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Examensarbete C, 15 hp Juni 2014 Uppsala universitet
Introductory physics students’ conceptions of algebraic signs used in kinematics problem solving Moa Eriksson Supervisor: Cedric Linder Subject reader: John Airey Divisions of Physics Education Research, Department of Physics and Astronomy
Abstract The ways that physics students’ conceptualize – the way they experience – the use of algebraic signs in vector-‐kinematics has not been extensively studied. The most comprehensive of these few studies was carried out in South Africa 15 years ago. This study found that the variation in the ways that students experience the use of algebraic signs could be characterized by five qualitatively different categories. The consistency of the nature of this experience across either the same or different educational settings has never given further consideration. This project sets out to do this using two educational settings; one similar to the original South African one, and one at the natural science preparatory programme known as basåret at Uppsala University in Sweden.
The study was carried out under the auspices of the Division of Physics Education Research at the Department of Physics and Astronomy at Uppsala University in collaboration with Nadaraj Govender, University of KwaZulu-‐Natal, who performed the original study while completing his PhD at the University of the Western Cape, South Africa.
This study is situated in the kinematics section of introductory physics with participants drawn from the natural science preparatory programme at Uppsala University and physical science preservice teachers’ programme at the University of KwaZulu-‐Natal, South Africa. The participating students completed a specially designed questionnaire on the use of signs in kinematics problem solving. A sub-‐group of these students was also purposefully selected to take part in semi-‐structured interviews that aimed at further exploring their experiences of algebraic signs. The students’ descriptions and answers were categorized using Nadaraj Govender’s set of categories, which had been constructed using the phenomenographic research approach. This approach is designed to enable finding the variation of ways people experience a phenomenon. The process of sorting the data was grounded in this phenomenographic perspective. From this categorization it was possible to identify four of the original five categories amongst the participating students.
The results suggest that these four categories remain educationally relevant today even if the context is not the same as the one for the original findings. Although one of the original five categories was not found, the analysis cannot be taken to definitely eliminate this from the original outcome space of results. A more extensive study would be needed for this and thus a proposal is made that further studies be undertaken around this issue.
The study ends by suggesting that physics teachers at the introductory level need to obtain a broader understanding of their students’ difficulties and develop their teaching to better deal with the challenges that become more visible in this broader understanding.
Sammanfattning På vilka sätt fysikstudenter föreställer sig och förstår användandet av algebraiska tecken i vektorkinematik har endast studerats i mindre utsträckning. Den mest omfattande av dessa få studier genomfördes i Sydafrika för 15 år sedan. Denna studie upptäckte att variationen av de sätt studenter upplever användandet av algebraiska tecken på kunde karaktäriseras genom fem kvalitativt olika kategorier. Hur solida dessa upplevelser är i en liknande eller helt annan utbildningsmiljö har däremot inte studerats vidare. Detta projekt ämnar till att göra detta genom att använda två olika studentgrupper; en liknande den ursprungliga gruppen i Sydafrika, samt det tekniskt-‐naturvetenskapliga basåret vid Uppsala universitet, Sverige.
Studien har genomförts med stöd från avdelningen för fysikens didaktik vid institutionen för fysik och astronomi vid Uppsala universitet i samarbete med Nadaraj Govender, University of KwaZulu-‐Natal, Sydafrika, som genomförde den ursprungliga studien under sin doktorandutbildning vid University of the Westen Cape, Sydafrika.
Denna studie är begränsad till den del av den grundläggande fysiken som behandlar kinematik och innefattade deltagare från det tekniskt-‐naturvetenskapliga basåret vid Uppsala universitet samt tredje års studenter vid physical science preservice teachers’ programme, University of KwaZulu-‐Natal, Sydafrika. De deltagande studenterna genomförde ett specialdesignat frågeformulär kring användandet av algebraiska tecken för att lösa kinematiska problem. En del av dessa studenter valdes sedan ut för att delta i semi-‐strukturerade intervjuer som syftade till att vidare utforska deras upplevelser kring algebraiska tecken. Studenternas beskrivningar och svar kategoriserades med hjälp av Nadaraj Govenders fem kategorier som tagits fram genom ett fenomenografiskt tillvägagångssätt. Detta tillvägagångssätt är framtaget för att kunna hitta variationen av hur människor upplever ett fenomen. Sorteringsprocessen grundades i detta fenomenografiska perspektiv. Från denna kategorisering var det möjligt att identifiera fyra av de fem ursprungliga kategorierna bland de deltagande studenterna.
Fyra av de fem ursprungliga kategorierna som föreslagits av Govender återfanns genom denna studie varför dessa kategorier föreslås förbli relevanta idag även om utbildningsmiljön skiljer sig från den ursprungliga. Trots att den femte kategorin inte hittades kan denna inte definitivt exkluderas från det outcome space som beskriver studenters upplevelser för algebraiska tecken. Det föreslås att vidare studier undersöker förekomsten av denna kategori.
Studien avslutas med att föreslå att fysik lärare på grundnivå behöver få en bättre förståelse för sina studenters svårigheter samt att de behöver utveckla sin undervisning för att bättre kunna hantera dessa svårigheter och på så sätt göra undervisningen mer anpassad för mångfalden av studenterna.
Acknowledgements To my mother and father for believing in me and encouraging me to fulfill my studies and doing what I believe in.
I would like to express my deepest appreciations to my supervising professor, Cedric Linder, for his continuous interest, guidance and engagement in the development of this study. All your encouragement throughout this project has been invaluable and has helped me complete my thesis! I would also like to thank Dr. Nadaraj Govender for his feedback on my study design, his encouragement and for his valuable support as my external collaborator during the project. Special thanks goes to Jonas Forsman for his interest and guidance from the very start of my project, which helped me to quickly progress.
I would further like to thank all members of the Division of Physics Education Research at the Department of Physics and Astronomy at Uppsala University, especially Anne Linder, for making me feel truly welcome as a part of your group.
Table of contents
1 INTRODUCTION ................................................................................................ 1
1.1 Problem setting ..................................................................................................................................... 1 1.1.1 Research question ............................................................................................................................................... 2
1.2 Object ........................................................................................................................................................ 2
1.3 Goal ........................................................................................................................................................... 2
2 BACKGROUND .................................................................................................. 3
3 METHODOLOGY ............................................................................................... 5
3.1 Theory ...................................................................................................................................................... 5
3.2 Method ..................................................................................................................................................... 6 3.2.1 Ethics ........................................................................................................................................................................ 6 3.2.2 Validity and reliability ....................................................................................................................................... 7 3.2.3 Questionnaire ........................................................................................................................................................ 7 3.2.3.1 Design ......................................................................................................................................................... 8 3.2.3.2 Pilot studies ............................................................................................................................................. 9 3.2.3.3 Translation ............................................................................................................................................... 9
3.2.4 Interviews ............................................................................................................................................................... 9 3.2.4.1 Design ....................................................................................................................................................... 10 3.2.4.2 Pilot interview ...................................................................................................................................... 10 3.2.4.3 Transcriptions ...................................................................................................................................... 10
3.2.5 Analysis .................................................................................................................................................................. 10
4 RESULTS ......................................................................................................... 11
4.1 Summary of findings ........................................................................................................................ 12
4.2 Evaluation ............................................................................................................................................ 13
4.3 Description of categories ................................................................................................................ 13
5 DISCUSSION .................................................................................................... 17
5.1 Identified challenges ........................................................................................................................ 18
6 RECOMMENDATIONS ..................................................................................... 20
7 CONCLUSIONS ................................................................................................ 20
REFERENCES ......................................................................................................... 22
APPENDIXES ......................................................................................................... 24
Appendix 1 ...................................................................................................................................................... 24
Appendix 2 ...................................................................................................................................................... 28
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1 Introduction Mathematically, when using vector notation the choice of an appropriate sign is largely an arbitrary decision, however, in physics problem solving, a set of conventions that have a strong conceptual base usually guide their use. For example, in problems involving the direction an object is moving when restricted to one dimension, a range of possible temperatures, the sign of an electric charge, and the gain or loss of energy from a specified system. All of these use algebraic signs, which have distinct conceptual meanings as a function of their context. Thus, getting to appropriately understand how to conceptualize the use of algebraic signs across contexts and phenomena is an important aspect of learning in introductory physics.
From an early stage, throughout school, students meet the use of the algebraic signs “plus” and “minus” across different contexts. However, these appear to become understood procedurally without an accompanying comprehensive and appropriate conceptual anchoring. Such procedural knowledge has limited sustainability in introductory university physics and little value for the study of more advanced physics. As students go from one problem solving context to another a lack of conceptual grounding can easily generate challenges both for introductory problem solving and for more advanced problem solving. For example, at the introductory level -‐6 may be considered to be mathematically smaller than -‐4, yet a velocity of -‐6 m/s may be considered to be larger than a velocity of -‐4m/s, and making sense of the sign convention for the basic Dirac equation using applications of the Minkowski metric can be challenging.
When dealing with vectors in introductory university physics the issue of appropriately understanding the signs that get used becomes more problematic. For example, students often have not been introduced to the use of vectors in more than one way and they get to think about component vectors in vector terms rather than scalar terms. When this happens the sign used is often taken to denote direction and basic scalar additions become convoluted with perceptions of vector direction. A further example is that students may want the sign conventions to continue to fit the conceptualization of the meaning that they have already constructed. For example, a negative velocity may be taken to mean “slowing down” rather than designating the labelling being used from the establishment of a coordinate system and the direction within one dimension of that coordinate system.
1.1 Problem setting This project builds upon previous research that will be further described in the Background section below. Only a few studies have investigated students’ use of algebraic signs in physics problem solving (for example, Viennot 2004; Hayes & Wittmann 2010). The most comprehensive of these are the studies carried out by Govender (1999; 2007). Govender (1999) used his analysis to generate qualitatively different categories of description of the experience of using signs in kinematics problem solving. Later Govender (2007) expanded upon these results. In both studies, Govender drew on a phenomenography framing (Marton & Booth 1997) to provide details of the outcome space for the qualitatively different ways students experience signing in introductory physics. This outcome space consisted of five different categories each reflecting the variation in the ways that signs get conceptualized in introductory kinematics physics. The outcome space together with Govender’s original study can be found in appendix 2.
Govender’s studies were carried out in a very different socio-‐economic setting to that of Sweden but with a programme that has many similarities to the natural science preparatory programme in Sweden known as basåret. The data set that Govender used is now nearly 15 years old and the changing educational experience in physics internationally (Henderson, Dancy & Niewiadomska-‐Bugaj 2012), could reasonably be expected to affect the current applicability of the results that Govender obtained. Using this as a starting point, the following research question was developed.
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1.1.1 Research question For Swedish and South African student populations who are enrolled in a physics introductory course which has a similar approach to that of the natural science preparatory programme in Sweden known as basåret, the specific research question for this study is the following:
• How relevant are Govender’s (1999; 2007) results for the variation in ways that the use of algebraic signs are experienced for one-‐dimensional kinematics problem solving for students in the natural science preparatory programme at Uppsala University, Sweden, and the third year physical science preservice teachers’ programme at the University of KwaZulu-‐Natal, South Africa?
1.2 Object To achieve a good quality of physics education, teachers must be aware of the challenges that students experience when being introduced to the use of vectors. If teachers can attain this awareness they will be able to, in a better way, address the diversity of the student group. The study performed by Govender (1999) shows in what different ways introductory physics students in KwaZulu-‐Natal, South Africa, experience the use of algebraic signs in one-‐dimensional kinematics problem solving. However, no further research has been carried out to study if the results can be generalized for use in explaining the understanding of signs among, for example, introductory physics students in Uppsala, Sweden.
Thus, the object of this project is to study the different ways students in the natural science preparatory programme in Uppsala, Sweden, and physical science preservice teachers at the University of KwaZulu-‐Natal, South Africa, experience the sign conventions used for describing three fundamental concepts in Newtonian mechanics (displacement, velocity and acceleration). These experiences will be categorised using Govender’s (2007) categories in order to, in the future, make it easier for teachers to address the variation of physics students’ experiences when learning algebraic signs.
Since Govender’s (1999) study, there has been no further research in the particular area of trying to categorise students’ different ways of understanding the sign conventions used in vector kinematics. Many studies have been made investigating students’ understanding of vectors and sign conventions in physics (see, for example, Aguirre 1988; Hayes & Wittmann 2010; McDermott 1984), however no one has further tried to generalise Govender’s discovered categories in the area of kinematics. Thus, it is of great interest to perform a study where the five qualitatively different ways students experience the use of signs in vector kinematics found by Govender (2007) are reviewed to investigate their generalizability.
1.3 Goal The goal of this project is to investigate how introductory level physics students at Uppsala University conceptualize the way they use signs in physical problem solving in the area of kinematics, and to use this analysis to compare and contrast results obtained at the University of KwaZulu-‐Natal in South Africa. This comparison will be used to identify generalizable learning challenges in this area in order to inform the development of associated physics education. Knowing the nature of the identified generalizability would provide a powerful platform to inform the teaching and learning of kinematics in ways that better accommodate the diversity of student population found in the introductory level of physics education in Sweden today.
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2 Background Different challenges that students face when learning physics have been investigated in many studies over a long period of time. For example, research has shown that students have difficulty understanding negative velocities and how to link them to a physical situation (Goldberg & Anderson 1989; Testa, Monroy & Sassi 2002). When presented with a velocity-‐time diagram many students failed to recognise the motion of an object when the diagram showed a negative velocity and could not with certainty draw a diagram of the motion by themselves. This difficulty, understanding the meaning of negative velocity, could stem from the fact that students often are only exposed to vectors in one way. For example, students are used to describing vectors through equations and drawing vector arrows in a coordinate system, where a negative velocity usually is directed to the left. However, by just drawing vector arrows, students fail to experience the link between the vector and a physical situation.
It has further been argued in the literature that the difficulties in understanding the use of algebraic signs that are commonly used in kinematics problems can often be traced back to the misuse of the correct signs in physics textbooks (for example, see Brunt 1998). When deciding on a sign for a quantity in physics problem solving there are some rules, which have to be followed, that textbook writers frequently do not follow or perhaps are not aware of (Brunt 1998). Such lack of coherence and systemization can easily generate ambiguities in what meanings are being signified in the ways the signs get used. In order to avoid such problems emerging, Brunt (1998; 242) proposes that teachers of physics incorporate two simple guidelines into their teaching practice when teaching at the introductory level: (1) when deriving an equation always “draw a diagram with all variables in a chosen positive direction”, and (2) to “never substitute a sign unless substituting a number, or its algebraic equivalence”.
The first of these two guidelines proposes that one should first decide on a positive direction in order to draw an appropriate coordinate system to situate the needed diagram. This is verified in a study (Viennot 2004) that argues that one often has to decide on an axis of reference in order to assign the appropriate sign to a quantity. However, this does not mean that the quantity has an inherent positive or negative attribute. Viennot (2004) further states that people often want to just put the sign in front of the quantity instead of assigning it a positive or negative sign according to a chosen coordinate system. An example of this can be seen in studies (Viennot 2004; Rebmann & Viennot 1994) where students were asked to write the equation for the force of a spring being contracted. In this example, students often “make up for” the contraction in the derived equation ending up with the incorrect equation 𝐹 = +𝑘𝑥 (Figure 1).
Figure 1: Presented with this image, students were asked to write the equation for the force of the spring in the three last pictures. Students noted that the force was directed to the right in cases 2 and 4 and incorrectly put a + sign in front of the expression. (Viennot 2004)
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Another example of the complexity in signing at the introductory level of physics is in two body problems where the motion is in different directions, for example Atwood Machine type problems. Figure 2 illustrates the type of approach that Brunt (1998) is critical of because it fails to systematically use paired coordinate systems to establish the assigned signs. Instead a derived rule of “bigger force minus smaller force = ma” is used.
From these examples it is possible to understand how students may not always have an appropriate conceptualization of the signs they are using, although their application of some rule may lead them to the correct answer!
In the area of kinematics there have been specific studies conducted, apart from the above mentioned, with the aim of identifying difficulties students experience trying to understand the physical concepts and how to connect this to real world phenomenon. For example, Trowbridge and McDermott (1980; 1981) investigated students’ understanding of the concepts of velocity and acceleration through interviews with introductory physics students and came to the conclusion that students often confuse position with velocity and velocity with acceleration. It has further been reported (Bowden et al. 1992) that a seemingly correct interpretation of concepts during undergraduate physics courses is not a guarantee that students understand the underlying principles. Bowden et al. found that the level of conceptual understanding often decreased as the problems students face become easier to solve. This tells us that students often are missing the conceptual understanding of physics in introductory courses, which becomes a large obstacle to overcome when learning more advanced physics.
Another challenge students encounter with introductory physics emerges when being introduced to the use of vectors. For example, Aguirre (1988) discusses students’ preconceptions in vector-‐kinematics that often remain with the students for a long time. The study investigated students’ preconceptions of vectors that often are not discussed by instructors because they are argued to be obvious. Aguirre suggests that simply telling the students the correct way of thinking will not change their beliefs but the instructor needs to be aware of the different understandings of vector conceptions students already have.
To address such learning challenges as described in the paragraphs above a physics teacher needs to have insight into the variation of perception that they can expect to find in their classes. This study aims to contribute to understanding this variation.
Figure 2: An illustrative Atwood Machine type problem taken from an online AP Physics B lessons (onlearningcurve 2012). In this solution no coordinate system is directly used.
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3 Methodology When planning and conducting a project of any kind, the methodology is an important part to consider. This section will describe the theory behind the method used for this project as well as the implementation of the theory.
3.1 Theory Govender’s original two studies used a research approach called phenomenography in order to find the qualitatively different ways in which students experience the use of algebraic signs in vector-‐kinematics. The sorting of the data for this project took on this phenomenographic perspective. Phenomenography is a research tool that aims to describe the qualitatively different ways individuals experience various phenomena or aspects in the world around them (for example, see, Trigwell 2000; Marton 1981; Marton & Booth 1997). It should be emphasized that phenomenography is not a research theory, nor is it a method even though it uses aspects of both. Instead, Marton and Booth (1997; 111) refer to phenomenography as “a way of – an approach to – identifying, formulating and tackling certain sorts of research questions, a specialization that is particularly aimed at questions of relevance to learning and understanding in an educational setting”.
Phenomenography seeks to find the variation in ways that people experience a specific phenomenon, and aims to sort the experiences into categories that are qualitatively different from each other. This is done in order to find the limited number of qualitatively different ways a phenomenon is experienced. The experiences of a specific phenomenon are called categories of description and are “the fundamental results of a phenomenographic investigation” (Marton & Booth 1997; 122). Further, the categories of description can be listed in a hierarchical way to form what is called the outcome space of the phenomenon (Trigwell 2000). What characterizes phenomenography is this ranking of the understandings of a particular phenomenon (Bowden et al. 1992), where a high ranking, indicates a better understanding of the phenomena.
The perspective of a phenomenographic study is of the second order, meaning that the researcher will report the experiences as described by others (Marton 1981). This approach is different to a first-‐order perspective where the researcher describes the phenomenon as experienced by themselves. It is important to remember that when having a second-‐order perspective, the researcher may not always agree with the experience of the phenomenon, but the experience is still “recorded as a valid experience” (Trigwell 2000; 6). In this study, the second-‐order perspective will be maintained through using the students’ experiences as the base of analysis, regardless of how much we agree with them.
Trigwell (2000; 3) gave an excellent summary of the phenomenographic research approach as follows:
it takes a relational (or non-‐dualist) qualitative, second-‐order perspective, that it aims to describe the key aspects of the variation of the experience of a phenomenon rather than the richness of individual experiences, and that it yields a limited number of internally related, hierarchical categories of description of the variation.
To investigate how relevant Govender’s original outcome space is for current introductory physics students, a phenomenographic perspective is used to sort experiences of the usage of algebraic signs in vector-‐kinematics from students in Sweden and South Africa.
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3.2 Method The data that was used for this study was collected through both a questionnaire and discussion based interviews. The questionnaire was given to students in the natural science preparatory programme (basåret) at Uppsala University, Sweden, as well as to third year students in the physical science preservice teachers’ programme at the University of KwaZulu-‐Natal, South Africa. The main part of the data was collected through the questionnaires whereas the interviews focused on obtaining a deeper understanding of some of the answers received from the questionnaires.
The total number of students who were registered on the Physics 2 course in basåret in Uppsala was 120, of which 60 answered the questionnaire. Unfortunately we have no data on how many students were actually present at the time when the questionnaires were handed out, hence there can be no statistics showing the actual response rate of the participating Swedish students. However, as 50 % of the total students enrolled in the Physics 2 course answered the questionnaire, it can be seen that a significant number of students from basåret did take part in the study. Among the South African students the response rate was 78 %; a total of 32 students were on the Physical Method 2 course of which 24 students answered the questionnaire. This gave us a total of 84 students completing the questionnaire.
Five Swedish students were selected to take part in semi-‐structured follow-‐up interviews. The five were chosen from the 28 students that provided their e-‐mail address and thereby accepted being contacted to take part in this interview. In total 14 Swedish students were contacted, however only five responded. Among the South African students, six were selected to take part in an interview. The selection was made by Govender and aimed at interviewing students from different races common in the KwaZulu-‐Natal province, to ensure equity and representativity of the student population. The distribution among the races was: one white student, one Indian and four black students.
To conduct this study, the method was divided into several phases where the first phase consisted of a literature review of previous research done with students’ conceptualizations when working with signs in physics. The second phase involved the creation of an appropriate ethical agreement to be used, which was based on the ethical guidelines set up by the Swedish Research Council (Vetenskapsrådet 2002). In the third phase, the work by Govender (1999; 2007) was considered in order to design a validated questionnaire to give to the targeted Swedish and South African students. The fourth and final phase involved conducting semi-‐structured follow-‐up interview discussions (Kvale 1996) with purposeful samples (Patton 1990) of participating students.
To be able to perform a scientific study of this kind, it is important to be familiar with the area of research and aware of any previous research that has been done. Thus, a detailed literature review was conducted in the beginning of the project to obtain a full background for the particular area of study. A thorough background in issues involved in performing qualitative research was also extremely important in order to be able to design the questionnaire and interviews in a suitable way. Before the questionnaire and interviews could be carried out many aspects had to be considered to make sure that the data being collected was appropriate for answering the research question. Govender agreed to act as an external expert to validate the questionnaire.
3.2.1 Ethics When performing scientific research it is important to consider the ethics of the study. To maintain the physical and psychological well being of the individuals being part of a scientific research in the area of humanities and social sciences, the Swedish Research Council has four
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main requirements that must be considered for conducting research of this kind (Vetenskapsrådet 2002). The four ethical requirements of the Swedish Research Council are:
1. The requirement for information states that the researcher has to inform the participants of the aim of the study.
2. The requirement for approval means that the participating individuals have to decide for themselves if they agree to be a part of the study.
3. The requirement for confidentiality tells the researcher that he or she has to handle all personal information from the participating individuals in a way that others cannot access.
4. The requirement for usage states that the information obtained during the study will only be used for research purposes and may not be used for non-‐research purposes.
All of these demands were met through the information given on the questionnaire and during the interviews.
3.2.2 Validity and reliability In order to argue that the result of this study would be of scientific value, the method used had to have a well-‐established credibility, meaning that the method had to provide a valid and reliable result.
Validity of the method means that the method used will provide the result that is wanted for this study. The result wanted for this study was answers from students explaining their experiences of the sign conventions used in kinematic problem solving. Thus, the method chosen for this study had to provide these kinds of answers. To obtain validity of the method used, Govender agreed to review the questionnaire, as well as the questions used for the interviews, during all the design stages, thus the validity of the method is argued to be satisfied.
Reliability of the study means that the result of the study will have to be able to be obtained again if the study is repeated using the exact same method under the same conditions. In qualitative research, difficulties can arise when arguing for the reliability of the method. For example, Merriam (1995; 2009) reminds the reader of the high improbability that one will obtain the exact same results twice, due to the qualitative research being based on the experiences of human beings. Thus, Lincoln and Guba (1985) replace the term reliability with the term dependability. The question to be asked is therefore that of “whether the results are consistent with the data collected” (Merriam 2009; 221). This means that from the data collected, the result obtained will have to make sense. In this report I provide a clear and full account of the research process so that the dependability of the study can be assessed and show how the research decisions were made and implemented.
3.2.3 Questionnaire The main data that was collected for this study was collected through a specially designed questionnaire that was given to basår students at Uppsala University, Sweden, and to students in the third year of the physical science preservice teachers’ programme at the University of KwaZulu-‐Natal, South Africa. The questionnaire can be found in appendix 1. Since the time for this project was limited to ten weeks, this had to be considered when designing the method for data collection. With the use of a paper based questionnaire many responses would be able to be collected in a relatively short period of time and thus this method was argued to be the most suitable to use. In order to have good data for the comparison with Govender’s categories, a special effort was made to attract as many students as possible to participate in the study. The design of the questionnaire was of extreme importance in order to maintain a good quality of the collected data and leave room for as little personal evaluation as possible (Robson 2002).
To act as the foundation for the study, the data collected from the questionnaire should provide a
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large amount of different explanations of how students use algebraic signs in vector-‐kinematic problem solving.. To limit the possibility of the questionnaire evoking cognitive overload and/or generating reluctance-‐to-‐complete, the number of questions had to be limited. The questionnaire thus consisted of two problems, each containing a number of questions, to get the students to reveal how they conceptualize the way they use algebraic signs in kinematic problem solving. Govender agreed to act as an external expert to validate the questionnaire and took part in all the design stages. Pilot studies were done as a pre-‐test of the questionnaire (van Teijlingen & Hundley 2001) to help evaluate the adequacy of the chosen research method, and where found to be necessary, modify the questionnaire outline (see Section 3.2.3.2).
The distribution of the questionnaires among the Swedish students was done by the author, whilst Govender distributed the South African questionnaires.
3.2.3.1 Design To obtain the desired data, the design of the questionnaire was critical. In order to be able to make a qualitative comparison with Govender’s five categories, the problems included in the questionnaire were chosen to be similar to the problems used in Govender’s original study. Two problems, which were based on the type of problems included in Govender’s study, were used for the questionnaire. The problems each consisted of several questions exploring how students apply algebraic signs across a set of one-‐dimensional kinematics problems dealing with displacement, velocity and acceleration.
The design of the questionnaire was based on the following:
• time for completion; • complexity of questions; • possibility to obtain theoretical result.
When administrating a self-‐completion questionnaire there are some aspects that have to be considered in order to obtain the best result (Robson 2002). The length of a questionnaire of this kind is a critical factor and should be kept short. Hence, a range of 15-‐20 minutes for completing the questionnaire was striven for. Since there was no or little possibility to clarify the questions after the questionnaire had been distributed, the questions had to be clearly stated. Because of this no problems containing calculations were included because this might scare some students off. Also, the questionnaire had to be designed in a way that, theoretically, it would be possible to find all five categories in the answers obtained.
The first problem dealt with a small ball rolling on a smooth surface. After rolling a certain distance the ball hit a barrier and returned to its original position. There was no frictional force acting on the ball as well as no energy-‐loss in the collision. Hence, the students should focus on the kinematics of the problem instead of dealing with the energy conservation of the collision. The students were asked to describe the displacement, distance, speed, velocity and acceleration of the ball (speed and distance were included to provide differential aspects for the analysis, if needed). They were also asked to describe the motion of the ball and explain if there was any difference in the motion of the ball before and after it hit the barrier. In all questions they were asked to explain the meaning of any algebraic signs they may have used. This was a critical aspect of the questionnaire since the study sought to understand the ways that students experience signs, and not if the students were right or wrong.
The second problem involved describing the velocity and acceleration of a police car chasing another car. The students were asked to sketch the velocity and acceleration for the police car, using signs and/or arrows, in five different parts of the sequence described, which involved the police car speeding up and slowing down as well as driving at a constant velocity. The problem dealt with the motion in two opposite directions with the aim of investigating if the students showed any difference in their use of signs in the two different directions. As in the first problem,
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the students were asked to explicitly explain the meaning of any signs they may have used.
Part of the aim of doing all the pilot studies (see next) was to create a questionnaire design that would yield the kind of data needed to effectively answer the research question. In the main study the questionnaires carried out this function extremely well.
3.2.3.2 Pilot studies Several pilot studies of the questionnaire were carried out to obtain as much feedback about the design, language, difficulty, and length as possible. These studies also provided feedback on the validity and the dependability of the chosen research method (see, Peat et al. 2002; 123). As part of this process, ambiguous questions were located, questions were re-‐phrased for clarity, and the time taken for completion were given due consideration.
The first pilot study was performed with two PhD students in the group and tested the very first draft of the questionnaire. Several changes were made to the problems used, the outline and the design of the questionnaire. The second pilot study was performed with seven engineering students. The comments obtained resulted in shortening the questionnaire from three to two problems because it took the students a longer time than planned to answer the questions. After these changes a final pilot study was done with four engineer students, and from this, smaller clarifications were made in the problem outlines.
3.2.3.3 Translation The questionnaire was originally written in English to make sure that everyone involved in the study would be able to follow the progress on the design of the questionnaire. Before distributing it to the Swedish students, the questionnaire was translated by the author. The translation was cross-‐checked by three independent people for validation before printing. The translation into Swedish was argued to be necessary for two main reasons; (1) the participating students’ current physics education is held in Swedish and a questionnaire in Swedish would help them understand the questions, and (2) students that are familiar with the physics included in the problems, might have difficulties answering because they are unfamiliar with the English language used.
3.2.4 Interviews Based on the analysis of the questionnaire data from both the Swedish and South African student populations, a group of students were selected for interviews based upon recognizable diversity and variation in the questionnaire results. The aim was to obtain interpretive clarifications that I had made of the descriptions that the students had provided in the questionnaire. At the same time they were also intended to provide deeper insight into the descriptions that the students provided. . Swedish students that were willing to take part in an interview provided their e-‐mail address with their questionnaire and were contacted with an enquiry to take part in an interview.
The Swedish interviewees were purposefully selected (Patton 1990) using one or more of the following criteria:
• showed interesting experiences in at least one question; • explained their reasoning carefully in a majority of the questions; • shared the same experiences throughout the questionnaire; • shared the same experiences throughout the questionnaire but deviated from this in one
or more questions.
Since the South African students came from many different schools throughout the KwaZulu-‐Natal province before they attended the university, the participating students were purposefully selected to capture a variation of these students and thus of answers to the questionnaire.
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The type of interview used for the study was semi-‐structured (Kvale 1996). Thus, questions were predetermined, however the order of the questions could be changed and additional questions added to follow up on the interviewee’s answers. During the interviews, the students were presented with his or her questionnaire allowing the interviewer to address specific answers or wording for clarification.
3.2.4.1 Design The interviews were designed to clarify and deepen my understanding of the students’ experiences of the use of algebraic signs in kinematic problem solving. The interviews were designed to be short, not involving questions that would give too much room for long descriptions. The interviews were kept in the order of 15-‐20 minutes, which put high demands on the interviewer not to let the interview lose focus. The interviews were kept this short also to attract students to participate.
The questions used for the interviews were chosen to give the students a chance to freely explain their experiences about algebraic signs used in the problems, however keeping the topic specific. The predetermined questions were of three different types: (1) “what sign would you use to describe displacement/speed/velocity/acceleration?”, (2) “how does the sign for velocity/acceleration change during the car chase?”, (3) “what does the sign for velocity/acceleration mean to you in everyday life/physics etc.?”.
The first of the three types of questions were mostly directed to the first problem included in the questionnaire while the second type was directed to the second problem. The third type were questions that would provide an explanation of the students’ overall experience of algebraic signs used in different contexts. Further questions asked during the interviews depended on the situation and varied between the different interviews but aimed to clarify interesting experiences from the questionnaire or to follow up on answers given during the interview.
During the interviews a commonly used sequence of questions for interviews was mainly used (Robson 2002), opening with an introduction to the study, which included the purpose of the study and explained the ethics involved. All interviews were recorded, after permission from the participants, which allowed the interviewer to focus entirely on engaging with the participants.
3.2.4.2 Pilot interview Before the first interview the interview protocol was tested on an independent person to identify any ambiguity in the questions and for the interviewer to feel more secure when doing the later interviews. During the pilot interview, the same questions and the same setup were used that were going to be used for the student interviews.
The pilot interview provided further understanding of how to conduct interviews and supported the planning and conducting of the student interviews. Difficulties with the interviews that were highlighted during the pilot interview included knowing when it was appropriate to move on with the next question as well as how to keep a neutral tone signaling that there was no prejudice towards the students’ understandings.
3.2.4.3 Transcriptions All Swedish and South African interviews were transcribed verbatim by the author. Transcribing the interviews very carefully ensured that no important information was omitted. The extracts from the interviews used within this report have been translated when necessary but without the meaning of the question or answer being changed.
3.2.5 Analysis The analysis of the data was done in line with Govender’s (1999; 2007) research where the obtained answers from both questionnaires and interviews were sorted in the five already
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existing qualitatively different categories (see appendix 2). However, possibility for the appearance of any new category was taken into account during the process. Following Govender, a phenomenographic perspective was used when sorting the data for the analysis (see Section 3.1). It is important to emphasize that throughout the analysis no references were made to a specific student, but rather to a specific experience of the use of algebraic signs.
The analysis process of the data was performed in many different stages and culminated in sorting the questionnaire answers and interview transcripts into boxes representing each of the five original categories. Each questionnaire was given a specific number instead of the students’ name to uphold the confidentiality and the interview transcripts were given the same number as the corresponding questionnaire for that particular student. This was done for both the Swedish and South African data. After being numbered, the individual answers on the questionnaires were cut apart leaving them each on a single piece of paper. The same thing was done for interesting excerpts from the interview transcripts. Throughout the analysis process this numbering, linking questionnaires with interviews, was crucial in order to make it possible to find the answers if one of the participating students asked to withdraw his or her answers.
To begin the analysis, the first set of data, being the Swedish questionnaire answers, were roughly divided into groups showing similar experiences of algebraic signs. These groups were further analyzed being split into new groups or conjoined with others, a process that was conducted several times. Next, the answers from each group were sorted into physical boxes representing the five qualitatively different categories. This sorting was remade through an iterative process until the categories had been saturated, meaning that no categories were placed in another box when resorted. The sorting and resorting of the questionnaire answers were made with a few days in between.
Once the first set of data had been analyzed, all other data, meaning Swedish and South African interview transcripts and South African questionnaire answers, was only analyzed through sorting and resorting directly into the physical boxes. This reduction of analysis stages was possible because through the first analysis process it had become easier to recognize the different categories and thus there was no need to make the first rough sorting. The Swedish and South African data were sorted in separate boxes to be able to make a comparison between the experiences of the two student populations.
In all stages of the data sorting and categorization, the same phenomenographic research perspective was used.
4 Results In the following sections the results of the study will be presented. The analysis from the questionnaires and interviews will be compared to the outcome space generated from Govender’s (1999; 2007) study (appendix 2).
First, the results of the questionnaire and interviews will be given as a summary of the findings, followed by an evaluation of the results. Later, example excerpts from the different categories will be presented in order to better describe the categories. Here, no differences will be made between the Swedish and South African students. The excerpts named “Question” and “Answer” originate from a questionnaire, while the excerpts coded “Interviewer” and “Student” shows examples of answers from an interview.
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4.1 Summary of findings From the analysis of the data obtained from the questionnaires and interviews, I was able to identify four of the five categories proposed by Govender (2007) to describe the variation of ways students experience the use of algebraic signs in vector-‐kinematics in both sets of data. No new categories were found. The obtained outcome space for my study is presented in Table 1 below. The names of the categories are the same as those in Govender’s original study.
Table 1: The obtained outcome space
A: Algebraic signs are not applied in vector-‐kinematics
C: Algebraic signs are applied as changing magnitude
D: Algebraic signs are applied as both magnitude and direction
E: Algebraic signs are applied as directions
The relative frequency of each categorization is shown in Figure 3 using a scale of 1-‐4. It is important to point out that a categorization of a piece of description is not to be taken to be synonymous with a single person; a person may have provided descriptions that fit into more than one category. The most commonly used description in both the Swedish and South African data sets fell into Category E followed by Category C. The two categories that were the next most common were Categories A and D. Here, Category A was slightly more common in the Swedish data set, while Category D was more common in the South African data. Govender’s Category B (Algebraic signs are applied as magnitude only) was not found from the analysis of either the Swedish or the South African data sets, hence this category is not included in the figure.
Some answers were not analysable for this study since they did not contain information about the use of algebraic signs, or were completely blank. Particularly in the second problem many answers could not be analysed. In this problem students were asked to sketch the velocity/acceleration of the police car during the chase with the use of signs and/or arrows. Here, many students only drew arrows without any signs and thus these answers were not analysable for this study.
0 1 2 3 4
A
C
D
E
Relative frequency in data sets
Category
South African data set
Swedish data set
Figure 3: The relative frequency of the categories occurring in the Swedish and South African data sets. The scaling used is 1-‐4 where 4 is the most common and 1 the least common. Govender’s original Category B (Algebraic signs are applied as magnitude only) is not included because it was not found among the analysed answers.
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Among the South African students almost all answers were analysed, while many Swedish students failed to answer all questions. The question that was left blank most often was question 2.5.
4.2 Evaluation The results obtained showed that not all of Govender’s original categories, which characterize the ways that students conceptualize how algebraic signs should be used in kinematics problem solving, could be found among the student populations involved in this study; Swedish introductory physics students and South African physical science preservice teacher students. The obtained outcome space of the qualitatively different ways students apply algebraic signs in kinematics problem solving can be found in Table 1.
The research question asked how relevant Govender’s categories were, that is if they were generalizable to another context. Being able to find only four of Govender’s five original categories, I can conclude that Govender’s results were not generalizable as an entire outcome space for the qualitatively different ways students experience algebraic signs in vector-‐kinematics.
When analysing the data obtained, it was found that the experiences among the Swedish and South African students was the same, which implies students in different contexts share the same experiences and/or challenges in the area of vector-‐kinematics.
Since this study has been able to identify four qualitatively different categories of description of how students conceptualize the use of algebraic signs in vector-‐kinematics, it is of interest to investigate how teaching and learning has developed during the 15 years that has passed since Govender’s first study. Although the reason for not finding one of the original categories (Algebraic signs are applied as magnitude only) is something that we presently cannot comment on, we would like to think that the development of teachers' and physics instructors’ teaching has been the foundation for this change in conceptualization.
4.3 Description of categories Next a presentation of the findings from this study will be made showing the qualitatively different ways students in the study experience the use of signs in vector-‐kinematics. As it is not possible to display all the different conceptions belonging in the qualitatively different categories, and thus the excerpts shown below will only highlight some interesting aspects found. The different quotes will show the variation of experiences in the same category to highlight various interpretations.
A: Algebraic signs are not applied in vector-kinematics The first category proposed by Govender states that students experience algebraic signs as not applied at all in vector-‐kinematics to describe the concepts of displacement, velocity, and acceleration. This category shows that students make no connection to the representation of vectors-‐concepts in terms of signs. Below are some interesting explanations for the absence of algebraic signs obtained from the questionnaires and interviews.
Question: Is direction important to specify the motion of the ball? Explain. Answer: Yes, the direction is important to specify because it tells you where the rolling ball
comes from. We need to tackle how to give its direction of motion, because positive or negative will not work since it could be moving at any angle.
This student is unable to connect signs as a description of directions of a vector. The connection to direction is made, however the student feels that plus and minus is not sufficient to indicate
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the direction of motion of the ball, since there can be many different angles. Here, students do not connect plus and minus to a coordinate system.
Another example of the unwillingness to use signs to describe vector-‐concepts is shown in the excerpt below. The student does not show any need or motivation to use signs to describe vector-‐concepts and cannot remember having seen this used before.
Question: Explain the meaning of any algebraic sign (+ or -‐) that you used [to illustrate the velocity of the police car].
Answer: You cannot use arrows or signs to describe the velocity or acceleration, only numbers. I have at least never encountered anything other than numbers to describe this within physics.
This clearly shows us that the student has not made the connection between algebraic signs and direction. Unfortunately this excerpt comes from a student who was not interviewed and thus a clarification of this experience could not be obtained.
Further, students often show willingness to express the direction of a vector (or the motion), however not particularly in terms of plus and minus.
Interviewer: When you say that 𝑎 = 5 left, you don’t say that 𝑎 = −5? Student: No. Interviewer: Why not? Student: It depends on me. As long as I in the beginning would specify that left means
something is going to the left.
Question: Describe the speed and velocity of the ball before and after the collision. Explain the meaning of any algebraic sign (+ or -‐) that you used.
Answer: As before, a motion to the right feels positive and to the left negative. I now realize that I think that + and – seems a bit unnecessary. Why don’t you just say a motion to the right or left?
This experience shows that students do not link algebraic signs with directions and feel that they are redundant to describe the motion of an object. The students see it as sufficient to state the direction in already defined terms such as left or right, instead of using signs that to the students have no explicit meaning in the problem solving process.
C: Algebraic signs are applied as changing magnitude A common experience of the use of algebraic signs in vector-‐kinematics was that signs are applied as denoting the change of magnitude. One student explained this explicitly as follows:
“I experience plus as something that is getting bigger, and minus as something that is getting smaller”.
This category was mostly used to describe the change in magnitude for velocity and acceleration. The different meanings of the signs applied for change in velocity and signs applied for change in acceleration is interesting to point out.
Question: Explain the meaning of any algebraic signs (+ or -‐) that you used [to illustrate the velocity of the police car].
Answer: For increasing motion the positive sign was used, regardless of the direction, while a negative sign was used for decreasing velocity.
Question: Explain the meaning of any algebraic signs (+ or -‐) that you used [to illustrate the acceleration of the police car].
Answer: (+) Increase in acceleration, (-‐) Decrease in acceleration.
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The above excerpts show that students use plus and minus as changing magnitude for both velocity and acceleration. An interesting similarity can be found between the use of signs for the two conceptions. Note that the signs denote the change in magnitude, i.e. the change in velocity and the change in acceleration. However, while it is easy to picture a change in velocity as an object is speeding up or slowing down, a change in acceleration is harder to picture. The meaning of this will be discussed in Section 5.
Another interpretation of signs linked to a change in magnitude is the next excerpt. Here the student was asked to illustrate the acceleration of the police car in the second problem of the questionnaire:
Question: Explain the meaning of any algebraic signs (+ or -‐) that you used. Answer: +𝑎 means the car is accelerating. −𝑎 means the car is decelerating.
Interviewer: [What sign would you use] for acceleration? Student: When it’s acceleration, it’s speeding up, I would say positive and when it’s slowing
down I would say negative.
From the first excerpt we see that students of this experience link a negative acceleration with deceleration, meaning a decrease in velocity. The interpretation of this is that to students of this experience, a negative acceleration will always imply a deceleration, regardless of the direction of the motion of the object. This interpretation is closely linked to the second excerpt above where a negative acceleration was used to describe an object slowing down. From this we see that for students, a negative acceleration can be experienced as a decrease in velocity, the object is slowing down, regardless of the motion of the object.
A final interesting excerpt for this category shows the explicit meaning of the use of algebraic signs as denoting change in magnitude. This student is consistent in his reasoning and does not differentiate between the use of algebraic signs in everyday life and physics.
Interviewer: What does plus and minus mean, in general? Student: I would say change in magnitude. Interviewer: And in physics, what does plus and minus mean to you? Student: The direct experience is that it means bigger or smaller. It is not linked to
directions.
D: Algebraic signs are applied as both magnitude and direction The category stating that algebraic signs are applied as both magnitude and direction is an inappropriate experience of the use of algebraic signs in vector-‐kinematics. This category shows that students experience algebraic signs in different ways for different concepts; often this difference is displayed between velocity and acceleration. In other words, algebraic signs are not used consistently in vector-‐kinematics.
In the excerpts below, it is shown how students interpret algebraic signs as both magnitude and direction.
Question: Is there any difference between the velocity and acceleration arrows or signs that you drew in the above questions? If so, explain.
Answer: Yes, in velocity the signs only specify the direction of motion, however in acceleration it means speeding or slowing.
Question: Is there any difference between the velocity and acceleration arrows or signs that you drew in the above questions? If so, explain.
Answer: Yes, when it comes to velocity + and – only show direction. When it comes to acceleration they only show the acceleration’s increase or decrease and doesn’t take direction into consideration. Why it turned out this way I don’t know!
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These excerpts exemplify how students have little understanding of the meaning of the use of signs in vector-‐kinematics. These students have the right interpretation of the use of signs for describing the velocity of an object (plus and minus are applied as directions), however in the second excerpt the student shows confusion about how signs can mean different things for velocity and acceleration. The interpretation tells us that students are unaware, or have little understanding, of the sign conventions used in vector-‐kinematics.
The excerpt below is another example, obtained from one of the interviews, illustrating how students describe the use of algebraic signs as both magnitude and direction.
Interviewer: What does the velocity sign mean to you in general? Student: The velocity signs mean directions. Interviewer: And acceleration? Student: For acceleration they mean whether speeding up or slowing down. For speeding up
I use plus, for slowing down minus.
E: Algebraic signs are applied as directions The only appropriate experience among the five categories described by Govender was the last category stating that signs are applied as directions only. As in the other categories, students may have different interpretations of signs, thus in this case directions have different meanings for different experiences. For example, “directions” may mean direction of the motion or direction of the vector quantity.
A common experience of signs applied as directions is that signs are dependent on the direction of the movement, rather than the direction of the vector quantity.
Question: Describe the acceleration of the ball before and after the turn. Explain the meaning of any algebraic signs (+ or -‐) that you used.
Answer: The acceleration will be positive during forward motion and negative during reverse motion.
We note that when assigning the signs the student only takes the direction of motion into consideration and does not link the signs to the direction of the vector. Unfortunately the student does not explain why the signs are applied this way. The following excerpts, obtained from a different student, further explain this.
Question: Explain the meaning of any algebraic signs (+ or -‐) that you used [to describe the acceleration of the police car].
Answer: Only + is used since there is only one direction, i.e. there is no change of direction to the opposite direction because the car slowing down doesn’t change the sign.
During the following interview with the same student, clarification was sought.
Interviewer: How does the sign of the acceleration of the police car change? Student: It is positive. Because as long as it is following the Volvo in one direction I take it as
positive which means it is going to the right. Interviewer: So when it accelerates and slows down you would give the same sign? Student: Yes
This, although an incorrect use of signs as directions, primarily shows that the student is consistent in his interpretation. The student allocates signs to describe the acceleration depending on the direction of the car. Thus, the experience is that signs are applied as directions, however not the direction of the vector quantity as would be the correct experience.
Often, in both questionnaires and interviews, students expressed that plus and minus applied as directions were just something they had been taught and they did not further think of the meaning of this.
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Question: Is there any difference to the motion of the ball before and after the turn? Explain the meaning of any algebraic signs (+ or -‐) that you used.
Answer: We usually determine motions to the right, the original motion, as positive, thus with a + sign. When the ball has turned, it travels in negative (-‐) direction compared to its original motion.
Question: Is there any difference to the motion of the ball before and after the turn? Explain the meaning of any algebraic signs (+ or -‐) that you used.
Answer: I have learned that right is indicated with a + and left with a -‐.
We see that students might not think about the meaning of the chosen sign convention, but accept it as something that is defined to be this way. They think that “right” is connected to + and “left” is connected to – in all cases without understanding why. That students accept the sign conventions presented to them by their teachers can be seen in the excerpt below. .
Interviewer: Why was the motion of the ball positive to start with? Student: I guess that it is just something I assume or presuppose. Interviewer: And then it becomes negative, why? Student: Well, after it turned it can’t continue have the same sign as before. I think that it
should be negative.
To the student, different directions should be described with different signs. However, no understanding of the meaning or the background to this is seen.
5 Discussion The analysis of the data obtained from students in the basår programme at Uppsala University, Sweden, and physical science preservice teacher students at the University of KwaZulu-‐Natal, South Africa, has revealed that four of the five qualitatively different categories of how students experiences the use of algebraic signs in vector-‐kinematics, reported on by Govender (1999; 2007), still are found today. The category that was not found among the participating students was Category B (Algebraic signs are applied as magnitude only).
Through analysis of the data I have been able to sort the descriptions into categories of the variation of ways students in Sweden and South Africa apply algebraic signs in vector-‐kinematics. This analysis led to identifying four of the five categories from Govender’s original study. The obtained outcome space of these qualitatively different categories of experiences can be found in Table 1. I have not only been able to show the existence of these four categories across these two contexts 15 years later, but can also say that the Swedish and South African data sets were similar. Hence the categories found is claimed to be consistent in different contexts. What also should be emphasized is that Govender’s study was carried out nearly 15 years ago, showing that students’ experience of signs has not changed significantly over a long time period.
Four of the five original categories were identified. This means that the full generalizability of Govender’s categories was not established by my study. This does not mean that the missing category no longer exist, a larger study could perhaps show that this category still exist.
Even though I was not been able to find Category B (Algebraic signs are applied as magnitude only) among the data obtained in this study, from a phenomenographic perspective we cannot with certainty exclude this category from the outcome space of how students experience algebraic signs in vector-‐kinematics. Since phenomenography seeks the variation of ways through a qualitative rather than a quantitative approach, this category, since it has been found in an earlier study, cannot be eliminated completely. It has not been found in this study, but there is a possibility that it could be found in future studies.
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A question that now naturally arises is why we were not able to identify all categories. The absent category was found 15 years ago in South Africa, but could not be found in either of the data sets. Since we have no research showing the development of the physics education in South Africa we cannot draw any conclusions about the elimination of the category.
Using the results of this study, we will be able to inform physics teachers about students’ difficulties when using signs in vector-‐kinematics, which will provide teachers with valuable pedagogical tools to improve physics education that will better suit the diversity of the student populations that can be found at introductory physics levels today. Since it has been shown, through this and other research (see for example, Govender 1999; Hayes & Wittmann 2010; Aguirre 1988; McDermott 1984), that students experience difficulties with vectors and vector notations it is important that teachers become aware of students’ struggles in order to help them overcome their challenges. If teachers are not familiar with the common problems that students may have, it will be difficult to design a physics education that meets their students’ need.
It is not hard to recognise that if teachers are not made aware of the struggles students have, there will not be a change in the teaching and learning. One of the goals with this study was to suggest different pedagogical tools for teachers, based on the results of the study, to improve physics education. However, unfortunately there has not been enough time to further investigate this. Nevertheless, I will below discuss common challenges I have identified and suggestions of what might be the causes of these interpretations of signs.
From doing this study I have come to better understand the challenges students encounter when being introduced to vectors. I could recognise myself in many of the answers which made me remember the difficulties I experienced when learning how to use vectors. To be able to recognise my own thinking in the data made me feel a strong link to the study and made me even more eager to obtain a result that would inform teachers. I have realised that when you use vector notation appropriately after several years of physics training, it might be difficult to recognise difficulties that students are encountering or to understand what causes these difficulties. This is why it is important that teachers are informed about the results of this study to help their students overcome their difficulties.
5.1 Identified challenges Below I will discuss some of the educational challenges that have been highlighted through this study and will briefly discuss suggestions of how to help students overcome these challenges.
Students are not consistent with their use of algebraic signs From Category D which states that algebraic signs are used as both magnitude and direction, it can be seen that students are inconsistent in their use of signs. On one hand signs indicate a direction and on the other hand, plus and minus are applied as signifying changing magnitude. When analysing individual questionnaires, it was found that the same student described the use of plus and minus differently in different problems, which might be an indication that students use different sign conventions in different problem contexts.
To help students overcome this incorrect use of algebraic signs, I believe it is important that teachers are clearer about the reasons for using signs but also that students get to meet vectors in different ways to help them understand the sign conventions used.
Students apply an incorrect use of signs in the same way for both velocity and acceleration As seen in the description of Category C (Algebraic signs are applied as changing magnitude), students often indicate that signs are used to describe an increase or decrease in velocity. From a physics point of view, this is an inappropriate use of signs in vector-‐kinematics which shows that
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students are not familiar with the common sign conventions used. Another common interpretation of signs in this category was that plus and minus indicates an increase or decrease in acceleration. This is also an inappropriate use of signs since it shows that for the students it is not clear how to interpret a change in acceleration in terms of signs.
Students’ incorrect use of signs in the same way (as changing magnitude) for both velocity and acceleration, might originate from an incorrect interpretation of signs applied for acceleration. If students have learned that the sign for acceleration indicates an increase or decrease in velocity, they might think that signs are used in the same way to describe the velocity. For example, if students have learned that plus is used for acceleration when a car is speeding up, they might think that the same sign will also be used for velocity in the same situation. Hence, to these students there is no experienced difference in the application of signs between velocity and acceleration.
The same approach might be the explanation for the interpretation that signs for acceleration are applied as direction of the motion. As we could see in Category E, many students assigned plus and minus signs for acceleration indicating in what direction the object was moving, instead of the correct interpretation that signs should indicate the direction of the acceleration vector. This might be explained in a similar way as in the paragraph above.
If students have learned that signs are used for velocity to indicate in what direction the object is moving, they might think that signs are used in the same way also for acceleration. Hence, the sign for acceleration indicates the direction of the motion. For example, if an object is moving to the right, a plus sign is used to describe its velocity, similarly a plus sign is used to describe the acceleration of the object, regardless if the object is speeding up or slowing down.
Students need to experience vectors in more than one way Many students described the use of plus and minus in a way that tells us that often teachers define the “correct” sign conventions without telling students this is an arbitrary choice. Examples of this could be found in Category E (Algebraic signs are applied as directions). By telling students that “right is positive” and “left is negative” they only come to see vectors as used in one way hence, the correct understanding of algebraic signs in vector-‐kinematics will not be clear to them even though they might be using signs as directions. To overcome this problem, I think it is important that students are tested more on their conceptual understanding, something that is also proposed by McDermott (1984).
“Direction” means different things in different situations The most common category that could be found among the student responses was Category E stating that signs are applied as directions. This is proposed by Govender as the appropriate use of signs in vector-‐kinematics. This category was expected to be the most common when starting the study, which would mean that students do make the connection between algebraic signs and directions for vector-‐concepts, just as we hoped they would. We need to discuss the many different ways students use signs for directions, not only the direction of the vector arrow, but the direction of motion or direction of change of magnitude.
As described above, signs applied as directions can be interpreted in different ways. A correct interpretation is that the sign for velocity is linked to the direction of motion. However this interpretation is not appropriate to be used for the concept of acceleration. Despite this, many students applied the same sign for acceleration that was used to describe the direction of motion. Another inappropriate use of signs applied as directions is when signs are applied as a direction of change of magnitude, as discussed previously. Students might, for example, use plus and minus to denote the direction of the change in velocity meaning that if the velocity decreases the change is directed in the negative direction and the velocity becomes negative.
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From the above examples it can be seen that “direction” can mean different things to different students, especially in different contexts. Even though students are aware that signs are used as directions, they might not realize that directions mean the direction of the vector. Thus teachers need to be more explicit in explaining that signs are used as the direction of the vector and not the motion or change in magnitude.
Students do not understand the reason for using signs Category A (Algebraic signs are not applied in vector-‐kinematics), describes students’ experience where they have not all grasped the sign convention used and do not understand the signs used in vector-‐kinematic problem solving. Teachers need to make clear to students that plus and minus is a tool for describing vector-‐quantities in a coordinate system. Hence, it is important that teachers help the students to differentiate between the directions in everyday life (right and left) and the directions in physics (plus and minus).
6 Recommendations It is proposed that this research is developed in future studies in order to verify the results obtained. This should include a close investigation to see if Category B (Algebraic signs are applied as magnitude only), which could not be found in this study, could be found using a different data set. A continuation of this project should also involve further studies of how this result should be implemented in the teaching and learning of introductory physics. It is important to find out how teachers can take this result and use it to develop their teaching. Also, it is of interest to find out more about the reason for the students’ conceptual difficulties, which can be found through learning more about how teachers explain and demonstrate the use of signs in vector-‐kinematics in the classroom.
It is important that teachers understand the results of this study in order for them to be aware of students’ difficulties and experiences of algebraic signs in vector-‐kinematics and also how common these are for them to be able to improve the teaching and learning of introductory physics. A central issue for teachers to grasp is the necessity of being clearer about the actual meaning of algebraic signs as the direction of the vector and not the direction of the quantity itself.
7 Conclusions In order to find the number of qualitatively different ways students of introductory physics experience the use of algebraic signs in vector-‐kinematics to be able to examine the relevance of Govender’s results, I designed a special questionnaire where students were asked to explain their use of signs in one-‐dimensional kinematics problems. The questionnaire was completed by 60 Swedish basår students and 24 South African physical science preservice teacher students. Based on the answers from the questionnaire, students were purposefully selected to take part in semi-‐structured interviews, where I sought clarifications to some experienced interpretations of signs. In total five Swedish students and six South African students took part in the interviews.
All questionnaire answers and interesting excerpts from the transcribed interviews were analysed using sorting based on a phenomenography perspective, where the researcher tries to find the variation of ways people experience a phenomenon. Through the analysis, I was able to identify four of the five original categories proposed by Govender (1999; 2007). The categories can be found in Table 1.
21
The study has primarily shown that the outcome space of how students experience algebraic signs in vector-‐kinematics found by Govender, has only slightly changed since the first study was performed 15 years ago. The category that could not be found in my study stated that students experience that algebraic signs are applied as magnitude only. The reason for this need to be investigated in future studies. Further, I have also been able to show that in this study no differences were seen between the experiences of algebraic signs among the students in Sweden and South Africa and the obtained outcome space could be used in different contexts.
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References Aguirre, J. M. (1988). Student preconceptions about vector kinematics. The Physics Teacher, 26,
212-‐216.
Bowden, J., Dall’Alba, G., Martin, E., Laurillard, D., Marton, F., Masters, G., Ramsden, P., Stephanou, A., & Walsh, E. (1992). Displacement, velocity, and frames of reference: Phenomenographic studies of students’ understanding and some implications for teaching and assessment. American Journal of Physics, 60, 262-‐269.
Brunt, G. (1998). Questions of sign. Physics Education, 33(4), 242-‐249.
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Hayes, K., & Wittmann, M. C. (2010). The role of sign in students’ modeling of scalar equations. The Physics Teacher, 48(4), 246-‐249.
Henderson, C., Dancy, M., & Niewiadomska-‐Bugaj, N. (2012). Use of research-‐based instructional strategies in introductory physics: Where do faculty leave the innovation-‐decision process? Physical Review Special Topics -‐ Physics Education Research, 8(2), 020104.
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Appendix 1 The questionnaire that was distributed among the participating students.
Problem 1: The motion of a rolling ball A small ball rolls along a smooth surface (ignore friction). When the ball has rolled 2m, it reverses when it hits a barrier (no energy is lost during the collision) and it rolls back to its original position. For the questions below, please explain your reasoning carefully.
1.1: Is there any difference to the motion of the ball before and after the turn? Explain the meaning of any algebraic signs (+ or -‐) that you used.
1.2: Is direction important to specify the motion of the ball? Explain.
1.3: Describe the distance and displacement of the ball before and after the turn. Explain the meaning of any algebraic signs (+ or -‐) that you used.
1.4: Describe the speed and velocity of the ball before and after the turn. Explain the meaning of any algebraic signs (+ or -‐) that you used.
1.5: Describe the acceleration of the ball before and after the turn. Explain the meaning of any algebraic signs (+ or -‐) that you used.
Before: After:
Problem 2: Velocity and acceleration of a car chase Imagine the following sequence: (1) A police car is standing by the side of the road at the intersection between Dag Hammarskjölds väg and Kungsängsleden when she sees a Volvo travelling at a constant speed through a red light. (2) The police car immediately starts chasing the Volvo, along a straight part of the road, accelerating from rest until reaching a maximum chasing speed. (3) The officer holds this speed until she is alongside to the Volvo. (4) She turns on the blue light signalling to the Volvo to pull over. The driver of the Volvo starts to slow down, the police car also slows down, staying alongside the Volvo. (5) Both cars finally stop by the side of the road. 2.1: For the different parts of the sequence (1)-‐(5) above, sketch the velocity of the police car using arrows, and signs if appropriate. (1) (2) (3) (4) (5)
2.2: Explain the meaning of any algebraic signs (+ or -‐) that you may have used.
2.3: For the different parts of the sequence (1)-‐(5) above, sketch the acceleration of the police car using arrows, and signs if appropriate. (1) (2) (3) (4) (5)
2.4: Explain the meaning of any algebraic signs (+ or -‐) that you may have used.
2.5 Supposed the police car turns around and follows the exact same sequence in the other direction.
2.5.1: For the different parts of the sequence (1)-‐(5), sketch the velocity of the police car using arrows, and signs if appropriate. (1) (2) (3) (4) (5)
2.5.2: Explain the meaning of any algebraic signs (+ or -‐) that you may have used.
2.5.3: For the different parts of the sequence (1)-‐(5), sketch the acceleration of the police car using arrows, and signs if appropriate. (1) (2) (3) (4) (5)
2.5.4: Explain the meaning of any algebraic signs (+ or -‐) that you may have used.
2.6 Are there any differences between the arrows and/or signs that you used to describe the velocity and acceleration respectively in the above questions? If so, please explain.
Appendix 2 Govender’s (2007) article describing the study conducted and displaying the result including the outcome space of the qualitatively different ways students experience the use of signs in vector kinematics.
This article is reproduced as part of this project report with the permission of Fred Lubben, Chief Editor and publishing coordinator of the African Journal of Research in Mathematics, Science and Technology Education.
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Physics student teachers' mix of understandings of algebraic sign convention in vector-kinematics: A phenomenographic
perspective
Nadaraj Govender School of SMTE, Edgewood Campus, University of KwaZulu-Natal
[email protected] Abstract Physics pre-service student teachers' mix of understandings of positive (+) and negative (–) algebraic sign convention in vector-kinematics was explored using three familiar demonstration contexts in physics. The students were interviewed about their understandings of algebraic sign convention as applied to high school Physical Science and Mathematics and, specifically, to fundamental concepts in vector-kinematics, namely, displacement, velocity and acceleration. The interviews were analysed from a phenomenographic perspective. Five different conceptions and hierarchical relations of the different ways student teachers' understand algebraic sign convention were identified. The data highlights the need for physics educators to take cognisance of both student teachers' limited conceptual understandings of vector-kinematics and its associated algebraic sign convention in first-year university physics. Introduction One of the important aspects of mathematical 'inscriptions' – numbers, graphs, signs, symbols etc. (Roth, Tobin & Shaw, 1997, p. 1075) are signs and sign convention, which serve as a tool or technique for expressing physical quantities and solving of physics problems. Sign convention is an arbitrary choice of mathematical representation used as a technique in problem solving, for example, algebraic signs (+) and (–) is used to denote direction of motion of object. By assigning the "+" and "–"signs to numbers one refers to what has been called directed numbers (Fischbein, 1994). In vector-kinematics this involves symbolic representations of algebraic operations and subsequent manipulation and interpretation of those symbols within the context of problem solving in one, two and three dimensions. However, research has shown that students in tertiary physics are often confused by the application of algebraic signs in vector-kinematics (Rebmann & Viennot, 1994; Romer, 1998). In this regard, Rebmann and Viennot (1994) argue that since algebraic language is one of the main tools used in physics, it is educationally beneficial to analyse students' comprehension of and skill in algebraic procedures including their notion of algebraic signs. Since physics teachers play an important role in the development of their students' understanding of fundamental concepts, it is therefore critical to develop insight into the different ways in which student teachers' as prospective physics teachers understand sign conventions in physics. A literature review of work in this area of physics has shown that qualitative studies in algebraic sign convention in vector-kinematics are rare. This study attempts to contribute to filling this gap by exploring the different ways that student teachers' conceptualise algebraic sign convention in vector-kinematics. Since the study was about identifying the range of perceptions of sign conventions in vector-kinematics, phenomenography as a research perspective was deemed the most appropriate method to analyse students' understandings. Phenomenography was developed by Ference Marton (1981) to discover the "qualitatively different ways in which people experience, conceptualise, perceive, and understand various aspects of, and phenomena in, the world around them." In this study, the phenomenon is positive (+) and negative (–) algebraic sign convention as applied to vector-kinematics concepts of displacement, velocity and acceleration.
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Tertiary pre-service physics student teachers were interviewed about their understandings of algebraic sign convention as applied to everyday Physical Science and Mathematics contexts, and to fundamental concepts in vector-kinematics, namely, displacement, velocity, and acceleration, using familiar demonstration contexts in physics. The interviews were then analysed from a phenomenographic research perspective. Literature review The role of symbols in scientific communication Positive (+) and negative (–) algebraic sign convention performs an essential role in physics, as it provides us with a succinct and efficient method for use in vector-kinematics. Reif (1995) elaborates on the use of symbols, of which sign convention is just one aspect, by pointing out that:
Formal modes of conceptions, using precisely defined symbols and explicit rules for their manipulation, are very well suited to facilitate long and accurate inference chains. Such formal modes of conceptions, exploiting mathematics and logic, are thus widely used in physics (p. 23).
Rebmann and Viennot (1994) investigated university student teachers' mastery of algebraic sign use in physics and added that correct usage and conventions are far from obvious to students. Brunt (1998, p. 242) stated that, positive and negative algebraic signs are frequently misapplied. He added that some authors of textbooks might be unaware of some of the basic rules of algebraic sign convention and, for clarification of readers, he discussed some of the rules for sign convention in physics. Romer (1998) also emphasises the latter point and says, referring to the confusion in sign convention that, "there are so many places one could go wrong that it sometimes seems that one has at best a 50-50 chance of getting it right" (Romer, 1998, p. 849). Algebraic sign convention in kinematics – a complex problem Students find great difficulty in relating the motion of an object to its variables and algebraic formalism. In an investigation with physics students in mechanics to determine correspondence between an observed motion and the algebraic formalism, Lawson and McDermott (1987, p. 811) found that students make little connection between the algebraic symbols of the formula and the features of the demonstration – unlike a physicist, who almost 'sees' the application of formula to real world situations. Newburgh (1996) also noted that first-year physics students find it difficult to understand the role of mathematics in making inferences about the real world. He suggested that this is partly because they have not developed much physical intuition. Such intuition comes from experience and that real, imaginary and complex numbers appearing as solutions in mechanics problems can pose difficulties in interpreting such solutions. There are many different symbols and notations to denote vectors, and students often get confused in using them in physics. Boldface signs, for example, +, – and = signs are used in vector equations in some textbooks to emphasis the distinction between vector and scalar operations with ordinary numbers. The importance of vector-scalar symbolism is highlighted by Arons (1997) who feels that if students do not distinguish between scalar and vector quantities in the notation, they will most likely not make the distinction in their thinking either. Allie and Buffler (1998) noted that, in South Africa, vectors are not discussed in the Mathematics school syllabus and are dealt with only in the Physical Science syllabus. Since most of the high school physics problems are dealt in one-dimensional physical situations, where vectors are labelled with positive and negative signs, "high-school students end up perceiving no practical difference between scalar and vector algebra" (Allie & Buffler, 1998, p. 618).
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Students experience confusion surrounding algebraic sign convention especially in the transition between one- and two-dimensional motion, with the notion of scalars and vectors coming into play. Scalars have a sign and a magnitude whereas vectors have magnitude, which is inherently positive, and has direction (Warren, 1979). The components of a vector in a rectangular one- two- or three-dimensional Cartesian coordinate system can have positive or negative signs. In a special case, the one-dimension Cartesian system component is taken to have direction that can have a negative or positive algebraic sign. From a mathematical perspective, a directed scalar can be considered as a one-dimensional vector. This notion is also accepted in the one-dimensional treatment of vectors in physics but the existence of both positive and negative values of a quantity in physics can sometimes be wrongly regarded as a vector. This is possibly one reason why some scalars, such as potential energy and temperature, which can have negative values, are often mistakenly thought to be vectors (Warren, 1979). Students also experience difficulties with the concepts of displacement, velocity and acceleration, and in interpreting algebraic signs for kinematical graphs of motion. Trowbridge and McDermott's (1980) study showed that there was confusion about the vector nature of velocity. The concept of acceleration, and its associated sign convention in vector-kinematics, poses a more difficult aspect for students. Warren (1979, p. 3) adds that difficulties caused by the incorrect use of the negative sign to acceleration, implying that the speed is decreasing, arise in cases where a body undergoes periodic motion, or thrown upwards, or in head-on collision of objects. Students, who are taught that rectilinear acceleration is a rate of increase of speed, and use negative rectilinear acceleration or 'deceleration' (not a formal physics term) for a rate of decrease, find it very difficult to accept, in the case of uniform circular motion, that a body is accelerating when its direction changes at a constant speed. Students' application of algebraic sign convention is further compounded by poor conceptual knowledge in mechanics. Halloun and Hestenes (1985) noted that the most common, and critical, problem for students was a failure to discriminate between various kinematical quantities. Research indicates that a significant number of students do not master the basic kinematical ideas in the first years of introductory physics (McDermott, 1984; Trowbridge & McDermott, 1980, 1981). Aquirre (1988) also suggests that teachers of introductory physics need to give explicit consideration to the study of vectors. The Mechanics Baseline Test by Hestenes and Wells (1992) reported that the lowest scores were generated by questions requiring an understanding of vector properties. Hestenes and Wells (1992) adds that while students face an immediate hurdle with the concept of force, inabilities to reason correctly about vector quantities, present a second, and less-studied, hurdle. First-year university and final year high school physics student teachers' understanding of fundamental concepts in vector-kinematics has been explored using the phenomenographic research perspective (Bowden et al., 1992; Dall'alba et al., 1993; Govender, 1999). From the analysis of the interviews from these studies, a hierarchical set of conceptions of the different ways students understand the concepts in mechanics have been developed and the relationship between different levels of understanding have been identified. More recently, Erdman (2004) citing students' difficulties with vectors devised outdoor activities to force students to apply the concept of vector in terms of length and direction. The above literature review suggests that the understanding of algebraic sign convention and concepts in vector-kinematics can be a complex issue. In physics, a sign convention is a choice of the signs (plus or minus) of a set of quantities, in a case where the choice of sign is arbitrary – meaning that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. Thus central to a thorough understanding of sign conventions in vector-kinematics is the comprehension of the notion of arbitriness or free choice of assigning signs. Although there are several articles on vectors and kinematics over the past ten years, a recent literature search (2007) has shown that no new
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studies were published on sign convention in vector-kinematics. This may possibly be due to a notion by physicists that there is little or no difficulty experienced by first-year students in vector-kinematics or an inadequate understanding of their students' school physics background. Thus this paper contributes to a better understanding at the school-university interface by providing a deeper analysis of student teachers' qualitative understandings of algebraic sign convention that can contribute to physics education. Methodology Detailed knowledge of the ways in which students understand the central phenomena and concepts within a domain prior to study, is believed to be critical for developing their understanding of the central phenomena, and, hence, of the mastery of the domain (Bowden et al., 1992). This study thus uses the phenomenographic research perspective developed by Ference Marton (1981) to discover the "qualitatively different ways in which people experience, conceptualise, perceive and understand various aspects of, and phenomena in, the world around them". According to Marton and Booth (1997), phenomenography uncovers not the 'inner world' of the students but how the students see their relation to the world. In this study students seek to recognise and structure relationships, in the mix of conventions and definitions in kinematics, linked to the real world phenomena with which they were presented. Thus phenomenography is an appropriate methodology to identify the distinctive different ways in which we organise our relations to the world. In all phenomenographic studies, it has been found that each phenomenon, concept or principle can be understood in a limited number of qualitatively different ways. In this study, it is also assumed that a limited number of ways of understanding positive (+) and negative (–) algebraic sign convention as applied to the vector-kinematical concepts in physics can be found. This study formed part of a longitudinal qualitative case study. The study involved interviewing 19 first-year college physics pre-service student teachers in order to determine their conceptualisations of positive (+) and negative (–) algebraic sign convention used in vector-kinematics. The pre-service student teachers are prospective physics teachers pre-selected for a four-year teacher-training course. An interview protocol was used to facilitate the elicitation and collection of student teachers' understandings of algebraic sign convention in vector-kinematics. Student interviews were carried out on an individual basis and were based on three demonstrations using common material and simple apparatuses that students are likely to have encountered in everyday life. The first was 'a ball rolled on a smooth surface', the second was 'a ball thrown up and returning to its original position' and the third 'a ball was rolled up an inclined plane and returned to its original position'. In all these situations the students were required to discuss the concepts of displacement, velocity and acceleration and the application of positive (+) and negative (–) algebraic sign convention to these concepts in the context of understanding vector-kinematics in mechanics. The transcripts that originated from different individual interviews made up the data to be analysed. The data was then analysed phenomenographically. Initially, the process involved looking for comprehensive structures of use and explanation of algebraic signs in all the data and, then identifying distinct ways of conceptualising positive (+) and negative (–) algebraic sign convention in vector-kinematics. Conceptualisations and not students were being sampled. There was no attempt to fit the data into predetermined categories. The categories of conceptions of algebraic sign that were drawn from the interview data were obtained from a lengthy process of iteration. To ensure reliability of data, the process of iteration was repeated, twice, with a month gap. To ensure validity of data, the 'cut-out' slips of interview data was re-worked together with a physicist with experience in phenomenography, ensuring that the categories of conceptions were consistent to the ones obtained by the researcher. The final result is expressed as an outcome space that is defined as "the pool of meaning, a complex of categories of description comprising distinct groupings of aspects of the phenomenon and the relationships between them" (Marton & Booth, 1997, p. 125).
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Results of study Table 1 provides a general description of categories of conceptions with examples to elucidate how student teachers assigned algebraic signs to vector-kinematical concepts of displacement, velocity and acceleration. Table 2 provides the outcome space ─ the qualitatively distinct and different ways of understanding algebraic sign convention in vector-kinematics. The conceptions captured are all possible ways of understanding algebraic sign conventions in vector-kinematics; some are conceptually incorrect in terms of accepted understanding by the scientific community. Table 1: Categories of conception for positive (+) and negative (–) algebraic signs applied to displacement, velocity and acceleration
A.1 Algebraic signs are not applied to vector-kinematical concepts of displacement, velocity and acceleration. In this case definitions of concepts suffice.
B.1 Algebraic signs are used as positive and negative integer numbers represented on a number line as magnitude (in Mathematics). B.2 Algebraic signs are used to denote magnitude of speed (v) as 'how fast or how slow' e.g. v = -5 means slow or v = +5 object going fast.
C.1 Algebraic signs are used for velocity (v) and acceleration (a) to denote changing magnitude, e.g. v = +5 means the object is going faster or v = -5 means the object is slowing down.
D.1 Algebraic signs for acceleration (a) applied as direction and magnitude, e.g. a = -5 means that the object moves left and is slowing down.
E.1 Algebraic sign for direction of gravitational acceleration is negative only but direction of acceleration for straight-line motion can have both positive and negative signs. E.2 Algebraic signs applied as directions for displacement using the number-line concept with zero as the origin, e.g. left of the origin is (-) and to the right is (+). E.3 Algebraic signs applied as directions to vector-kinematical concepts without reference to zero as an origin, e.g. (+) is motion to the right and (-) motion to the left.
Table 2 provides the outcome space (the qualitatively different ways of understanding) of algebraic signs in one-dimensional vector-kinematics. Table 2: Outcome space for algebraic signs in one-dimensional vector-kinematics
Conception Outcome space for algebraic sign convention in vector-kinematics A Algebraic signs are not applied in vector-kinematics B Algebraic signs are applied as magnitude only C Algebraic signs are applied as changing magnitude D Algebraic signs are applied as both magnitude and direction E Algebraic signs are applied as directions
The hierarchical arrangement in Table 2 of the different ways of understanding algebraic signs in one-dimensional vector-kinematics provides a basis for analysing conceptions about how learning of algebraic sign convention occurs at first-year physics level. The categories of conceptions allow for the differentiation among ways of conceptualising a particular phenomenon, is this case, algebraic sign convention. Category A reflects a narrow understanding of algebraic signs in vector-kinematics whereas Categories B-D reflects mixed understandings and Category E reflects the correct scientific conception. In this study the outcome space (see Table 2) for the phenomenon of positive (+) and negative (–) algebraic signs for one-dimensional kinematical motion, provides five qualitatively different ways in which students assigned algebraic signs to vector-kinematical phenomena. This constitutes the main outcome of the research. The next section discusses in detail examples supporting these categories of conceptions.
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Examples and analysis of interviews supporting the categories of conceptions The discussion to follow examines in detail the different ways students understand algebraic sign conventions in vector-kinematics as evident in Table 1. Each category of conception is supported by sample evidence obtained from student interviews. The main purpose of matching interview extracts with categories of conceptions was to provide strong evidence linking fieldwork, that is, student teachers' conceptions from interviews of the phenomenon of algebraic sign convention, with the invention of the outcome space (Table 2). The outcome space is the second-order conceptions derived from the phenomenographic iteration process of the phenomenon of algebraic sign convention as interpreted by the researcher. A.1. Algebraic signs are not applied to vector-kinematical concepts of displacement and velocity. The student applies the definition of displacement and since the answer obtained is correct, the student does not use algebraic signs.
Interviewer: What algebraic sign would you use for displacement? Student: No sign. Because it is 5 m from starting point. It does not matter whether
it's up or down. When the object returns, the displacement is zero. Interviewer: Why zero? Student: Because from the definition, if the object returns to its starting position,
displacement is zero. The concept of displacement is possibly treated synonymously with the concept of distance. Since distance is always positive in magnitude and includes just the length and not direction, displacement is also assumed by the student to be positive and therefore cannot have a negative sign although for one-dimensional motion we do assign a negative sign for direction.
Interviewer: What algebraic signs would you use for speed and velocity? Student: No signs also. Speed is just a number and velocity is speed plus direction
that can be a compass like North. To this student, the concepts of speed, displacement, and distance are seen as inherently positive, as magnitude with a positive quantity while velocity is understood correctly as having both magnitude and direction hence a vector. The difficulty the student faces is translating the cardinal compass directions into signs, for example, representing North as (+). The student also indicates that the direction of acceleration is not represented by a sign as evident in the following extract.
Interviewer: What does the sign for acceleration mean? Student: We don't have to use sign convention for acceleration. You don't
necessarily have to use it for acceleration because it's not an actual vector thing for direction or something.
In this category, there is little evidence to show that students understand the significance of symbolic representations in physics. Thus, a shift towards a deeper conceptual understanding of algebraic signs to be interpreted as directions in one-dimensional vector-kinematical motion is necessary. B.1 Algebraic signs used as positive and negative integer numbers on a number-line as representing magnitude of vector–kinematical concepts The notion of negative numbers has confused mathematicians for centuries (Fischbein, 1994). The understanding of negative numbers now has a theoretical mathematical basis and is presented with different understandings at different levels of education. The everyday understandings of (+) and (–) signs may start from very early childhood but in South Africa,
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the formal conceptual development of positive integers (+1, +2 etc.) start in primary schools in Grade 3 (9-10-year-olds) and negative integers (–1, –2 etc.) in Grade 7 (12-13-year-olds) where a number-line extending in both right and left directions with positive and negative integers are taught in Mathematics. The number-line is taught as a mathematical representation of magnitude of numbers and is reinforced from Grades 3-9 (9-15-year-olds), a period of approximately six years.
Interviewer: Where did you use + and – signs? Student: Mainly in Mathematics, the numbers where the other side of the number
line is less than zero. In the number-line system, the numbers and their signs are examined from a theoretical perspective as defined in Mathematics. Conventionally, the number +10 is drawn on the right of the number line implying a positive number. The –10 implies a negative number and smaller in magnitude. Integers have magnitude only and no direction (scalars) but in one-dimensional vectors, the signs of the numbers (–1 or + 1) are assigned directions and the number, a positive magnitude. In calculating displacement in one-dimensional vector-kinematical problems, students commonly use scalar subtraction rather than vector addition to determine the displacement of an object. While we are familiar that in one dimension, scalar subtraction has the same result as vector addition, students lack the conceptual understanding that calculations with scalar quantities use the operations of ordinary arithmetic, but vector addition requires a different set of operations and, in particular, that subtraction of vectors means to add vectors considering both their magnitude and direction. B.2 Algebraic signs are used to denote magnitude for speed as magnitude of 'how fast or how slow'. For speed, the student interprets algebraic signs as magnitude (how fast or slow) rather than as a direction. Thus a value of +10 m.s-1 simply means a positive magnitude (related to size or quantity) that the object is moving with a certain speed. The (–) sign is interpreted incorrectly as 'slow' whereas one should compare different values of speed. For example, –5 m.s-1 is incorrectly interpreted as the object moving slowly whereas one should compare the speed of 10 m.s-1 to 5 m.s-1 .
Interviewer: Would you use signs for speed and velocity? Student: It might confuse them if I give them for speed – using direction they
would think that speed also has direction. They would be confused because if I am using sign convention for speed, I will be using plus (+) for ... how fast or how slow minus (–)
C.1 Algebraic signs for velocity and acceleration are used as changing magnitude.
Interviewer: What does the sign for acceleration mean? Student: We can use it as negative or deceleration. If the answer came out to be
negative then that is deceleration, it's slower in that sense. This category implies that 'going faster' is assigned a positive (+) sign and 'going slower' a negative (–) sign. The sign for acceleration is understood by the student as obtained from the changing magnitude of the velocity. Thus the sign of acceleration is often incorrectly interpreted as follows; a positive sign if the object is going faster or negative sign if the object is going slower. This seems to be a common interpretation by students and this notion is sometimes applied to both the concepts of velocity and acceleration; hence the vector-kinematical concepts are not clearly differentiated.
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Interviewer: What about the acceleration of the ball as it is going up? Student X: If they did not want to say acceleration or deceleration they can use signs
but they must state the signs. There was a change in velocity, initial velocity starts from zero, and then it increases, once it stops, the final velocity will be zero again.
Student X: The acceleration of the ball is decreasing as it is going up, because here it becomes zero, it is decreasing.
Student X: Coming back, it is increasing because its speed is increasing. Although there are some elements of correct thinking by the student in that acceleration is zero at maximum height and a change in velocity occurs, however acceleration is NOT decreasing but constant and it is the velocity that decreases uniformly as the object goes upwards (see category D.1. for in-depth discussion). D.1. Algebraic signs for velocity and acceleration assigned as direction and / or magnitude For velocity, the student correctly states that the algebraic sign is assigned for direction.
Interviewer: What does the sign for velocity mean? Student: It might confuse them if I give them the velocity using sign convention
for direction. But for velocity because of its direction I will be using sign convention there. If I am going to use negative or positive, ok, this is for direction.
Interviewer: What about signs for acceleration? Student Z: Its acceleration is directed upwards, so it's plus, when it comes to a
certain point there is no acceleration, it will be zero, and when it is coming down, the acceleration is also a plus, the object is going faster.
The student incorrectly states that the direction of acceleration is upwards for a falling object but allocates a correct (+) sign for upward motion. The student also incorrectly says that at maximum height, the acceleration (g) is zero instead of g = 9,8 being directed downwards. The velocity and NOT acceleration is zero. For upward motion, the sign for acceleration refers to the direction of motion but for downward motion, the (+) sign for acceleration is chosen to mean a change in magnitude. Hence, the algebraic sign for acceleration for falling bodies is assigned both as direction and as magnitude. Of significance, students have difficulty in the concept of acceleration as the rate of change in velocity especially in understanding the motion of falling bodies. At maximum height, the acceleration g = Δv/ Δt = (0 - 9,8)/1s = - 9,8. Here the magnitude of acceleration is 9,8 and (–) sign implies acceleration directed downwards (where initially displacement upwards was selected as (+)). E. 1 Algebraic sign for gravitational acceleration cannot be positive but acceleration for straight-line motion can be. This category implies that students assign only a negative sign for acceleration due to a falling body in its downward motion. Thus the notion of an arbitrary allocation of algebraic signs is not fully comprehended.
Interviewer: Can the acceleration of a body thrown up be positive? Student: No, gravitational acceleration can't be positive but normal acceleration
can when you are talking about straight-line motion.
The possibility that a positive sign could be assigned for acceleration for downward motion as well may not be reinforced in school physics – possibly because learners are not explicitly taught about the arbitrary choice of sign convention.
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E.2 Use algebraic signs of the number line as directions for displacement. The mathematical drawing of a number line, a positive (+) integer to the right of the zero origin and a negative (–) integer to the left is often interpreted incorrectly as directions for displacement. This notion is different from B.1. The zero origin here is explicitly specified and a single integer is incorrectly used to specify the displacement instead of the position of the object. This is an incorrect conception of interpreting signs for displacement as the definition of displacement is the change in position and the sign resulting from this change gives the direction of the object and hence a sign for displacement. Arons (1997) states that the concepts of position and displacement must be first firmly established before more difficult vector concepts are introduced. For example, the integers +5 and +10 are position points on a number line. The change in position = (+ 10) – (+5) = +5 indicates motion to the right if the + sign is arbitrarily chosen as motion to the right. If the change in position = (+5) – (+10) = –5, the displacement is to the left. E.3 Algebraic signs applied as directions to vector-kinematical concepts This category corresponds to the norm in physics of assigning algebraic signs to vector-kinematical concepts of displacement, velocity and acceleration.
Interviewer: Where did you use + and – signs? Student: In science, it had to do with the direction…a bit confusing.
The student understands that for one-dimensional motion, signs are used to express direction of motion of object. From Grade 11 (16-17-years-old), students are taught for the first time, in school physics, that a vector has magnitude and direction and that signs (+) and (–) can be interpreted as directions, for example, a positive sign (+) means the object is moving to the right and a negative sign (–) means its moving to the left and the number, its magnitude. The conceptual shift from the early understanding of signs as only as positive and negative 'mathematical' numbers with magnitude only in Grade 9 and 10 mathematics was possibly confusing to students at the first introduction of signs as representing directions in Grade 11.
Interviewer: Would you use any signs for acceleration? Student: There will be a sign because the ball is moving in an opposite direction.
Acceleration will be negative because it will be moving downwards. Interviewer: Why? How come? Student: Gravity acted on it, acceleration going up is positive so going down will
be negative. For this student, the sign for acceleration is taken to be the same as the sign allocated for the direction in which the object is moving. For example, if the object is moving up, it is allocated a (+) sign for the direction of displacement and a (+) sign for the direction of acceleration as well. The sign for acceleration as representing its direction is often incorrectly interpreted together with the initial sign chosen for displacement. If initially, the upward displacement of the object is chosen as (+) then the sign for acceleration is (–) implying that the acceleration is directed downwards. What needs to be understood conceptually is the acceleration of a falling body is always directed downwards irrespective whether the object is moving up or down. Hence, if (–) sign is chosen for upward displacement then (+) sign representing the downward direction of acceleration should be chosen and understood. Discussion Phenomenographic analysis of this study (see Table 2) has revealed five core understandings of algebraic sign convention that physics students hold. Two broad categories of algebraic sign convention in the form of integer signs can be discerned from Table 1, algebraic signs are conceptualised in terms of the number-line system used in Mathematics, namely, as algebraic
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signs (+ and –) is used in the context of understanding integers as positive and negative numbers as having magnitude (category B1) and as having directions in Physics (category E3). The study reveals that these student-teachers hold a mix of algebraic sign conventions that are inherently linked to their vector-kinematical conceptual understanding. In particular, the concepts of scalars and vectors and their associated signs are not clearly differentiated. Scalars are physical quantities having magnitude only whereas vectors have both magnitude and direction. The vector-kinematical concepts of displacement, velocity and acceleration are vectors and are connected by their definition in physics; hence the algebraic signs, used for direction of motion, have consistent application in their meaning when solving vector-kinematical problems. In physics, the quantities are defined as follows: displacement is the change in position; velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. To illustrate the confusion in the interpretation of signs for concepts of velocity and acceleration, consider the example of a car speeding up from 5 m.s-1 to 10 m.s-1 in 5 s. Let us arbitrarily choose the direction of displacement to the right as (–), then the direction of velocity to the right is also (–). The acceleration is a = Δv/ Δt = [(-10) – (-5)]/5s = –1 m.s-2. This is interpreted as the magnitude of acceleration of 1 m.s-2 and the (–) sign as the direction of acceleration to the right. Students often interpret the (–) sign of acceleration together with its magnitude in most cases to be 'slowing down' without examining their choice of sign convention. If the car was slowing down from 10 m.s-1 to 5 m.s-1 in 5s, and if displacement to right is (+) sign, then a = [(+5) – (+10)]/5s = –1 m.s-2. In this case the direction of acceleration is to the left. Hence the algebraic sign for displacement, velocity and acceleration is to be interpreted as a direction only. Some physic textbooks discuss different ways of interpreting signs that may cause some confusion to students. Numerous incorrect preconceptions, particularly with regard to the role of the component of the velocity vector have already been observed among physics students (Aquirre, 1988). The special case of allocating positive and negative signs for vectors in one-dimensional motion confuses them with scalars, which can often have two signs (e.g., charge, potential energy, total energy and time). Bauman (1992) raises an important point in this area of research that we often do not explain to the novice that we freely shift between vectors and their components. Many students, as well as school teachers, are not, however, aware of this important distinction, possibly because only one-dimensional vector-kinematical motion is taught for Grade 11 and 12 learners in South African schools. Warren (1997) believes that most difficulties are caused by the introduction of only a one-dimension analysis of motion in physics. He suggests that it is conceptually better to start with motion in a plane, i.e. in two dimensions and in this way the essential nature of a vector, having magnitude and direction, is explained at the beginning of vector-kinematics. When students study tertiary physics, algebraic signs must now be interpreted as scalars in one, two and three-dimensional Cartesian co-ordinate system so that complex physics problems can be solved efficiently. Within the context of physics, one-dimensional motion can be interpreted as a special case of motion where algebraic signs understood to be scalar is now interpreted as direction for displacement, velocity and acceleration The notation of vector symbols must also be emphasised in the study of vector-kinematics. Students should be encouraged to use arrows, boldface type, or underline the vector symbols. If they do not distinguish between scalar and vector quantities in the notation, they will probably not make the distinction in their thinking either (Arons, 1997). In the case of velocity, some students interpreted the sign for velocity as a change in magnitude rather than in direction (one-dimensional motion only). This perception could possibly stem from the number line system where positive numbers to the right of zero are increasing in
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magnitude and negative numbers to the left of zero are decreasing in magnitude or from the confusion of acceleration and its effects on velocity. Another interpretation is that the concept of addition uses a plus sign that increases the magnitude of a number, while a subtraction uses a minus sign that decreases the magnitude of a number. Thus the sign of velocity as positive is experienced as evidenced by interviews as motion 'faster' and velocity negative as motion 'slower'. Thus we may infer that velocity may not be understood as an instantaneous quantity. The interpretation of 'instantaneous velocity' as a number referring to a single instance is a conceptual 'hurdle' for many students (Rosenquist & McDermott, 1987; Trowbridge & McDermott, 1980). Velocity is not correctly interpreted as how fast the object is moving but rather as 'faster' or 'slower', which is acceleration. In this study, some students, confusing the concept of velocity with acceleration, presented the incorrect notion that velocity meant increase in speed. For falling bodies and incline planes, some students incorrectly stated that as the ball fell, acceleration would increase and, as the ball was falling, it was moving faster and therefore, the acceleration was increasing. Some students experienced an increase in speed as directly proportional to an increase in acceleration. These students thought that the ball when thrown upwards already had acceleration upwards when it started from rest and it continued through top speed. The students did not realise that a constant force acted on the object and, hence, a constant acceleration was produced in a downward direction even though the ball was up going and coming down. Some students, who knew that acceleration due to gravity (g) always acts downwards, incorrectly insisted that the sign of g is always negative. Students in this study do not seem to be aware that the choice of a coordinate axis is itself arbitrary. The analysis of motion of a point can be in any arbitrary direction. We can always consider the motion along definite straight lines, that is, along the coordinate axis. Upward direction can be positive or negative and the choice of coordinate axes dictated by the conditions of the problem and that g can be allocated a positive sign if displacement downward was originally taken as positive. For falling bodies, students in this study said that if a particle's velocity is zero then its acceleration is also zero at maximum height. A possible reason for student teachers' responses is that they memorize rules and definitions without discrimination and often retrieve and remember knowledge in fragments without considering the context (Rief & Allen, 1987). This study, and other research, confirms that students bring from their schooling experience to the formal study of physics, naïve and mixed understandings of the common concepts associated with motion in the study of vector-kinematics. The concepts of position, displacement, speed, velocity and acceleration and their associated algebraic sign convention are not distinguished clearly, not applied to real life motion and often exist as memorised definitions used mainly for tests and examinations and thus may soon be forgotten. Implications for teaching This study suggests that algebraic signs as a sign convention and concepts in vector-kinematics which are central to students' scientific understanding in a wide range of topics should be analysed more in-depth and be given an appropriate amount of teaching time. This can be achieved by first exploring student teachers' alternative conceptions of algebraic sign convention in vector-kinematics, and then, by engaging student teachers' perceptions, using the outcome space for sign convention in vector-kinematics invented in this study (Table 2). Furthermore, the introduction of motion in two and three dimensions first, and then simplifying for one-dimension of motion would provide a better and deeper understanding of vector-kinematics. Conclusion The results of the study, for sign convention in vector-kinematics, conclude that there are five qualitatively different ways of conceptualising sign convention in vector-kinematics (Table 2). The results of the study also concludes, for everyday understandings of algebraic sign
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convention, sign convention in the form of integer signs are experienced in terms of the number-line system used in Mathematics, and, in Science, to indicate direction of motion of an object (Table 1). The study also confirms that students hold a mix of algebraic sign conventions, which is compounded with poor understanding of vector-kinematics concepts. Furthermore, the study reveals that students, on their own, fail to formalise a consistent idea of algebraic sign convention in vector-kinematics. Acknowledgements This paper is based on work supported by the National Research Foundation towards a PhD obtained from the University of Western-Cape (UWC). I acknowledge the assistance of Prof. Cedric Linder in this regard. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Research Foundation or the UWC. References Allie, S., & Buffler, A. (1998). A course in tools and procedures for Physics 1. American Journal of
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