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	The Role of Complex Mathematical Tasks in Teacher Education
	Gilbert J. Cuevas
	University of Miami

The NCTM Curriculum and Evaluation Standards promotes increased emphasis on problem solving, mathematical communication, thinking, and reasoning. The resulting gap between a "traditional" and a reform-based approach to mathematics instruction poses a challenge for teacher educators as they develop strategies to help both preservice and classroom teachers implement the Standards. To help teachers become thoroughly familiar with the kinds of instructional activities that reflect the Standards, mathematics educators in teacher preparation programs can focus on processes similar to those recommended for students. The use of complex mathematical tasks can play a very important role in closing the gap between what teachers have traditionally done in the mathematics classroom and the approaches emphasized in the Standards.

     A primary purpose of such tasks is to engage students in meaningful and worthwhile activities that lead to understanding of mathematics as a subject matter that has real-life applications. Complex mathematical tasks have several noteworthy aspects. First, they can be thought of as instructional activities that focus on specific sets of ideas and skills. Second, they demand active involvement by both students and teachers. Finally, such tasks should provide opportunities for expansion and reinforcement of learning. This learning should focus on the exploration of concepts being addressed in class, on the reinforcement of skills and ideas, on connections between ideas, and on the promotion of student communication through discussion, justification of solutions, and explanations of mathematical processes.

     My personal experience in teaching mathematics methods courses has convinced me that the use of complex tasks helps teachers become oriented toward a Standards-based approach. Such tasks encourage classroom teachers to explore instructional strategies that reflect a problem-solving approach to mathematics education. In addition, they help teachers to see the value of classroom discourse for student learning and also to develop strategies that will assist students in improving their communication skills. Just as students need time to acquire the knowledge, strategies, and skills needed to deal effectively with tasks at different levels of complexity, teachers also need opportunities to develop a mental picture of how such tasks can be integrated into typical classroom activities.

     Teachers who have limited experience with complex tasks regularly raise certain questions primarily concerning their desire to know how to implement these instructional activities in their classrooms effectively and how to help students get the most out of the experiences. Teachers ask questions such as the following:

• How can I know if the level of the mathematics addressed by the task is too difficult for my students?

• How can I help my students see that a task may be approached in more than one way?

• In what ways can I incorporate complex tasks into the curriculum that I am supposed to teach?

• How do I grade students on this?

• How do I make sure that all the skills and concepts in the curriculum are addressed?

     Other questions have focused on helping students learn from the tasks:

• How do I know that the approaches the students have used are appropriate and lead to correct solutions?

• How do I provide feedback to students on their performance?

• What do I do if students cannot begin the task or are not able to describe what they have done, to draw conclusions, or to justify their solutions?

     In the search for the answers to these questions I have found five guiding principles for the preparation of teachers. They are as follows.

     Model a problem-oriented classroom environment. In methods courses for preservice teachers, complex tasks and "big problems" can be used throughout. They can be integrated into classroom activities to begin the study of particular mathematical topics. For example, the following task introduces the number concept:

Suppose a friend told you she had a suitcase large enough to hold one million one-dollar bills. She asked you to help her bring the suitcase to a bank. Could you lift such a suitcase?

     Complex tasks also can be used to summarize and reinforce ideas and concepts dealt with at different times during the course. Some tasks are used for assessment purposes. Preservice students work in small groups on these tasks and, upon completion, present their solutions to the class. Members of the class provide the group with evaluation feedback through the use of a predetermined rubric.

     Provide experiences with tasks at all levels of mathematical complexity. Some secondary students will find certain tasks difficult for reasons ranging from inexperience with the activities involved to lack of appropriate mathematical background. Teachers should develop strategies to identify tasks that provide a challenge to students without being impossible to complete. Such strategies and sensitivity to the level of task difficulty can be acquired through exposure to a variety of mathematical tasks of different levels of complexity. Preservice as well as classroom teachers need to have numerous opportunities to engage in mathematical tasks, to analyze their mathematical content, and to develop solution strategies. Throughout these experiences, teachers must be guided to develop a framework by which they make decisions about the mathematical content of a task and its difficulty level. I have found that this is best accomplished when teachers reflect on and discuss the tasks.

     Promote discussion of mathematical tasks, their content, and solutions. I have found three successful approaches to promote classroom discourse: small-group exploration and discussion of a given task, individual or group presentations of solutions to the whole class, and class discussion of students' approaches and solutions to tasks. For the latter, I present the class with samples of student work that I have collected in the local schools. The preservice teachers first analyze the work individually, then discuss it in small groups, and, finally, present their comments to the whole class. These experiences provide teachers with opportunities to explore a variety of student approaches to the tasks--some more effective than others--and to identify errors in mathematics or reasoning. Also, these exercises allow teachers to construct feedback as if they were communicating with the students whose work was examined.

     Emphasize development of communication skills. Communication skills and the promotion of classroom discourse in mathematics should be approached developmentally. We cannot assume that if we give students an unstructured task such as "use data from current newspaper advertisements to examine the economics of buying a car versus leasing" that they will give complete explanations of procedures, solutions, and conclusions. Teachers should guide students in the development of mathematical communication until students achieve the skills and comfort level to communicate mathematical ideas effectively. A teacher using the buying versus leasing task might structure communication with a framework such as the following:

• Describe the facts stated in the newspaper advertisements.

• Describe the differences and similarities in the facts among these advertisements.

• Describe the factors you need to take into account to begin your comparison of buying versus leasing.

• Describe how you decide whether it is more economical to buy or lease a car.

• Write your conclusions and your reasons that support your conclusions.

     Provide opportunities for reflection about the tasks and their implementation with students. Teachers need time to reflect on the mathematical content, thinking and reasoning requirements, student solutions, and communication demands of each task. In this way, teachers will develop strategies to address the concerns posed earlier in this essay. In methods courses, I require students to write reflections about tasks they have completed. These reflections are then shared in small groups during class time. In workshops, I give participants opportunities to reflect on task features and possible implementation strategies.

     Rather than concluding with a personal thought on the role of complex mathematical tasks in teacher education, I will share a comment made by a teacher on this matter: "By working through the tasks, I became confident of my mathematical ability, developed ideas for using them with students, and I am now more sensitive to the difficulties and obstacles students may have in their learning of mathematics."

Gilbert J. Cuevas is a Professor of Mathematics Education in the School of Education at the University of Miami. He has directed a number of professional teacher development projects in bilingual, mathematics, and science education. He has also served as a member of the Mathematical Sciences Education Board.