1. Incommensurability and Multiple Models: Representations of the Structure of Matter in Undergraduate Chemistry Students FERNANDO FLORES-CAMACHO, LETICIA GALLEGOS-CÁZARES, ANDONI GARRITZ*, ALEJANDRA GARCÍA-FRANCO Centre for Applied Sciences and Technological Development *Faculty of Chemistry Universidad Nacional Autónoma de México,

2. Introduction The recognition that students construct multiple representations or models has reopened the debate on the construction of school science knowledge, especially about the conditions that support the coexistence of different models in students’ conceptual structure.

3. We think that incommensurability help us to understand the coexistence, without contradiction, of different representations during the students’ process of knowledge construction.

4. In these work, we present an epistemological approach to the problem of incommensurability, using translatability criteria, to account for the different models of the structure of matter used by students. We also present a profile of the use of these models for a group of students as an example.

5. About incommensurability T. Kuhn found support for incommensurability in the historical analysis of science development and determined that it is an implicit element in the construction of scientific knowledge. However, incommensurability is present also in the individual process of knowledge construction as was formulated by G. Bachelard and more recently by authors as Carey, 1992; Hernández & Ruiz, 2000 or Chi & Roscoe, 2002

6. The question that arises is: Can we use incommensurability to rationally evaluate and compare students’ multiple models of the structure of matter?

7. Elements used to analyse the incommensurability of students’ models It is necessary to define a set of rules or criteria that allow us to determine incommensurability among different models.

8. I. Regarding the issue of non-translatability on basic concepts 1. Identity and overlap of basic concepts 1. Identity and overlap of basic concepts. There is no set of theoretical elements that can make the meaning of concepts become equivalent in two different models. Basic concepts cannot be comparable through relations such as identity or overlapping. 2. Translatability rules for basic concepts 2. Translatability rules for basic concepts. There are no translation rules, such as mathematical, that can make the meaning of concepts equivalent. 3. Relation of basic concepts with the terms of reference 3. Relation of basic concepts with the terms of reference. There is no set of referents that can be related to the basic concepts of two incommensurable models which could make them acquire the same meaning.

9. II. Related to the meaning of referents 4. Comparability of phenomenological referents 4. Comparability of phenomenological referents. There is a set of phenomenological referents that is shared and can be compared in both models. 5. Different meanings for comparable phenomenological referents 5. Different meanings for comparable phenomenological referents. Phenomenological referents acquire different meanings, relations and properties according to the model they belong to. 6. Translatability of phenomenological referents’ meaning 6. Translatability of phenomenological referents’ meaning. There are no translational rules for the meaning of phenomenological referents from one theory to another as described in 1, 2 and 3.

10. Models and profiles of chemistry students for the structure of matter and its incommensurability In order to know the different models of the structure of matter held by the undergraduate chemistry students, we applied two questionnaires and an interview. The population was a sample of Chemistry students composed of 106 students (25 students in first semester, 21 in third, 16 in fifth, 21 in seventh and 23 in ninth) at the Faculty of Chemistry of the Universidad Nacional Autónoma de México.

11. Models of the structure of matter used by chemistry students Analyzing students’ answers, we could establish five models of the structure of matter: Continuous (C) Substantialist (S) Molecular I (I) Molecular II (II) Electronic (E)

12. Continuous Model of Matter (C ) In this model we grouped together all the students’ statements that contain no reference to particles, atoms or molecules and where the conception of continuous matter is explicitly stated.

13. Substantialist Model (S) Students explicitly involve particles. The particles conserve the same macroscopic properties of the substance. The particles have movement and properties such as variable size, colour, or phase.

14. Molecular Model I Is formed by particles. Molecules in the gas phase are free to move, they have velocity, kinetic energy and collide with each other continuously. External agents, such as heat cause interactions among the particles by modifying their spatial arrangement or movement.

15. Molecular Model II Is formed by particles and these interact through electrostatic forces. Interactions are intrinsic to the particles (not caused by external agents).

16. Electronic Model (E) Matter is formed by atoms which contain electrons. Macroscopic changes are determined through electrical forces between atoms and electrons and variations in electron energies.

17. Incommensurability of the models The analysis is performed between models, not between concepts.

18. Example. Substantialist model vs. Molecular model I 1. Identity and overlap of basic theory or model concepts. In Molecular model I, all particles have movement and elastic collisions. In the Substantialist model, particles have properties inherited from the substance (form, density, weight, etc.). Therefore, there is no possibility for identity or inclusion.

19. ModelsCSIIIE CMinimum incommensurabilit y Incommensurable SMinimum incommensurabilit y Incommensurable I IIIncommensurable Commensurable EIncommensurable Commensurable

20. Students’ model profiles In the figure we present the model profile of a group of students from different semesters. This group was determined from a hierarchical cluster analysis.

22. Final considerations and implications for teaching The profile presented reasserts the fact that students, even those who are actually pursuing a degree in chemistry, have not developed a unique model to represent the structure of matter. On the contrary, they have constructed a set of diverse models that are applied according to the context of the question (Flores et al. 1999; Ivarsson 2002).

23. Incommensurability does not imply that there is a contradiction between different models; rather, that there is a problem of non- translatability. Students do not recognize contradictions between their own models because they are not aware of the need to establish the core differences between their models.

24. If some contradiction exists between core concepts in the same model, the students are regularly able to recognize it. Recognizing non-translatability between models as well as the foundational elements of every model is not an easy task and demands a deep analysis of the limits and scope of each model and need the teacher assistance.

25. The fact that students have constructed different models, denotes that they are able to interpret several phenomenological referents at different levels, showing progress in the comprehension of the structure of matter. We consider that it might be helpful for students to be aware that in the history of scientific knowledge incommensurable models were built.

26. Why do students build such incommensurable models? This is an issue that needs to be investigated.