Inquiry Based-Computational Experiment, Acquisition of Threshold Concepts and Argumentation in Science and Mathematics Education

Introduction Inquiry based learning and modelling in science and mathematics education Inquiry based learning has officially been promoted as a pedagogy for improving STEM learning in many countries (Bybee et al., 2008) and can be defined as "the deliberate process of diagnosing problems, planning experiments, distinguishing alternatives, setting up investigations, researching conjectures, sharing information, constructing models, forming coherent arguments, collecting and analyzing data" (Bell et al., 2004). Significant parts of scientific research are carried out on models rather than on the real phenomena because by studying a model we can discover features of and ascertain facts about the system the model stands for. This cognitive function of models has been widely recognized in the literature, and some researchers even suggest that models give rise to a new form of reasoning, the so-called "model based reasoning" (Magnani & Nersessian, 2002) while modelling ability is also associated to model-based reasoning (Chittleborough & Treagust, 2007). It is well known that scientific theories are developed through a process of continuous elaboration and modification in which scientific models are developed and transformed to account for new phenomena that are uncovered. Similar processes are involved in students' learning of STEM concepts when they develop conceptual models (e.g., Bell et al., 2010). In a similar fashion, inquiry based learning requires from students to make successive refinements to their mental models in order to transform them to conceptual models that align to scientific theories. The Computational Experiment (CE) Computational Science can be considered as a third independent scientific methodology (the other two are the theoretical science and the experimental science), has arisen over the last twenty years and shares characteristics with both theory and experiment, while it demands interdisciplinary skills in Science, Mathematics, Engineering and Computer Science. According to (Landau, Paez, & Bordeianu, 2008) Computational Science focuses on the form of a problem to solve, with the components that compose the solution separated according to the scientific problem-solving paradigm: (a) Problem (from science); (b) Modelling (Mathematical relations between selected entities and variables); (c) Simulation Method (time dependence of the state variables, discrete, continuous or stochastic processes like e.g., Monte Carlo simulation); (d) Development of the algorithm based on numerical analysis methods, (e) Implementation of the algorithm (using Java, Mathematica, Fortran etc.); and (f) Assessment and Visualization through exploration of the results and comparison with real data in authentic phenomena. In this framework, being able to transform a theory into an algorithm, requires significant theoretical insight, concrete Physical and Mathematical understanding as well as a mastery of the art of programming. In order to describe inquiry based learning as a search process, (Klahr & Dunbar, 1998) introduced two spaces, the hypothesis and the experimental spaces. (Psycharis, 2013) added one more space, the prediction space, in order to address the introduction of modelling and the comparison of data produced by the model with real data. In the "prediction space" the Computational Science methodology is implemented through the development of models of simulations that favor the so called "Computational Thinking (CT). " According to (Psycharis, 2013), the three spaces of the Computational Science methodology should include issues form (CT), namely: (a) logically organizing and analyzing data; (b) representation of data through abstractions such as models and simulations; and (c) algorithmic thinking. In this context, the three spaces of the Computational Science methodology include: * The hypotheses space, where the students, in cooperation with the teacher, decide, clarify and state the hypotheses of the problem to be studied, as well as the variables included in the problem and the possible relations between the variables. …