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Discovery Approach

The Discovering Algebra, Discovering Geometry, and Discovering Advanced Algebra textbook series covers the topics offered in traditional algebra, geometry, and advanced algebra courses. But the Discovering Mathematics approach may be different from what you experienced when you studied these courses.

In the past, students were asked to spend a lot of time in symbolic manipulation before they were given the opportunity to develop a solid understanding of what they were doing. For example, you may recognize this scenario: After going over homework, your teacher showed a new type of problem and a method for solving it. You worked alone with pencil and paper to practice solving problems of that type. For homework there were more problems of the same type. The next day the class went through it all again with a new type of problem. At some point there was a test with a bunch of problems on it. You had to remember the methods and figure out what method to use for each problem. If you did well on all the tests, you were “good at math.” If you didn’t do well, you may have thought you “just couldn’t do math.”

Many students cannot succeed in such an environment. The teacher and text cannot furnish enough examples to carry over to a new situation or problem. As a result, many students are limited, unable to do more than mechanical manipulations. They don’t know when to apply what problem-solving strategy. Even students who pass are reluctant to continue on in mathematics.

But all students can learn math better. Discovering Mathematics works to have all students reach deep understanding of math through investigating interesting and novel problems in cooperative groups using technology appropriately and practicing skills.

All students: The Discovering Mathematics authors know from their own teaching experience that all students can experience more success in mathematics. When the focus is on understanding concepts and problem-solving strategies instead of just memorizing formulas, all students can be more successful. To say that all students can learn math does not mean the courses have been watered down. In fact, even very successful math students will find they are challenged, learn more, and remember longer with the Discovering Mathematics approach. That’s because the concepts and methods are not isolated from real-world applications, and the mathematics that students study is closer to what is needed by both employment-bound and college-intending students.

Deep understanding: In your own math classes, you may have been told “Just do it; don’t ask why.” But there are logical reasons behind math methods and ideas, and the people who understand these reasons succeed at math. Discovering Mathematics books help more students understand the reasons. Because the concepts make sense to students, they remember the methods and can apply them to new problems. To help with that understanding, Discovering Mathematics books offer a more visual approach and acknowledge the need for a gradual development of mathematical ideas. Students are asked to demonstrate comprehension orally and in writing. Understanding the math can make it more fun and increase the chance that students will use math in their lives.

Investigating: The way something makes sense to one person, though, might not make sense to someone else. The heart of Discovering Mathematics lessons is the investigations. Students work together on the activity. They each develop their own understanding and benefit from sharing ideas and suggestions offered by others. Students learn there are many approaches to solving problems. They also learn they are individually responsible for describing what they have learned.

Interesting problems: Students are more interested in class if the problems they investigate are related to the real world. Many of the hands-on investigations involve problems that students might see in their lives outside school.

Novel problems: In life we all need to be good at solving problems that don’t exactly fit into a model we know. To help prepare students to use math in their lives, many investigations in Discovering Mathematics pose problems that students haven’t already been told how to solve. Thus they learn skills of problem solving, rather than learning only how to solve particular types of problems.

Cooperative groups: Students are not expected to do all this learning by themselves. Many students make sense of mathematical ideas best in interaction with other people. They think best out loud, or they get ideas from others. And they understand better from seeing other students’ viewpoints. The Discovering Mathematics series supports cooperative group work, in which the teacher works as a partner to student groups. Group work not only helps students learn better; it also teaches essential teamwork skills.

Technology: Computers and calculators surround us, so working with them in these classes teaches students skills that will be useful later on. Technology also helps keep students interested. Used appropriately, technology can make mathematics more visual, more logical, and more fun. Most importantly, technology tools allow students to investigate many more situations than they can explore by hand, thus helping them see patterns that lead to deeper understanding of concepts. Technology helps shift the responsibility of learning mathematics to the learner.

Practicing skills: As students investigate, they are practicing basic skills. After students have figured out a concept, they will continue to apply their skills to additional practice problems.

The Discovering Mathematics books support an approach to math that brings about better understanding of concepts and skills. Instead of solving one type of problem after another, students engage in investigations, examples, and exercises that allow them to build up their own bank of skills and concepts. Students learn to describe how and why something is true. Instead of working alone, students bounce ideas off their peers. Because they have been personally involved in the development of skills and concepts, students can successfully approach test problems even if they have forgotten a particular method or formula.

On other pages of this site, you can see advice for working with your student. You can also find details about how each book in the Discovering Mathematics series carries out these ideas.