recitation section) setting.

3.4 Assessment

We attempted to measure the effectiveness of interventions by comparing pre- and post-test datafor the past five years. Only the last two of these years incorporated the intervention describedabove. Of the six years, the years 2009-2010 and 2010-2011, which had the benefit of theintervention, scored among the three highest on the SCI (see Table 1). However both these yearshad a relatively low pre-test (FCI) scores, indicating that the higher achievements were not likelyattributable to initial conditions. During the six-year interval student enrollment in the courseincreased from about 350 to more than 600.

Because the instructional sections on static equivalence (included as parts of three subsequentworksheets) were most directly affected by the outcomes of the interviews, we specifically lookedat the average post-test results of the three test items comprising the static-equivalence subsection(stat-eq) of the SCI (see last column in Table 1). In contrast to the overall post-test results, studentachievement on the subsection was by far the highest in the 2009-2010 and 2010-2011 cohorts,even if compared to the relatively strong result of 2005-2006.

Year Instructor FCI (pre) SCI (post) stat-eq[%] [%] [%]

2005-2006 A 50 42 172006-2007 B 45 36 122007-2008 A 51 30 112008-2009 B 49 29 132009-2010 B 47 43 272010-2011 B 43 40 25

Table 1: Summary of FCI pre and SCI post-test results over the last five years, as well aspost-test results of static equivalence subsection of SCI.

While these results are encouraging, there are at least two additional factors which could explainthe relatively good performance in the latest two years. First, in 2009-2010 the curriculum wasslightly changed, with the net effect that student contact hours with a teaching assistant effectivelydoubled, thereby increasing overall contact hours by one third. Second, instructor B (Table 1)who taught the course in the two most recent years followed a format that was introduced byinstructor A in 2007-2008 to incorporate active learning, and which instructor B had followed forthe first time in 2008-2009. However, in the year 2010-2011 student enrollment had increased bymore than 20% over the previous year due to shortening secondary education by one year. Henceadditional contact with teaching assistants and improvements in the lecture instruction, includingthe implementation of active learning techniques, do not provide plausible alternativeexplanations for the remarkable gains after the use of the materials described. In the currentsituation, given the many factors involved, no unique cause can be identified.

4 Summary and Conclusions

A program has been implemented to advance engineering education at our institution. It follows athree-step process that consists of (1) identifying student difficulties, (2) designing instructionalmaterials to address these difficulties, and (3) using assessment data to test the effectiveness of thematerials. In this paper, we have illustrated this process in the context of a first-semesterengineering mechanics course. Through analysis of answers on diagnostic tests and in interviews,we have identified student difficulties with the concepts of forces and moments and concerningthe choice of systems and its implications.

One specific misconception, which to our knowledge has not been previously recognized, is thebelief that the effect of force couples depends on their point of application. This is somewhatironic because the lack of such a dependence is what distinguishes couples from moments ofindividual forces and hence motivates the introduction of this concept in the first place. Thiscommon misunderstanding of students was revealed when exploring student conceptions of staticequivalence that appeared from an analysis of the FCI/SCI sequence of tests. Although persistentamong the body of students we interviewed it is not identified in the FCI/SCI sequence, and onlyemerged serendipitously through the course of our interviews. Upon reflection many factors maycontribute to this misunderstanding, and instructional material may be readily developed toaddress it.

The results of this investigation have then been used to guide the development of curricularmaterials that foster conceptual understanding of the key concepts in engineering mechanics.Specifically, we have designed instructional strategies that help students overcome the difficultieswe have found. By considering incorrect statements of fictional students, for example, thestudents in our courses are given the opportunity to rethink their understanding of relevantconcepts. The materials we develop take the form of collaborative-group worksheets that can beimplemented in small-group instruction as well as large lecture settings.

Data obtained from standardized post-tests after two implementations of the curriculum stronglyindicate that student conceptual understanding has been substantially improved. The fact that thisis particularly true for concepts that have been explicitly addressed by the worksheets suggeststhat an extension of our research and development effort to other topics has a realistic chance ofmaking an impact in undergraduate engineering education.

ACKNOWLEDGEMENTS This research has been supported by the NORDMETALL Foundation.

References

1. Sheila Tobias. They’re Not Dumb, They’re Different – Stalking the Second Tier. Research Corporation, Tucson,1990.

2. Ulrich Heublein. Zwischen Studienerwartungen und Studienwirklichkeit – Grunde fur den Studienabbruch:Ergebnisse einer bundesweiten Befragung von Exmatrikulierten in Maschinenbau-Studiengangenangen.Hannover, 2009.

3. Wiebke Derboven and Gabriele Winker. Ingenieurwissenschaftliche Studiengange attraktiver gestalten.Springer, 2010.

4. Lillian C. McDermott and Edward F. Redish. Resource letter: PER-1: Physics education research. AmericanJournal of Physics, 67:755–767, 1999.

5. Richard R. Hake. Interactive-engagement versus traditional methods: A six-thousand-student survey ofmechanics test data for introductory physics courses. American Journal of Physics, 66(1):64–74, 1998.

6. Andrea Brose and Christian Kautz. Research on student understanding as a guide for the development ofinstructional materials in introductory courses. In Edmond Byrne, editor, Proceedings of the 3rd InternationalSymposium for Engineering Education ISEE2010, June 30 - July 2 2010, University College Cork, Ireland,pages 239–245, 2010.

7. David Hestenes, Malcolm Wells, and Gregg Swackhammer. Force concept inventory. The Physics Teacher,30:141–158, 1992.

8. Paul S. Steif and John A. Dantzler. A statics concept inventory: Development and psychometric analysis.Journal of Engineering Education, 94:363–371, 2005.

9. David M. Brohn and John Cowan. Teaching towards an improved understanding of structural behaviour. TheStructural Engineer, 55(1):9–17, January 1977.

10. Jeffrey L. Newcomer and Paul S. Steif. Student thinking about static equilibrium: Insights from writtenexplanations to a concept question. Journal of Engineering Education, 97:481–490, 2008.

11. Luanna G. Ortiz, Paula R. L. Heron, and Peter S. Shaffer. Student understanding of static equilibrium:Predicting and accounting for balancing. American Journal of Physics, 73:545–553, 2005.

12. Lillian C. McDermott, Peter S. Shaffer, and Mark D. Somers. Research as a guide for teaching introductorymechanics: An illustration in the context of the atwood’s machine. American Journal of Physics, 62:46–52,1994.

13. Russell C. Hibbeler. Technische Mechanik 1 Statik. Pearson Studium, 10 edition, 2005.

14. Figure force couple: Man in inner tube, 03 2011.

15. Figure force couple: Beam, 03 2011.

16. Jean Piaget. The Child’s Conception of the World. Routledge and Kegan Paul, 1929.

17. Lillian C. McDermott. Millikan lecture 1990: What we teach and what is learned – closing the gap. AmericanJpournal of Physics, 59(4):301–315, 1991.

18. Lillian C. McDermott and the Physics Education Group at the University of Washington. Tutorials inIntroductory Physics. Prentice Hall, Upper Saddle River, New Jersey 07458, 1 edition, 1998.