## Mathematical Modeling

INTRODUCTION

ABOUT THIS BOOK

This book is written primarily for teachers of mathematical modeling in upper secondary schools or in high schools. Students in a teacher training program at a university or studying mathematical modeling in an introductory course at the university may also want to explore the possibilities that GeoGebra can afford. The book was conceived from the standpoint of the Swedish curriculum, which regards mathematical modeling competence to be one of seven competencies that should be taught and assessed in upper secondary school.

As a school subject, mathematics is no longer only about calculation. Some parts of mathematics, of course, relate strongly to procedures and counting, but altogether this part of the curriculum has less emphasis today than it used to have. Today, mathematics is treated as a tool, as an aid, as a language, and as logic. The curriculum in many countries is nowadays expressed in terms of competency objectives. The competencies are general and not related to a specific mathematical content. Yet, the competencies are developed in levels by students' processing specific content. The modeling competency is one of these competencies that draw heavily on functions and differential equations.

Mathematical models and other mathematical representations such as diagrams, histograms, functions, graphs, tables, and symbols normally make it easier for abstract mathematical concepts to be understood and for other phenomena to be described in mathematical terms. Educators today are facing a world that is shaped by increasingly complex, dynamic, and powerful systems of information that are meet through various media. Being able to interpret, understand, and work with mathematical models and other complex systems involves important mathematical processes that become discernible and obvious when teaching mathematical modeling.

In mathematics education, as seen from the K-12 perspective, teachers work with different representations in order to help students understand mathematical objects and concepts. Models such as geometrical constructions, graphs of functions, and a variety of diagrams are used to introduce new concepts and to show relationships, dependency, and change. Mathematical models, structures, and constructions are also used in different scientific fields, such as in physics and the social sciences. To be able to construct, interpret, and understand mathematical models is becoming increasingly important for students all over the world.

Our main academic position is that once modeling competency is acquired in the classroom, all other competencies will be addressed automatically. With training in mathematical modeling, instead of always asking "Why are we doing this?" students will find classroom work to be interesting and related to reality, and then concepts, procedures, problem solving, reasoning, communication, and relevance will follow without much effort. If you, the teacher, try to do it the other way around, you may soon discover that in sticking with too many routine calculations you will end up without time to address the modeling and reasoning competencies.

There were some basic considerations that we needed to address in writing this text on mathematical modeling. We could have chosen to only focus on the process of constructing and developing models or instead on the evaluation of already produced mathematical models. We decided to try and address both situations in this book. However, for those of you teaching mathematical modeling in upper secondary school, it may be a good idea to start with existing and well-developed models. Then, as students become familiar with the mathematical modeling concept, they could be started on constructing their own mathematical models.

To place mathematical modeling into a particular branch of mathematics, one could consider it as applied problem solving using data