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By Lyn Riverstone

In a *Mathematics for Business and Social Sciences* course, 60 students in small cooperative groups discussed the graphs of 6 functions they had been given on cards labeled (A) through (F), which they sorted into piles, according to common characteristics of their own choosing. When they finished, they described how the functions they had grouped together were *different*. The students had to explain their thinking to the whole class at the end of a 15-minute discussion with their group. Each group's discussion was lively as students discussed which characteristics to focus on, how many different "types" the graphs could be sorted into, and which ideas from class were relevant to this task.

Several of the groups began their sorting process by looking at the entire graph of each function, putting those with a similar "shape" together, while others focused on the point x = *a* in each graph, noting that some have a relative extreme point at *a*, but others don't. Several groups noted a common characteristic between several of these functions is the change in concavity at *a*. Next, the students looked for differences among the graphs they had grouped together. Throughout these discussions, students used the powerful language of calculus (slope, derivative, concave up, relative maximum, etc.) to describe the graphs and explain their reasoning.

Throughout, the students were struck by the number of possible ways to sort these functions, as well as to describe their differences. After a few groups presented their work, I ended the class period with a brief summary of what had been said and pointed out how all three student presentations were connected. For example, one group had discussed the inflection point of graph (C) while another had pointed out that the second derivative at a in this graph is equal to zero.

Reflecting upon this lesson, I felt I had met my goal of providing students with an opportunity to review recently introduced concepts without doing their thinking and organizing for them. What was most striking to me about this particular day in class, however, was how drastically different a "review" might have looked only a few years before. Probably, students would have sat quietly in rows, watching me draw graphs of functions that all had some common characteristic, already sorted for them, as I noted differences. They would have taken notes, hoping to get down what might be on the test. There would have been little opportunity for them to use new vocabulary, to construct meaning for themselves, or to even voice their confusion or questions.

**Reflections on Those Beginning Years**

As a new college instructor, I taught the way I had been taught. I lectured every day, assigned practice problems from the textbook, and gave regular quizzes and exams. Due to my inexperience and lack of confidence, I relied heavily on the textbook for content, sequencing, and choice of exam questions. I often chose test questions that were very similar to ones students had solved in homework or had observed me solve during a lecture, and modeled what I expected from my students in terms of showing work and justifying answers. In short, I showed or told my students how to do something, they practiced being good at doing it, and finally I tested them to be sure they could answer questions and justify their answers.

During those early years, I had excellent teaching evaluations. Students commented that I had a strong understanding of the mathematics I taught; I put mathematics into plain words that were easy to understand; I explained concepts in different ways; I helped them understand concepts they had never before been able to grasp; I was patient and available; and I cared about their success. Most of my students learned the mathematics I taught them, and came away from my class with what I thought of as a deeper understanding of mathematics. Based on all of this, I thought of myself as a good teacher. Then, I began teaching the *Foundations of Elementary Mathematics* courses, a year-long sequence for pre-service elementary school teachers, which transformed my idea of what it means to be a good teacher.

**The Process of Change Begins**

In these courses, I lectured about standards-based pedagogical practices, but did very little modeling of that way of teaching. One day, as I demonstrated how to use fraction bars to model fraction addition, I began to realize that something did not feel right. Why was I telling my students how to use manipulatives with children rather than giving them an opportunity to have their own hands-on experience? Were proven instructional strategies effective in teaching school-aged children also appropriate at the college level? At first, I thought the answer had to be "no." Otherwise, why were so many of my colleagues teaching in the traditional lecture mode?

Fortunately, one professor in my department was sympathetic when I approached him with my ideas for moving away from a lecture-based format toward a collaborative learning model. I wanted my students to engage in serious mathematical thinking, rather than passively watch me solve problems on the chalkboard. His enthusiastic response prompted me to take the leap-stepping out of my comfort zone and beginning to try some new pedagogical approaches. It was scary, but I was excited about improving my teaching!

With very little knowledge of what researchers in the field of mathematics education have found to be the "best practices" for teaching math, I relied on my instincts at first. My gut and experience told me that students learn more when they are allowed to work collaboratively with others to investigate mathematical ideas and solve problems. I also realized that talking about mathematics (what I later learned is referred to as "mathematical discourse") engages students in the learning process and hence has a more profound impact on their achievement. Incorporating 5 to 10 minute small-group discussions of homework problems before my lecture each day, I made my first baby step toward my goal of changing my teaching practice. In the six years since then, I have made a lot of other changes that have, I believe, improved my teaching practice immensely.

**What I Learned**

*Peer observation and discussion is a catalyst for change. Talking with and observing experienced instructors was the single most important factor influencing my ability to make lasting and worthwhile changes to my practice. Working in isolation makes change difficult, while working with the support of a community makes it easier to face the fear, frustrations, and, even fun, of many a significant change. I sought out colleagues, with a shared interest in change. I attended conferences with a focus on teaching, which also helped transform my teaching practice. Improving my practice became not so overwhelming when I realized I didn't have to be afraid to use other people's ideas. *

*Incremental changes make the process of transformation less daunting.* In the beginning, I decided I wanted to incorporate more active learning and student discourse into my practice, so I started with the homework discussions only. Continuing with this goal the next year, I began devoting entire class periods to collaborative learning assignments, leaving lectures to only one day a week. These days, I rarely lecture; instead I try to guide my students, providing tasks I believe will be worthwhile, facilitating their discussions about the mathematics, and asking questions that may help extend their thinking. These changes have had such a positive impact on my students' learning and attitude toward mathematics that I've become even more open to change.

In recent years, I've experimented with assessment (e.g. rewriting textbook exam questions to better reflect the kinds of activities students had done in class, choosing problems that require more original thought than simply regurgitating), assignments (e.g. assigning less drill and practice and more problem-solving tasks, written reflections and journals), and collaboration (e.g. having students work together in pairs and outside of class on group projects.) Some of these changes have been successful while others have not fit well with my personality or comfort level. Nonetheless, I continue to try innovative practices as I learn about them.

*Revisiting and attending to old habits are crucial to maintaining change*. It can be challenging and even frustrating to break old habits and to form new ones, especially when the old habits "work." I have had a hard time letting go of "control" of my students' learning and have not believed they could really do mathematics. I was ready to jump in at the slightest sign of difficulty. Nearly every day, I must remind myself that students will learn more if I allow them to struggle with the process of doing mathematics. Even though the process might feel easier (no caring teacher enjoys seeing her students in the midst of struggle), I must remember that the outcome for my students is so much more rewarding for everyone if I don't tell them what to do.

*Focusing on those things that are in my power to change is essential*. Many argue that NCTM-recommended changes are not possible at the college level. Class sizes can reach 100 to 300 students, there is not enough time to cover the material in the syllabus, students are underprepared, etc. I have no control over these situations, but rather than give up, I focus on the pedagogical pieces I can manage, such as requiring students to show and explain their thinking on some exam questions, rather than relying only on multiple-choice. Online discussion boards provide students opportunities to discuss concepts and problems that may not be possible on a daily basis in large sections. Another idea for engaging students, which I have yet to try, is a wireless response system in which students use a remote control to respond to questions posed by the professor. Once students have responded, the anonymous results can be displayed for all to discuss. This helps meet the needs of students in large sections because the pace of the lecture can be adjusted depending on student responses.

*Transparency can help to involve students in the change process*. Being honest with my students about the changes I am trying to make has helped everyone feel more comfortable. I may say, "I learned this at a conference last week and would like to try it, because I think it will really help organize our conversations around this concept. This is new to me, so please bear with me as I figure it out." I've found that students are very understanding and will often let me know when a new technique or assignment is working or not, and may offer advice for improving it the next time. I've even provided students an opportunity to comment anonymously about something new I have tried in the classroom. Students appreciate the opportunity to give feedback about their course.

Teachers often expect perfection from themselves, and appearing to be anything less than an expert is sometimes uncomfortable. However, we serve as models of life-long learning when we are up front about our own attempts to improve our teaching practice.

**The Next Phase**

Making changes to my teaching practice has not been easy; nonetheless, I look forward to the challenges of continuous self-improvement that comes with each new school year. I will continue to collaborate with my colleagues, attend conferences and take advantage of other professional development opportunities. In fact, I am organizing a faculty learning community in my department as a means for improving my own teaching practice as well as helping my colleagues to make changes themselves. Through inquiry, dialogue and reflection, we will help one another solve problems related to teaching, examine the "best practices" of teaching mathematics, look collaboratively at student work to better understand student thinking, establish informal mentoring relationships, and work together on peer observations.

This year, I also plan to focus on issues of equity in my classroom by trying to create a safe environment for discourse to occur, thinking more carefully about how to group students, using group roles and enforcing them, allowing and honoring private think time, and relying on protocols that encourage every member of a cooperative group to contribute in meaningful ways. I know that not every new thing I will try will be perfect - I will see not failures, but invitations to try again. Breaking the habits of a lifetime is a huge challenge, so I will continue to set attainable goals and make incremental changes that help me reach those goals as I work to get better at this complex work of teaching. Though I will continue to face the challenges that go along with teaching large classes at a university, if I am honest with my students and myself, my teaching practice can do only one thing - improve!

*Lyn Riverstone teaches in the Mathematics Department at Oregon State University. She has been interested in mathematics education and the preparation of teachers for many years. The Oregon Collaborative for Excellence in the Preparation of Teachers (OCEPT II), funded by a National Science Foundation grant 0222552, provided the financial and collegial writing support through a WRITE ON! writing retreat to help make this article possible.
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