Much of the classroom time in a MathScape lesson is devoted to one of two things: student exploration or discussion. Because one of the underlying principles of the curriculum is that students learn best when they are most actively engaged, each lesson places a strong emphasis on giving students time to work individually or in groups on a particular investigation.
The expectations is that students will work actively to learn mathematics, rather than being more passive recipients of information from the teacher. The teachers' crucial role is then to set students up for investigations, to facilitate those investigations, and most importantly, to bring the class back together to help students make sense of the mathematics they encountered in the investigation and informally assess their understanding. In the context of these investigations and discussions, teachers have an opportunity to teach certain terminology, skills and procedures that are linked to the investigation. In this way, MathScape makes room for students to incorporate their own mathematical thinking and experiences with necessary information the teacher needs to impart. Through the use of these materials, teachers can foster a strong learning community among students.
The investigations in the MathScape curriculum are designed to:
Each investigation in MathScape is followed by a discussion. This discussion is intended to provide an opportunity for students to formulate, extend and express their understanding of the mathematics. The discussion provides an opportunity for teachers to not only assess whether the goals of a lesson have been attained but to draw out and illuminate the point of having spent time in this investigation and to plan future instruction.
In completing an investigation, students experience mathematical information and processes. The follow-up discussion, though, is an essential tool in helping students to synthesize that information and to make connections to other contexts and topics. In fact, for many learners, an opportunity to discuss the experience fosters understanding and retention.
The investigations in MathScape are designed to address the diversity of learners that exist in any classroom. Investigations begin with a simple problem or task that is designed to be largely concrete and straightforward. The goal is that any student can get started with the task.
For example, in the grade 6 unit The Language of Numbers, students begin the unit by constructing a "mystery device," a circular "abacus" of sorts with both large and small beads on spokes. Students are asked to assign values to different beads and then to use their mystery device to make the numbers 7, 24, 35, 83 and 112. Investigations then build to require more sophisticated mathematical thinking by the end of the lesson. With the mystery device, students go on to design and test out a number system and begin to discuss properties of their number system with questions like: does position of the beads affect the value of their number?; is there a way to show zero?; is there more than one way to make particular numbers in your system? and other questions. Some students develop a very straightforward system while others develop very mathematically efficient systems that involve both positive and negative numbers. Investigations are designed to allow students to explore a topic at different levels of complexity. One student in a 6th grade classroom came up with an addition algorithm for adding positive and negative numbers to arrive at a total, and was unknowingly previewing the rules of integer addition. Anecdotes from teachers of MathScape are full of similar stories of students of all ability levels posing their own questions that they explore in the context of some of the investigations and making interesting and mathematically relevant discoveries.
The MathScape curriculum was designed with the view that the success of mathematics education depends upon the interaction of the teacher, the students, and the curriculum materials. While the curriculum can contain good content and well-designed lessons, only the teacher knows the individual students, can monitor their day-to-day progress, create a productive classroom atmosphere, engage students' interest, and adjust lessons to his or her own students. Since we believe that the success of the curriculum depends completely upon the teachers, an important goal of the materials is to convey to teachers a vision of an effective MathScape classroom. The Teacher's Guides were designed to provide ideas ad recommendations for the teacher and to provide extensive support for teachers, knowing that, in many cases, the curriculum was asking them to teach new content and to employ new approaches in their classrooms.
The Math Background for each phase of each unit is designed to more deeply explore the underlying mathematics content in each phase. These entries are intended to extend and deepen teachers' thinking about the mathematics.
In addition, the lesson plans and teacher support materials were designed to help teacher to do the following: