

A MODEL FOR THE DEVELOPMENT OF LEADERSHIP AND THE ASSESSMENT OF MATHEMATICAL AND PEDAGOGICAL KNOWLEDGE OF MIDDLE SCHOOL TEACHERS*  
Carole B. Lacampagne, Northern Illinois University Thomas R. Post, University of Minnesota Guershon Harel, Northern Illinois University Merlyn J. Behr, Northern Illinois University** 

ABSTRACT
OVERVIEW The basic goal of the Mathematics Leadership Project is to formulate and test a model for developing professional leadership abilities in intermediate grade (46) teachers of mathematics. To achieve this goal, the project was partitioned into three phases: Phase 1 (September 1987 through March 1988). A teacher profile was compiled and administered. It consisted of two written instruments and an interview instrument (De Kalb N = 51; Minnesota N = 167; 15 teachers were interviewed at each site). Phase 2 (March through July, 1988). A fourweek, fullday staff development institute for teachers of mathematics in grades 46, was planned and executed. A significant portion of the agenda resulted from needs identified by the preliminary teacher profile in Phase 1. Phase 3 (September 1988 through June 1989). Leadership and followup activities will be conducted. Teachers in the workshop will attend half day twice monthly, or, monthly full day followup sessions throughout the school year. Teacher leadership teams will plan and carry out at least one inservice workshop for their colleagues at their building site as well as provide a variety of inclass consultative services. They will also begin to involve parents in the curricular renewal process. Our approach here is systemic in nature; teachers, students, administrators, and parents all will become involved in the change process. THE STAFF DEVELOPMENT MATHEMATICS LEADERSHIP PROGRAM The major purpose of this phase of the project was to develop teacher leadership teams to function at the local building level. We attempted to accomplish this by strengthening teachers’ mathematical concepts, enhancing the quality of their teaching strategiesespecially by encouraging teaching for discovery learning and for student use of manipulatives to build mathematical conceptsby acquainting them with the politics of educational change and by consideration of leadership strategies related to the tasks to be accomplished (Price 1981). To meet these ends, careful attention was paid to the selection of participants and to the development of curriculum and materials. Selection of participants. At the Northern Illinois site, twentyfive participants were selected from among sixtyfive applicants. All were teachers in grades 46. Selection was made on the basis leadership team potential, and some academic background in mathematics and mathematics education. This varied widely among participants however. Special consideration was given to those applicants who had participated in Phase 1 of the project. In Minnesota, all participating schools were selected from a single large urban school district; actual institute participation was limited to those schools whose entire 46 faculty participated in Phase 1. In Illinois, one or two teams of two to three intermediate grade teachers were selected from seven North Illinois school districts. Two additional small districts were represented by a single teacher. It should be noted that participants from the Northern Illinois University site differed considerably from those from the University of Minnesota site in that the NIU participants were geographically diverse, traveling from 5 to 70 miles from school to the university. Most came from relatively small school districts in somewhat rural settings. The number of years of teaching experience of participants differed from 5 to 30 years, with the median number of years being 13. The number of semester hours of course work in mathematics and mathematics education of the teachers ranged from 3 to 27 with the median number of hours being 9. Although diversity of participants was not a criterion for selection, the group was diverse: 17 females, two of whom were black, and 8 males with 9 fourth grade, 8 fifth grade, and 8 sixth grade teachers. In Minnesota, 9 threemember schoolbased teams were selected from 17 schools which applied. All but one participant taught mathematics in grades 4, 5, or 6. There were 25 females and 2 males. Participants were given a daily stipend of $40, 6 graduate credits in mathematics education, and, in addition, received a wide variety of printed instructional materials, a classroom set of calculators, and assorted manipulative materials, most of which were constructed by the teachers. Research findings relevant to the workshop. As of this writing, data collected from the three instruments administered in Phase I of the project have not yet been completely analyzed. However, certain characteristics of teacher perceptions of mathematical concepts emerged early in the analysis. Teachers lacked a clear understanding of the partwhole/partitioning concept. Specifically, given a model representing a part of a whole, they had difficulty representing the whole. They could generate an area model for a fraction but had trouble representing that fraction when given various sets of discrete objects which defined the whole. In this respect, teacher conceptions of partwhole/partitioning were found to be similar to those of children in the grades they teach (Behr, Post, Lesh, 1986). See also accompanying paper for examples and discussion of fraction test items (Cramer & Lesh). Teachers had difficulty solving division problems when both divisor and dividend were less than one. This was true whether numbers were expressed in fraction or decimal form. When encouraged to draw a picture to model a story problem involving such division, however, they were often able to solve such problems. Nevertheless, many teachers had great difficulty composing a story problem involving division of nonintegral rational numbers. In fact, when asked to do so, they often wrote a story problem that required the inverse operation of multiplication. In short, teachers held certain primitive models of multiplication and division such as "multiplication makes bigger, and division makes smaller." These tended to dominate their strategies. Similar domination of thought processes by such primitive models was also found in children (Bell, Fischbein & Greer, 1984; Fischbein, Deri, Sainatinello & Marino, 1985). A disturbingly large percentage of teachers were not successful with relatively routine categories of rational number knowledge. Part I of the teacher profile contained 58 short answer or multiplechoice items. Part If contained 6 problems with directions to "solve and to explain how you would describe your solution to children." Table 1 below lists the main categories of items which comprised Part I, along with the mean percentage correct by site. 

*The first number refers to Minnesota, the second to De Kalb. Table
1 Teacher Profile  Part 1 Results 

Other aspects of the teacher profile dealt with single operation multiplication and division problems with "awkward" fractions and decimals. Results related to that aspect of our investigation are discussed elsewhere in this volume (Harel, Post, & Behr). Although space here does not permit an expanded discussion of these results, a cursory inspection of Table I suggests that achievement levels across sites is relatively stable and that overall approximately 30% of the items were missed by teachers who are to various degrees involved in classroom instruction in the majority of these topical areas. A closer inspection of the distributions of teachers’ scores suggested like percentages of teachers achieving a score of less than 50% on the instrument. Surely this is cause for concern and a topic in need of serious (and constructive) discussion by our profession. Use of research findings in planning workshop. At the NIU site, topics to be covered in the workshop were determined by the instructional team on the basis of their past experience in observing mathematical strengths and weaknesses found among local teachers, recommendations for mathematics curricular changes (NCTM, 1987), and the research finding noted above. The goal of the workshop was to enhance teacher perceptions of certain mathematical concepts by increasing their background in the areas addressed by the profile, place value, multiplication and division of whole numbers, nonnegative rational numbers, proportional reasoning, geometry (both 2 and 3dimensional), probability, statistics, counting principles and elementary number theory. Leadership participants will be expected to consult with other teachers about some of these issues during 198889. The Minnesota site did little with place value, multiplication and division, statistics and number theory. More effort was devoted to part whole, order and equivalence, ratio and proportion. These concepts were taught in a way suitable for use in the classroom; namely, a variety of manipulatives were used to introduce concepts, instruction was learner centered and discovery oriented, and emphasis was placed on problem solving activities, mental arithmetic, estimation, and the appropriate use of the handheld calculator. Because of research findings related to other aspects of the profile, much time was spent in De Kalb developing the concept of division of whole numbers, using both the measurement and partitive models, to ensure sufficient conceptional background for division of rationals. Later, a variety of physical models, both discrete and continuous, were used to develop the concepts of order and equivalence of fractions, and operations with fractions. The units on geometry, counting and number theory, probability and statistics complemented and extended the understanding of rational number. These units also provided numerous concrete examples and problems related to proportionality. Estimating and checking for reasonableness of answers was stressed at both sites. At Minnesota, translations within and between modes of representation were stressed. Understanding was defined as the ability to make appropriate translations. This has enormous implications for the ways in which mathematical knowledge is assessed and also for the general nature of mathematics instruction. Much more will be done with this idea during the upcoming year. Participants were encouraged to verbalize and later to write as a group generalizations, rules or algorithms which they had discovered. They were also asked to develop story problems on topics covered, suitable for use in their classrooms. Four days each week were to be spent in classroom activities at the NIU campus. The fifth day was to be used to develop individual or team teachinglearning units to be used in classrooms in 198889. Three units would be required of each team: one in fractional numbers, one in geometry, and one in probability, and/or statistics, permutations, or number theory. These units were to be student centered, including activities using a variety of physical models. The fifth day was to be used also to develop outlines for the workshops that teams would present to their colleagues. These activities were to be carried out at school or community sites, with members of the NIU instructional staff available at various school sites and at the university. In Minnesota, five days per week were spent on site; however, two to three afternoons per week were spent by teacher teams developing and synthesizing materials for classroom use during 198889. This time was coveted and appreciated by teachers and, in retrospect, is felt to be an important element in any such future undertaking. Leadership Institutein retrospect. Participants varied considerably in the breadth and depth of their mathematical knowledge. For example, at NIU, several participants had not mastered the concept of place value and had difficulty identifying a unit. Very few had any knowledge of geometry or had even the most fundamental perceptions of number theory, counting, probability, or statistics concepts. Most participants followed the textbook rigidly and seldom got to the back of the text where noncomputational topics were discussed. This not uncommon phenomenon is probably more a function of systemic expectation than it is the result of individual teacher decisions. Very few at the De Kalb site assigned story problems because, "Children don't like them." Most had had little experience in writing story problems and found it difficult to do so. Almost all worked well in their teams of two or three and often spontaneously joined together to form double teams of four or five when appropriate. At NIU, the long carpooling commute that several teams faced each day had the unplanned benefit of uniting teams and providing time for informal problem solving sessions. Although participants varied in their willingness to shed old styles of learning and teaching and to try new ones, all were enthusiastic learners. At the Minnesota site, about one half of the participants had been involved in a previous oneweek leadership institute (summer 1986 or summer 1987) designed primarily to acquaint them with current thinking in the field and to suggest, by example, alternative instructional formats. As such, these individuals were already enthusiastic about prospects for the new project. The teachers were eager to learn about profile results and, in particular, to "fill in" identified conceptual gaps. This was important since it is these individuals who will serve as leaders in the next cycle of this project during the summer of 1989. Phase 3 of the program will begin in September of 1988. It is there that the fruits of these efforts will be assessed. As mathematics educators continue to attempt to improve presecondary mathematics programs, principally by expanding the scope of appropriate components (problem solving, geometry, estimation, computers, etc.), we must realize that we are asking a significant percentage of teachers to teach concepts which they themselves have never (or so far in the distant past so as to be unretrievable) been exposed to as students. Add to this the variety of additional "addons," which further complicate the ever expanding role of the elementary school teacher, and the results presented here become perhaps more understandable although certainly not more acceptable. It seems feasible for us to urge consideration of alternative delivery systems as well as alternative school organizational patterns and to carefully identify the implications each would have on the structure and substance of elementary school mathematics programs. We are convinced that teachers are also very interested in substantive progress. REFERENCES Behr, M., Lesh, R., & Post, T. (1986). Representations and translations among representations in mathematics learning and problem solving. In Janier (Ed.), Problems of representation in the teaching of mathematics. Hillsdale, NJ: Erlbaum. Bell, A., Fischbein, E., & Greer, B. (1984). Choice of operation in verbal arithmetic problems: The effect of number size, problem structure, and context. Educational studies in mathematics, 12, 399420. Curriculum evaluation standards for school mathematicsWorking draft. (1987). Reston, VA: NCTM. Fischbein, E., Deri, M., Sainatinello, M., & Marino, M. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 316. Price, J. (Ed.) (1981). Changing school mathematics: A responsive process. Reston, VA: American Association of School Administrators, Association for Supervision and Curriculum Development, and National Council of Teachers of Mathematics. *This research was supported in part by the National Science Foundation under grant No. TEI8652341. Any opinions, findings, and conclusions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation. **The authors wish to acknowledge important contributions made to this project by Anne Bartel, Kathleen Cramer, Sarah Currier, and Nancy Williams. 
