July 1999

Elegance. Culture. Habits of mind. Such phrases are usually reserved for literature, philosophy, or fine arts. But in the case of EDC's newest curriculum, they describe geometry. While covering the basics of high school geometry, Connected Geometry discusses ways to build elegant bridges among mathematical ideas, create a lively culture of mathematical investigation, and develop students’ abilities to inquire and think.

Connected Geometry, developed with funding from the National Science Foundation, is published by Everyday Learning Corporation. The fruit of six years of development and field testing, the package includes a textbook, four resource guides, and a CD-ROM. It has been piloted in working class and professional communities; in urban, suburban, and rural settings; in tracked and untracked classes; in classes where limited English-proficiency students are the majority; and with honors-level classes.

Connected Geometry is not only a course of study but an approach to developing a "mathematical culture." As the developers say, "you need more than a good text with good problems." What you need is a culture of investigation, that:

• values deep thinking about mathematical problems
• gives students the time and the space to do that thinking alone as well as with others
• encourages them to articulate their ideas clearly, give precisely reasoned arguments, and discuss and refine their ideas with other students

Along with correctness and completeness of solutions, curiosity and playfulness are valued highly. "Connected Geometry is different from traditional high school geometry. We want kids to work on problems, but we also want them to understand what mathematics is all about," says Michelle Manes, a developer. "Students need to feel safe to expose their thinking and admit their puzzlement," say the developers. The curriculum honors partial solutions, new ideas, and different solutions to a single problem. "Students are groomed to be thinkers: pattern hunters, experimenters, describers, tinkerers, inventors, visualizers, conjecturers, guessers, and seekers of reasoned argument and proof," says Al Cuoco, one of the developers.

The curriculum aims to make explicit what is too often implicit. "What’s really important in mathematics is often left unsaid: the thinking that goes into it," says Cuoco. Lessons in Connected Geometry are full of prompts such as, "Describe your way of approaching this problem," or "How are problems 1 and 2 related?" or "Explain how you drew the picture." Each lesson closes with a section called "Take It Further," where students can extend their learning.

Mathematical Habits of Mind

In its six modules, Connected Geometry focuses on developing 10 mathematical habits of mind, among them:

• Picturing the visible, picturing the invisible
• Analyzing and interpreting a figure
• Using precise language to describe and analyze
• Tinkering with problems
• Mixing deduction with experimentation
• Looking for invariants
• Reasoning about processes
• Reasoning by continuity
• Proving
• Proof as a research tool
• Using different systems

During pilot testing, developers discovered that the  nontraditional and nonlinear approach of Connected Geometry could sometimes be difficult for teachers. In the typical geometry curriculum for example, a textbook introduces theorems and then provides students with problems that illustrate or use the theorem. In Connected Geometry, theorems are often introduced later in the process—after students have wrestled with the mathematics underlying the theorem. To help teachers understand and use this approach, the developers adapted the teachers guide to provide more information and guidance about the progression of the activities and the relationships within the mathematics.

Field Tests

As the curriculum developers encouraged teachers to experiment with their approaches to geometry instruction, the teachers provided developers with valuable insights about what worked in the classroom—and what did not. In one case, the developers reluctantly decided to scrap an entire unit when teachers reported that the content failed to engage their students.

Students participating in the field test had the opportunity for a rare—and valuable—glimpse into the process of curriculum development and field testing. Michelle Manes remembers receiving a call from a field test teacher who was struggling with one of the activities. Halfway through the call, Manes realized that the teacher was calling her directly from his mathematics classroom, in the midst of the class. "It was really wonderful," she recalls. "The students saw the teacher explain what he couldn’t understand, and heard the questions he asked. They saw him draw pictures of what he was hearing," says Manes. "Usually students see only the results of the thinking, not the process."

In other cases, student contributions found their way into the final curriculum. In a unit of "Optimization," the curriculum poses the following problem for students:

Suppose your tent is on fire. What is the fastest route to run to the river with the bucket and back to the tent with water to put out the fire? One student took the situation a step further by raising this question: Can you run faster on the way to the river with an empty bucket than you can on your way back with a full bucket? "That whole extension got added to the lesson," said Cuoco.

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For more information, visit the Connected Geometry Web site or EDC's online catalogue.