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Mathematics: What's the Big Idea?

Series Overview



Most math teachers have read, talked about, or heard about the NCTM Standards and, in general, what they recommend. Many of you are changing the way you teach mathematics or considering how you might. Change is always intimidating, especially if you have worked hard on the way you currently teach. This workshop series will support your efforts to change in three ways: (1) by helping you learn some mathematics yourself in new and exciting ways; (2) by suggesting how you might teach mathematics in new and exciting ways; and (3) by showing you some classrooms that are changing, and letting you watch and talk to the teachers who work in these classrooms.

Some of you probably didn't enjoy your mathematics education. For many people, math class involved memorization, timed tests, and quiet, solitary work. The picture of successful math classrooms today is different. Students may work in groups, find several different ways to solve a problem, create their own representations of mathematical situations, do math in conjunction with language arts or science, and share their strategies with the rest of the class. They work with geometric figures, collect and analyze data, make graphs, and write their own stories to give meaning to numerical relationships.

In this workshop, we hope that you will enjoy doing mathematics, that you will learn about yourself as a learner and as a teacher, and that you will work with colleagues both at your school and at other sites in discovering the "Big Ideas in Mathematics."

This workshop series is based on an approach to teaching and learning that emphasizes students doing mathematics themselves and constructing their own mathematical strategies, rather than following rote recipes that someone else has designed. Much research has shown that students who approach math in this way develop a deeper and more flexible understanding that supports their ongoing mathematical learning. Students learn to describe their strategies to their peers and to prove that their approach is sound.

This approach to learning math does not mean that students neglect basic mathematical relationships such as addition and multiplication. Rather, students begin by establishing a basic understanding of an operation, often through the use of manipulatives. They then expand their knowledge by looking at more and more complex number relationships, explore the connections among these relationships, and develop a fluency that enables them to work with more complex mathematics. If students develop this kind of conceptual understanding, they will learn and retain the basic facts, as well as develop the foundation for more complex mathematics.

What kinds of problems fit with this approach? A good problem leads students to consider important concepts in mathematics. It should have more than one approach, so that students can share a variety of strategies and so that students with different mathematical strengths are able to approach the problem. While it is important for a problem to be somewhat open-ended, it is also important that there be enough structure to give students direction. A good problem is an opportunity for students to explore and use a variety of forms of representation (for a single mathematical model, equation, relationship, etc.) including verbal, pictorial, numerical, tabular, graphic, and symbolic. During these workshops, we will include as many such problems as possible to give you a chance to see how this works.* For example, as we interact with sites around the country, we will see diverse approaches to both mathematical and pedagogical problems.

We will also be exploring assessment. By assessment, we mean discovering and evaluating what students know at a given time and using that information to plan future lessons. (The problem of actually assigning grades is separate from assessment in this discussion.) With problems that invite individual approaches and discussion, teachers can learn much more about their students than when strictly numerical tests provided most of the information. With such rich information, ranking students is not the point; mathematical understanding is much more complex than a single grade can convey. Assessment should take place within the context of actual lessons, as well as during explicit assessment problems. In this workshop series, we will consider opportunities for assessment within specific content areas.

Technology-calculators and computers-has an important place in the math classroom. The key to solving a numerical problem with a calculator is knowing what numbers to enter, what operation to perform, and how to interpret the answer. Students will work with calculators all their lives, no matter what career they choose-it's important that they don't just take the answer that appears on the calculator at face value without considering whether or not it is reasonable.

Our hope is that by participating in this interactive series of math workshops you will gain a better appreciation for the meaning, value, and importance of mathematics in school and in society, and you will have a better idea of where mathematics came from and, perhaps more importantly for your students, where it is headed.



-- The Content Guides






* To solve a good problem, you need the right tools. We will be asking you to come prepared to do activities that require some supplies. For every workshop you should have available pencils, paper, scissors, rulers, calculators, tape, and a variety of colored markers. Specific workshops may require a few additional supplies.



Mathematics: What's the Big Idea?

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