Introductory physics students' conceptions of algebraic signs used in kinematics problem solving

(1)

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

(2)

(3)

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.

(4)

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.

(5)

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.

(6)

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.

(8)

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.

(9)

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

(10)

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.

(11)

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.

(12)

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

(13)

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

(14)

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,

Introductory physics students' conceptions of algebraic signs used in kinematics problem solving