Released: September 2014
Welcome to Fostering Mathematical Thinking in the Middle Grades, another in the Casio series of books designed to promote meaningful discourse and deeper understanding of ideas in the mathematics classroom. Though the investigations in this volume are designed for the middle grades mathematics classroom, the sample solutions are written for the middle grades mathematics teacher, not the student. In addition to providing sample solutions, we have included discussion of areas of frequent difficulty, connections to other ideas, and suggestions for approaches to classroom discourse. For those who have not studied mathematics in depth or for whom mathematics was taught as a "bag of tricks" or procedures to be mimicked, the investigations may challenge your own thinking about what mathematics is and what our responsibilities to our students truly are. We hope you embrace this challenge.
The investigations are based on the Common Core mathematical standards. The focus is on expanding students' understanding of the number system, ratios and proportional relationships, expressions and equations, and statistics and probability. Though the PRIZM has outstanding geometric capabilities, we did not target the geometry in this volume.
Most of the "big ideas" contained in the Common Core Math Standards for grades 6-8 are addressed in these investigations. However, in addition, teachers will likely wish to create their own mini-investigations, both to promote facility with the calculator and to help students learn to think and explore for themselves, albeit on a smaller scale. No matter the investigation, 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.
In a similar vein, we encourage teachers to view contextual situations as a means to aid students as they grapple to make sense of the world. Too often, context suggests the dreaded word problem, an overly complicated context that confounds learning as students try to apply mathematical ideas they are only just learning. Instead, teachers should employ context to help students build on their own knowledge and experiences; by investigating and talking with friends, students can develop deeper understanding and see mathematics as sense making. One of the many challenges that teachers face is finding tasks that will engage students and that they can connect to, and yet perturb them sufficiently so that they create new knowledge for themselves. We hope and believe this book will help in this process.
We caution teachers that the investigations are not written as lesson plans. While we hope that the investigations can be the foundation for many worthwhile lessons, teachers must determine their specific objectives and assessments, as they are in the best position to know what will best meet their students' needs. Teachers may wish to use only parts of some of the investigations and break others into multiple lessons. Furthermore, some investigations require more direct guidance than others. Again, teachers know best what their own students need.
Fostering Geometric Thinking with CASIO Technology is a research-based, university-developed book that provides classroom teachers with meaningful investigations using the PRIZM.
- Math Investigations
- PRIZM-based instruction