DIFFICULTIES ON INFERENCIAL PROCESS. A STUDY ON THERMODINAMIC PROBLEMS
Marta Massa, Marta Yanitelli, Susana Cabanellas, Facultad de Cs. Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Rosario, Argentina
1. Introduction
Differences on problem-solving performance between experts and novices have been studied during more than two decades. Attention has been focussed on mental processes to organise knowledge (Chi, et al., 1988; Chi, et al., 1982) on specific areas such as: categories that the subjects use to organise knowledge; relationships between these categories and problem-solving abilities; identification of similarities between problems; degree of mental processing required to produce the categorisation tasks (Smith, 1992). The latter has received less attention and the degree of mental processing has been analysed by measuring the time required to complete the tasks. Results have shown that there is not a direct linkage between expert behaviour and successful solving procedures, as experience sometimes may be associated to automatic processing rather than to problem-solving ability (Schoenfeld and Herrmann, 1982).
In physics courses at University, students perform problem-solving activities that demand similar conceptual knowledge with quite different success. Frequently, teachers ask themselves if the observed differences are due to features that Chi, Feltovich and Glaser (1981) attributed to the problem’s “surface structure” (literal physics terms, objects, described physical configurations), the structure of the statements, the language and style, or to the cognitive process that are demanded for the situational modelling.
Among the procedural abilities required during problem solving, the production of inferences has not been extensively studied. Its relevance lies on the fact that it is an important activity of information processing: internal representation of data, integration to previous knowledge, transformation in order to construct new meanings and relationships (Riviere, 1986).
Many researches have shown that a procedural knowledge of algorithmic is a necessary condition but not a sufficient one for the comprehension of physics concepts (Solaz et al., 1995; Chi et al., 1981; Kempa, 1991). So, an effective and successful problem-solving activity would lie on the way of representing connectives between concepts involved in mental models.
Specifically, a remarkable different performance was observed when students solved problems related to the model of an ideal gas if: (a) it is applied in the analysis of a process or (b) it is required to determine its validity in a real situation. Although both activities imply the same concepts, the former is related to an algorithmic procedure, while the latter involves inferences and concepts processing. In fact, it was significant that 37% of the students that were successful when solving task like (a), failed on solving (b).
The purpose was to analyse patterns of reasoning when students make inferences to solve problems expressed in conditional statements. The research sought data to answer the following questions:
- Which is the reasoning structure employed by a student to solve the reference for a conditional statement?
- How does a conceptual model, such as the ideal gas one, relate to mental models representing a real or possible situation involving a specific gas?
2. Method
Subjects. The subjects of the study were 64 students that attend an engineering basic physics course dealing with Thermodynamics. Classes were developed according to the conceptual structure presented in Statistics Physics and Thermodynamic (Jancovici, 1976). They were asked to solve a set of classical problems dealing with the framework of the kinetic theory of gases and the first law of thermodynamics, as a partial evaluation of the learning process. The problems were similar to those presented in the class and the textbooks that the students used as a complement reference for their studies. We specifically analysed one of these problems stated in a conditional form, extracted from Fishbane (1994):
1 mol of helium gas is contained in a cubic recipient of 50 cm each side. In this condition its internal energy is 3600 J. If it were possible to apply the model of ideal gas to the air in normal conditions, ¿would it be possible to do the same with the described helium system?
Data collection. We used as protocols only 26 pencil-and-paper tasks (41 %) belonging to the subjects that intended, successful or unsuccessfully, to solve the problem. During the first stage, we analysed the protocols considering: i) selected data and their initial transformation; ii) principles and laws applied to calculate the relevant variables; iii) conclusion statement; iv) justification.
In a second stage, we attempted to identify similar features in order to establish possible categories of resolution style. Finally, we made an interpretative analysis of possible inferential processes involved in the detected categories.
3. Results
The protocols were examined in order to find similarities in (a) the identification of data, (b) variables that were calculated to arrive to a conclusion and (c) the justification stated to make the conclusion consistent. As a preliminary result, 14 students concluded that “the model of ideal gases is applicable”, showing the three mentioned stages while reasoning; 7 students considered that “the model was not applicable”, after the three stages, and 5 did not arrive to any conclusion, with scarce calculus of variables. Table I shows the resultant categories for the students that arrived to a conclusion.
Category
/Category Analysis
Model validation /Justification characterisation
/Frequency
p-V-T / Yes / He – Air comparison using the functional relationship between volume, pressure and temperature(V-T or p-T). In some cases the comparison is made only between pressure values. The criterion is based in the proximity of the stated numerical values. / 5No / He – Air comparison using the p-T variables but looking for coincidence with the normal conditions values. One case takes into account the different molecular composition. One case presents no justification. / 4
Density
/ Yes / It is stated that the density is low without establishing a comparison parameter. In one case the comparison is made between molecular density of He and Air. / 4Internal energy
/ Yes / Comparison of Internal energies only. The functional relationship between p-V-T is ignored. The criterion is based in the proximity of the stated values. / 2No / Comparison of Internal energies looking for coincidence between numeric values. The functional relationship between p-V-T is ignored. Misconceptions are detected. / 2
-p-T / Yes / In one case the comparison is made between the three variables. The other one resorts to the qualitative definition of ideal gas stating “if air is an ideal gas at normal T and p, then it is a low density gas”. / 2
U-T / No / He – Air comparison using the U-T variables but looking for coincidence with the normal conditions values. Misconceptions are detected. / 1
-U / Yes / He – Air comparison based on y U . Confusions over the molecular composition of the air are detected. / 1
Table I: Description of the identified categories.
4. Discussion
The analysed protocols suggest the existence of different stages to arrive to a conclusion. Only few subjects, belonging to p-V-T (Yes) category, proceeded as Fishbane (1994) did. Consequently, their reasoning may be assumed as an “expert” one. As a first approximation, the stages involved in arriving to a conclusion may be interpreted as following:
1º) Representation of the problem statement initial models. A representation of the content is constructed by integrating the text components and refining specific information of previous knowledge related to the molecular structure of gases. It may be supposed that “bridging” inferences are made. They do not strictly derive from the linguistic properties of the text but contribute to solve the reference, i. e.: mol Avogadro´s number; helium monatomic.
2º)Interpretation of the question stated as a conditional clause. This premise comprises three elements: air at normal conditions, ideal gas and a conditional statement. The disposable information does not allow to establish in which direction the process is made, but the subject is supposed to organise progressive models and to make new “bridging” inferences: air mixture; normal conditions pressure = 1 atm; temperature = 0º C = 273 K; ideal gas low density, diluted; U = Ek av. and other related concepts. A more complex task would be, perhaps, the analysis of the syntax to discover the conditional clause and to transform the premise to a logic format “if p then q”.
3º) Substitution of transformedpremises. Substitution of (a) an entity by the class[1] it belongs to and (b) the normal condition by valuesproximal to it. In this way the premise extends its meaning as shown in the figure presented below.
4º) New model construction which implies “renewing” the interpretation of the first premise to integrate it to the model of the second one and a “re-ordering”. In addition, a complex selection process of the necessary and sufficient conditions related to the “proximal to normal conditions of the gas” is performed to define a bridge to compare the class entities. Then, the helium state is evaluated in order to configure the following modus ponens figure, from which the required conclusion arises:
If a gas state is near to normal conditions then it is an ideal gas.Helium state is near to normal conditions.
Helium may be considered an ideal gas. / If p then q
p*
q
5. Conclusion and implications
A significant proportion of the students (see Table I) consider Density as a relevant variable. This fact shows an availability bias when, probably, the qualitative definition of an ideal gas is recuperated: It is said that an ideal gas is the state to which all gases tend when their density is very low (Jancovici, 1976). The subjects would be moving through the mentioned stages to elaborate an “helium model” making use of their previous knowledge of ideal gas as a diluted one.
Likewise, it may be interpreted that subjects included in the Internal energy category select this variable because of the influence of its presence as a datum in the first premise of the problem. Then, this fact shows a bias of representativeness that takes them to set up the following configuration:
If a gas has U equal or proximal to that of air in normal conditions then it is an ideal gas.
Helium has U=3600 J.
Helium may be considered an ideal gas.
Students belonging to -p-T category would seem to reason as those included in the p-V-T category, but considering density as relevant variable. This may be attributed to the influence of the textbook definition. Finally, the two latter categories, with a scarce number of individuals, share the characteristic features of Density and Internal energy ones.
In summary, the different types of reasoning identified seem to be related to the processes required by the third and fourth stages to transform a premise in spite of a proximity criterion and to determine the necessary and sufficient conditions to validate the model application. The study has allowed the identification of some organisational patterns for the information and elaboration of a conditional statement, which deserve further analysis.
Nevertheless, the evidence suggests that students are more apt to look beyond the logical form of a proposition and consider alternate hypotheses in contexts that they are able to restructure concretely. Teaching should be geared toward assisting students to achieve this concrete restructuring. Likewise, considering the interplay among a variety of variables in a given context is crucial for the generation of viable hypotheses and reasoning about the situation. Students should be taught to carefully consider the relevant variables in a given situation and the necessary and sufficient conditions for the applicability of a given conceptual model.
References
Chi M. T., Feltovich P. J., Glaser R., Categorisation and representation of physics problems by experts and novices, Cognitive Science, 5, (1981), 121-151.
Chi M. T., Glaser R., Farr M.J., The nature of expertise, Hillsdale, NJ: Erlbaum, (1988).
Chi M. T., Glaser R. Rees E., Expertise in problem solving, In Sternberg, R. J. (Ed.), Advances in the psychology of human intelligence,.1, Hillsdale, NJ, Erlbaum, (1982).
Fishbane P., Gasiorowicz S., Thornton S., Física para Ciencias e Ingenierí, 1, Prentice-Hall, (1994).
Jancovici B., Physique Statistique y Thermodynamique, Premier Cycle, Mc Graw Hill, (1976).
Kempa R. F., Students' learning difficulties in Science. Causes and possible remedies, Enseñanza de las Ciencias, 9, (1991), 119 – 128.
Riviere A., Razonamiento y representación, Madrid: Siglo XXI, (1986).
Schoenfeld A. H. and Herrmann D. J., Problem perception and knowledge structure in expert and novice mathematical problem solvers, Journal of Experimental Psychology: Learning, Memory and Cognition, 8, (1982), 484 - 494.
Smith M.U., Expertise and the Organisation of Knowledge: Unexpected Differences among Genetic Counsellors, Faculty, and students on problem Categorisations Tasks, Journal of Research in Science Teaching, 29(2), (1992), 179-205.
Solaz J., Sanjosé V., Vidal-Abarca E., Influencia del conocimiento previo y de la estructura conceptual de los estudiantes de BUP en la en la resolución de problemas, Revista de Enseñanza de la Física, 8, Nº 2, (1995), 22-28.
[1] The class pis an abstract entity that represents a group of elements p* having a common property. In this case, the two entities (helium and air) are interpreted in the general sense of a gas.