**Modellus** is a freely available application that enables students and teachers (high school and university) to use mathematics to create or explore models interactively.

Modellus is used to introduce computational modeling for allowing an easy and intuitive creation of mathematical models using just standard mathematical notation, for having the possibility of creating animations with interactive objects that have mathematical properties expressed in the model, for allowing the exploration of multiple representations and for permitting the analysis of experimental data in the form of images, animations, charts and tables. Modellus focuses on modeling and on the meaning of models.

It has been published in different languages (Portuguese, English, Chinese, Greek, Spanish, ...) and it is used in all over the world with samples ranging from Physics to Mathematics, going through Mechanics, Chemistry, Statistics, Algebra, Geometry, among others.

### Features

- Creating and exploring mathematical models is a fundamental task in many sciences. Modellus can help students experience minds-on learning while creating, simulating, and analyzing mathematical models interactively on the computer.
^{} - To set up a model, users enter conventional mathematical equations and expressions (functions, differential equations and iterations). No programming language or special commands are used.
- To create an animation of the model: choose objects, like images or vectors, and assign properties, like position or size. The interface includes charts and tables.
- Modellus can be used as an exploratory (the user explores models made by others) or as an authoring environment. Models or any other window can be hidden and/ or protected.
- Modellus comes with pre-constructed interactive examples that illustrate many scientific concepts. There are also many others available on the website or from curriculum projects.
- Modellus was designed by science education researchers to offer a software package with pedagogical basis. It can help teachers and students become familiar with the mathematics (the language of Nature, according to Galileo).

### Background

Physics and mathematics is a key subject in the science and technology education curricula. In spite of such central role, introductory mathematics and physics continues to be extremely difficult for students both at the secondary and at the university level. Due to a lack of understanding of fundamental concepts in physics and mathematics, the number of students unable to pass on course examinations is usually very high. What is worse is that many of those students that eventually succeed also reveal several weaknesses in their understanding of elementary physics and mathematics.^{}

A solution for this problem requires changes in the processes by which exact sciences are taught and learnt. In recent years (2000 to 2011), many results of physics education research have shown that the process of learning is effectively enhanced when students are involved in the learning activities as scientists are involved in research. Scientific research in physics, chemistry or other exact science is a dynamical process of creation, testing and improvement of mathematical models that describe observable phenomena. This process is an interactive blend of individual and group reflections based on a continuously evolving and mutually balanced set of theoretical, computational and experimental elements. It is from this cognitive frame of action that an inspiring understanding of the laws of the physical universe emerges. The process of learning exact sciences turns out to be more successful in research inspired environments where students are helped to work as teams of scientists do. In this kind of class environment, knowledge performance is promoted and common sense beliefs as well as incorrect scientific notions are fought. ^{}

In the scientific research process, computational modeling plays an important role in the expansion of the mathematics cognitive horizon through more powerful calculation, exploration and visualization capabilities. Modeling with computational methods and tools is an important aspect of research inspired learning environments. In this context, it is crucial to achieve an early integration of scientific computation in a way that is balanced with theory and experiment. Only then the learning and teaching processes can be in phase with modern scientific research, where computation is as important as theory and experiment, and with the rapid parallel development of technology.

Modeling physics, chemistry, mechanics (and other exact sciences) in computer learning environments started with an emphasis on programming languages. Using, for example, Fortran, Pascal and Python, this approach requires students to develop a working knowledge of programming. The same happens with scientific computation software such as Mathematica or Matlab. To avoid overloading students with programming notions and syntax, computer modeling systems such as the Dynamic Modeling System, Stella, Easy Java Simulations and Modellus were developed to focus the learning activities on the understanding of the concepts of physics and mathematics. ^{}

In spite of these important advances, a proper integration of computation with theory and experiment in physics, mathematics and other exact sciences curricula and learning environments has not yet been achieved. Activities based on computational modeling with Modellus in general physics courses have been given to engineering students. During computational modeling classes, activities were successful in identifying and resolving several student difficulties in key physical and mathematical concepts of the course. Of fundamental importance to achieve this is the possibility to have a real time visible correspondence between the animations with interactive objects and the object’s mathematical properties defined in the model, and also the possibility of manipulating simultaneously several different representations. Students react positively to the new component of the courses, showing clear preference for interactive and exploratory group work. Students have used Modellus in materializing abstract concepts ^{}in the learning process of mathematical and physical models.