Isabel M. Araújo

Exercise Composition: From Environment Properties to Composed Problems

It is well known that (1) problems have a structure, (2) often this structure comes from its input, output or internal objects, (3) this structure organizes the problem resolution. Decomposition of a problem by its structure is a typical path in problem resolution. In this article we will do the inverse: composition of simple problems using structure-builder operators. We will start by (1) discuss problem structures, (2) select a set of important structure-builder operators, (3) study their properties, (4) apply them to a simple math case-study.

New algorithms for smart assessment of math exercises

This paper deals with the field of mathematics education where the aim is to generate exercises with randomized data. The process of exercise generation involves, first, identification of common errors that may be performed when solving the exercise, then, modeling of these errors by appropriate functions and recognition and distinction of errors through algorithms. Thus, although randomized, the exercise parameters must be chosen in such a way that it would be possible to understand the error the student committed and with it to guarantee a proper feedback on his answer. We call this process smart assessment. In this paper we present two algorithms for exercise generation admitting smart assessment.

Gröbner bases and mathematical exercises generation with nondetermined structure

Given a concrete scenario, there is frequently a huge quantity and variety of exercises that we can write about it. In this article, we present an algorithm which allows us to automate this process. To generate mathematical exercises, we start from the relations between the variables present in the scenario, we calculate the possible exercises, we analise the complexity of its resolution process and its viability. The algorithm uses Gröbner bases to determine if the exercise is solvable and a possible path of resolution. Based on the analysis of the obtained results, its possible to create an exercises base related to that scenario.