in learning engineering mechanics and theifl influence on the enrollment [PDF]

ANALYSIS OF DIFFICULTES IN LEARNING ENGINEERING MECHANICS AND THEIFL INFLUENCE ON THE ENROLLMENT

IN A COURSE OF STATICS

Olga Lebed M.A., Civil Engineering University Dnepropetrovsk, Ukraine, 1984

THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF Master of Science in the Faculty of Education

O Olga Lebed 1998

SIMON FRASER UNIVERSITY August 1998 A11 rights reserved. This work may not be

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ABSTRACT The purpose of the study is to investigate the difficulties encountered by students in leaming the Engineering Mechanics course offered by the Department of Mathematics

and Statiçtiks, S N . The study was designed to investigate h o questions:

- What are the typical difficulties for students in leaming Engineering Mechanics? -

\niy is the enrollment in the course low and how do the above mentioned

diffkulties influence student enrollment?

The dificulties were analyzed and classified.

The following groups of

difficulties were discussed: difficulties related to problem-solving process, dificulties related to mathematics and difficulties related to the textbook. The study revealed the typical rnistakes related to the problem-solving process.

Analysis of results of

homework, midtems and final exarn is presented The ways to overcome the dificuities are suggested. Some suggestions are made

regardinç the implications for teaching and students' motivation. The relation behveen the difficulties and enrollment are discussed.

The study miçht be used to prescribe remediation to overcome difficulties in the

course and for developing student interest in Engineering Mechanics which may result in increased enrollmerit in this course. The research might be of generd interest to those mathematical teachers in coileges who prepare students for an engineering career.

Acknowtedgrnents

While wrïting this thesis, 1have received assistance fkom many people.

Thank you, Harvey Gerber. I greatly appreciate the time you spent with my work. Your constant encouragement and outstanding thorough job helped me to produce this work. 1have leamed a lot fiom you.

niank you, Tom O'Shea, for your guidance dunnç two semesten. Your classes helped me to decide my topic. Your encouragement in the early stages of my work were so important to me.

My sincere thanks to you, Edgar Pechtaner. Your kindness and willingness to

heip are greatly appreciated. I really enjoyed your lectures.

My thanks go to al1 students who participated in this research. My warm thanks to my family for their patience, heip and understanding during

the many hours required to prepare this thesis. A very special thanks to my dad, [gor

Kamovsky. 1 hated your cnticism, dad, but it was essential to the production of this wo rk.

Table of Contents

-.

Approval ........ .... ............................................................................... Abstract ...................................................

11

---

A. ......................................

111

AcknowIedgments .................................................................................

iv

Table of contents ..................................................................................

v

List of Tables .....................................................................................

WI

...

.-

List of Figures.. ................................... ............................................... .--VIII

Chapter I .

introduction ......................................................................... 1

Statement of the problem .............

. .................................. ... .... ........ - 3

Si@ficance of the study ................................................................. Limitations ................................................................................ Justification for selecting Engineering Mechanics .................................. Organization of thesis ..............................................

Chapter 2.

...................... ,

Literature Review ................................................................. - 8

CIassification of seIected literature ....................................................

8

The use of instructional stratedes to motivate

students in Engineering Mechanics ............................................

9

Difïiculties connected with teaching and learning Engineering Mechanics ..............................................

16

Chapter 3 . Research Design ................................................................ ..19

Method ...................................................................................

19

Subjects ...................................................................................

20

Course content ..........................................................................

21

Activities and instruction ........................................................

22

Evaluation of the course ........................................................

22

Data coI1ection ..........................................................................

23

Assessing students' performance ...............................................24 Questionnaires .................................................................... 24 Interviews .........................................................................

26

Data anaiysis and reliability of results ........................................ 27 Chapter 4. Analysis of Data ................................................................. -29 Classitication of difficulties ...........................................................

29

Difiiculties related to the problem-soIving procedure ............................. 30 Problem statement ............................................................... 32 Selection and substantiation o f appropriate rnethod o f solution ... ........43 Solution ...........................................................................

-47

Presenting of conclusion ........................................................ 54 AnaIysis of the results related to the problem-solving procedure ................ 57 Difiicuities reIated to the textbook ................................................... 61

Difficulties related to mathematics ................................................... 64 Analysis of rnidterrn and final exarn results ......................................... 73

Chapter 5. Discussion and Conclusions ...................................................... -53 Review of the sîudy .................................................................... - 8 3

Suggestions t i r conceptual and procedural development .................. 84 Enhancing students motivation ................................................ 89 Enrollment in the course MATH 262 ................................................ 90 Further research .........................................................................

92

Conclusion ..............................................................................

-93

References ........................................................................................

94

Appendix A .Consent Fonn ...-...................,...............*................*........... 97

Appendix B .Questionnaire 1 --...-........................ .. ............*..*......*.......... 99 Appendix C .Questionnaire 2 and tabulated results- ....................................... 103 Appendix D - Contents ofthe textbook (Beer, 1990) .................... . ................. 111

Appendix E .Supports for ngid bodies ......................................

. ............... 116

Appendix F - Midterms and fina1 exam ...................................................... 118 Appendix G .Solution of some problems ...................................................

122

Appendix H .Problems 2.35-2.54, the textbook (Beer, 1990) ........................... 135 Appendix 1 - Retèrence car& ................................................................

138

List of Tabfes Table 1 .Distribution of difficulties in the prob lem-so Iving procedure ................. -60

Table 2 .Midtenns and final exam results ...................,........,....--...*............ 74

Table 3 .Commonality in probIems of Statics ...............................................86 Table 4 .Enrollment figures in mathernatics and physics courses ....................... 91

List of Figures

Figure 1 .Difticulties wvith mathematics prior to Engineering Mechanics ...............61 Figure 2 .Students' performance on the First Midterm .................................... -75 Figure 3 .StudentsSperformance on the Second Midtenn ................................. -77 Figure 4 .Students7performance on the Final Exam ....................................... -79

Fi-me 5 .Distribution of tiequencies of "ditficult" problems .............................. 80 Figure 6 .Statics as a tbundation for other courses ............................. .... ........... 90

Figure 7 .Prerequisites for MATH 763, 36 1 and 470 ......................................... 91

Chapter 1 Xntroduction Applied mathematicians are in demand.

Since its foudation, Simon Fraser

University (SFU) has had one of the strongest groups of applied mathematicians in

Canada At present, there arc ten regular faculty whose research interest is in applied mathematics. A student wishing to focus on a specific a r a of this subject has a broad range of applied mathematics courses to choose from. Students interested in specializing

in the areas of physics and engineering find Applied Mathematics makes an excellent bais for M e r studies in these areas. The Department of Mathematics and Statistics offers a wide variety of Applied Mathematics courses. These courses uiclude those with various essential mathematical and computational techniques, as well as those more oriented towards specific areas such

as fluid or solid mechanics.

Mechanics provides an excellent introduction to

mathematical modelling for those students who are interesîed in pursuing kinesiology (the study of mechanics of human movement), biology (particularly in modelling molecular dynamics), medicine (particularly in heart valve design, blood movement in vessels) and especially in engineering (robotics, aircraft building, shipbuilding etc.). Despite the diversity of problems that arise, their sohtion, at least in p a c is based on certain general principles common to al1 of them, namely, the laws goveming the motion and equilibrium of material bodies. An understanding of the laws of mechmics is necessary in studying, understanding and interpreting a wide range of important

phenornena in the surrounding world. Mechanics is divided into statics, Ignematics and dynarnics. The Department of Mathematics and Statistics offen Statics (MATH 262) and Dynamics (MATH 263), which incfudes Kinematics.

These are required courses for the BSc in Apptied

Mathematics (SFU Calendar, 1996, p. 171). Both these courses, which are known as Engineering Mechanics, combine engineering problems and mathematical methods for solving these problems. Engineering Mechanics is a necessary subject, it is the link between mathematics and practical engineering. Unfortu~tely,-dents

do not learn this

subject easily. There are several reasons for this. For exarnple, there are difficulties connected with the application of strong mathematical techniques (Shaw & Shaw, 1997; Targ, 1976). However, there are also specific dificulties inherent in the subject which are comected with the number of new concepts, ideas and solution methods. At the end

of this course students see Engineering Mechanics without a clear understanding of how different parts of the course are connected (Van Heuvelen, 1991b). Finaily, students do not see the interdisciplinary links and, therefore, do not see the necessity of this subject for his / her firmre career, which can lead to the loss of motivation to study. Overcoming these difficulties and developing students' steady interest in leaming Engineering Mechanics is an important and interesting problem which 1investigate in my thesis. Although mechmics is the scientific bedrock of modem engineering, there is evidence of decreasing enrollment in the course. According to the report by the Office of Analytical Studies at SFU, enrollment figures over the last five years for the course

MATH 262 are as follows: 1994 - 20 students; 1995 - 19 students; 1996 - 17 students;

1997 - 11 students. Analyzing these figures, I looked at a prerequisite for MATH 262,

MATH 152 (Calculus 2). Since thïs course prepares potential students for MATH 262,

one rnay expect proportional enrollment in both courses. There was a steady increase in enrollment in MATH 152: 1993 - 446 students, 1994 - 539 students, 1995 - 561 -dents, 1996 - 575 students. However, enrotIrnent in MATH 262 decreased. There is concen that the situation rnay worsen rather than improve.

In my proposal I planned to examine students' attitudes towards Engineering Mechanics, their performance and difficulties connected with mathematics and special areas of Engineering Mechanics in order to identiQ and understand low student enrollment in this course at Simon Fraser University. However, during i n t e ~ e w sand after questionnaire surveys, I noticed that students paid little or no attention to questions related to attitude towards the subject, but repeatedly mentioned that they have dificulties in leaming i t Thus, I understood that students consider difficulties related to the course as a cardinai factor which influences their attitude towards the course and, as a

result, the enrollment. nius, understanding the difficulties which students experienced in l e h g Engineering Mechanics and suggesting ways of overcoming these difficulties became the main focus of my thesis.

Statement of the problem 1. What are the difficulties encountered by students in leaming Engineering Mechanics? 2. How do they influence the low enrollment on this course?

Si.PJiificanceof the study:

There is a stable tendency of decreasing enrollment in the course in the Department of Mathematics and Statistics at SFU. There is concern that this tendency may lead to the discontinuation of this course.

The study might be used to prescribe remediation to overcome difficulties in the

course and to develop student interest in Engineering Mechanics which may result in increased enrollment in this course.

The research might be of general interest to

mathematics teachers in colleges who prepare students for an engineering career.

Limitations Although Engineering Mechanics is offered at ùie University of British Columbia

(UBC) and British Columbia Institute of Technology (ECIT), this çtudy is based only on the results obtained £tomthe sample at SFU. There is no evidence if students from UBC

or BCIT experience the same difficulties. The study dso did not compare the textboob used in the other educational institutes. A major limitation is that there were no discussions with students who took

Engineering Mechanics in previous years at SFU. So there is no indication whether

students in previous years experienced the similar difficulties in leaming Engineering Mechanics.

There was no pilot study, since this course, MATH 262, is offered once a year in Septernber. It was not clear if this course would be offered in the fdl 2998 because of the low enrollment,

Another limitation of the study is the small sample of students. The study is based on only one ciass. It is not clear whether this group is typicd and can represent students who take Engineering Mechanics during many years at SFU.

The i n t e ~ e w swere a fieeflowing conversation between the students and the researcher. These conversations were not tape recorded.

Even though I made notes

s &er each conversation 1had to rely on my memory. reIated to i n t e ~ e w immediately The researcher cannot base this study on studies of other authon and compare the

results since little research has been conducted in this topic. The influence of other factors on enrollment, such as budget, advertisement, etc., were not coasidered,

It is believed that the above mentioned limitations did not

influence the resuits of the study to any significant degree.

Justification for selecting Engineering Mechanics

The selection of the course was bas& on the following criteria: 1) 1 have an engineering background. 1 graduated from the Civil Engineering

University in Ukraine; my major subjects were: Mathematics, Engineering Mechanics, Mechanics of Matenals and Structural Mechanics. Afier getting an honom diplorna in 1984, I worked as a practical eogineer for four years at the State Design uistitute. Then I enrolled in the pst-graduate program and worked on my PhD thesis devoted to

dynamical behavior of complex mechanicd systems. Even though my research work was interrupted by my emigration, I retained my interest in engineering subjects.

2) I taught Engineering Mechanics at the University level in Ukraine for several yean, so 1 feel confident in questioning students and making conclusions related to the subject matter.

3) 1see this course to be intereshng, very important and beautifid. It saddens me that the enrollment in this corne is iow. 1 want to understand why this coune is not popular among students at SFU. 4) I hope to r e m to teaching this subject again and I intend to use my research as a

teaching toof in the future.

Organization of the thesis

The shidy has been organized into five chapten and a set of appendices. Chapter 1 describes the importance of Engineering Mechanics for engineers and the significance of MATH 262 offered by Department of Mathematics and Statistics at

Simon Fraser University. The coune enrollments for the pst few yean is shown. An overview of the midy is presented: the research problem is formulated md its significance is shown, and the limitations of the study are outlined The reasons that led the researcher to this particdm topic are given. Chapter 2 contains a Iiterature review with an indication of how the Iiterature was selected. The chapter discusses the literature in the light of the research questions.

Chapter 3 presents the methodology of the study, describes how the data were collected and the methods used to analyze the dataChapter 4 defines two goups of difficulties for investigation. Dificulties related to problem-solving procedures are discussed and analyzed in detail. Difficulties related

to mathemaîics and the textbook are also discussed. Analysis of results of midtenns and final exam is presented. Chapter 5 suggests ways to overcome the difficulties, discusses the relation

between the difficulties and enrollment, and draws conclusions focused on the research questions formulated in Chapter 1. The fina1 section contains the Appendices. questionnaires and solutions of some problems.

These inctude consent hm,

Chapter 2

Literature Review This chapter presents a review of the literature devoted to pedagogical issues in Engineering Mechanics. A classification of articles in the light of research questions is given. The articles which consider instnictional stmîegies and difficulties related to leaming Engineering Mechanics are discussed

Classification of selected literature The literature devoted to a complex discussion of questions regarding leaming

and teaching Engineering Mechanks is not extensive. The literature presented in this çhapter concentrates on articles which consider instnictional strategies and difficulties related to leaming Engineering Mechanics. The articles I found related to pedagogical issues in Engineering Mechanics can be divided into three main groups: 1) articles devoted to implementing different instructional strategies to motivate students and to increase their interest in Engineering Mecbics

(Brilihart,1981;

Cumming & Mclntosh, 1982; Eisenberg, 1975; Faulher, 1986; Heggen, 1988; Heller & Hollabaugh, 1992; Kulik & Jaksa, 1977; Plants & Venable, 1975, 1985; Reif, Larkin, & Brackett, 1976; Rosati, 1985; Van Heuvelen, 1991, a & b; Van Heuvelen, 1995). 2) articles focused on meîhods of teaching specific questions of Engineering Mechanics (Freudenthal, 1993; Jong & Crook, 1990; Kondratyev & Sperry, 1994;

Newburgh, 1994; Rosati, 1985; Sperry, 1994). 3) arîicles concentrated specifically on difficulties connecteci with teaching and

learning Engineering Mechanics (Brown, 1988; Brown & CZement, f 987; Clement, 198 1; Larkin & Brackett, 1974; Larliin & Reif; 1979; Morgan, 1990; Reic Larkin, & Brackett, 1976; StathopouIos, 1989).

Some artades are at the intertaces behveen the above mentioned goups.

The use of instmctiona1 strategies to motivate students in Engineering; Mechanics

Developing students' interest in Engineering Mechanics is an important task. Heggen (19823) considered the question why engineers of diffèrent specialities need to learn Statics and Dynamics. He noted that engineers need commonality. The laws of

mechanics are universal, they are the sarne for diftërent matenal bodies regardless of their nature (building structures or human skeletons), and regardless of their size (extra

large or extra srnall). The solutions of diffèrent engineering problems are based on general cornmon principles and Iaws. Laws of mechanics apply equally to the earth's

motion around the s u , to the flight of a rocket or to human motion.

Its Iaws and

methods are necessary in studying, understanding and interpreting a \vide range of important phenornena in the surrounding world. Weggen also indicates that engneers need awareness o f their hentage. Mechanics

is amazinç since it is both an ancient subject and a modem one. Development of rnechanics retlects the evolution of humanity in architectural and technical rnernorïaIs: temples, bridges: ships and planes. Modem mechanics is the development of robotics, rocket and space technique. Heggen argues that 'the content of mechanics traces the thoughts of great

thinkers. Statics and dynamics can be more than a study of forces, it can be a window to our evolution (p.3 18)." Van Heuvelen (1991a) reviews instructional strategies and pedagogy in

introductory physics courses.

The author views Engineering Mechanics as a good

example to represent certain difficulties. He indicates that students do not understand the meaning of basic concepts and lack the organization of knowledge. Van Heuvelen says that one of the c'objectives of our instruction is (1) to help students l e m to form a knowiedge hierarchy about a srna11 number of basic concepts at the top with detailed application below and (2) to learn cues for accessing that knowledge from top down, that

is, kom the generd to the specific (p.894)."

Van Heuvelen (1995) suggests a strategy for problem-solving which should contribute to students' concephial understanding.

He suggests that *dents

make a

hierarchical chart to help them organize their own knowledge structure around the basic concepts of engineering mechanics.

In both these articles Van Heuvelen gives one example of a hierarchical chart related to Newtonian Mechanics. However, this chart is not suflicient and in no way gives a clear picture of the structure for the course of Engineering Mechanics as a single whole.

In another article Van Heuvelen (1 99 2b) considers different instructional strategies in introductory physics cornes which enhance the students' problem solving ability. He describes a new form of instruction which produced significant increases in student scores. He suggests a Study Guide md a set of Active Leaming Problem Sheets

as a supplement to a standard textbook. nie main idea of this strategy is to break chapter objectives into a srnail number of conceptua1 groups and to present each group as a diagram with an ovenriew as to where basic concepts are built in the block of knowledge. Students who understand the construction of the basic concepts ofthe subject benefit in the problem-solvinp part. This study indicates that students who were taught by this method understood more than students taught by a conventionaI method- A sirnilar idea

was presented by Eisenberg ( 1975). HeIler and HoIlabaugh ( 1992) suggested an instructional approach tor improving the problem-solving pertormance of pre-engineering students. One of the factors having

an eftëct on the students' success is the type of problems they must solve. The authors considered three types of problems: standard textbook problems, real world problems and specially designed Lbcontext-ricK'problems which were presented as short, sometimes humorous, stones. The statement of the problem did not always clearly speci- the unknowns, some intorrnation was rnissing and the students may have had to make some assumptions. The authors recommended such "context-rich" problems for cooperative group work to force the groups "to discuss physics issues while practicing efIèctive problern-solving techniques (p.639)." However, it is not clear whether the students acquired suficient knowledge to soIve classical problems before they were given "mntextnch" problems.

Although it is usehl to recognize such aspects of teaching, 1 cannot agree that such an approach is reasonable for promoting probiern-solving ability. In my opinion, it is better to solve a simpler problem and to present a situation where the student experiences

success. Ln the beginning of each study a clear idea of the assumphons, objectives and goals should be presented The student needs to work on many basic problems. Once a snident cm solve a bdk of typical problems, he is able to shitt to a hiJher stage ofthinking to make assurnptions. andyze and make conclusions. I see the main principle in leamïng appropriate to al! students, going tiom "simple to difIicultf'. Cumrning and McIntosh (1982) introduced a Persona1 System of Instruction (PSI) into a course on engineering mathematics tor a class of about 200 students. Seventy tive percent ofstudents expressed overa1I pretërence for the method and thouçht it should be continued. The authors believed that the course made a worthwhile step thanks to the

PSI approach.

Kdik and Jaksa (1977) expIained that the PSI method is so efiective

because of the use of such instructional strategies as tutor assistance, freedom ot'student pacing and rnastery quines, which include tiequent quizzing, immediate tëedback on quiz performance and a requirement of mastery on quines in order to go on to more advanced material: "Frequent quizing insures that students demonstrate what they are leaming while they are Iearning; immediate quiz fèedback provides students with guidance at each step in their work; and a mastery requirement insures that students do not push on to new matenal without a sound foundation (p. 19)." Eisenberg (2975) discussed difirent instructional strategies employed at the University of Floricia for teaching Statics, Dynarnics and Mechanics of Materials. His strategies included a combination of modular curricu1m packagïng, variable pacing, programmed learning rnaterials, cornputer managed instruction and conventional lectures. The author gave a definition of diffèrent instructional systerns: "hfodzdurinstructional systems are individualized systems in that they facilitate the construction of course syllabi to meet ïndividuaI needs. Programmed 12

imtructiom~systems are individualized systems in îhat they are addressed to individuds rather than a public. Selfpaced system are individualized systems of instruction in that they try to accommodate individual differences in ability and are subject to a significaot extent to individuai control (p.257)"

Eisenberg decomposed Engineering Mechanics into 34 modules and discussed the effectiveness of the different instructional systems employed for the course. He said that "it would be felicitous to report evidence that the modular, flexibly paced instructional system results in supenor student rnastery of engineering mechanics in cornparison with students educated by traditional rnethods. Although there is reason to believe this to be so, evidence for such a sweeping conclusion is difficuIt to produce (p.259)." B~llhart's(198 1) results are in contrast with Eisenberg's (1975). Considering the Engineering Mechanics course structure, Bdlhart illusirateci a rnoduIar ins~rucfiond systern which includes development of additional materials supplemental to the textbook.

She discussed what critena such matèrids must rneet. The author described handout modules covering approximately four lectures. Each module consisted of objectives, major lectures in partial note form and solved problems (about fifieen problems per module). The solution of problems should explain the theoretical b a i s and the stepbystep procedure used in the solution. Each module contained a test. To evaluate students performance the author suggested the testing procedure schematic diagram which is given in Figure 1. The first test was given in class. A student muld rewrite the test for a better grade and was required to rewrite the test if he failed it. Before a second re-test

was taken, the student had to complete remedial work, which included solving assigned problems. Before a third re-test was taken tutoring may be given by the inçtructor or

IN u M S TEST

Figure 1 . Schematic for cornputer testing (Brill hart, 198 1 )

Brill hart compared the students' performance in traditional and experimental statics courses offered in Triton College. The results, shoivn for 243 students, indicated that in the experimental course, A and B grades increased 15% and 10%compared with the grades in traditional courses while the C grade decreased 13% in the experimental

course. The author said that "student input has been consistent and extensive. They leam in many ways and like the options of many paths to achieve competency. The development of altemate materials to the text allows students to have multiple paths to success in the course without relying on the instructor (p.349)." The idea o~progrunzmii7gund self-puced insfnrctioin in Engineering Mechanics was given by Plants & Venable ( 1 975, 1985).

The authors described programmed

instruction at West Virginia University for Statics, Dynamics and Mechanics of Materials.

Some fiaction of the course is taught by traditional methods and a bit of

instruction is based on specially designed computerized material. The system is based on fiequent quirzes over each unit of programmed matenai. Students may rake a rnakeup quiz or take quines ahead of schedule in a self-çtudy center. The stuciy indicated that the performance of the students using programmed instruction is much better than the

scores of -dents

taught by traditional methods. However, there is a possibility that the

students who used programmed instruction may be better students and this factor may

account for the observed improvement. Rosati (1985) describes a computerized problern-solving process employing

Polya's strategy (1973). Based on the results of study of a pilot group of seven students

h m a f i e s b a n Engineering Statics course, the author found that the students' opinion about computerized tutorid program \vas not very positive. The students found the

cornputer tutorial program less helpFul then traditional classes. Cornputer problemsolving tutorials could supplement the corne, but in no way couid it replace the teacher.

Students would appreciate a lesson on problem-sofving strategies given in a smail class tutorial. The author said that ''the students do need help in developing a problem-solving strategy and for the well-structured problems of the Statics course, the Polya strategy was suitable (p-Zl)." However, Polya7sstrategy presented in this article does not reflect the specifics of Engineering Mechanics.

It shouid be modifieci in accordance with the

specifics of the subject. This will be considered in detail in Chapter 4. Another group of articles focused on methods of teaching specific questions of

Engineering Mechanics. Jong and Crook (1990) introduced the concept of displacement center in statics. This proposed concept is not found in mechanics textbooks, but the authors proved a theorem for locating displacement centers and stated two corollaries. The authors used the concept of displacement center in the statics course at the University of Arkansas and stated that the use of the concept "considerably enhances the effectiveness of the rnethod of virtuai work as a technique in solving certain equilïbrium problerns (p.477)." Other articles of this goup are not discussed in this review, since they are not related to the odined questions of research.

Difficulties connectai with teaching and learning Engineering Mechanics Clement (1982) identifies some factors considered as a source of difficulties, such

as abstractness of the materiai and mathematical skills required. The main focus of his paper is difficulties comected with key concepts and fundamentaI principles.

He

indicates that "many science-oriented students have difficulty understanding these concepts at the qualitative level (p.do)." Brown (1988) analyzed the misconceptions students have about force.

He

indicated that the misunderstanding of Newton's third law (the mutual forces of action and reaction between two particles are equal, opposite and collinear) is the key to these

misconceptions. He said that many students view a force as a property of an object rather than arising as an interaction between objects. A similar conclusion was made by Brown

and Clement (1987), who indicated that the common misconception -dents objects in equilibrium are unabie to exert forces.

have is that

Stathopoulos ( I989) descrïbed several demonstration models for students who have difficuity understanding the basic concepts of mechanics, such as the moment of force wlth respect to a point or an axïs. The author indicates that the models were wdcomed by the students since such devices helped them grasp the concept. They aIso asked the teacher to make more models tor other concepts inîroduced in the statics

course. The author mentioned that very limited infOrmation exists on demonstration devices related to the principles of statics. Larkin and Reif-(1979) addressed their study to finding effective problem-solving processes that students mi&t use so that problems could be solved proficiently. They considered two models of the e4upert?sand novice's problem-solving processes. The expert problem solver does not immediatdy constnict a mathematical description of the problem, as the novice problern-solver does. "Instead he selects a general method and uses it to work by successive retinements, first constructing a low-detail qualitative physical description of the problern and using this description to assure that there will be no intractable difficuities in applying hxs selected method. OnIy afienvards does he elaborate the mathematica1 details of his solution (p. 1%)." Reit; Larkin and Brackett ( 1976) taught students a simple problem-solving strategy consisting of the ibIlowing major steps: description, planning, irnplementation and checking. The authors said that although this strategy is quite simple, "it addressed some the key dificulties of many students. In particular, it provides a systematic approach which encourages students to examine a problern before blindly calculating and to check their answers afterwards. Furthemore, the steps in this simple strategy remain essential steps in a more complex stmtegy designed to deal with more sophisticated problems (p.2 16)"

Some studies (Larkin & Brackett, 1973; Morgan, 1990) have indicated that having the essential mathematical skilIs is a vital necessity tor taking a mechanics course. Students without a knowledge of mathematics cannot concentrate on new ideas and concepts, because they just hying to tollow the necessary mathematics. Morgan (1990) discovered that "students very often experience difticulties, not with the engineering pnnciples, but due to Iack of competence in the necessary mathematical techniques and the application of them to the engineering problems (p.975)". As is obvious fiorn the Iiterature review, questions reIated to dificulties for students studying Engineering Mechanics were not considered in depth and there is no ~Iassifcationof such dificdties. From the Iiterature 1could not obtain an answer to the tùndarnental question: "What are the typical ditticulties students have in Iearning Engineering Mechanics?"

Thus, the aim of the current research is to classi@ and analyze the typical difiiculties encountereci by students studying Engineering Mechanics.

Chapter 3 Raearch Design This cbapter describes the research procedure and defines the method employed

in the study to understand difficulties encuuntered by students in the course. The description of the students wvho participated in the research and the coune contents are given. The structures of questionnaires and i n t e ~ e w are s presented.

Method

The method employed in this research is an ethnographic study on a Engineering Mechanics course, MATH 262, offered by the Department of Mathematics and Statistics, Simon Fraser University Romb e g (Grouws, 1992) states that an ethnographic "approach usually requires a participant observer to examine artifacts, conduct clinicd interviews, and do clinical observations. Clinical observation is used to study goup behavior. The observer is ofien

a participant in the situation (p.57)." In the clinicd i n t e ~ e w the s researcher prepares a set of carefully structured questions and 'Wie sequence of questions varies with each respondent, depending on prior answers (p.56)." I conducted fieldwork observations in which one class was monitored. Dr. Edgar Pechianer, the instnictor of hWTH 262, agreed to provide his classroom for my observation and he had no objections to my atîending his classroorn lectures and tutorials. Al1 -dents

were aware of my purpose for being in the class. The aim of rny

observations was not to judge or criticize the teaching rnethods, but to analyze the

*dents7

perceptions of the course, students' performance and particular difficulties

related to the course content. Thus, the general objecti-ve of this obsewation was to analyze students' performance in Engineering Mechanics for the purpose of detennining difficulties comected with the subject, lack of understanding and any weak points so remedial action could be taken,

In addition to observing, in order to obtain a better picture of the students' perception of the course, I conducted a survey by using questionnaires, analyzed artifacts (homework assignments, midterms and final exam), and conducted interviews.

Participants Eleven students were registered in MATH 262 for the Fail of 1997.

The

participants in this study were nine students who signed the consent form (Appendix A). Two were female and seven were male. Eight of the participants were nineteen or twenty years old These students entered SFU directly from hi& school. One participant, age twenty five, had graduated from college. Al1 students exhibited a good knowledge in mathematics and science in high

school. Five students had an A in both mathematics and science, two students had a B in both subjects and two students had an A in mathematics and B in science. Only one student indicated that he did not like mathematics and science and the rest indicated that they enjoyed these subjects in school. AI1 the students had taken Calculus courses (MATH 151, 152) and one student

even took the Honors Supplement Calculus (MATH 161, 162). During the study five

students were taking Linear Algebra (MATH 232) and Vector Calculus (MATH 251 or 252). Al1 students had taken the following courses in physics prior to the current

research: Modem Physics and Mechanics (PHYS 130); General Physics (PKYS 101); and Optics, Electricity and Magnetism (PHYS 121).

Course content

The content of the course is based on the textbook "Vector Mechanics for Engineers" (Beer & Johnston, 1990). The topics covered in the course are as hllows: 1. Basic concepts and principles: Force vecton; auiorns; constraints and their

reactions; tiee-body diagram. 2. Concurrent hrce systems: Composition and resolution of torses; equilibrium of a system of concurrent forces; Varignon's thearem. 3. Generat case of hrces in a plane: Moment of a force

-

scalar and vector

formulation; moment of a couple; reduction of a coplanar system to a given center and to the simplest possible forrn; distributed loading; conditions for the equilibnum of a

coplanar force system. 4. Equilibriurn of a rigid body: Conditions for ngid-body equilibnurn; moment of

a force about axis; equiIibrium in plane and space; scalar and vector solutions. 5. Structural analysis: Definition and cfassification of stnxtures; misses - method ofjoints and method ofsections; h m e s and machines. 6 . Center of gravity: Center of gravity of a rigid body; composite bodies; theorem

of Pappus-Guidinus; fluid pressure.

7. Interna1 forces: Bearns

-

shear and bending moment diagrams; relations

between distributed Ioad, shear and mcrnent. 8. Friction: Laws of static Bction; angle of tiicrion; equilibrium with tiiction. 9. Principle of vimial work.

Activities and instruction The only technologïcal too1 for instruction in Engineering Mechanics was an overhead projector.

During the Fall semester the students had twelve weeks of

instruction in MATH 262. The students had three 1-hour lectures in a week (Monday, Wednesday, Friday) and one tutonal (Thursday). During lectures the students wrote notes and asked questions.

Hornework was given every Friday and it \vas due on

following Friday. Marked hornework was returned every Monday with a discussion of the ditficult problerns. Assignments were tkom the text. Teamwork was permitted. There were two midterms during the semester. The tirs%midterm covered topics 1-3, and the second midterm covered topics 4-6. The tinal exam covered al1 topics.

EvaIuation of the course The students' achievernent in MATH 262 was evaluated on the basis of the homework assignments and their results on the midterms and final exam. There were usually 6- 10 problems for homework. The marking scales on the homework assignments were usually différent; it depended on the dificulty of the assigned problems. Two midterm exarns included questions showing problem solving ability; they were "'open

book" tests. The final exarn had an "open book" part and a "closed book" part. The closed book part, consishg of ten questions, was designed to evaluate understanding of

the main concepts and ideas. Tt was the theoretical part of the course. The open book part consisted of eight problems which covered the whole course. The gniding scheme

was as follows: 10% for homework a~si~gments, 30% for each midterm exarn and 50% for the final exam (25% for the closed and 25% for the open part).

Data colIection The data was collected over a period of thirteen weeks, begiming on Septernber 2

and ending on December 16, 1997. I attended all lectures and tutorials making detailed observations of how students participated in the class, how they asked questions, whether

they paid attention to a lesson and what was their reaction on new concepts (whether they were confùsed, interested or indifferent). A total of 36 lectures and 17 tutorials were

observed At the very

fim lectures when l introduced myself and explained the purpose of

my being in the course, I offered help to students as a Teaching Assistant so we could discuss difficult probIerns and ideas of the course. hactically once a week students

asked me to help them with homework assignrnents. In such sessions 1 did not provide solutions, but I tried to determine the students' weak points. On this basis we built the understanding of the concepts which led to the solution of the problem. During such

meetings informal i n t e ~ e w were s conducted Also, informal conversations took place before and after lectures. A total of twelve meetings with students were conducted-

Each meeting took approximately 20 to 40 minutes; these meetings included explanation of difficult problems and interviews. On each session 1 met wïth one to four students; more ofien ttiere were iwo students.

Assessino; students' perf'ormance Documentation for the students' problern-solving performance consists of log of a weekly record of the students' perfOrmance related to solving probIems and organizing homework assignments.

The log contains the date, problems, perccentage of

performance, totai mark given by prokssor and my evaluâtion. I re-read each honework assignment afier it had been marked. Marks given by the prokssor belped me to identiQ cornmon errors and difiiculties in the problem solving process. The tocus of my own evaluation included the tollowing: identiwng the given and the unknown; drawing and Iabeling of design diagram; pIan for the solution; stepby-step explanation of the solution;

use of units during solution; answer; explanation of the obtained results and conclusion,

Questionnaires Two questionnaires were given; one at the very first lecture, in September, and the other at the end of the course, in November. The questionnaires are presented in Appendixes B and C. The first questionnaire was developed in order to determine the students'

background, their mathematics and science achievements in high school, their level at

university, their knowledge about MATH 262 pnor to their taking the course, and their expectations fiom this course.

This questionnaire was divided into four parts.

In the first part of the

questionnaire, respondents were asked for a personal description. ln the second part respondents were asked about their mathematics and physics levels before coming to the university. In the third part they were asked to use a five-point scale to rate difficulties

they have with mathematics at univenity level. Nso, in this part of questionnaire they used a four-point scale to explain what influenced them when were applying for

mathematics courses at the univesity.

In the fourth part of questionnaire students

indicated briefly their expectations and beliefs before they started MATH 262. The second questionnaire was developed in order to determine how students

understand the hierarchy of the course, what problems they have related to the subject matter. Students responded on a five-point scale fiom "Always" to "Never" and fkom

"Strongly agee" to "Strongiy Disagree". The first scale was chosen in order to gain an insight into the confildence of students' problem solving ability and to understand their style of working with the textbook The second scde was chosen in order to receive their opinion pertaining to the course. There were several open-ended questions which were categorized according to fiequently occming comments.

interviews

There were no f o n d interviews. However, a fieeflowing form of interview took place. The questionnaires which students completed pnor to the i n t e ~ e w sserved as a guide, and conversation was allowed to flow. Questions were asked in such a manner

that the shident codd expand on the topic. A total of twelve informal interviews were conducted. Each interview took approximateiy 10 to 20 minutes. The main focus of the questions durhg i n t e ~ e w in s the first half of the course

were questions related to their reasons for talcing the course, what factors intluenced them in their choice, and what mathematical difficulties they had. The questions during the interviews conducted at the end of the course were related to the -dents'

attitude

towards the course. These questions helped to understand the low enroliment in the course, and what difficdties they have in leaming the çubject. Kowever, students paid littIe or no attention to questions reIated to their attitude towards the subject, but repeatedly mentioned that they had difficulties in learning it. 1

then understood that students consider diEcdties related to the course as a cardinal factor which influences their attitude towards the course. This in turn influences the enrollment Thus, the main focus of the questions was shifted to the questions regarding the difficulties students experienced in leaming Engineering Mechanics.

The first interview was tape recordeci, but it was unsuccessful, since students

were not willing to discuss almost any question. When there was no tape recorder on the second i n t e ~ e w ,&dents felt more free and an interesting conversation took place.

Thus, conversations were not tape recurcied, but notes of d l the discussions were made right &ter each interview.

In generai, 1 did not have problems with i n t e ~ e w s .Students asked me for help many brnes; I was always avaitable and students were satisfied with my answers, so when it was my tum to ask questions they were open and frank discussions took place.

Data a d y s i s and reliabiiitv of results General difficulties and common errors have been identifie& The evidence cornes from a data bank which inchdes homework, midtems, final tests, questionnaires

and interviews. Protocols of dialogues, shidents' sketches and interesting kdgments of homework were selected, copied and sorted by the author. Al1 these data were coff ected for the analysis stage and presented in the text in relevant places as a supportîng evidence

of researcher's statement. The statistical analysis consisted of simple descriptive statistics: mean valses for

homework and tests. Grmping and coding techniques were used to examine students' responses to questionnaires.

On the basis of the homework assignments and the questionnaire survey, the percentage of difficulties on each step in the problem-solving process was determinedDetailed description of how these percentages were obtained is given on pages 58-6 1.

On the basis of rnidterms and the final exarn, the average score for each test was calculated. To generalize students' performance on midterms and the final exam, a mean

value was used to represent a typicd data value in each set of data. This is the average

achievernent tbr each problem in each test. On this bais, the diagrams showing students performance !vas made using the standard program Microsoft Excel. This information is presented in Table 74 and Figures 2-4.

The validity of answers given in the questionnaires is difticult to justity. So in order to ensure truthkl responses, similar questions were phrased ditferently on the interviews. A stable match in responses during the interviews and in responses given in

the questionnaires were observed.

Chapter 4 Analysis of Data

This chapter is devoted to an analysis and classification of the difficulties that

students have when learning Engineering Mechanics.

The substantiation of such

classification is given and two groups of dificulties for investigation are defined DiEcdties related to problem-solving procedure are discussed and analyzed in detail. Difficulties related to mathematics and the textbook are also discussed.

Classification of diEculties The analysis of data (homework assignments, tests results, questionnaires and

interviews) revealed the typical groups of mistakes. Mistakes of the first group occurred

when students did not use the necessary problem soiving strategies. Mistakes of the

second group occurred because of difficulties with mathematics and the textbook As the first group 1 conçidered specific difficulties inherent in the subject of

Engineering Mechanics. Since developing problem solving abilities is the dominant aim

in the Engineering Mechanics course, and acquiring the appropriate strategies is applicable to many other subjects and in hime engineering practice, the investigation of difficulties of the first group is of great importance. These are difficulties related to problem-solving process (PSP) ody. This group of difficulties I called the dificulties of problem-solving process (DPSP). The second group of difficulties \vas related to Engineering Mechanics to a lesser extent than the difficulties of the first group, and the nature of those dificdties varied

(related to textbook, to mathematical background, etc.). Difficuities of the second group

did not reflect the specific subject matter to such extent as DPSP and that is why they are considered a11 together.

DifficuIties Reiated to Pro blem-SoIving Procedure The problem-solving procedure is a combination of certain steps which lead to the solution of the problem. Polya (1973) suggested an approach for the solution of nonroutine problerns. The problems *ch

were considered in the course MATH 262 were

typical, classical problems. These were not "nomroutine" problems according to the Polya's strategy. However, I decided to adopt Polya's strategy and then to adapt it to the solution of the "routine" problems in Engineering Mechanics. My decision was based on

my beliefs that Polya's fundamental strategy c m be applicable for the solution of al1 problems in every subject, because each problem required understanding of it, designing

a plan, -ng

out the plan and lookïng back It is obvious that these steps are necessq

for the solution of routine problems as well. But since each subject has its own nuances, inherent to this padcuIar subject, then Polya's strategy should be rnodified in accordance with the specific of the subject.

I suggested the foltowing modification of Polya's strritegy to the solution of problems in Engineering Mechanics: 1. Problem statement: a) understanding the problem; b) visualizing the problem;

c) identifjring the given and unknowns; d) drawing and labeling the diagram

2. Selection and substantiation of appropriate methods of solution:

a) discussion of

diEerent methods of solving problem; b) plan of solution 3. Solution: a) choosing the body whose equilibrium shouId be examined; b) necessary

information which is to be dispiayed on the fiee-body diagram; c) mathematical mode1 of the problem (equations of equilibrium, principie of virtud displacement, or special

theorems) and its soIution; d) expianation 4. Conclusion: a) units control; b) analysis of results; c) answer

The solution of problems in lectures and tutorials fitted this scheme. Examples of

solving some problems in accordance to this scherne are shown in Appendix G. We will consider difficulties encountered by students on each stage of PSP in

Engineering Mechanics.

Each stage wilI be discussed separately. To evaluate the

problem-solving skills in accordance with the above mentioned scheme, the tollowing data were analyzed: hornework assignments, midterm results, Questionnaire 2, hgments

of interviews and intormd conversations. For the purpose of analysis, the data were compiled in tabuIar t o m and the final results are presented in Table 1 of the current chapter.

1. Problem statement

The goal at this stage is to make the statement of the problem clear and precise. It shouId contain the aven data and indicate what is required to determine or prove. A neat diagram showing al1 quantities should be presented.

To gain a complete

understanding at this stage, one shou1d follow the toilowing steps:

a) Understanding the problem DifficuIties in this step were mainly reIated to new concepts and corresponding terminology.

They were related to a lesser extent to the wording of the problem

conceming the description of the object and principle of its action. Nav concepts: Students had problems in understanding the basic concepts: such

as action and reaction, constraints and their reactions, how the forces exerted on a body

are related to the forces which the body exerts on other bodies. Misunderstanding of these concepts would lead to serious mistakes. For example, 1 suggested the tollowing problem to one student: Knowing the magnitude of the vertical torce P and al1 geometry, determine: a) the tension in the cable CD; b) the reaction at D; c) the force exerted by the cable on rod at C. d) reaction at B; e) the force exerted by the rod AC on support B. The ffagment of the dialog between me and the student was as follows:

R: S:

R: S: R:

- is the statement of the problem clear? -Yeah. - How will you start? - Show free body diagram. - OK, do it.

The copy of the student's work lor cases (a) and (b) is given below.

The studeni drew [fie free-body diagram correctly to detentiinc the [ension of the

cable CD, however he did not know how to find the reaction at D. The future dialog with a student revealed that he knew that action and reactions are equal and opposite. However he could not apply this concept to solve the problem.

Wording of the problenr: The students asked some questions related to

terminology. They were about the types of supports (rollers, collar, hinge, fixed support), types of constnictions (trusses, frames and beams) and mechanisms (pdleys, slender rod, etc.). For example, one student asked me "how does hinge look like

iii

reality?". 1

referred hirn to the textbook (Beer & Johnston, p. 51), where real view of different type

of supports are given. Example of schematic representation of hinge and its view in the reat world is as folIotvs:

Similar types of questions arose several time during lectures. Usually questions were about where such supports are used in the real world, how they look in the real world, etc. After the first explanation such questions did not anse. The obsewation of

lectures showed that students were completely satisfied aRer they found out the

connection between schematic representation of supports and their real view. Schematic representation of different supports and connection and their real view are displayed in Appendix E. Another group of questions was related to the name of mechanisrns or their parts.

For example, afier one lecture the student asked me for help to understand the concept of dismernbenng mechanism. Afier I explained the main concepts to hirn briefly, I gave him a problem. None of us had the textbook, so I drew hirn the following picture

d\-7

O --

3 O" ------------------

F

f-----

&r3;

and asked the question of the problem. A fragment of our dialog is given below.

R:

- The piston rod applies a force F.

#

The system in equilibnum. Calculate the

moment acting on the crank a m . S: - What's îhat? R: - What is what? - Piston rod and crank a m . S: Questions related to rnechanisms arose several times, whicli is natural since the

names of the mechanisms are not familiar to students.

The observation of students' dialogs and tutorials revealed that students had

di fiiculties in visualking problems, especially if they were three-dimensional. I noticed that several students tned to look at a problem in the text using different angles and

moving the textbook away from them. ARer the tutonal 1 asked these students if they had some problems. AII of them told me that they had difticulties understanding 3-D

pictures. The following pictures from the textbook were confusing for siudents (p.50, 51):

Fig. P2.87 and PZ89

Flg. P2.75 and P2.76

P

FI^. ~2.77,P2.78, and P2.79

omission of some of the relatively unimportant factors. For example, if we think that the size and the shape of the object are unimportant, we may imagine it to be a point. Such a simplified and idealized version of a comples engineering problem is cal!ed a model of the original situation.

The ability to visualize the problem and make assumptions

concerning the information that should be omitted to permit an analysis of the situation, is very important for engineers. Students should be aware of the process of constmcting models as idealizations of reality. The followihg dialog shows that students have some problems in making models (this dialog refers to the previous probIem and it was behveen the researcher and one student, while tsvo other students listened):

S: R: S: R: S: R: S:

Let's show the picture first. (Started to copy picture in a11 details) What are you doing? Just copy from the tex?. Why are you rrying to show ail picture? Why do you need so many details? To show where forces are applied. Make a model. (ConFusion)

A mode1 for this problem is represented in this picture:

Visliaking the concepts: As indicated in the questionnaire and informa1 discussions, students had dificulties with constmcting images for the following concepts: moment of

the force and couple as a vector; moment of inertia; intemal forces in beams and cables.

For example, four students indicated in the Questionnaire-:! that they have difficulty with the concept of moment, because "it's different", "hardto visualize", '-hard to visualize

and get right siy","don't understand how it works".

C)

Identifjrinc the given and unknovms

Homework results showed that students were not in the habit of identibing the giveii and unknowns before writing the equations of equilibrium. Probably, students did

not have difficulties here, they just did not see the necessity of representing the problern

and introducing the suitable notation. One example o f the student3 work is given below.

From this shident's work it is not clear what is @en and what has to be determined.

4

d) Drawing and labeling the diagram It is important for students to sketch the problem with all the given quantities and geornetry. This step helps in the visual imaging of the problem, before the actual solution. It is not just a pretty picture - it reminds the student of the initial system, which

contains al1 the given infomation. The aim of this stage is twofold: 1) Sketching the picture from the text helps students to notice a11 the details,

which rnay be missed without this drawing.

2) Since engineering problems contain a lot of information which may not al1 be used in the solution, it is necessary to make an initial sketch after disregarding the

irrelevant information. Students should be aware that there are no strict requirements at this step. There is flexibility in drawing, except one must make the sketch clear (tu get rid oîùnnecessary information which distracts his attention) so as to analyze the problein further. For C

example showing models for Crames, trusses and mechanisms using lines which have "'bodies", may distract the student's attention h m the main idea.

On the example

s h o w below we can see that one student tned to solve the problem using lines with ccbodies",however he switched to the "single line sketch'' because it more suited him to understand the problem.

Kowever, the model he made was not clear and was not helpful for the solution the problern. Another student felt cornfortable using lines with "bodies", fragments of his work are given below.

Construction of a meaningful model is a very important experience and ofien the most dificult phase in the problem-so1ving process. This initial sketch is a first step leading to drawing a free-body diagram. A carefully drawn diagram helps to choose the correct method of solution and prevents errors in stating the conditions of equilibrium or

using special theorerns or principles. Understanding and conceiving of the problem starts

at this stage.

Students should be aware that in real engineering practice, models of mechanisms

or another type of consîructions are rarely represented using lines with "bodiesn. So it is beneficial for mident to be used to "single line" representation of a model. For example instead of Form 1,I would recommend to use Fonn 2.

Form 1

Fonn 2

The results of this study are represented on Table 1, page 6 1. The biggest problem in Step 1 is understanding and visuaiizing the problem. According to students, they did not have major difficulties in identifjring the given and unknowns. Shidents7 difficuities at this stage were close to my expectation, which is not surpnsing. Indeed, afier the conceptual understanding and visualigng the problem, the procedure o f identimng the given and unlaiowns is not a problem for the students. However, the rnajority of students did not identifjr the given and unknowns in a clear manner in their work. Sorne difficulties arose when students needed to show a sketch, since this required a decision of what information was necessary, what had to be s h o w and what could be omitted. For beginners it is quite often a dificuit decision. Obtaining the percentage of the difliculties for this step was not applicable, since ofien this step can be combined with the drawing of a fi-ee-body diagram, which will be discussed later.

2. SeIection and substantiation of ap~ropriatemethds of soiution

Problems in Engineering Mechanies may be solved by using different methods. However, the selection of a proper rnethod of solution for the begimers is ofien quite

difficult The goal at this stage is to understand what methods may be used in pnnciple and decide which rnethod is better (easier and faster) for solving the given problem, and

make a clear plan for setting up a mathematical mode1 for the problem. At this stage, one should follow a number steps.

a) Discussion of different rnethods of solving the problem Under "different methods" we understand the following cases:

1. Different approaches which are based on different fundamental ~rinciples.For

example, a complex mechanism can be calculated using the principle of vimial displacements or by dismembering the rnechanisrn and considering each part separately.

For instance, these approaches are applicable for the following problem: Detemine the vertical force P whîch must be applied at C to maintain equilibrium of the linkage (Beer & Johnston, # 10.4). Both solutions are aven in Appendix G. 100 N

A

2, Using the condition of equilibriurn in analytical or graphical form, or their

wmbination For example, for concurrent forces on the plane, there is a possibility of using both methods for the solution. Lf there is a general force system there is only the

analytical method. For exarnple, problem A cm be solved analytically or graphically, while problem B cm be solved only anaiytically: (A) Two cables are îïed together at C and Ioaded as shown. Determine the tension in AC and BC (Beer & Johnston, # 2.36).

(B) Determine the reactions at A and B for the beam and loaduig shown (Beer Johnçton, # 4.13). D

Bath solutions are given in Appendix G.

3. Representation of forces in scaIar or vector form. The vector form is advisable

for three-dimensional problems. However, dependuig on the form of presenting the given data ( a force can be defined in a vector form or using angles with coordinate axis),

the student shouid be able to choose the most appropriate way of solvuig the problem. For example, we discwed the following problem, whose sketch was given in Form 1: The 12-131pole is acted u p n by an 14-kN force as shown. It is held by a bal1 and sucket at A and by the two cables BD and BE. Neglecting the weight of the pole, determine the tension in each cable and the reaction at A (Beer & Johnston, # 4.73). 14kN

z

&-

The -dent

:. Y

Form 1

Fom 2

had no problem to solve it in a vector form. Then I presented the

same problem in the Form 2, which assumed the solution in a scaiar fom. The fkagment of our dialog is given below.

- How will you solve this problern?

- Same way, there is just angles are given instead of coordinates. - So, will you use vector solution again?

- Yeah, why not? - Do you b o w other way of solving, like ushg projections?

- Oh, sure it wodd be easier.

- So why did you want to use vector solution? - Well ...,1don't know.

This problem cm be solved easily in vector f o m if its sketch is given in Form 1

or in scalar form if its sketch is given in Form 2. Both solutions are given in Appendk G. Students shouid be able to explain why it is desirable to use one method of solution (or form) instead o f another. I saw fiom the study of homework materials that d e n t s quite

often used a more complicated methods or approaches to the solution. This may indicate that they did not see alternative ways of solving the problem. Results of

study are

represented on Table 1, page 60.

b) Plan of the solution M e r the method of solution is chosen and substantiated, one has to make a plan

for the solution To make the plan of the solution a -dent

shouid see the hierarchy of

the solution process. This means that the main theorem or method which is used for the

solution çhould be stated firsf then the governing equations or equations of equilibrium, then define the necesçary information such as dimensions and angles and so on. As seen fkorn a study of their homework, the students did not have clear and

easily followed plans for solution. For example, they rnight start to detennine the missing angles or dimensions wiîhout showing the general line of the solution. They mi& obtaïn the final result correctly, but their way of thinking was rather complicated

and the general culture of setting a mathematical model was rnissing.

ln more

complicated problems, the absence of a clear plan may often lead to mistakes.

Results of this study are represented on Table 1, page 60. A student has to have sufficient knowledge to be able to discuss different methods and approaches. When a

decision regarding an appropriate method of a solution has been made, making a plan of

solution using the chosen method is less diflicult. However, the increase of dificulties in the step 2(b) shows that even though a student may choose the method of soiution, he has

difflculties in setting up a pian for the solution.

3. SoIution

Only on the basis of carefully drawn free-body diagrams can the correct solution of a given problem be obtained. ' A thorough undersîanding of how to draw a fiee-body diagram is of primary importance for solving problems in mechanics (Hibbeler, 1995, p. 183)". The free-body diagram is the most important single item used in the andysis of engineering problems. So the goal at this stage is to Iearn how to constnict a free-body diagram and on this basis provide the solution. The following çteps are necessary for this stage:

a) Choosing the bodv whose equilibrium should be examined The object under study mu-t he chosen in accordance with the problem. The same problem can have different fiee-body diagrams depending on what information has to be obtained. Making decisions regarding what object or system of bodies should be isolated is ofien a serious task, Complete extemal boundary should define the isolated system corn ail contacting or surrounding bodies. The isolation of the system under consideration is a crucial step in the formulation of the mathematicat modet-

Observation of the homework and discussions inciicated that the students found difficulty in making decisions concerning which system had to be isolated for further study. For exarnple, the students asked many questions conceming the following problem:

L

A 200-kg crate is supported by several rope-and-pulley arrangements as shown. Determine for each arrangement the tension in the rope (Beer & Johnston. #2.49).

Several students asked me for help in this problem. While they decided what system should be isolated, the question "How do you know what part of the systern to consider?" arose several tirnes. This important question indicated that choosing the system for investigation is a serious problem for students.

Some copies of students'

work with free-body diagrams for this problem are shown below.

These sketches show that students do not undentand which object or system of objects has to be isolated for consideration. They do not show a clear boundary and this 48

resulted in erron while showing al1 forces which act on a body. The free-body diaagrams

and solutions for each case are given in the Appendix G. During the semester the students deveioped the ability to isolate the object for investigation and by the end of semester these difficulties greatly decreased. So the percentage of difficulties at this stage is not presented.

b) Information which needs to be disp1aved on the fiee-body diagram The free-body diagram is a sketch of the isolated body or system of bodies considered as a single body showing al1 forces acting on it by other bodies that are imagined to be removed. It is imperative that the force system under study be clearly defined and represented. The correct representation of forces on the fiee-body diagram requires knowledge of the characteristics of contacting bodies. It is vitally important to understand the symbolism involved in the given draiying. Often the students' misunderstanding of the representation of constraints and supports resulted in incorrect fiee-body diagams. Some students were tempted to omit fiom the fiee-body diagram certain forces that may not appear, at first ghnce, to be needed in the calculations. The omission or inclusion of a force which does not act on the body wilI give wrong results. AIthough statics generaily studies force systems, it is recommended that students show the actual fiee body and systern of forces applied. Students often showed only the system of forces, which did not give a rneaningful picture for analysis. For example, the following problern \vas given to students:

1

The coetricients of €fiction between the 20-kg block and the incline are A = 0.40 and pk = 0.30. Determine whether the block in equilibriurn and End the magnitude and direction of the Friction force when P = 600 N (Bcxr & Johnston, # 8.3).

In drawing the free-body diagram the student omitted the actual body and showed

the force system only, which does not provide a meaningful picture for analysis. A copy of student's work is presented below.

The choice of coordinate axis should be indicated directly on the diagpm. Quite b

oAen students did not show coordinate axis or their origin and direction were not clear. For example, in the following copy of the student's work there is no clear indication of the ongin of the coordinate s i s .

1

Free-body diagrams are designed to show ail forces acting on a body and some additional information, such as angles, dimensions, etc. However, excessive information,

such as given quantities, may distract attention.

Force arrows should be clearly

distinguished from dimensions arrows or coordinate axes. To avoid such confusion a different color is recommended. Only after a free-body diagram bas been carefully

drawn shouId the equations of equilibrium be wn-tten. As this study indicates, students underestimated the necessity of careful l y and neatly drawing free-body diagram. None

used color or niler, and no clear indication of coordinate mis were given. Sometimes students did not even show a free-body diagram at all, for example

Fragments of a student's work with reasonable free-body diagrams is shown below.

@ 4-

- -

C)

a -

î3..

Equations of eauilibrium and their solution When the previous steps are completed, the setting up of goveming equations (in

statics

- equations of

equilibrium) is straightfiorward. Before applying equations of

equilibrium, one of the most useful steps is an analysis of different choices of moment

centen. Generaliy, the best choice of a moment center is one through which as many unknown forces pass as possible. Students did not experience major problems here.

d) Explanation

Many problems that at first may seem difficult become clear and straightfonvard once they are attacked in a Iogical and disciplined marner. It is highly recommended for students, especially for beginners, to explain each major step in the solution of a problern. A solution wi-thout explanation, with just formulas, is of little or no value to engineers. As the study showed, students rarely explained their solution. (Results of this

study are represented in Table 1, page 6 1).

3. Presenting of conclusion

Once the solution has been cornpleted, each problem has to be concluded by a brief section showing that the results are reasonable. The goal at this stage is to analyze and explain the obtained results.

a) Units controt Solutions can be presented in numerical or general (symbolic) form.

The

cornputations should, as a rule, be witîen out in general form. Such a presentation has several advantages over the numerical one.

By using symbolic representation of

quantities, the student is focusing on the connection behveen the physical situation and its related mathematical description.

This provides formulas for determining the

unknown quantities which can be used to analyze the results. Solutions in the general form make it possibie to catch mistakes by checking the dimensions, whereas dimensional homogeneity may be lost when numerical vaIues are usedIf obtaining the final result in algebraic fonn is complicated, then it is almost

always advisable to carry along the n i t s of the quantities as well as numbers. Such practices ofteri facilitate the detection of errors; if at any stage in the computation two terms are dimensionally inconsistent, an error has been made. As the homework results indicated, in most cases students represented a final

answer along with correct dimensions, however, they did it as a mere fonnality wïthout checking the units while solving the problem, which is recornmended, or at the end of

calculation, which is required. Students rarely used the analfical formula obtained as a

final solution for units controI. Sornetimes students iorgot to convert uni6 to rnake al1 dimensions in the problem consistent

For example, they calculated using mived n i t s such as millimeters and

meters, Xewtons and kiloNewtons.

b) Analvsis of results

Once the solution has been completed, each probfem has to be conctuded by a

brief section showing that the results are reasonabte. Providing only numerical answers to an engineering problem is not enough. The answer has to be studied with technical judgment and cornmon sense to determine whether or not it seems reasonabIe. Even simple judgments (for example, understanding that the answer makes sense for the given problem) are important for the engineering student and in engineering practice. Sometimes intuition can help in the analysis of the results. However, in some problems a student cannot rely only on intuition. For example, in the following problern analysis of the results is necessary: Two cables are tied together at C and loaded as shown. Determine the tension in AC and BC (Beer & Johnston, # 2.35).

The answer of this problem is TAC= T, = -. It is necessary to understand 2 sina that the tension in the cable becomes infinitely large when a approaches zero and thus

I

such construction cannot be designed.

Such analysis is necessary especiaily For

engineers.

This study indicates that students can analyze the final answer, but they do not show it in their work or do it rarely. For example they never explained what was the

meaning of a negative result.

CI Answer Since the study of statics is directed toward the description of forces that act on

engineering structures in equilibrium, then answers should include the followiving information: numerical values, dimension and direction of the Lbrces. As seen in this

study, students provided the numerical value and ofien forgot to indicate the direction of the reaction. Quite oflen students did not indicate the final answer in a clear inanner,

example:

!or

Analvsis of the results related to the Problem-Solving Procedure

The calculation of the percentage of difficulties in the Problem-Solving Procedure was performed on the basis of three independent assessments given by the professor, students and researcher. For this the following materials were used: 1) students? self-evaluation - Questionnaire 2

2) professor's evaluation - a mark given by professor 3) researcher's evaluation - homework assignrnents

This combination of materia1 allowed one to achieve a full picture of the dificulties which students had in the problem-solving process.

Shidents' seIf-evaiuation - Ouestionnaire 3 Part A in the Questionnaire-2 (see Appendis C) was developed in order to determine what dificulties the students have related to the subject matter. responded on a five-point scale from "Always" to "Nevei'.

They

This scale was chosen in

order to gain an insight into their confidence in their problern solving ability For the analysis of the questionnaire, the results were p p e d as follows: "Always-Often" was defined as '-No problem" and "Sometimes-Se:dom-Never" was considered as "Have problerns".

The number of students with "Have problems" were calculated and

represented for the appropriate step in Table 1: Step 1, Problem statement - (a), (b), (c); Step 2, Selection and substantiation of appropriate method of solution - (a), (b); Step 3, Solution - (b), (d); and Step 4, Conclusions - (a), (b). These numbers were expressed as percentages.

Professor's evaluation The professor evluated students homework assignments on a regular basis. The

mark given by the professor was based on the midentYsability to show a free-body diagram, to apply a mathematical solution of the problem and to expiain the obtained results.

Even though partial credits were given for the gee-body diagram and

explanation, most of the credit was given for the semng up and solving equations of

-

equilibrium. So 1assigned this mark to position (c), in Step 3 Equations of equilibrillm and their solution. The average mark for ail hornework has been obtained, expressed as

a percentage and presented in Table 1.

Researcher' s evaluation I re-read each homework assignment after it had been marked by the professor.

The focus of my evaluation included the fol(owing positions: 1) identifying the given and the unknown; 2) drawing and Iabeling of design diagram; 3) plan for the solution; 4)

stepby-step explanation cf the solution; 5) use of

during solution; 6) answer, 7)

explanation of the obtained results and conclusion-

1 evaluated each homework

-

assignment on a four-point scale "Very good - G d Fair - Poor7'. For the analysis of the resuits I gouped "Very good" and 'Good" and defined them as "No problem".

"Fair" and POO^^^ were considered as "Have problems". The number of shidents with "Have problerns" were calculateci for each hornework assignrnenf and this number was expressed as a percentage. On this basis the average percentage for al1 homework was obtained. These numben were expressed as

percentage and presented in Table 1 for Step 1, Problem statement - (c); Step 2, Selection and subsîantïation of appropriate method of solution - (b); Step 3, Solution - (b), (d); and Step 4, Conclusions - (a), (b), (c).

Step 1. Probiem statement

DifficuIties

Based on:

a) understanding the problem

25%

Students' self-assessmemt

b) visualizing the problem

25%

StudentsfseIf-assessmemt

O%

Students' self-asessmemt Researcher's evaluation

C)

identifying the given and unknowns

9 1% d) drawing and iabeling the diagram

N/A

Step 2. Selection and substsntiation of appropriate method of solution

a) seeing of difFerent methods of solving proble

50%

Students' self-assessmernt

b) plan of soIution

50% 69%

Students' seIf-assessmemt Researcher's evaluation

Step 3. Solution

a) choosing the body whose equilibrium shodd be examined

N/A

b) necessary information displayed on the fiee-body diagram

O%

43%

Students' self-assessmemt Researcher's evaluation

c) equations of equiIibrium and their solution

71%

Professor's evaluation

d) explanation

0%

74%

Students' self-assessmemt Researcher's evaluation

a) units control

88% 36%

Students' self-assessmemt Researcher's evaluation

b) anaIysis of results

13% 75%

StudentsfseIf-assessmemt Researcher's evaluation

48%

Researcher's evat uation

Step 4. Conclusion

Table 1. Distri'bution of difficulties in the Problem-Solvinç Procedure

Difficulties related to the textbook

The important role of textbooks in education is obvious. The purpose of this section is to see what dificulties students studying Engineering Mechanics have related to the textbook.

For this purpose Questionnaire 2 and fragments of informal

conversations were analyzed. The data was compiled in mbular f o m and the final results are presented in Appendix C.

Textbook desion The textbook used for the course is "Vector Mechanics for Engineers. Statics" by Beer and Johnston (1990). The textbook is organized into ten chapters, 472 pages (the table of contents For the textbook appears in Appendix D). Each chapter is divided into small theoretical units. Many examples with soiutions are presented throughout the text. 4 number of problerns are çiven at the end of each unit for practice. The answers to the

even-numbered problems appear at the back of the book. At the end of each chapter a short sumrnary of the chapter is presented. The textbook contains many ilIustrations relevant to the text.

Research resul ts Part A in the Questionnaire 2 (see Appendix C ) was developed in order to determine how students view the textbook, how much they use the textbook, how dependent students are on the textbook and how they understand it. responded on a five-point scale fiom "Always" to "Never'?.

The students

Three students indicated that they read the textbook carefully and on a regular

basis. Two students indicated that they use the textbook seldom or never. Three midents use the textbook quite often, but not on the regular basis.

Ali students indicated thaf for the most part, tL,y understand the material in the textbook. If they have dificulties with the materid in the textbook, five students read it again until they understood it and only one of them used another source; three students

asked fiends or the professor for explanations. OnIy two students used the textbook for taking notes and for additional exercises

to reinforce the problem-solving technique and these students received excellent grades for the course-

As for general comments about the textbook, three students indicated that it was

hard to Say whether the textbook was good or bad, two students evaluated the textbook as very good, and three students evaluated it as bad.

As became clear from informal

conversations, some students viewed the textbook as badly organized Some snidents çtated that they had trouble using the teabook to search for information.

Discussion

In my opinion, the textbook has some limitations, however 1will discuss only that which influences the learning process to the greatest extent The biggest problem at the beginning of the course was drawing a fixe-body diagram. Why was this so difficult?

What concepts does a student need to know to be able to draw a fiee-body diagram? Statics considers only constrained bodies. So, in drawing a free-body diagram students

need to know such fundamental concepts as constraints and their reactions. On page 32, unit 2.11, in the textbook, the authoa introduce the notion of fiee-

body diagrarn as -'a diagram showing the particle and al1 the forces acting on it." In order to show al1 forces which act on particle, we m u t imagine that we remove the supports

and replace them by the reactions which they exert on the particle.

In other words, the

equilibrium of constrained bodies is shidied on the basis of the following axiom: any construined body can be treated as afiec body relievedfiorn its construints,

provided the latter are represented by thezr reactiom

However, the notion of constraints and their reactions are introduced only on page 127, unit 4.2. The authors give a table with the description of supports (rollers, cables,

short link, hinge and fixed support) and their reactions on page 129. However the problerns which contain these supports are given much earlier in the text (pages 36-39). So students working on page 36 do not yet know how to show al1 the forces acting on the particle, because they do not know the notion of "reactions". That c m be why they have

difficulties in showing a free-body diagram. Al1 the problems from 2.35 to 2.54 (see Appendix H) are very simple, but the students experienced difficulties in solving them, because for the solution of these problems students need to know concepts which appear much later in the textbook This supports the researcher's beliefs that such important concepts as actions and reactions dong with the principle of isolation of surroundings should be introduced together with the "Fundamental concepts and pnnciples7' (pages 2-6) at the very beginning of the

textbook.

Difficu Mes reiated t s mathematics The aim of this section is to see what difficulties students studying Engineering Mechanics have related to mathematics. For this purpose homework and tests were analyzed. In order to identify dificulties students had in mathematics prior to Engineering

Mechanics, Questionnaire 1 was given at the beginning of the course.

The data was

compiled in tabular form and the distribution of their difficulties in different areas of mathematics is presented on Figure 1.

Figure 1. DificuIties with mathematics prior to Engineering Mechanics

64

Mathematical skills are essential for Engineering Mechanics. The prerequisite tor Engineering Mechanics is MATH 152 (Calculus

a) or MATH

155 (Calculus LI for the

Biologïcal Sciences). As indicated in the Pretace ofthe textbook which students used for the course of Engineering Mechanics (Beer & Johnston, 1990), "thematerial presented

in the test afid most of the problems require no previous mathematical knowledçe beyond algebra, trigonometry, and elementary calculus, and a11 the elements of vector algebra necessary to the understanding of the text have been caretully presented (pxv)." And it is really so. The hornework and tests were examined in detail and it tvas discovered that students did not have major problerns related to mathematics. However, some common areas where the students experienced ditflcuIties were discovered. Tnese difficuIties are presented in accordance with ditlfèrent areas of mathematics.

It appeared some students experienced difficulties in solving inequalities. For example, the following problem was given as hornework.

Two cables are tied together at A and Ioaded as shown. Determine the range of values of P for which both cables remain taut (Beer & Johnston, #2.44). (The soiution of this problem is presented in Appendix G)

Some students did not quite undemand the meaning of "range". Even thou-

the

equations of equilibnum were set correctly, representing the final result was difficult. Unfominately, 1 do not have a copy of a student's work related to this problem, since this problem was given at the very beginning of the course and 1 did not understand yet what kind ofdifficulties 1 would look at,

DiRerentiation and Intemation: Al1 the students had taken Calculus courses (MATH 151, 152) and one student

took the Honoe Supplernent Calculus (MATH 161, 162). However it seemed that the students could not apply the mathematics they leamed in other courses to the solution of engineering problems. The following problem was given to the students: A 300-N force is applied at A as shown. Determine: a) the moment of the 300-N

force about D; b) the magnitude and sense of the horizontal force applied at C which creates the same moment about D; c) the srnaIIest force applied at C which creates the same moment about D (Beer & Johnston, ti 3.7). The solution of this problem is given in Appendix G.

Al t students solved this problem correctIy using the theory of moments.

However, none of the snidents recalled that there is a mathematical procedure For dealing with maximum-minimum problems. Most likely, the students did not have problems 66

with difierentiation itself but they probably did not see the relation of that mathematical procedure to engineering probIems.

One of the bigçest problems students experienced in the course was related to the qualitative construction of shear and bending moment diagrams (the graphical variation of V and ibf as a tünction of the position -Y along the beam's axis). This drawing is based on the diffërential relations that exist between the load, shear and bending moments. They are the tollowing:

ciP-

-= - w ( I ) du

- Slope o f the shear diagram is equal to the negative of the intensity of the dÏstrïbuted load.

dfLd --

- Ci - Slope of bending moment diagram is equal to the shear.

ds: dM In particular, if the shear is equal to zero, -- 0, and then a point o f zero shear dx

corresponds to a point of maximum (or possibly minimum) moment. Thesr relations can be represented in inte_@ km: AV

-

(

) d

- Change in shear between points B and C is equal to the

negative of the area ind der the distrïbuted-Ioading curve between these points. A M,=

IV dr - Change in the bending moment between B and C is q u a 1 to the

area under the shear d i a m within repion BC. Students experienced great dificulties in the qualitative construction of shear and bending-moment diagrams for beams of diffèrent types (simply 67

supported, cantilever beam) and under different loads (di*buted

load,

concentrated force and moment).

For example, one student told me that he had difficulty in sketching shear and bending-moment diagrarns and asked me to explain it before the fina1 exam. 1drew him a beam and the part of our conversation proceeded as follows:

R:

-Let's do the shear diagram. (The student constnicted the shear diagram correctly, his work is shown below) R: - Now let's do a bending-moment diagram. (The student drew a horizontid line under AB on the axis of beam, which is wrong since it indicates that the moment in this section is zero) - Why? - Because there is no moment on this interval. - How do you know that there is no moment? (Confusion) - What do values on shear diagram mean? (Confusion) - OK, what is the relation between moment and shear?

- -=CC C

M

ak

- Correct. - Slope.

So what does each value of the shear diagrarn mean?

- Slope where?

- In the moment diagram. - So show the moment diagramHe showed it incorrectly. Then I drew the bending moment diagrarn and he

expIained relation between shear and bending-moment diagrams on each interval. Our work is given below. This dialog suggests that the student knows the relation between shear and

bending moment However, he was not able to sketch the graph of the derivative of funclion in a qualitative marner.

Problems of this type caused major dificulties in the course. This is surprising because in the physics course students met similar problems (such as sketching acceleration curve knowing the velocity curve).

Even when the shear and bending

moment diagrams were constructed by means of an analytical solution using the method of sections, the students in most cases could not apply the appropriate relations for the control of the numerically obtained solution. The solution of a typical problem of this type using analytical and qualitative method is represented in Appendix G.

Some students had difticulties with geometry, particularly with the problems conceming three-force bodies.

Çuch problems can be solved graphically or

mathematicafly from simple trigonometric or geometric relations. Some students forgot the sine law. Sometimes the students intuitively knew the solution for the problem, however they did not know how to prove it using pometrical properties. The students had difficulties to substantiate geometrical constructions from the mechanical point of view. For example, the following probtem caused d i f h d t i e s in t e m s of proof. The uniform rod AB lies in a vertical plane with its ends resting against the frictionless surfaces AC and BC. Detennine the relation between the angles 8 and a when the rod is in equilibriurn (Seer & Johnston, # 4.57).

The students knew that there are three forces acting on the body (RA, Ru and W). They also knew that for equilibrium of three-force body all forces must be concurrent.

So, they found the point of inkrsection of Ruand W, point D, and then connected point D with A to define the direction of the reaction at A. A copy of the student's work is given

In this case the reaction at A was s h o w incorrectly, since it is not perpendicular to the surface. For the correct geometrïcal construction the student should take into account the properties of the frictionless surfaces and show reactions RA and Rn perpendicular to corresponding surfaces and then prove that force W (weight) Iies on the diagonal of the rectangle ACBD. The solution of this problem appears in Appendix G. Thus, dificulties in geometry anse, mainly, not because of the lack of Lmowledge in geornetTy, but because of not using the mechanieai conceptions for the georneirical constructions.

Vectot algebra During the sîudy five students were taking Linear Algebra (MATH 232) and Vector Calculus (MATH 251 or 352). Even though the students codd solve the rnajority

of the problerns using a vector approach, quite ofien they did not use the correct notation (used scalars instead of vectors and vice versa). For example, the followïng problem was gïven to the students:

The square steel plate has a mass of 900 k g with mass center at its center C. Calculate the tension in each of the three cables.

A student's work with the typical rnisusing vector and scalars is presented below.

The distribution of difficulties in different areas of mathematics on the basis of hornework and tests was dificuit to express as pewntage. However, the frequency of the most common mistakes in diferent areas of mathematics corresponds to Figure 1. As a conclusion to this section we can Say that mathematics is a tool for solving

Engineering problems.

However, having a good knowledge in rnathematics is not

enough to be able to solve engineering problerns. Only the combination of a deep knowledge of mechanics and rnathematics will give positive results.

Analvsis of midterrns and final exam results

This section analyzes the students' performance for the c o m z MATH 267 on the midterms and final test.

The purpose of this section is to determine what type of

problems as a whole are the most difficult for students. A short description of how the students were evaluated for the course MATH 262

is given in Chapter 3. Copies of both midterrns are given in Appendix F. The first Midterm took place on October 1, 1997. This midterm consisted of eight probkms covenng the following topics: coplanar force systems; three-dimensional force systems; moment of a force; moment of a couple and reduction of force system. The average achievement for this test was 71%. The mean percent correct for each problem was calcuIated and the results are presented in the Table 2. On this basis, the diagram showing students' performance was made and presented as Figure 2. The more difficult problems are s h o w Figure 2. These will be considered in the order of decreasing dificulty. Problem (7) - system "block-rope-pulley", \vas the most dificult for students. The mean of students' achievement for this problern was 44%. This problem required a

sound understanding of what system should be isolated for consideration, the students also had to know how the pulley transferred the tension of the rope. Students had dificulty in deciding what system should be isolated, and as a consequence they had difficulty in drawing a fiee body diagram.

Midterm 1

8) Thc icnsiie forcc in ciblc AB h i , 2 kN msgniiudc Foi the force from the cibla AB on thc point B find iho IMWIa i ~ O~(IP~ BIVL t X,Y,Z C O ~ P O M ~ U or RI.

Problem &

Figure 2. Students' performance on the First Midterm

Problem (4) - reduction of torce system. The mean achievement tor this problem was 58%. This problem invoived the use of the concept of resultant couple moment and

the procedure of moment summation. ProbIem (5) - moment of a force about a point The mean achievement tor this problem was 67%. This problem required an understanding of the concept of moment of a force about a point and the use of Varignon's theorem. Problem (8) - moment of a force about a point. The mean achievement for this probtem was 68%. This probtem required a knowIedge of a moment of a force with respect to specified point in vector tom.

The second Midterm took place on November 14, 1997. This rnidterm consisted ot-eight axoblerns covering the toilotving topics: beâms: trusses: system of bodies; water pressure and centroids. The average for this test \vas 67%. The mean value for each problem \vas calculated and the results are presented in the Table 3. On this basis, the

diagram showing studenrs' performance was made and presented as Figure 3.

The

problems with percent correct less than average are s h o w on the diagram. Problem (7) - equilibrium of systems of bodies - was the most dificult problem for student. The mean achievement Cor this problem was 3 1%. To solve this problem a student must know how do dismember the mechanism, and show the free-body diagrams for each part separately.

.

Midterm 2 . a ,

50% -----

i s hingcd ai A. Rcactioi Find ruciions ni A and

I

\

5 ) R n d forcc in c~bk.cÉ,A i O ii a roiter.

-

--- -.

. .

..

.

-

.

/

--

- .--. -

----i t

--

- - 1

I

I

I

l

-

7) The bOir b his a rwss of 50 kg, the bar AC Ir rmsrlcrr. lha ropo ODC ir fioxible. The wh:cl i l D 1s frictionlcss. i)Draw FBD't of 1) h a b, ii) bar AC, b) Plnd forces on bar i t

rn

Problem b

Figure 3. Students' performance on the Second Midterm

7.---

-

-Average for midterm +Average par problem --. . -- - - . . -- - -- -- ... - --.

-. .

-,

Problem (5) - equilibnurn of systems of bodies. The mean achievement for this problem was 50%. This problem involved the same procedure as probIem (7). SC~dents had diffTculties in disrnembering the mechanism and applyiiig forces tor each part. Problern ( 3 ) - water pressure. The mean achievement tbr this problem was 57%. Students had difiiculties in tinding the resultant force created by the distnbuted loadProblem (4) - tnisses. The mean achievement for this problem was 65%. Using the method ofjoints, students made mistakes when moving ttorn one joint to another.

The final examination took place on December 12, 1997. It consisted o f two

parts

- open-book

and closed-book.

1 will consider the results of the open-book part

only, which consisted of eiçht problems. These problems covered the following topics: moments of inertia; systems of combined bodies; forces in space; internat forces in a beam; friction and the principle of Mrtual work. The average achievement for this part of the test was 39%. The rnean percent correct for each problem was calculated and the results are presented in Table 2.

On this bais, the diagram shawing students'

petiorrnance [vas made and presented as Fi-me 4 The probtems uith mean less than 39% are shown on the diagramProblem (8) - principle of virtual work.

The results for this problem were

extremely tow, the mean achievement for this problern was 2%. The most ditticult part kvas to find rdations between virtual dispiacements.

Problem (7) - equilibrium of systems of bodies. The mean achievement for this

problern was 26%

Dismembering cf the syysteem and applying forces to each part was

the most ditticult part of the problem. Problem (4) - equilibrium of systems of bodies and fiction. The menn achievement tor this problem was 26%.

ProbIem ( I ) - moment of inertia of a symmetncaI body. The rnean achievement for this problem was 26%.

Analysis of student's performance on the midterms and final examination

revealed the typical problems which were the most dificutt for the students. The distribution of tiequencies of the problems which were below the average line is shown

on the histogram below (Figure 5).

1- " b i a c k ~ u l l e y "

I

2 - reducüon of farce systern 3 - moment of a fwce about a point 4 - quilibriurn of systems of bodies 5 - water pressure 6 - tnisses 7 - priricipieof urtual vmk

O

8 - fiction

a - a

&2

-'

u.

Type of problern

Figure 5. Distribution of "dificult" problems 1 - "block-rope-pulley". The problems of this type are found in Midterm 1: # 7.

2 - reduction offorce systern. The problems ofthis type are iound in Midterm 1: fC 4.

3 - moment of a torce about a point. The problems ofthis type are ibund in Midtem 1:S

5, 8. 4 - equilibrïum of system of bodies. The problems of this m e are tound in Midtem 2: #

4, 5,7; Final Exam: ff 4,7. 5 - water pressure- The problems of this type are %und in Midtem 2: fi 3. 6 - tnisses. The problems of this type are found in Midterm 2: !! 3. 7 - principle of virtual work. The problems ofthis type are tound in Final Esam: $ 8 .

8 - friction. The problems of this type are tound in Final Exam: iC 4. 9 - moment of inertia. The problems of this type are found in Final Exam:

1

Of course, such classitication is relative, since some problems c m be considered

as "hvo-Q~e" problems.

For example, problem 3 i?om the Final Exam could b e

considered as a problem with fiction or equilibnurn of system of bodies. I reiërred to this problem hvïce, because difticulties were tound in parts dealing wïth hction and disrnembering the system. Each problem @en on the midterms and final examination retlected specitic topic and a represented a whole class of problems which were considered during semester.

Since these problems were typical I concluded that the students had

difficulties with corresponding types of problerns. As clear tiom Figure 5 , the problems dealing with equilibn'um of systems of corn bined bodies caused dificulties more tiequently. This type of problem is based on dismemberïng a structure into separate bodies

and writing an equiIibrïurn equations for each as a fiee body. In solving problems by this

method it should be remembered that fthe action of one bu& on a n o h is denoted by a force R or its reguIar cuntponents X and Y,then uccordzng to the principle of action and reaction, the action of the second bo& on the first m u t be denoted &yaforce R ' e q d to

R in magnitude and opposite Nt seme or by its recfunguIar comporients X'

and Y'

respecrfliv equd to X and Y in magnitude and opposire s e m

The following mistake was often made in solving this type of problem: having

drawn the forces X' and Y' correctiy, the student assumes in solving the equations that X' = -X and Y' = -Y, obtaining a wrong answer-

Another "difficult" type of problem dealt with a moment about a point. These two types of problem are based on the fiindamental concepts and notions, presented at the beginning of the course. Frankly speaking, they are not reully diff~cultproblems,

wàich require a strong mathematical procedure or some special knowledge; these problems require basic laiowledge of mechanics. Tbese results indicate that students had

to practice more on simple problems and only when they solve a sufficient amount of basic problems with full understanding of them, c m they move to a more difficult step.

Chapter 5 Discaission and Conclusion

This chapter is devoted to the discussion of the resuits arising from the study. The general difficulties that students have when leaming Engineering Mechanics are

summanzed. Some suggestions are made regarding the implications for teaching and students' motivation The enrollment in MATH 262 is discussed and questions for further research are outiined.

Review of the Study

This study was designed to analyze difficulties students have in Ieaming Engineering Mechanics. The following groups of difficulties were discussed: difficulties

related to problem-sotving process, difficulties related to mathematics and difficulties

related to the textbook. The study indicated that a nurnber concepts in rnechanics were diEcult to understand (action and reaction, constraints and their reactions, moment of the force and couple). The study revealed difficulties related to the problem-solving process. The

students had problems wiîh visualking the problem (types of supports and mechanisrris: how they look and whsre they are used in the real world). Students had a great difficdty in deciding what system shouid be isolated for consideration and drawing a free-body diagram. This is the most important step in the solution of problems in mechanics. By means of the fiee-body diagram we define the system of forces with which we must deal in the investigation of the conditions of

equilibrium of any constrained body. Any error in this step wïll reflect on al1 subsequent work.

The midents, in general, did not have a clear plan for the solution. It was too ofien the case that the student7s attention was focused ody on the solution of purely mathematical probierns so he or she did not see the connection between this and the true

engineering probtem. They dso had dificulties in analyzïng the obtained resdts. The main difficulty students experienced in mathematics was calculus. In generaI students were not able to sketch the graph of the derivative of function and its antiderivative in a qualitative marner.

Although the textbook for the course is badly organized and it is f i c u i t to search for the information, students involved in this study did not use resources other than the text,

Suggestions for conceptual and procedural develooment An engineer is continually facing new problems which do oot always yield to

routine methods of solution. However, the person who can successfuily cope with such non-routine problems must have a sound understanding of the fundamental principles that apply and rnust be very farniliar with the solution of typical problems and various methods of their attack.

In my opinion, overcoming difficulties in the problem-solving process is based on two main principles:

1. Start with simple problems before considering difficult ones.

2. Follow the cornmon steps in the problem-solving process. These steps are following 1) choosing the object under investigation 2) drawing a fk-body diagram 3) analyzing system of forces

4) choosing the method of solution 5) writing conditions of equilibrium

6) solving the equations 7) providing the answer

8) analyzing the results

Table 3 represents solutions of different types of probtems according to these steps. These steps are based on the probiem-solving approach descnied in detail in Chapter 4. Of course, not every problern could be solved in the h

e of this approach.

However, the majorïty of problems in Statics cm successfully be solved using this

standard approach.

If a mident gets used to it he / she wilI benefit in subsequent

courses, such as Dynarnics, where the sarne approach with minor modification will be used. As it appeared from the study students were not familiar with the idea of cornrnon

steps in the problem-solving approach in the Engineering Mechanics problems. Table 3 might help students to see that the same algorithm is applicable to the solution of different problems of Statics.

3. System of forces 4. Method of solution 5. Conditions of equilibriurii 6. Solution

Concurrent force system on the plane Graphical and analytical

Concurrent t hne-dimensional force systemc Analytical

General farce system on the plane . Analyticiil

General force system in the space Analytical in vector forni

The force polygon drawn with al1 forces has to be closed

Draw a closed triangle with forces P= 17< NIRA,Rn as its sides, starting from the given force P.

tF=F+F,4+Tc+TU=0 CM.\=rllx( F+Tc+TI,)=O I ) Express each force in Cartesian vcctor fom: F=(- 1000j) N; F,,=A,i+ A,j+ A,.k TI .= TJ.iic.= Tc (rllc.1 rllc)

707Tck =0.707Tci-0. T l l = T ~ h l )TD = (rlll)/ rlll)) =-0.333Tl)i+0.667T1ij-0,667TDk 2) CF=(A,+0.707Tc-0.333Tl))i t. (- 1 000+A,.+0.667TD)j -t(A,-O. 7 0 7 ~ ~ - 0 , 6 6 7 ~ ~ J k = 0

DI,=O: -4Tl)t6000=0 ZM,=O: 4.24Tc-2Tn=0

7. Answer

8. Aiialysis

R3=P 1 sina sincp R2=P(1 -cotc.p)1 sina RI==P[1 +tan ((p/2)] cola R3=231 N, R2=85N,R1=273N the triangle we see tliat the The results show that at (p