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Welcome to Fostering Mathematical Thinking through Music, another volume in the Casio series of books intended to make mathematics intriguing in a way that engages them in meaningful, thought-provoking investigations that lead to deep understanding of key mathematical ideas.

Mathematics is everywhere. It involves a fundamental way of thinking that supports learning in every discipline, from physical sciences to social sciences, from physical education to psychological and social education, from language arts to fine arts. The mathematics classroom can and should be the most fascinating of all classrooms, as we can draw upon all other walks of life to lead students to develop their abilities to think and reason mathematically and to solve problems as they connect mathematical ideas to their experiences and prior knowledge. Sadly, the mathematics classroom that accomplishes this is the exception, not the rule. Despite the research on how we learn, despite the emphasis on inquiry-based instruction and learning, and despite the fact that so many people accept (erroneously) that they, along with others, simply aren’t good in math, the typical mathematics classroom remains one in which teachers demonstrate a skill, students practice it in an effort to automate it even without an understanding of the underlying ideas, teachers test the skills, and students forget what they have learned shortly after the lesson. Quite simply, we have to change.

So with so many disciplines to choose from, why is this book focused on music? Music is an area with intriguing connections to mathematics. Research is beginning to shed light on some of the cognitive connections that exist, and, in fact, many mathematics professors and teachers have excellent musical talent. In addition to being one of our passions, music also is an area of interest for the vast majority of students. Not only does it provide motivation for learning, it also provides a means to help even those who do not consider themselves overly musical to learn important ideas in a way that they will retain the ideas as they become stronger mathematical thinkers.

The investigations presented in this volume are compatible with both the processes and content espoused by the Common Core State Standards for mathematics. Our goal is to expand students’ understanding in a wide variety of topics, from fractions to algebra to trigonometry. Some of the investigations are intended for students as young as fourth or fifth grade, while others are intended for students perhaps in a pre-calculus class. Nevertheless, we believe the majority of investigations are accessible to, and would be valuable for, the majority of students.

The sample solutions are written for teachers. They are not meant to be an exclusive way for conducting the investigating, but instead suggest to teachers a possible way that they might guide their students, with discussion of the underlying ideas and some of the pitfalls that students might encounter. Thus, these investigations are not written as lesson plans. Teachers can take the investigations and structure them into lesson plans in a way that will be the most useful for their own students. They may choose to use only parts of some of the investigations, while with others they may want to extend the investigation over several class periods.

As we have discussed in other editions of the Fostering series, we believe for truly deep and meaningful learning to occur, students must first explore, trying to make sense of things for themselves, prior to an explanation; this, to us, is the heart of inquiry. In fact, the teachers with whom we have worked have heard us say “Explore before Explain” probably more times than we would care to admit. Because few of us were taught mathematics this way, this seemingly simple idea turns the traditional mathematics classroom on its head. It is not an easy transition for most teachers; it is much simpler (and often more comfortable) to tell students how to perform procedures.

At the heart of inquiry, we should begin with context. To many math students, context suggests the dreaded word problems at the end of each section of the text. Though we believe firmly that students should be challenged to solve difficult problems, we also believe that contexts that students understand and have experience with should be used so that students can connect what they’re learning to what they already know. In this way, the mathematics becomes more intuitive, and students are far more likely to retain the underlying ideas.

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