Translation of P = kT into a pictorial external representation by high school seniors
Igor
Matijašević
*,
Jasminka N.
Korolija
and
Ljuba M.
Mandić
University of Belgrade-Faculty of Chemistry, Studentski trg 12–16, 11 000 Belgrade, Republic of Serbia. E-mail: igormat@chem.bg.ac.rs
First published on 22nd March 2016
Abstract
This paper describes the results achieved by high school seniors on an item which involves translation of the equation P = kT into a corresponding pictorial external representation. The majority of students (the classes of 2011, 2012 and 2013) did not give the correct answer to the multiple choice part of the translation item. They chose pictorial representations of the other gas laws (P = k/V, or V = kT) instead. Failure to choose the correct answer was surprising considering that the symbol for volume was absent which should have been the key clue. Through the analysis of students' explanations (the classes of 2011 and 2012) and interviews (the class of 2013) we considered the reasoning applied by students who chose the correct answer or distractors for the multiple choice part of the item. Among the students who answered correctly there were explanations which contained misconceptions. Several factors that lead to the unsuccessful translation between external representations have been discovered. Students interpreted the change in one quantity based on the notation for the change in another one because of deep rooted cognitive schemas about changing two quantities (volume and pressure, pressure and temperature, temperature and volume), without consideration that for such changes to be valid for gases all three quantities need to be considered for a certain amount of substance. Those cognitive schemas interfered with mathematical reasoning, i.e. students possessed limited understanding of the equations.
Introduction
The purpose of our research has been to better understand students' abilities to translate from one external representation (ER) to another, especially from equations (formulae)† to the corresponding pictorial external representations (ERs). This article presents the results of a pilot and exploratory study on high school students' abilities to translate between the equation P = kT and a pictorial ER. The terms external representation and translation between external representations (TBER) were defined and then research questions are presented.“External representations” (i.e. physical notations) are “knowledge and structure in the environment” while “internal representations” (i.e. interpretations) are “knowledge and structure in memory” (Zang, 1997, p. 180; Izsák, 2004; Kozma and Russell, 2005; Rapp and Kurby, 2008). It is thought that ERs guide, constrain and even determine cognitive behaviour and the way the mind functions (Zang, 1997). As Eisner (1997, p. 349) pointed out: “It is an old idea (…) that the forms we use to represent what we think (…) have an impact on how we think and what we can think about”. Kirsh (2010) has emphasized that ERs are significant for human cognition because they “reduce the overall cognitive cost of sense making” and “change the domain and range of cognition” (p. 442). However, it's necessary to keep in mind that the internal and external cognitive resources, i.e. representations, are “coordinative partners” and that a study of cognition requires examining both social interactions among participants and their interactions with material structures (Xu and Clarke, 2012, p. 508). The importance of social interactions and representations on cognition is recognised from the evolutionary point of view. Some researchers (e.g.Mercer, 2013) consider that collective intellectual activities have a significant influence on the development of individual cognition, which is accomplished through language and other forms of ERs. As a result of connecting individual cognitive resources, powerful problem-solving tools are created, and, according to Mercer (2013, p. 151), “it is likely to be this, rather than individualized competition, that has ensured the dominance of our species.”
ERs lie along a continuum depending on their abstractness (Pozzer and Roth, 2003). For example, equations are more abstract than pictorial ERs. According to Schnotz (2002) equations and pictures belong to the different classes of ERs: descriptive and depictive, respectively. “Descriptions are more powerful in representing different kinds of subject matter”, while “depictions are better suited to draw inferences” (Schnotz and Bannert, 2003, p. 142). For example, pictorial ERs of the P = kT gas law directly provide (some) information on the meaning of the symbol k, but the equation P = kT does not.
The concept of ERs is one of the cornerstones of the scientific practice (McGinn and Roth, 1999; Strickland et al., 2010) and, from the history of science, it is evident that the inventions of the ERs are of fundamental importance for advances of science (Kozma et al., 2000; diSessa, 2004). For example, in chemistry, Kozma and Russell (2005) argued that there had always been a strong relationship between chemists' understanding of chemical phenomena and the ERs that they use to represent them. This is because the cores of scientific knowledge are concepts, which are articulated across ERs (Lemke, 1998). Therefore, learning about new concepts cannot be separated from learning about both how to represent these concepts as well as what ERs used mean in the world (Hubber et al., 2010; Waldrip et al., 2010). This implies that all attempts by learners to understand concepts in science entail representational work (Hubber et al., 2010; Waldrip et al., 2010). This is why using ERs has been referred to as the (meta)representational competence (diSessa et al., 1991; Kozma and Russell, 1997; diSessa, 2004, Michalchik et al., 2008). Better representational abilities correspond to better scientific understanding (Stieff, 2011). In addition, according to Gilbert (2005) working with ERs can have distinct metacognitive character, which he denoted as metavisualization skills. Generally speaking, in the past three decades research studies showed that representational competence of middle/high school students as well as university students is generally low. Also, researchers noted that students' representational competence with one ER does not imply competence with different but related ERs (Kohl and Finkelstein, 2005; Madden et al., 2011).
The concept of TBER is an important aspect of the (meta)representational competence. It is at the core of medium and higher levels of abilities which constitute representational competence (Kozma and Russell, 2005; Michalchik et al., 2008). Lemke (1998) has noticed that to understand and use a scientific concept, someone has to be able to translate back and forth among ERs of that concept. For Grove et al. (2012, p. 844) students' abilities to translate are necessary in order for them to “truly grasp many scientific ideas”. In addition, ability to re-represent a problem, i.e. to translate from the ER in which the problem is given to another ER is an important aspect of problem-solving (Lesh et al., 1987; Mayer, 2003). Thus, problem-solving involves abilities to coordinate multiple representations, and according to Ainsworth (1999, 2006) coordination is based on students' translation abilities. Due to its high complexity studying construct representational competence is difficult. However, since the operational definition for TBER has been relatively well established (see below), TBER, according to the literature, has been commonly used for measuring the representational competence (see, e.g.Madden et al., 2011; Nitz et al., 2014; Taskin et al., 2015; Stieff et al., 2016).
The concept of TBER is not new. Years ago, Bloom et al. (1956) defined translation as one of three types of comprehension behaviour “which means that an individual can put a communication … into another form of communication” (p. 89). In a revision of Bloom's taxonomy translation has a similar meaning to the conversion of information from one representational form to another (Anderson et al., 2001). This “conversion” in previous research studies in math and science education was operationalized with items in which students were asked to recognise the correct ER, among several offered, for the given ER (e.g.Gagatsis and Shiakalli, 2004; Kohl and Finkelstein, 2005), as well as to construct the correct ER for the given ER (e.g.Keig and Rubba, 1993; Madden et al., 2011). In this article, the given ER will be called the source representation, and the recognised or constructed ER the target representation (cf.Keig and Rubba, 1993; Gagatsis and Shiakalli, 2004).
Many questions about TBER can be asked as a subject of didactical research. We categorize them into three groups. The first group consists of the questions about students' abilities to execute translations. The second group of questions is related to the instruction (educational environment) which can promote this process. Finally, the third group consists of questions about the nature of this cognitive process.
Regarding the first group of questions, there are studies in math and science education which support the opinion that TBER is difficult both for young and older students (Ainsworth, 1999; for example: Clement et al., 1981; Wollman, 1983; Keig and Rubba, 1993; Kozma and Russell, 1997; Hitt, 1998; Zazkis et al., 2003; Gagatsis and Shiakalli, 2004; Kohl and Finkelstein, 2005; Kurt and Cakiroglu, 2009; Hadfield and Wieman, 2010; Madden et al., 2011). However, previous research studies covered a relatively small number of topics. In the area of our research on gas laws there are only a few studies. Madden et al. (2011) showed poor scores of university students on translation tasks. They concluded that students of lower representational competence tended to view the equation P = kT (n, V = const.) as not being meaningfully connected to other ERs (graphical and pictorial). On the contrary, Becker and Towns (2012) showed that the majority of their sample of university students could give an answer to the next request: “Write an expression [by using differentials] for how the pressure of the container below [given picture] would change if the container were heated while volume remains constant” (p. 213). Finally, although Levy and Wilensky (2009) were satisfied with the achievement of high school students to translate from a scatter chart to the corresponding equations for gas laws (success was 52% for P = k/V and 69% for P = kT), a substantial number of students were still unsuccessful.
Research on the relationship between instruction and translation showed that working with multiple ERs in a software environment facilitated the development of translation abilities (Wu et al., 2001; Stieff, 2011; Nichols et al., 2013; McCollum et al., 2014). The research by Hwang and Roth (2011) showed an extremely important role of TBER in the realisation of communications between a lecturer and students during classes. In addition to the importance of TBER in the “one way” communication Becker et al. (2015) also pointed out the importance of TBER during a whole class discussion, especially in connecting ERs of different phenomenological levels: macro, submicro and symbolic. They also emphasised the importance of the teacher in promoting TBER in such circumstances. It is not surprising, therefore, that the protocols for the evaluation of the quality of using ERs in the teaching of chemistry (see e.g.Philipp et al., 2014) put emphasis on TBER. In the domain of textbooks Dávila and Talanquer (2010) showed that the three top-selling college general chemistry textbooks in the US allocate a small percentage of their items to questions and problems that require students to TBER. In addition, Kumi et al. (2013) showed that better spatial organisation of 2D and 3D ERs of organic molecules in organic chemistry textbooks would enhance translation between them in order to facilitate the understanding of 2D chemical formulae. Finally, it should be pointed out that some didacticians (e.g.Schönborn and Bögeholz, 2009) give an important role to TBER, together with the concept “knowledge transfer”, when creating curriculums.
When combining the first and the second group of questions we are in agreement with the views of some didacticians (see Fredlund et al., 2014) that the difficulties students have in solving ERs are, at least partially, due to the so called process of “rationalization of a representation” (creation in a sense of more abstract ERs) as a way to create communication tools with the potential for faster and more comprehensive communication. This creates significant interpretation problems for beginners due to the use of more abstract notations as constitutive elements of such ERs. Hence, for successful teaching, Fredlund et al. (2014) advocate necessary “unpacking” (explicit explanations, translation into less abstract ERs). Although the authors did not mention TBER in the context of the “unpacking” process, from the description of the process it is evident that TBER was its constitutive element.
Regarding the nature of translation, there is agreement that conversion from one ER of a concept/situation to another is a type of cognitive process (Keig and Rubba, 1993; Kozma and Russell, 1997; Anderson et al., 2000, Gagatsis and Shiakalli, 2004). TBER is an indicator of deeper achievement in comparison to the achievement resulting from recall or execution of algorithms (Keig and Rubba, 1993).
There are several theoretical considerations about the nature of TBER. The educational psychology literature suggests that TBER happens at two levels: at conceptual sense making of connections (reasoning about how different ERs show the same concept, and how such ERs still give somehow different information) and at perceptual fluency (quickly and effortlessly making connections among ERs, as the result of nonexplicit reasoning), and that both levels play a role in students' learning (see Rau, 2015). These two levels should not be associated with Ainsworth's (1999) assertion that translation can occur through direct mapping between symbols (surface elements, notations which constitute ERs) or through domain understanding. Another account is related to the mechanism of TBER. One level is about the abilities to interpret the source and target ERs individually, and the other is related to the integration of those interpretations (see Seufert, 2003). The interpretation of ERs involves, first of all, the recognition of notations (for example, if pressure is represented with weights, students need to recognise that drawn objects are weights) and then link recognised notations with the knowledge about concepts and concepts' ERs (what is pressure and how drawn weights on a movable piston actually represent pressure of the gas below the piston). The number of recognised notations as well as the extent of linkage to the knowledge about concepts and concepts' ERs will provide smaller or bigger cognitive activity. This means that a dominant theoretical framework for the explanation of students' achievement in translation tasks lies in information processing theory (see Wu et al. 2001). As can be expected from this theory knowledge about concepts and their ERs has a major influence on success (Keig and Rubba, 1993; Kozma and Russell, 1997; Superfine et al., 2009). For example, Superfine et al. (2009) showed that translation as a cognitive process involves conglomeration of knowledge elements about concepts and their ERs, and that complex interaction between them determines the output in translation tasks. In addition, influence of the implicit knowledge is also important. For example, research studies in math education (Kurt and Cakiroglu, 2009; Kop et al., 2015) showed that heuristic strategies play an important role in translation between graphical and algebraic ERs of functions, while Becker and Towns (2012) showed that translation between a pictorial ER of the P = kT gas law and the corresponding equations (see above) is strongly supported by knowledge elements about the structure of equations (formulae), named symbolic forms (Sherin, 2001). Finally, Wollman (1983) in math education, and Padalkar and Hegarty (2015) in chemistry education, showed that students' self-reflection and teachers' feedback as an initiator of students' self-reflection have an impact on the achievement of translation tasks.
Research questions
At the beginning we stated the broad purpose of our research (to better understand students' abilities to translate from one ER to another) and the specific aim of this article (translation from equations (formulae) to the corresponding pictorial ER). Three research questions guide our methodology and interpretation of results:(i) Are high school seniors successful in translation between the equation P = kT and pictorial ER?
(ii) How do students explain choosing the correct target representation?
(iii) How do students explain picking the incorrect pictures which they thought are representing source representation?
Pertinent to the second question, we are interested in seeing if the students who chose the correct picture gave correct explanations, and what was the nature of those explanations. Namely, we assume that the following reasoning (“strategies”) could be applied by students:
 | ||
| Fig. 1 The instrument used. Correct answers: statements 2 and 4 for item 1 and Picture 2 for item 2. | ||
The first strategy is mere reproduction, with minor consideration of the structures of source and target representations (i.e. that the volume is constant). However, the second and the third strategies are based on those considerations. It is important to note that it is expected from students to at least tacitly know that P, V and T cannot have negative values.‡ We are more interested in seeing the extent to which Strategy c was applied by students. By applying this strategy we interpret it as an indicator of one aspect of students' understanding of the equation P = kT as a mathematical object. Namely, equations (ERs of mathematical relationships among physical quantities) are inseparable components of many science topics, since they are used to better describe or explain scientific phenomena or knowledge (Park and Choi, 2013). It is not possible to learn chemistry and physics without learning equations. Although equations are very abstract in nature, they are powerful ways for communicating ideas. However, many high school and even university students can use equations mainly as a means to find numerical values of unknown variables by simple manipulation of equations. Thus, researchers argued that many students do not understand equations. It seems that this problem is the consequence of students' inability to see formulae as algebraic equations, i.e. they are unable to exploit important information from such mathematical objects and thereby contribute to their own reasoning. For example, Park and Choi (2013), who examine how well high school students understand the pH concept and the possibility of expressing this concept mathematically and graphically, found that only a small percentage of students explain the pH concept in light of dependent and independent variables, as constituents of algebraic equations. At the same time those students showed the greatest conceptual understanding.
Knowledge of how to exploit information from equations (formulae) was termed the symbolic forms (Sherin, 2001). Sherin (2001) argued that knowing how to use equations and knowing in what situation it is appropriate to use them is one aspect of understanding of equations. He underlined that there is another aspect, which is dominantly independent of content represented with the equations. He believes that “students learn to express a moderately large vocabulary of simple ideas in equations and to read these same ideas out of equations” (Sherin, 2001, p. 482). As mentioned above Sherin named the elements of this vocabulary as symbolic forms. This knowledge depends on the arrangement of symbols in equations, i.e. their particular arrangement expresses a meaning that could be understood and successful students can learn to understand what equations say in a fundamental sense. What does equation P = kT say in a fundamental sense without any knowledge about gas laws? The equation P = kT contains only one independent variable and one dependent variable, and the change in the independent variable leads to the change (directly proportional) in the dependent variable (so-called sole dependence symbolic forms). Letter k indicates that for the more comprehensive interpretation of the law it is necessary to include a constant. The equation does not reveal the meaning of the constant (so-called coefficient symbolic forms). We think that the abovementioned Strategy c could be seen in light of Sherin's symbolic forms: sole dependence and coefficient.
Sherin's theoretical frame has become a very useful tool for the interpretation of students' interactions with equations and concepts in areas such as mechanics, electrodynamics and thermodynamics (e.g.Meredith and Marrongelle, 2008; Nguyen and Rebello, 2011; Becker and Towns, 2012; Hu and Rebello, 2013; Kuo et al., 2013). In addition, it was shown recently (Schuchardt and Schunn, 2016) that Sherin's symbolic forms can be used as a theoretical basis for the creation of lessons targeted for developing qualitative conceptual understanding and quantitative problem solving.
An example of difficulties students have in utilizing information from the equations describing the gas laws can be seen in the research by Lin et al., (2000), Kautz et al. (2005a), and Leinonen et al. (2012).
The shared objective of these research studies is that students have to explain or predict gas behaviour. For example, Lin et al. (2000) demonstrated a few experiments about the behaviour of gases to the students and their teachers, and then asked them to make explanations and to give predictions about changes in variables. One well known experiment was “the chemical fountain experiment”. Although Boyle's gas law was at the heart of explanation for observations and predictions of changes in volume and pressure, many high school students in an advanced program, as well as their teachers, couldn't give correct explanations and predictions. As an explanation, one student said among other things: “Based on PV = nRT, when T is decreased, V should decrease too” (p. 237). Besides the fact that the student used an irrelevant variable (T), from our standpoint an even more serious problem is that his explanation is based on the relationship between two variables, but he used an equation that specifies a relationship among four variables. Lin et al. (2000) commented “although the subjects memorized the formula, they did not understand its meaning” (p. 237, our italic).
In a study by Kautz et al. (2005a), college students, who had already been taught algebra or calculus based introductory physics or even thermal physics, had a task to predict how would gas law variables change based on drawings which showed properties of gases. For example, a syringe that contains an ideal gas and has a frictionless piston of mass m is moved from an ice-water bath to a beaker of boiling water, where it comes to thermal equilibrium. Students were asked if the final pressure and volume of the gas were greater than, less than or equal to the initial pressure and volume, respectively (p. 1056). One student's explanation was: “P increases because P = nRT/V. So as T increases, P increases” (p. 1057). Besides the fact that the student used an irrelevant variable (P), the student ignored that the relationship between P and T is valid only if the amount of substance and volume remain unchanged, although, later in the interview, he correctly noted that the volume increased. Thus, the problem lies in the fact that the student quoted formula P = nRT/V, which implied that P, n, T and V were variables, but then in his response mentioned only two variables. This explanation was a typical example of reasoning. Consequently, Kautz et al. (2005a) argued that many students cannot properly relate “mathematical formalism to the real world” (p. 1058), and that “conceptual, reasoning, and mathematical difficulties [that students have in interpreting and applying the ideal gas law] often are intertwined and cannot be completely separated” (p. 1057, our italic).
Further elaboration of problems in reasoning (i.e. contradiction or inconsistency in explanations) of university students in a situation which included the behaviour of gases was given by Leinonen et al. (2012). They asked students to explain and predict what happened to the temperature of an ideal gas when adiabatic compression had been applied. Many students used gas laws, which alone cannot provide an explanation. One of them wrote: “The temperature of the gas increases because the increase of pressure inside the system causes the temperature to rise. PV/T = const” (p. 1173). These students inaccurately assumed that one of the quantities (volume) stayed constant (Leinonen et al., 2012) but their explanations (“PV/T = const”) implied that the volume was not constant. As Kautz et al. (2005a) argued this is another example of interaction of conceptual, reasoning, and mathematical difficulties.
In summary, these examples show problems in students' poor understanding of the equation PV = nRT. However, we want to emphasize that these three brief reviews do not mean that we seek to find if students used the equation PV = nRT to explain their answers. We are looking to find whether students considered quantities, which are not represented in the equation P = kT, as a potential indicator of students’ misunderstanding of the algebraic structure of the equation.
Methods, results and discussion
A combination of quantitative and qualitative methodologies was applied in three school years (2011–2013). Part I contains Methods, results and discussion for the 2011 and 2012 research years, while Part II contains Methods, results and discussion for the third year (2013) of research.Part I
In every school year students have, among other subjects, the following ones: chemistry, physics and mathematics. A brief review of physics, chemistry and mathematics curriculums for all four high school years is as follows. Freshmen learn general chemistry, sophomores inorganic chemistry, juniors organic chemistry and seniors biochemistry. Physics teaches kinetics to freshmen, fluids (including gases), thermodynamics and electrostatics to sophomores, acoustics, optics and electromagnetism to juniors and atomic and nuclear physics to seniors. In math freshmen and sophomores learn algebra (with the emphasis on algebraic functions), juniors mostly geometry and seniors integrals, differentials and probability. It is important to point out that the gas laws were not included in the chemistry curriculum, but only in the second year of physics (two years before this research took place). Also, the curriculum for seniors did not oblige teachers or text-book writers to review gas laws. In most cases physics and chemistry teachers teach their subjects to the same students in all four school years.
We did not collect information on how the students were learning about gas laws in the first two years of research. However, some information has been collected indirectly in the third year of research by asking the students during interviews about how the gas laws had been taught (see Part II). Taking into account the students' responses and the fact that most physics (as well as chemistry) teachers had not received (adequate) didactical training gave us a reason to believe that the dominant approach to teaching physics is based on rote memorisation, reproduction and algorithmic problem solving.
All three pictures can be classified as hybrid ERs (Gkitzia et al., 2011). The role of hybrid ERs in teaching/learning chemistry has not been sufficiently studied. However, their use in textbooks has been criticised primarily because they can possibly cause creation of alternative conceptions, especially about the particular structure of matter (see e.g.Harrison and Treagust, 2002; Taber, 2002). There have also been remarks about the usefulness of the hybrid ERs in developing representational competencies (e.g.Michalchik et al., 2008), as well as their recommended use for development of metavisualization abilities (Cheng and Gilbert, 2009). Another area without much data is the role of hybrid ERs in the evaluation of internal representations, despite the fact that research studies frequently use such representations to evaluate internal representations (in the area of gas properties see: Lin et al., 2000; Çalik and Ayas, 2005; Madden et al., 2011).
Since fixed response questions do not give insights into why certain answers were chosen (Johnstone and Ambusaidi, 2000) students were asked to justify their choices. In the revised Bloom's taxonomy, the explanation is, in addition to translation, another kind of cognitive process from the same category of cognitive processes, category “understand” (Anderson et al., 2001).
This instrument has been considered by several faculty members, among them two physics professors, three chemistry professors, and a few teaching assistants, as well as by most teachers of chemistry from the participating high schools. None of them had reproached for technical or didactical aspects of the instrument. The instrument has also been presented to the international science educator community (Matijašević et al., 2012). In addition, one of the professors mentioned, with more than ten years of experience teaching college general and inorganic chemistry, underwent the think-aloud protocol for the instrument in 2013. Her thought process about the ways to solve item 2 agreed with our expectations (see Strategies a–c in the Introduction).
In this paper some results of a more comprehensive research study are presented, which included 1228 high school students. Namely, 615 students were tested in 2011 and 613 in 2012. About one sixth of the students (207) had the test in the exact form as described in this paper (version A) and another sixth had the equation P = kT within the following sentence: “Dependence of pressure (P) of a certain amount of gas (n) on its absolute temperature with constant volume can be expressed as P = kT” (version B). The similar versions were given for other two gas laws, P = k/V and V = kT. Thus, students from all schools in every year were divided into six groups, and for each group one instrument was prepared. The results presented in Table 1 are based on 207 students' responses for item 2 of the instrument presented in Fig. 1, in the first two years of research (105 students in 2011 and 102 students in 2012).
In our educational system, after enrolment in high schools, students are grouped in such a way that 25–30 students make one “cluster”, which without significant changes in its composition stays together until completion of the high school. On average four such clusters were in each school. The distribution of all instruments for every cluster of students was as follows: the first student got, e.g. version A of one gas law, a student next to him/her received version B of the second gas law, third student, who sat behind them, got version A of the third gas law, next student version B of the first gas law and so on. This resulted in 15–20 students (out of 100–120) from every school having the version A of P = kT. Therefore, our sample of 207 students represented students from 10 different schools. The test time was limited to 15 minutes during regular chemistry classes.
The research has been performed in the same manner in two consecutive school years, for the evaluation of the reliability of items. Reliability, which is a measure of the stability of results in solving multiple choice questions, means that the same or similar group of test subjects achieves similar success, i.e. a similar percentage of chosen answers if the same multiple choice questions were used on more than one occasion (Johnstone and Ambusaidi, 2000, Sanger and Phelps, 2007). The biggest difference in results between the two years was <7% (Table 1), fulfilling our expectations. It is an acceptable variation, considering the number of participating schools and likely difference in academic levels. Both distractors (pictures 1 and 3) had been selected to a considerable extent (Table 1), which is the requirement for multiple choice questions to be considered suitable for testing (Schmidt and Beine, 1992).
In the school system of the Republic of Serbia access to the schools, particularly access to the students and realisation of research which includes students, is contingent on agreement of four parties involved: the school principal, teachers (one or more), school psychologist and/or pedagogue and the school parent council or parents of the students to be enrolled in the research. An established way to start a research study, which we followed, is a written request addressed to the school principal outlining the purpose (usually with a general description, such as testing of knowledge of certain topics from the school curriculum) and the research methodology (the testing of knowledge to be conducted anonymously) and how the results of the research will be used (it was stated that the data would be used for a doctoral dissertation material). Upon the approval, several days before presenting the instrument from Fig. 1, the students were informed, by their respective chemistry teachers that they would be given some problems to solve during the first part of their chemistry class. The students were also informed about the institution conducting the research and that the testing would be voluntary and anonymous. On the day of testing the primary author again delivered the above information to the students, emphasizing that the test results would not be used for their academic evaluation (school grades).
Results and discussion
Approximately one fifth of the students chose picture 2 and another fifth picture 3, significantly less than the number of students choosing Picture 1 (Table 1), i.e. most students performed the translation incorrectly. Based on the applied methodology of sampling of students and schools, and because results were stable, i.e. reliable, we are confident that similar results would have also been achieved for the entire Belgrade high school population (15 schools). Thus, it seems that the answer to the first research question is straightforward: the last year Belgrade high school students were not successful in translation between the equation P = kT and pictorial ER (however, see the section Final considerations).In order to answer the next two research questions, we are presenting an analysis of students' explanations. About 70% (71% in 2011 and 67% in 2012) of the students (145/207, Table 2) explained why they picked a certain picture.
Before going into detail about explanations given by the students, here is a short description of how we organized the results. Analysis of the students' explanations was divided into two groups:
(i) Analysis of students' explanations for choosing the correct picture (when picture 2 was chosen) and
(ii) Analysis of students' explanations for choosing the incorrect picture (when picture 1 or 3 was chosen).
For group (i), all explanations were analysed using the three strategies described in the introductory part of this paper. Also, examples of students' explanations were given to support our analysis.
As for group (ii), all explanations were considered depending on how the students had used the physical properties: pressure, volume and temperature (as explained with the third research question), to justify choosing either the first or the third picture. The results and the analysis were divided into two subgroups:
(iia) analysis of the students' explanations for choosing picture 1
(iib) analysis of the students' explanations for choosing picture 3.
Analysis of students' explanations for the correct selection of the picture (answer to the second research question)
As can be seen from Table 1, only 21 students in 2011 and 19 in 2012 chose the correct picture. The majority of these students (15 in 2011 and 14 in 2012) gave explanations (Table 2). The explanations were very diverse, and we were not able to categorise them dichotomously as correct (according to the given strategies) or incorrect. Namely, there were explanations which were at least partially based on the presented strategies (with or without misconceptions), there were also explanations which were clearly incorrect, and finally, there were some ambiguous explanations making it difficult to figure out what students had in mind. Therefore, it is evident that the correct answer to the multiple choice problem was not a proof that students could properly explain translation.We will quote a few students' explanations as examples of the first group. In some of them it was stated that P and T were variables, and that the volume was constant:
Charles Law states that increase of absolute temperature increases gas pressure. It describes isochoric processes, with constant volume, which is represented only in Picture 2. (1, our italic)
The explanation given indicates that Strategy a was applied, and the word only most likely also indicates that the student was aware that other pictures showed variable volume (possible application of Strategy c). Along the same lines is the next explanation, which did not mention all three quantities:
Because there will be no change in the volume amount. (2)
indicating that Strategy c might have been applied. However, even when students stated that the volume remained unchanged, there is a reason to question whether they completely understood the relationship between P, V and T (see the italicized portion of the next student's explanation):
Pictures 1 and 3 show volume changes which directly affect the pressure. Increase in volume decreases pressure and vice versa. For the formula given, volume is not a relevant, and volume is unchanged only in Picture 2. Since V is the same in Pictures 2a and 2b [the student labelled the left side of Picture 2 as a and the right one as b], and in 2b higher mass is applied consequently the temperature is higher and the pressure is higher in Picture 2b than in Picture 2a. (3)
Nevertheless, it seems that this student recognised an important feature of Pictures 1 and 3 – changeable volume, which was the guide in solving the item (application of Strategy c). One interesting feature of this explanation is that the student obviously thought that the change in the gas volume is always linked to the change in pressure. It seems that the student did not pay attention to the notations for volume and pressure.
Incorrect explanations, given for the accurately selected picture, are also analysed. Here is one of them:
If pressure increases volume increases too. (4)
This explanation indicates that the student correctly interpreted numbers of weights, i.e. the change in pressure. However, the underlined relation (pressure increase leads to volume increase) was not only invalid, but it was not presented in any of the pictures. Therefore, we see this explanation as student's failure to think about information from the equation, as well as from the picture, and to integrate them, which is a defining characteristic of TBER.
The next three explanations are quite different, but they all had the same conclusion that pressure was constant:
We can conclude that pressure in the vessel did not change despite extra weight. (5)
Pressure is constant (there is no value for temperature) → it increase or decrease. (6)
With the same temperature pressure is constant. (7)
How can we evaluate the thought process of those students? In spite of the fact that volume was not mentioned in the students’ explanations, in our opinion, those examples show that the students thought that the notation for one quantity also represented another quantity, i.e. that there was no change in pressure because volume did not change, which was recognised by the students (no change in the piston levels on both sides of Picture 2). We also interpret explanation (7) in this way. The change in temperature is obviously shown in Picture 2. If a student cannot interpret the notation for temperature, but sees that volume is constant, this can lead that student to assume that pressure is constant, and consequently that temperature is constant. Hence, explanations 5–7 indicate that the students' explanations were based on recognising the volume and using the volume to make conclusions about the other properties (see the discussion later in this paper). In addition, it is possible that students do not understand that the equation P = kT, because of its algebraic base, could be read in two ways: as an object (pressure is a product of constant and temperature) and as a process (change in temperature will cause a change in pressure) (see Sfard, 1991). The last reason can be quite possible based on the next example:
Well, the equation tells us only that the pressure is equal to the product of the temperature and the constant, whereas the pictures show changes in the pressure. Picture 1 – the piston in the cylinder went down compared to the first picture because the pressure decreased. Picture 2 – the pressure stayed the same except if those objects on the piston are weights, then the pressure increased a bit. Picture 3 – it is obvious that the pressure increased since the piston went up, and the reason is only because of the pressure. (8, our italic)
The third group of students' explanations given for the correct choice is distinct from the previous two, as we are uncertain what reasons (i.e. strategies) students applied. Here is a typical example:
According to the formula P = kT, pressure of a gas is proportional to the absolute temperature of the gas, which means that increase of the absolute temperature will cause increase in gas pressure. (9)
which raises the question whether students really understand how notations represent P and T (Strategy b), or they applied Strategy a, for example? Our dilemma comes from the fact that students neither gave explanations for notations nor commented on constant volume. Students' explanations, such as:
Particles become faster with an increase in temperature and since temperature and pressure are directly proportional and the pressure increased, the correct answer is number 2. (10)
can indicate that they recognise how temperature is represented (slower/faster movement of particles), and it could mean that students applied, at least partially, Strategy b.
Analysis of explanations when distractors were chosen (answer to the third research question)
First, we are going to comment on explanations for Picture 1, which was picked by more than half of students. In both test years the most frequent explanations included three quantities P, V and T. The majority of students explained that temperature and pressure were increasing and volume was decreasing:With decrease in volume pressure increases, there are more collisions in smaller space, kinetic E increases and with it the temperature increases too. (11)
Among other explanations the following two stand out.
When volume decreases gas pressure on given surface increases because the temperature is constant. Therefore, two weights are necessary to keep the system in equilibrium. (12)
When pressure increases absolute temperature will increase too and it will cause faster movement of particles and collisions.; Next to Picture 1 it was written “P↑ T↑ k = const.”, next to Picture 2 “V = const. k = const. T↑”, and next to Picture 3 “P↓ T↓”. (13)
Explanation (12) states that the temperature was constant, which was correct for Picture 1, whereas explanation (13) states that the volume is constant in Picture 2 and that the temperature is rising, which was also correct (although this student could have not correctly interpreted notations for Pictures 1 and 3). However, despite the correct findings students chose the wrong picture. Because of that, these two explanations are unexpected if students understand the equation P = kT as the mathematical object which contains variables and constant. We strongly believe that problems with mathematical components in understanding the equation P = kT (see previously reviewed Kautz' et al. work) are illustrated using these explanations.
The discussion below considers explanation (14) which claims that the pictures did not show a change in temperature. This could mean that students tried to apply Strategy b. However, this explanation raises the question: Is it possible that a greater number of students have such trouble? We will later say more about this. Nevertheless, students did not see that Picture 1 (and 3) presented varied volume.
The figure does not show temperature increase but only additional mass on top of the movable piston which compresses gas. The only logical figure is Picture 1. If increase of temperature was mentioned, then Pictures 2 or 3 would have been possible answers. (14)
Other explanations, almost entirely can be classified into two groups, ones with T and P mentioned, and others with P and V mentioned. In the first case, explanations were just verbalization of the equation:
According to the formula increase in pressure causes more collisions of gas particles creating great amount of energy which is manifested in temperature increase. (15)
At the first sight the explanation was not very informative, but this also could mean that those students neglected the information given by the equation and Picture 1, and that they were not aware of their own inconsistency in reasoning.
When Picture 1 was selected explanations mentioning P and V were frequent. They almost exclusively stated that an increase of pressure causes a decrease of volume, or vice versa so the argument is similar to the previous one:
With increase of pressure volume of gas decreases. (16)
It should also be mentioned that a few students only commented that Picture 1 shows a change in pressure, or volume.
In regard to the explanations given for choosing Picture 3, almost all explanations can be categorised as those in which P, V and T were mentioned and those mentioning T and P only. The latter ones were the most frequent. Students mostly verbalized the formula (as it was in the case of Picture 1 explanations) although some of them mentioned weights and piston, for example:
With increase of temperature gas increases pressure on the piston with weight and pushes it upward. If temperature decreases the pressure on the piston is lower and the piston with weight returns to the original position. (17)
If P = kT and T increases, than P will increase too. Therefore gas will push the piston with weights upward. (18)
The last example is interesting because the student explicitly made reference to the equation. One more explanation also contains reference to the equation, but the student thought that the formula shows inverse proportionality.
Pressure is dependent on temperature. When temperature decreases pressure increases. (19)
Among the second most frequent explanations (mentioning quantities P, V and T), the prevalent ones stated that an increase in temperature causes an increase of pressure and volume:
If increase of T increases pressure, according to the formula, it is logical that T increase will cause higher P and higher force on the piston carrying 2 weights, pushing it up and expanding the volume. (20)
As a variation of the previous explanation it was stated that pressure was dropping:
When temperature raises particles move faster and gas expands, volume increases and pressure drops. (21)
However, one explanation from this group deserves to be especially mentioned. Namely, one student linked an unchanged number of weights with symbol k:
P = kT, k = constant. The only possible constant in these pictures are weights. They are the same (number) in Picture 3. If weights are constant, then, as seen in Picture 3, volume increases and consequently pressure decreases, and according to the formula given decrease of pressure leads to decrease in absolute temperature and vice versa. (22)
This explanation can raise multiple questions and discussion. For example, the student made reference to the equation, but still explained the answer with quantity (V) which was not present in the formula, while Picture 3 showed that the volume changed. Furthermore, the student observed an equal number of weights and correctly interpreted that it signified constant pressure. However, the equation shows pressure as a variable. If the student correctly interpreted the notation for constant pressure, how did he/she not observe the same for volume? It is also evident that the student thought that the change in volume was always connected to the change in pressure, as we discussed previously.
Similarly to the incorrect choice of Picture 1, some explanations for selecting Picture 3 stated that volume is irrelevant to the equation, but students still chose the wrong picture. For example, in explanation (23) a student expressed such a conclusion directly, whereas in explanation (24) another student did it indirectly stating in what cases volume changed and when it was constant.
The formula given shows only relationship of pressure and temperature independent of volume and other effects of the environment. Picture 3 shows that if temperature increases pressure increases too. (23)
Pressure is higher because temperature increases; next to Picture 1 it was written: “V↓”, next to Picture 2: “V = const.”, and next to Picture 3: “V↑” (24)
In a few explanations students only wrote that pressure is constant. The connection between pressure and volume was only stated by one student.
Finally, none of the students, who chose the wrong picture, gave the correct explanation.
Generally speaking, results of previous research studies, cited in the Introduction section, show low success rates in solving translation tasks. Furthermore, it is well documented that high school and college students and even teachers have limited understanding about properties and behaviour of gases (Lin et al., 2000; Niaz, 2000; Taylor and Coll, 2002; Çalik and Ayas, 2005; Kautz et al., 2005a,b; Coştu, 2007; Liang et al., 2011; Madden et al., 2011; Bak Kibar et al., 2013). Despite all that, results of our research are still surprising, considering that the problem could have been solved in multiple ways, even without knowing the gas laws. The results obtained showed that none of the three strategies to solve the second item were used to a significant extent.
We did not explicitly check the students' knowledge of the P = kT gas law and it wouldn't be surprising if many students forgot about it and didn't apply Strategy a. They were studying gas laws two years earlier. Gas laws were not mentioned in the following years in physics (or chemistry) curriculum. Some students interviewed confirmed that they did not remember the equation P = kT (or gas laws in general).
Strategies b and c of item solving assume the correct interpretation of notations from the pictures and understanding of the equation. Since many students did not explicitly link graphic elements with the corresponding physical properties and representation of their changes, it is uncertain whether the students knew how to interpret lines as symbols for movement of particles and, consequently, the (changes in) temperature and weights as symbols for pressure. For example, from explanation (14), stating that temperature was not shown, it could be deduced that (some) students did not know that temperature can be represented the way it was shown in the pictures. That was not completely surprising because even college students can have problems with notations for temperature (Madden et al., 2011). Furthermore, from some explanations, e.g.(15), stating that more collisions cause higher temperature,§ it is almost certain that the students do not have a “deeper” understanding of the concept of temperature as well as pressure. Kautz et al. (2005b) argued that the misunderstanding of microscopic causes for pressure and temperature has a detrimental impact on students' ability to explain or predict macroscopic behaviour of gases, pertinent to the change in pressure, temperature and volume.
We have also noticed that some students had problems identifying weights as pressure, for example, those who stated that Picture 3 shows a change in pressure. However, it is more likely that even those who stated that weights signify pressure is a result of everyday use of word pressure than it is a result of understanding it (Eylon and Linn, 1988).¶ For example, some explanations contain expressions such as “when we press onto something, we exert higher pressure”. Showing gas pressure indirectly with weights pressuring the piston makes the right conclusion more difficult. Kautz et al. (2005a) came to such a conclusion (p. 1058, our italic):… many students seemed unable to relate the mechanical equilibrium of a piston to the force exerted on it by the enclosed gas and, hence, the gas pressure.
Therefore, there is a substantial difference in interpreting weights as pressure by the students and understanding that the number of weights is related to changes in pressure. Such a conclusion was supported by many explanations for Picture 3 stating that pressure changed despite the same number of weights being applied. One can conclude that the students did not understand the concept pressure and its ER in the pictures. Educational semiotics points out that the construction of a concept cannot be detached from learning of its ERs (Lemke, 1998; Hubber et al., 2010;Waldrip et al., 2010). As Hubber et al. pointed out conceptual difficulties with the concept of force are fundamentally representational in nature.
However, our findings show that the interpretation of notations is just a part of the problem. Namely, the problem also occurred when the notation for a change in one quantity was used as a sign for a change in another quantity. That's especially evident in explanations of Pictures 1 and 3 where the students linked changes in volume with changes in the pressure and temperature. We believe that it is caused by deep rooted cognitive schemas, which can be represented as: “ΔV → ΔP”, “ΔT → ΔP”, “ΔT → ΔV”, and vice versa (regardless of whether the effect is valid). Hence, the students did not realize that the relationship between any two values out of the three listed could be viewed only if all three are considered. The similar results were obtained in other research studies (Lin et al., 2000; Kautz et al., 2005; Leinonen et al., 2012; Hernández et al., 2014).
The interpretation of one quantity using the other one was in many instances done with volume. Explanation (13) confirms that, stating that temperature rises for Picture 1 and decreases for Picture 3. It is clear that the student did not come to a conclusion by interpreting the notation for movement/temperature but by indirectly observing a change in volume, and, consequently, a change in pressure too. (At the same time, it is evident that the student's reasoning is extremely inconsistent as he/she interpreted correctly the volume and temperature in Picture 2.) From the explanations it was obvious that almost all students had no problem determining when there was a volume change and when there was none because:…volume (…) [is] quite straightforward and can be immediately perceived – volume as a bigger and smaller container – and no reports in the literature have shown students' difficulties in understanding [it]” (Levy and Wilensky, 2011, p. 568).
Due to the difficulties in interpreting notations for temperature and pressure and deducing changes in one quantity by the notation for another quantity, Strategy b of solving the second item could not have been used to a significant extent. On the other hand, Strategy c expects that students interpret only the notation for volume, which did not cause problems. Therefore, a high percentage of the distractors chosen is surprising, since we expected that Strategy c (using elimination) would have significantly improved the translation of the equation into the corresponding picture. In such a situation, selection of distractors by the students made us to consider if the students understood the equation P = kT.
Selection of distractors may point out that the students did not understand that if a quantity was not present in an equation, then its value (volume in this case) did not change, and it could not be considered as a cause for changes in other quantities (pressure and temperature here). The same goes for symbol k which cannot include changes in a quantity (i.e. V). In other words, in the equation P = kT, the value for pressure changes solely as a result of the change in temperature which is explicitly noted with a single symbol for the independent variable. The results listed show that most of the students stated that all three variables were changing. Also, students who mentioned pressure and temperature in their explanations but chose distractors disregarded data given with the equation and pictures. It is obvious that the students who chose Picture 1 and then mentioned a change in pressure and volume disregarded the incompatibility of the picture with the equation. Thus, either the students, in spite of the equation P = kT given, listed more variables than the equation contained, or they just listed the variables that were more convenient. This is what Lin et al. (2000), Kautz et al. (2005), and Leinonen et al. (2012) have observed too. It is also obvious that Sherin's (2001) symbolic forms were not applied either because students did not possess them, or such knowledge could not have been activated.
In conclusion, the results from Part I suggested how to answer the research questions. It is reasonable to assume that the entire population of the final year high-school students from all 15 high schools in Belgrade (although only students from 10 schools were included in the study) would have achieved poor results on the applied item. Also, although students probably forgot about the gas laws and had poor understanding of concepts of pressure and temperature and their ERs, they did not use obvious information (about volume) from the equation and the pictures in order to solve the translation task. Instead, most students chose distractors, ignoring the fact about the types and numbers of physical quantities represented in the equation and the pictures. This was concluded on the basis of the number of students who in their answers mentioned all three quantities as variables, or mentioned variables V and P (which is contradictory because the equation does not contain V), or verbally restated the equation, which contradicted the picture, as it showed constant V.
Part II
At the end of the interviews (including the interviews for other gas laws) most students were asked to describe their experience with learning about gas laws during physics classes because we had not collected that type of information during 2011 and 2012 tests. All students agreed that their teachers had not used computer animations or simulations in teaching gas laws. Most students did not remember seeing figures similar to the ones we used. In two schools (the students from one of them participated in the interview shown here) the students had not attended the physics lab (one session was about the Boyle–Mariotte Law), although it had been a part of the curriculum and mandatory. The students were studying from the textbook and notes taken during the class sessions. Inspection of the corresponding textbook showed that the material was presented only by text, formulae and graphs. We believe that we would have gotten similar responses from the students during 2011 and 2012 in schools that did not have students interviewed.
Results and discussion
The students worked on solving item 1 first. Neither student A nor student B correctly solved item 1, although both students more or less made reference to the equation: (i) student A commented that constant k could not have had a negative value, because P and T could not have had such values; (ii) student B explicitly made a point that the equation did not contain symbol V. Now, we present the interview excerpt for solving item 2.Interview with student A
The student correctly solved the second item (she was the only one who correctly solved item 2 during the interview without prompts), by recognising that symbol V was not given in the equation, and that two out of three pictures represented a change in volume (lines 1 and 2). It seems that the student applied Strategy c.A||: [Reading the complete instructions for Item 2]
I: What are you looking for in the pictures, i.e. how do you evaluate them?
A: I don't know, I see they are different, I think that gas pressure is different.
(…)
I: Does it mean you are showing me Picture 2?
A: Yes.
(…)
I: Does any other picture show change in pressure, or which picture has the same meaning as the formula presented?
A: Actually, it can be Picture 1, but it is not related to the formula. (1)
I: Why do you say that Picture 1 is not associated with the formula?
B: Because, actually, I don't see volume anywhere in the formula. (2)
I: Does Picture 1 represent change in volume?
A: Yes. Then, Picture 2 could describe the formula.
(…)
I: There are some graphic elements signifying changes in some physical properties in Picture 2 for which you said it was the correct one. (…) How is volume represented?
A: Well, with the different level of the piston on the top.
I: Very good. Is there a graphic element describing change in temperature? (3)
A: Well, no actually, I think I don't see it. (4) (…)
I: There is something around those little pellets, like little tails. (5)
A: Oh, yes.
I: I don't know if you see them? (6)
A: Oh yes, I see it on some. Yes, that would mean, that if it is, if it has a tail that it is warmer, i.e. that it is moving more. (7)
(…)
When considering the notation for temperature, it seems that the student was not able to activate appropriate schemas, although she possesses them (lines 3–7). Therefore, this interview facilitated further discussion initiated by the explanation No. (14) given in Part I. Namely, it is possible for students to know that temperature can be represented the way it was shown in the pictures, but the second item did not activate such knowledge. We observed (including students in interviews for other gas laws; see also the interview with student B) that after reading item 2, students would usually remain silent. They were then asked to describe the graphic elements (notations) they see and their meaning. At that time, many students did not mention lines around spheres. It seems that the students “did not see” them. When asked to describe what they saw around spheres they responded – lines. After an additional question to describe what those lines represent, the students did not have a problem in responding that lines represent the movement of particles and (change in) temperature (mainly after a question if that was related to gas temperature). The students related higher/lower particle speed with higher/lower temperature. The challenge was that many students were not able to independently link the observed notations (lines around spheres) with temperature.
Interview with student B
Unlike student A, student B had a lot of difficulties in solving item 2.B: [Reading the complete instructions for Item 2.]
I: [Asked what he was thinking about.]
B: I am just looking and counting those pellets to see if the amount of gas increased, but since I can see it did not, Picture 1 is correct because if we increase weight we exert on the gas that has certain constant pressure and volume, the gas will compress.
I: O.K. Is this result of thinking about formula PV = nRT or not?
B: Well, yes, partially, but let's say it's not, because I used basic logic. If we have, as it is shown here, a cylinder and certain volume of gas in it, if we don't change the volume of the cylinder or temperature, just change pressure we exert on the piston, gas will compress, so, for that reason, it's Picture 1.
I: Was change of volume part of the formula? (1)
B: Change of volume was not part of the formula.
I: Good, but Picture 1 tells you that the volume changed.
B: Yes, Picture 1 is telling. /I: You need to find a picture with the same information as the formula. That means that the picture tells you that the volume changed, but the formula doesn't show change in volume. How do you know that the picture shows no change in volume?
B: Well, the formula is not containing volume, so if we look strictly at the formula (2), then. /I: Those were your instructions, to match the formula with a picture. /B: Then, I think it's number 2.
I: Picture number 2 because?
B: Because, like in the given formula there is no change in volume, pressure is product of the constant (sic) temperature, because the picture doesn't show in any way change of temperature, which means that pressure should also stay. /I: Be careful, though, if you say that the picture doesn't show temperature change, but the formula shows change in temperature, does that mean that something in the picture must be showing temperature change?
B: Well, it should be something in the picture, but in this case. /I: If you look closely around the spheres, do you see anything?
B: Yes, I see. For example spheres are moving in some pictures, I think it is shown that spheres are moving and in some pictures it is more pronounced, which means they move more, and in some pictures they don't. [He thought about left and right part of Picture 2]
I: Is that related to the temperature?
B: Yes, with heating, gas starts, gas molecules begin to move faster.
I: Good, and Picture 3, which means we eliminated Picture 1 – it is not correct. /B: Picture 3 is correct, Picture 3 is actually correct
I: But, what does Picture 3 show? (3)
B: Picture 3 shows that if we have the same weight on the pistons and if we heat that gas, its molecules start to move, it will increase volume and consequently raise the piston. /I: Which two values are variables then?
B: Well, pressure and temperature.
I: In Picture 3? [Surprised]
B: Well, in Picture 3, right in Picture 3 since temperature is not shown, I think. /I: Look closely around those. /B: In Picture 3, right pressure, but in this case volume of the gas has changed.
I: Good. What else changed? Did pressure change?
B: Yes, with it pressure changed too. (4)
I: But here you have two weights, which signify the same pressure in both cases.
B: Yes, but when we heat a gas or something, I mean objects and gasses expand with heat which means that gas volume will increase and exert higher pressure on the piston. (5)
I: I see. In that case as we change temperature, we also change volume and pressure?
B: Yes.
I: But, the formula shows only two physical variables.
B: Shows temperature and pressure.
I: Where is the third one then, why we didn't put the third variable in the formula? (6)
B: Well, I don't know, maybe because it is assumed (6), I think with logical observations, using logic, we'll get to it, but, I don't know.
As is obvious from the excerpt, the student had considerable difficulty in accepting that the second picture was true. It seemed that for student B the interpretation of the equation can be somehow loose (line 2). He recognised the absence of V in the equation (after prompt, line 1), and chose the correct Picture 2. However, the student then changed his statement, and chose Picture 3 (line 3). Even when he made correct observation about the equation, and the notation for temperature, he was still thinking only in light of schema that “objects and gasses expand with heat” (“ΔT → ΔV”, line 5), although other schema, “change in volume lead to change in pressure”, were present too (“ΔV → ΔP”, line 4). At the end of the interview the student revealed that his understanding of the equation (its structure and function) was still limited (line 6).
Final considerations
The aim of our pilot and exploratory study was to study high school seniors' ability to translate between external representations, specifically between equation P = kT and the corresponding pictorial ER. Literature search provided only one article (Madden et al., 2011) which had a similar aim. Authors concluded that university students could have poor abilities to translate from the equation to pictorial ER. Thus, our study is a contribution to the better understanding, in our case, of the high school seniors' abilities to translate between the equation P = kT and pictorial ER. The students' task was to choose one out of three offered pictures which had the same meaning as the equation P = kT. The distractors represented equations V = k/P and i V = kT. The participants were enrolled in the high schools with curriculums rich in science (chemistry, physics, biology) and math courses primarily preparing students for science and technical colleges.The three research questions guide the collection and interpretations of results in this article.
The results of the first research question, about students' success in translation, showed that high school students were generally not successful in translation.
The reasons for choosing the correct answer or distractors were our second and third questions.
For the second research question one can say that the students' explanations and interviews gave us a reason to believe that the full interpretation of notations as a way to solve the translation task would have been difficult to expect from most of the students. Another, may be a more serious problem, is that we cannot expect that the applied translation item would be solved to a greater extent by students by exploiting information from the equation based on its algebraic structure. Namely, from the algebraic formula structure it can be seen that volume is not present in the given equation, while the distractors contain a change in volume. It seems that students are not accustomed to mathematical reasoning, to see an algebraic equation as the foundation of the formula.
The results from the third research question revealed that the students did not notice that their reasoning was not consistent with the information contained in the equation and/or contained in the pictures. This was the case even when students made explicit reference to the equation and/or picture. Namely, students included quantities which were not represented in the equation and/or pictures, as it had been stated earlier in the literature. Since students mainly chose distractors, we concluded that a significant number of students may have had limited understanding of the equation. Use of notations for one quantity to indicate the change in another quantity is mainly caused by deeply rooted cognitive schemas. For this reason, students consider changes only in two quantities without proper understanding that for such changes to be valid for gases all three quantities need to be considered. In our opinion, such cognitive schemas had a significant impact on translation from the equation to the picture.
The conclusions were drawn from the analysis of results of three classes of students (2011, 2012 and 2013). The results obtained from the multiple choice item, students' explanations and interviews were similar for all three school years. This indicates the consistency of the results, which were not random but a rather consistent phenomenon, most likely to continue if certain changes in teaching are not implemented. The validity of the tasks applied was viewed from the aspect of its formal agreement with the curriculum (which included gas laws) and from the aspect of agreement from the fellow chemists and physicists about the usability of the item demands (recognizing the connection between a picture and the corresponding gas law) and the aspect of potential strategies to be applied for solving the item. One element of the validity needing an additional comment is the validity of the usage of our pictures to research the process of translation by high school students.
The first element refers to the question if the “hybrid nature” of the target ER compromised the validity of the item. As we pointed out in Part I, to our knowledge, there are almost no research studies on the role of hybrid ERs in the evaluation of internal representations. Despite that, hypothetically, the validity of use of selected target representations for the translation task, and, hence, the validity of the conclusions presented, could have been questioned if the students made comments about an absurd or unsustainable situation where weight(s) are kept in place by the “force” of only several particles, that only several pellets were shown when actually describing a gas, or that the students had a hard time understanding the pictures because the graphics overemphasized certain gas properties, etc. None of those were observed in students' comments or during the conversations with the students (including all interviews, regardless of the gas law in question). For those reasons we don't doubt the validity of hybrid ERs used, for the purpose they were used. It is very likely, although we don't provide specific proof, that the students have experience with hybrid representations, primarily through teaching of chemistry.
The second element questions justification for applying pictorial ERs we used for the research of translation considering that pressure and temperature were shown indirectly, which made the interpretation of the ERs more difficult. The question of validity depends on whether the students had an opportunity to get familiar with such notations and if they had not, would that fact diminish the validity of the conclusions drawn. Another question is what would the results have been if pressure and temperature were represented with instruments (pressure gauge and thermometer, respectively) for the applied ERs?
In the Introduction, when explaining the expression TBER, we mentioned at least two possible translation mechanisms:
(I) Can students interpret notations within external representations (source and target ERs)? and
(II) Can students integrate them?
The first aspect has two levels:
(Ia) Can students recognise the notation (e.g. if weight is shown can students recognise it as weight)?
(Ib) Do students know in what way the notations are related to the representation of the relevant concepts and how much they know about the concepts in general (how weights represent gas pressure and what is pressure)?
If (Ia) is not accomplished, the application of item 2 will not be valid. The data collected did not indicate such a problem, which we pointed out in our discussion.
As for the second level (Ib), it was known that the students had studied relevant topics (e.g. about pressure, temperature, etc.). However, we did not have information on whether the students learned about the notations representing pressure and temperature in a pictorial way, as it was in our case, using weights and dashes. In addition, the students themselves stated that they had no previous knowledge about the ERs from the task, although there was a possibility that the gas property lessons included examples of a cylinder with a moving piston. Also, it is for certain that the students got familiar with the notations for moving particles in their chemistry classes (e.g. chemistry textbooks contain such displays, which are fairly universal indication of “speed”). Can we justify use of such ERs and the results obtained, aside from the above statements, since no solid evidence existed, if we cannot claim for sure that the students previously learned how certain notations represent pressure and temperature? We believe that the answer is yes and we offer the following reasons.
The interpretation of weights (achievement of the criterion Ib in the area of knowledge of notations) depends on the successful transfer of previously acquired knowledge about concepts and notations, especially in the area of Newton's Laws of physics. Although the students had not seen similar pictures, it can be claimed for sure that they had been exposed at least to the free-body diagrams, where bodies are represented more or less in a pictorial way and mutual interactions formally with arrows. Therefore, it can be expected that students, when observing a cylinder filled with gas with a piston and weights on top of the piston, recognize that the gas pressure is a result of interactions between the objects and gas (weights and piston, piston and gas, gas and cylinder walls) and apply their knowledge about concepts of force and pressure. Hence, when students would see a cylinder with a piston and weight(s) for the first time, one could expect them to achieve the “far transfer” (Barnett and Ceci, 2002), to recognise the same concept in completely new situations, i.e. recognise the deep structure from the interaction of surface features (Chi and VanLehn, 2012). The situation described above can indicate how it is possible from one situation, where a concept is learned (e.g. pressure) together with its representation (notations for interaction between bodies), to transfer to another situation (a cylinder filled with gas and a movable piston) in which, by recognizing the relevant concept, a student can learn its representation (weight). For this reason, even if explicit teaching of how the weights represented gas pressure inside the cylinder with a movable piston was missing, we don't consider that it affected the validity of the application of the ERs used. However, as we noted above, although the students had no problem connecting the weights with the pressure, the basic understanding of how the weights were connected to the gas pressure was missing. Such an outcome was indubitably a result of insufficient understanding of the concept for force and interaction between objects.
As for the notations for temperature (dashes), we mentioned that the students both as a part of the formal education and non-formal contexts had learned notations for the speed of moving objects and, consequently, moving particles. Even if that was not the case, there was still a reasonable possibility that the students, seeing such notations for the first time, indirectly realise what they represent. Namely, it can be expected that the hybrid ERs (graphic emphasis of particles) should initiate a connection to one of the basic postulates of the kinetic theory of gases: that all particles move in straight lines, that the speeds of particles vary and that macroscopic properties (such as temperature) can be connected to the speed of particles. We consider this case also as an example of far transfer and learning the notation in an indirect way. The explanations from the students indicate that some students did not connect dashes with the movement of the particles and temperature. However, as mentioned in the discussion of Part II, it was apparent that the students had cognitive schemes (dash – particle movement) but that they could not have been activated. The interviews with students A and B illustrate that finding very well.
Finally, we would like to point out that even if the described indirect understanding of the role of the chosen notations was not possible to achieve (in situations when students had not learned about force and kinetic theory, which was not the case in our research, which included only students who had learned about those topics), from the aspect of testing the possibility of solving the task using Strategy c, it seems almost necessary (if not required) to make the task more difficult that way. That brings the question what kind of results would have been obtained if instrumental notations were used for pressure and temperature? We think that the task would have been too easy and explanations mostly represented verbalisation of the equation (which was a frequent result in our research). The interpretation of such results would have been very different: in our case the conclusion was that the translation was limited, but in the other case it would have been satisfactory. Such results would have been of little value and even misleading. The ERs chosen enabled an insight into cognitive resources, which is doubtful to have been achieved if the instrumental versions of ERs were applied.
Implications for teachers
The classroom implications of the results are that they clearly show significant deficiencies in teaching of both physics and chemistry. If we expect students to be successful in solving translation problems then students should be engaged in such activities. Furthermore, inability of students to perform translation based on understanding an equation means that teachers should teach about what information can be provided by equations, which is in agreement with the research studies emphasizing the importance of symbolic forms both for evaluation of students' knowledge/reasoning and as a theoretic framework for creating a curriculum (see the Introduction).Implications for further research
The next step will be to conduct a similar research with the first year high school students. They will be presented with the second item. Our goal is to see if students will be able to solve item 2 by translating an equation into a picture by applying Strategy c, even before they were taught the gas laws. That would give us an opportunity to decide if learning of the gas laws causes deep rooted cognitive schemas discussed above. Those cognitive schemas, which had caused poor results described in this paper, in some ways interfere with mathematical reasoning.Acknowledgements
Our special thanks to Dragan Marinković for help with translation and some valuable comments, and also to Ljiljana Solujić, Ivana Jeremić, Goca Savić, and Jelena Savić. This paper is the result of working on the project “The Theory and Practice of Science in Society: Multidisciplinary, Educational and Intergenerational Perspectives”, no. 179048, the realisation of which is financed by the Ministry of Education, Science and Technological Development of the Republic of Serbia.Notes and references
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Footnotes |
| † The term equation and/or formula instead of the full name physical equation or physical formula will be used in this paper to simplify our writing. However, it is important to keep in mind that we are dealing with physical equations/formulas for at least two reasons. The first one is the scope of the paper – ERs used to mathematically express the relationship between physical properties (therefore, physical equations/formulas) and not ERs for visualisation of the chemical composition and/or structure at the particulate level and mutual interactions (empirical, molecular and structural formulas and reaction equations). The second reason is that if a physical equation is labelled only as an equation, it can create a wrong impression that such ERs can be interpreted solely as mathematical objects. For example, a complete algebraic interpretation of P = kT, in terms of mutual dependence of T and P, is possible only if possible numerical values P and T can have are known. |
| ‡ Taking Note 1 into account, for example, one student made a comment that the solutions for the first item could have been both 1 and 3, depending on the positive or negative value for the constant k. |
| § The idea that thermal energy is generated as the by-product of collisions of individual particles is highlighted in the science education research literature as an obstacle to learning (Scherr and Robertson, 2015). However, these researchers recently showed, although on the sample of teachers, how that idea can “initiate intellectual progress”, and thus should not be considered as an obstacle to learning (Scherr and Robertson, 2015). |
| ¶ The word “pressure” is one of many dual meaning vocabulary words. Such words have scientific and everyday meaning. Students at the beginning of the teaching/learning process hold everyday meaning. Formation of scientific meaning is a long and very hard process. Thus, it is a common result that even after a long period of teaching students still have only everyday meaning for such words (see Song and Carheden, 2014, for further discussion and some results in the chemistry domain). |
| || I–The interviewer, /–interception, (…)–omitted one or more sentences, [ ]– our comments. |
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