Chazan, Daniel (2000). Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom. Reviewed by James Telese, University of Texas, Brownsville

Chazan, Daniel (2000). Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom. N.Y.: Teachers College Press Pp. xvi + 200. $22.95 (Paper)       ISBN 0-8077-3918-9 $48.00 (Cloth)       ISBN 0-8077-3918-7 Reviewed by James Telese University of Texas, Brownsville/Texas Southmost College August 18, 2000         Daniel Chazan has provided teachers, parents, researchers, or policy makers, a window into mathematics education. Based on his experiences teaching algebra, he offers the reader background into the issues associated with the teaching of high school algebra, motivation of students, curriculum, and the "craft" of teaching. The author describes his transformation from a traditional mathematics teacher to a reform oriented mathematics teacher. The metamorphism occurs through reflection on his teaching experiences at a parochial school and later at a suburban public high school.         The author presents a brief teaching autobiography in the first chapter. He describes his early experiences of teaching algebra at a northeastern suburban private parochial school; where, he found teaching highly motivated and eager eighth graders, an enjoyable experience. In contrast, a decade later, he returned to the pre-college classroom at a public high school. He taught students who were unmotivated and who saw little use for the learning of algebraic concepts. In each case, he reflects on his teaching practices, which were very traditional in nature.          At the parochial school, students described his class as a typical mathematics class where each day homework exercises were examined and problems he deemed important were emphasized, followed by introducing new material, practiced problems, and then assigned the next set of homework exercises. He found this to be a very unrewarding method of teaching. The textbook and he were the authorities in the class. Students were not encouraged to get involved in class through discussions and sharing of ideas. He taught his students that algebra has a series of steps and procedures to follow rather than teaching algebra for understanding. Since the students at the parochial school were college intending, self-motivating, and eager to follow the teacher's instruction, this method worked well.          His mathematics autobiography includes background information of his teaching experiences in a suburban Professional Development School. He had a strong desire to work with lower track classes. This opportunity occurred from1990 to 1993 where he taught one lower track class each year. He did not teach alone. He was assisted during his first year by a student teacher in this first semester and by the mathematics department head during the second semester. A mathematics teacher from the school was a co-teacher during the last two years.          The students he encountered needed to know why algebra was important. His students were unmotivated and expressed their unhappiness with school. Students in different social groups varied in their level of motivation. The social groups included "stoners," "preppies," "smokers," and "nerds." The "stoners" are described as being involved in gang activity and the use of illicit drugs. The "preppies" are students who were affiliated with school through membership in sport teams. The "smokers" were the majority of the students in his class who spent time before and after school hidden in far corners smoking cigarettes. The "nerds" are student who were separated from the other three groups and not associated with academic pursuits. This background provided the context for the pedagogy used to motivate the students.          Getting to know his students was his key to designing a curriculum that would engage the students with algebraic concepts. He found the concept of " trajectory" during his research on motivation. Chazan used the metaphor of trajectory as described by Berryman (1987). A trajectory is the pathway students expect for themselves as adults in society. Students will engage in curricular tasks in direct relation to their views of the future. Students develop an idea of their niche in the adult world and have basic notions of the future. The notion of trajectory is seen by Chazan as movement from the past to the future. Students make decisions that may keep them on a trajectory or change their path. Moreover, students may feel that they are incapable of changing a trajectory or may not have the will to do so. Chazan describes an interview with a "smoker" who characterized a "preppie" as someone who does their homework, gets good grades, and goes off to college and gets a good job. In contrast, some of the "smokers" don't care about their grades or homework, won't go to college and do whatever they can for a job. The trajectory of the "preppie" is graduating from college and getting a good job, and the trajectory of some "smokers" is to do whatever they can. Chazan believes that the 'laziness' exhibited by students is due to a mismatch between the students' sense of trajectory and the curricular content of the course. In order to engage students in the tasks, it then becomes necessary to align the trajectories of the students with their teacher.          His attempt at aligning the trajectories was the use of real world problems in his classes. Chazan uses Nemirovsky's (1996) view of "real" as describing the relationship between a problem and the students' experiences. "Real contexts are to be found in the experience of the problem solvers" (Memirovsky, 1996, p. 211). A complication arises in that not all students share the same experiences. He designed tasks related to calculating tips at a restaurant, which was of particular interest to a female student working as a waitress. Also, since most of his students were "smokers," he designed an investigation that examined the cost of cigarettes under a proposed taxation plan. The "preppies" did not see this as such a motivating task. Not surprising, Chazan contends that using real world problems is not a panacea and has limitations.          His attention then focused on teaching algebra conceptually. The use of real world problems was not totally abandoned. The meaning of conceptual understanding is not clear. For Chazan, conceptual understanding involves psychologizing the subject matter based on Dewey's work (1902/1990). Psychologizing the subject is the process of developing content using knowledge of the students' experiences, motives, attitudes, and organizing the subject matter around the students. In order to accomplish this, he had to understand algebra and his students' experience. As a foundation for addressing the question, what is school algebra, Chazan offers a brief history of algebra and describes two views of mathematics, which led him to use a relationship between quantities approach to teaching algebra. In this view, the concept of functions takes center stage. For example, the relationship between the costs of long distance calls to the number of minutes talking. The output variables (such as the cost) depends unambiguously on the input variables (the number of minutes). These relationships are studied through formal ways like the use of tables of values for the quantities, symbolic expressions, and Cartesian graphs, and through informal ways such as, diagrams, graphing devices, written or spoken words. The idea of x as a variable, Cartesian graphs, and functions were taught early in the course. He attempted to use real world examples based on the students' experiences. Three representations were discussed, the procedural aspect of getting outputs from various inputs, a focus on the values of the inputs and outputs rather than the how one arrives at an output from an input, and how do the outputs change as the inputs change. Consequently, Chazan began to implement reform- oriented strategies such as classroom discourse, real world problems, and technology.          Symbol manipulation and working with expressions are other topics in algebra. According to Chazan, the function approach offers the opportunity to develop meaning for symbol manipulation. It is accomplished within a context rather than in isolation using practice exercises. Expressions are seen as "arithmetic procedures and not names for numbers" (p. 89). He used graphs and tables of values to help students see the similarities of two algebraic expressions.          The students needed to know how algebra is helpful in the real world. To help students to see algebra around them, Chazan describes an activity where the students go out in the workplace and hunt for examples of mathematics. Students responded to a series of questions about the quantities used, the procedure for computing the quantity, and the reason for, and the importance of the computation. The activity helped the students realize the value and use of algebraic computations.          Another feature of Chazan's pedagogy for teaching algebra is classroom discourse, which he describes in chapter four as "classroom conversation." Students in lower track classrooms saw themselves as not having the ability to do mathematics. This hindered their participation in classroom discussions. In reform-oriented pedagogy, it is important that students develop conjectures, revise, agree, disagree and prove mathematical statements, in an environment where students respect each other's ideas and where a whole-class discussion results.          However, conducting whole-class discussions is a difficult task. Chazan believes that in order for a classroom conversation to occur, there are requirements for both students and the teacher. For students, the requirements include clear articulation of students' mathematical point of view, and contribution to a class atmosphere where each student feels safe sharing "wrong" answers. Students should consider them as steps along the path toward a deeper understanding of a problem. The task for the teacher is to lead the discussion and to keep students' ideas developing and growing. When disagreements occur, the teacher should support students in revising their initial conjectures. Although answers to questions are important, in Chazan's classroom, the focus was placed on the students' rationale.          Chazan closes the book with a chapter that attempts to describe the complicated process of teaching. His reflection on his practice led him to reconceptualized the subject matter, which produced other changes in his relationships with students and his understanding of the subject matter. He takes the stance that a teacher's own knowledge of subject matter is central to dealing with motivation and relevance. He contends that the culture of the school places the responsibility of motivation on the student rather on the teacher, "…the constructs of ability, motivation, and students' achievement-which encompasses a variety of relations between student and subject matter, teacher and subject matter, and student and teacher-becomes solely a function of the student" (p. 150). This results in the traditional view of mathematics teaching with an emphasis on getting answers to exercises, and the curriculum is the sole foundation for future courses. His solution to the many problems faced by mathematics teachers is to share teaching assignments, so that members of a department can observe each other. The shared assignments allow the teachers to identify problems of practice, try something, share the results of their attempts at practice, and develop trust and reflection.          The strengths of the book include a background into issues faced by mathematics teachers for those readers who are unfamiliar with issues such as motivation, student peer groups, school culture and curriculum and how they affect the teaching process. Although the author provides only brief lesson descriptions, other mathematics teachers may learn from them. The mathematics lesson description are presented in a way that readers with some understanding of algebra could grasp his point. This may have been due to the fact that he did not want the book to look too "mathematical." He did provide some background knowledge for the concept of slope when he described how he taught the lesson. The author names the important concepts of algebra and mentions how each fits into the functions-based approach. For example, slope was emphasized as a rate, linear equations where presented in contexts, expressions were seen as equivalent procedures.          Another strength is his illustration of the various components of mathematics education reform such as, discourse, student-centered instruction, and the use of application to teach concepts and procedures. The description of teaching algebra with a function approach was well done. The author provides background information as needed to support his argument such as the additional information about the history of mathematics related to algebra teaching and the two philosophical views of mathematics. This allows the reader to put into perspective algebra teaching today. His book is a model for how teachers can reflect on their practice.          The book does not have many weaknesses. Experienced teachers are very well aware of the aspects of mathematics pedagogy and working with lower track students that the author brings to the forefront. The author never really had the responsibility of teaching 'alone,' except at a private parochial school with motivated students, and for three years, he taught only one class per day. The reality of being the sole teacher of 90 to 150 public school students was not described, which may have provided additional insights into the trials and tribulations of school mathematics teaching. As a former lower track algebra teacher, this reviewer recognizes the hard work necessary to motivate students to achieve in the algebra classroom. Classroom management is one area the author did not mention, which is an important issue for both beginning and experienced teachers. This topic may be beyond the scope of this book, but is an important aspect to discuss.          However, perhaps this observation misses the point of the book, presenting the views that include mathematics teaching in high school should center around functions, that the teacher is responsible for motivating the students, and the importance of a shared teaching assignment, whether within the school or a university researcher/school teacher partnership. Many teachers do not have the luxury of shared teaching assignments. The author had, what many readers may regard, an ideal situation. Overall, the book offers those readers interested in the improvement of mathematics education ideas of what should be changed in mathematics pedagogy. The examples are valuable enough for use as models that other teachers or staff developers could emulate. References Berryman, S. E. (1987). Breaking out of the circle: Rethinking our assumptions about education and the economy. (Occasional Paper 2). New York: National Center on Education and Employment. Dewey, J. (1990). The school and society: The child and the curriculum. Chicago: University of Chicago Press. (Original work published in 1902) Nemirovsky, R. (1996). Mathematical narratives, modeling and algebra. In N. Bedharz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 197-200). Dordecht: Kluwer. About the Reviewer James Telese James Telese holds a Ph.D. in Mathematics Education. As an Associate Professor, he currently teaches mathematics education, secondary education, and educational research at the University of Texas, Brownsville/Texas Southmost College. His research interests include algebra classroom teaching research, and the mathematics teaching of language minority students.