posing worthwhile mathematical tasks;

engaging teachers in mathematical discourse;

enhancing mathematical discourse through the use of a variety of tools, including calculators, computers, and physical and pictorial models;

creating learning environments that support and encourage mathematical reasoning and teachers' dispositions and abilities to do mathematics;

expecting and encouraging teachers to take intellectual risks in doing mathematics and to work independently and collaboratively;

representing mathematics as an ongoing human activity;

affirming and supporting full participation and continued study of mathematics by all students.

Elaboration

I think that an investigative nature is essential. We can't make students into seekers if we aren't seekers ourselves. (A university mathematician)

The experiences that mathematics teachers have while learning mathematics have a powerful impact on the education they provide their students. Prospective and practicing teachers spend many hours in mathematics and mathematics education classes, workshops, seminars, and other structured learning environments. Through these experiences, they develop ideas about what it means to teach mathematics, beliefs about successful and unsuccessful classroom practices, and strategies and techniques for teaching particular topics. Those from whom they are learning are role models who contribute to an evolving vision of what mathematics is and how mathematics is learned.

My students were passive and docile. I felt I was perpetuating those attitudes in them. I wanted to devise a non-lecture teaching format, but I wasn't trained for such a search. They don't teach pedagogical techniques in mathematics graduate school. (Macrorie 1984, p. 66)

Instructors of mathematics and mathematics education in any and all learning situations need to address the major components of teaching - tasks, discourse, environment, and analysis of teaching - as detailed in section 1 of this document, "Standards for Teaching Mathematics." This vision of teaching redirects mathematics instruction from a focus on presenting content through lecture and demonstration to a focus on active participation and involvement. Mathematics instructors do not simply "deliver" content; rather, they facilitate learners' construction of their own knowledge of mathematics. Sometimes they stand back, letting students puzzle and come up with their own solutions. Sometimes they push and lead, helping students to reach particular sensible conclusions. And sometimes they help students by modeling or telling. Mathematics instructors do this through their choice of tasks and tools for instruction and their attention to the nature of the mathematical discourse that occurs in the learning environments.

I was once questioned by a colleague, " But you seem to imply that you are not doing much lecturing. Isn't lecturing necessary to cover the content?" You know, it really isn't anymore, and it doesn't seem to be really important to my students. (A university mathematician)

Mathematics and mathematics education instruction should enable all learners to experience mathematics as a dynamic engagement in solving problems. These experiences should be designed deliberately to help teachers rethink their conceptions of what mathematics is, what a mathematics class is like, and how mathematics is learned. Instruction should be organized around searching for solutions to problems and should include continuing opportunities to talk about mathematics. Working in groups is an excellent way for learners to explore, develop mathematical arguments, conjecture, validate possible solutions, and identify connections among mathematical ideas. In such experiences, teachers should be encouraged to generalize solutions and communicate results from their explorations of mathematical ideas visually, in writing, or through dialogue and discussion.

Representations are crucial to the development of mathematical thinking, and through their use, mathematical ideas can be modeled, important relationships identified and clarified, and understandings fostered. Physical models, materials, calculators, and computers help provide the array of rich and substantive experiences needed to build a deep and comprehensive knowledge of mathematical concepts and procedures. The experiences teachers have in these learning environments form expectations - implicitly or explicitly - of what constitutes good mathematics instruction. Such experiences provide the core from which teachers will eventually build learning environments for their own students.

For a long time my undergraduate courses in differential geometry have profited from slides and films and videotapes, but I wasn't prepared for the tremendous advance that came when the students were able to work interactively with computers. (Banchoff 1986, pp. 8-9)

Such instruction requires substantial changes in the philosophy and strategies of mathematics and mathematics education instructors at the college level and beyond who are involved in the preservice and continuing education of teachers of mathematics. Instructors need to experiment with new tasks, tools, and modes of classroom interaction and share and model new instructional strategies. This necessitates collegial interaction and support, as well as participation in professional development opportunities. Similarly, such changes necessitate changing the recognition and reward systems in colleges and universities. Also, school districts need to revise their perspectives on the kind of in-service support needed to effect substantive change. Finally, such changes place new expectations on teachers as students in their participation and engagement in learning. This challenges mathematics and mathematics education instructors to foster changes in their students' preconceived and generally traditional views about the way learning occurs.

Vignettes

1.1 Prospective middle school teachers are coming to the end of a yearlong mathematics course. By now they take it for granted that they are expected to make sense of mathematics, develop their own problems, make connections, and come up with further questions to extend their thinking. This hasn't happened automatically; the mathematics professor has worked hard to achieve her goal of encouraging students to develop greater reliance on their abilities to make sense of mathematics.

"It hasn't been easy to shift these students' expectations from wanting answers from the instructor to a point where they accept, and in fact demand, that they have a chance to make sense of a situation themselves."

Students confront their own perceptions about what it means to learn mathematics. Their level of discomfort is an indication that change is happening.

The instructor has identified long-term projects as worthwhile tasks. In this instance, students have been able to define their own investigation, making it relevant to their own lives and interests.

Students must define the task, work independently and collaboratively to accomplish their work, and use a variety of tools.

The students have calculators and computers to help with graphing and statistical analyses. These tools encourage conjecturing and "what ifing" as they investigate the data.

A different environment has been created that supports and encourages mathematical reasoning. Such an environment influences students' dispositions to do mathematics. For these prospective teachers, it can have an impact on how they think about their own teaching.

Early in the year, a student who was interviewed about the class noted, "Up until the university placement exam I just plugged numbers in and always got good grades. It had been a long time since I had math. I couldn't remember the way to do lots of the problems or appropriate formulas. I had no ability to tackle problems if I didn't know the formula. I like to plug numbers into formulas. This math class is very upsetting. This is the first time I ever thought about why. In high school algebra we plugged in the numbers...just waited for the formulas. I realize I am going to have to learn to think about it [why] if I expect to teach math."

Now, later in the year, students are working on a research project that allows them to apply and extend the ideas they been have studying in probability and statistics. They have been challenged by their instructor to design and carry out an investigation to better understand some situation that they find interesting. The class decided that they wanted to know who Mr. and Ms. Typical Student were on their campus. They have spent several days designing questionnaires and gathering information.

"Oh, no! Now we don't know which data we have already entered! We will have to start all over again. We'd better get ourselves organized this time." From another group, "Would you like to know how we are keeping track of our data?"

"This is looking good! Would you look at that! Students who are working seem to have higher grade-point averages. Let's do a graph of grades versus hours of work each week and see if there is a relationship here. Let's go to the computer to do this."

Students find this kind of involvement in the class different from other experiences they have had in mathematics. One of the students pointed out a contrast between problems in this class and assignments in a typical math class:

"I still get frustrated a lot, but I am more satisfied when I can figure things out rather than just doing busy work. I used to think that math was just busy work. Do thirty problems on the distributive property when they're just different numbers...all the same thing, mindless by the thirtieth problem. Here's thirty problems, do them. Turn them in tomorrow. Here's thirty more problems. This class doesn't have any busy work."

The instructor explores ways to enhance mathematical discourse through the use of technology.

The instructor helps students realize that mathematics is useful in understanding a variety of real-world relationships.

Creating a learning environment that encourages students to think for themselves, not simply copy and then memorize the words of a lecture, is critical in helping them learn calculus and also in providing a model of how they, in turn, can teach others.

1.2 Ron Adams, professor of mathematics at a large university, uses computer numerical, graphic, and symbol manipulation programs to support a new calculus course that emphasizes concepts, principles, and applications of the subject. He engages students in active exploration and discussion of these ideas while they learn. He plans his initial class sessions to establish these directions.

In his overview comments, Ron suggests to his students that calculus provides an array of methods for studying relations among numerical variables. As an example, he identifies three pairs of related variables:

SAT scores and freshman GPAs
Oil supplies and gasoline prices
Rainfall and mosquito populations

He encourages students, working with partners, to discuss each of these situations and then share their ideas about how the variables in each pair may relate to each other.

Following this, Ron introduces methods for representing quantitative relations that can help in identifying the critical properties of those relations. He selects two other pairs of related variables:

A table of student absences from school during a flu epidemic - data are reported by days of the week.
A graph of the growth of the fish population in a lake - data are shown in units of time.

Following a discussion of observations, Ron extends this awareness activity, asking students to construct tables of values and graphs that they believe represent the shapes of likely relations between the depth of tidal water and the time of day or between the profit for a resort hotel and the average price charged for rooms.

Ron encourages the students, working again in small groups, to talk about such relations and their representations, avoiding an emphasis on exact "right" answers.

Ron moves to a third representation, symbolic rules, that can be used to study significant properties of functions. He demonstrates how computer programs use algebraic rules for functions to produce tables of values and graphs and then to identify key points and overall properties of those functions. Using examples of familiar situations, Ron shows students how to produce computer tables and graphs for the following:

A linear demand function and the related quadratic revenue function.
The periodic functions describing voltage in an alternating current circuit.
An exponential population growth function.

Rita: I don't believe that I'll ever go back to my old ways of teaching undergraduates. I used to spend most of my time presenting formulas and going over homework. My students used to repeat back what I taught, always seeking the quick rule. Now, after a quick introduction to a new topic, Bill and I focus on problem solving. Students are really thinking; they create their own problems, explore extensions and elaborations, and are willing to do a lot of "what if " thinking. What helps is both the way I have changed my teaching in general and the use of the graphing calculator as an instructional tool.

Tom: Bill stopped me in the hall the other day. He indicated that you two had worked together quite well throughout the semester.

Bill: Hello, Rita. Hey, Tom. Are you talking about our work this semester?

Rita: Yes. I was just going to say that what really helped was spending a great deal of time rethinking how to present material so that students were more involved. Sometimes you sat in the back of the room and, after class, shared your observations about student participation.

Bill: And at other times we team taught a lesson. And sometimes I would teach, and Rita observed.

Through posing worthwhile tasks and carefully selecting questions to facilitate discourse, the instructor encourages students to reason mathematically and take intellectual risks. In this environment, they begin to make connections between the mathematics they are learning and life applications. They are disposed to reason about real-life problems and investigate their solutions.

The instructor has support from her institution for trying changes in the way her students do mathematics.

Rita: Yes, this all helped me become much more reflective about my teaching. I began to focus on ways to ask more engaging questions.

Bill: We also used cooperative learning strategies and relied on a great deal of class discussion to promote talking and thinking about mathematics. Rita's students became quite involved, bringing in problems from their own experience. For example, one student said her parents were willing to loan her $8000 at 9% interest for her to purchase a car. She explained that her parents only wanted her to pay them $50 each month. After some work with their calculators, other students told her that she would never repay the loan with that size payment!"

Rita: I was worried that I might not be able to cover the syllabus because of the extra time required to teach students how to use the calculator and the time needed for students to figure things out and discuss and work. Instead, we actually got more done. Taking the time to help students make sense of the mathematics enabled them to make connections between topics. New topics seemed more logical.

As some faculty change and experience success, others become interested and consider similar changes themselves.

Tom: The changes you and Bill have made in your teaching are affecting the preservice teachers in the mathematics courses we are teaching.

Rita: Even other faculty! After looking at the midterm exam for this pilot course, Dave Smith commented to me, "Your students can do this?" He and I actually are planning to work together next year.