Overview

For many experienced and prospective teachers, tasks like those in High School Mathematics at Work pose several inter-related challenges involving curriculum, pedagogy, and assessment.

• In planning the class: How can I tell if a task is appropriate for my students? How would such tasks fit in my curriculum? After choosing a task, what are the mathematical concepts and big ideas that can be approached with the task?

• During class: How do I get students working on the task I've chosen? What can I expect from classroom discussion? How can I engage all students in the big mathematical ideas?

• After class: What can I expect from student work? What should I expect from student work? How should I provide feedback?

     These questions speak to the broad demands that today's students and curricula place on teachers. To respond adequately to these demands, teachers must be very resourceful and must have the skills and inclinations to create an intellectual community in their classrooms. What is needed is teachers who are mathematically confident and have the tools to learn mathematics as they need it, so that they and their students may thrive when either the curriculum or the students take them into uncharted territory. Furthermore, teachers need professional support in creating learning communities of teachers in their schools, districts, and states.

     Working with both preservice and inservice teachers, the authors of the essays in Part Four find that tasks like those in this document have changed teachers' ideas about students' capabilities, about how a curriculum might be organized, and about what it means to do mathematics. In each case, such change requires time and support.

     The Professional Standards for Teaching Mathematics (National Council of Teachers of Mathematics, 1991) acknowledges the important role that teachers have in choosing tasks in their curriculum. Under the heading "Worthwhile Mathematical Tasks," it asserts that

     The teacher of mathematics should pose tasks that are based on--

• sound and significant mathematics;

• knowledge of students' understandings, interests, and experiences;

• knowledge of the range of ways that diverse students learn mathematics. (p. 25)

     In her essay, Glenda Lappan suggests that "teachers are architects of curriculum," for what is learned depends upon the context in which it is taught. She acknowledges that the use of complex problems creates more complexity in classrooms, and she notes that if preservice teachers work through complex problems themselves, they receive some of the background and confidence they need to handle such complexity. After describing problem-centered teaching and some of the issues that it raises, Lappan suggests that "teachers will find that learning alone is unlikely to be as powerful as engaging in dialogue with other teachers."

     Like Lappan, Gilbert Cuevas acknowledges that with scant experience solving complex mathematical tasks, many preservice teachers are uncertain about using such tasks in their classrooms. He presents five principles for the preparation of teachers, such as providing "opportunity for reflection about their tasks and their implementation with students," and emphasizing communication, discussion, and orientation toward problems.

     Paul LeMahieu and Marshá Horton note that assessments alone are not effective agents of educational reform. When extended, open-ended tasks are included in assessment, however, there is an opportunity for a different and powerful role, if teachers are involved in the evaluation of student work responding to these items. LeMahieu and Horton discuss how teachers develop consensus about quality and rigorous expectations for quality through discussions of student work on assessments. Furthermore, they note, inservice teachers' expectations for students change when they participate in such scoring.

     The tasks in Part Four might be used in the professional development of teachers, both as a site for discussion of student work, as suggested by LeMahieu and Horton, and as a complex task for their own exploration, as suggested by Lappan and Cuevas. Because these tasks are more open-ended than most in previous sections, the mathematical analysis sections do not include complete solutions, but instead suggest some of the mathematical and pedagogical issues and some sources of data or other useful information. Estimating Area (p. 145) brings to light mathematical ideas such as the distinction between distance, area, and volume, scaling factors, and estimation, possibly leading to calculus ideas such as limit and integration. Like the tasks in Part Three, this is a task that may be fruitfully revisited several times in a student's career.

     Timing Traffic Lights (p. 147) concerns a workplace situation usually considered by town and city planners. The ideas are similar to the ideas behind scheduling trains, airplanes, and canal-boats. The potential interest for students is that city planners' solutions (both the good and the bad) can be seen in everyday life. This task may be used to explore mathematical ideas such as distance, rate, time, velocity, modeling, and representation.

     Buying a Used Car (p. 153) is an everyday situation about which people do not often think mathematically. Yet, by considering in the analysis not only estimates of the purchase price and repair costs but also insurance, taxes, depreciation, interest on a loan, and inflation, there is high potential for rich mathematical discussion.

Reference

National Council of Teachers of Mathematics. (1991).

Professional standards for teaching mathematics. Reston, VA: Author.