Patterns of reasoning about mathematical models: a case study of high school mathematics teachers.

This case study illustrates the results of instruction in a mathematical modeling course for high school mathematics teachers. In this course the teachers were introduced to the process of mathematical modeling. They explored various problems and data sets. The study examines the course outcomes in terms of two teaching goals: (1) helping participants learn to build, and to reach consensus about plausible mathematical models, (2) influencing participants' thinking about appropriate mathematics curriculum and instruction. The development of participants relative to each of the course goals was traced. Most participants were successful in achieving both goals. The findings of the study highlight the need for design of professional development activities that take into account the diverse mathematical backgrounds of teachers.

Calls for restructuring learning environments that build around genuine mathematical inquiry have increased substantially over the course of the past few years. Explicit in the current recommendations for reform in school mathematics is the notion of helping learners develop the disposition to engage in activities similar to those of mathematicians (NCTM 2000). Current convictions shared by many mathematicians and mathematics educators about the nature of mathematical knowledge and how it is best learned include inquiry into real life questions and planning problem and data driven curriculum that motivate the search for meaning and the use of mathematics (National Research Council 2000). There is also the belief that mathematical concepts and theories are tools that are based on our collective experience in the world, and that we use these tools to make sense of our experience (Bishop 1988; Davis & Hersh 1981).

Mathematics teachers have traditionally recognized the dialectic between mathematical theories and concepts and using mathematics to solve problems by affirming two types of goals in their instruction. The first goal involves helping students to control the mathematical theories and facts. The second goal involves helping students use these algorithms in solving application problems. This controlled approach to teaching is problematic because it transforms the complex, socially embedded process of mathematical inquiry into traditional school tasks--bits of curriculum that can be easily managed in teacher-centered classroom.

Many reports of recent efforts to reform mathematics teaching have described classroom environments which are less teacher centered and which model many of the ideals and processes of the mathematics community (Borasi 1992; Manouchehri & Enderson 1999; Wood, Cobb & Yackel 1991; Yackel & Cobb 1996). Significant feature of this kind of classroom include valuing of student ideas as currency for classroom interactions, the teacher taking the role of co-learner, and evolving standards for collective validation of ideas. Often called a learning community (Lampert 1990), this kind of classroom has been recently described by Cobb and Bauserfeld (1995) as a setting in which the goal is to create a community of validators. Their description of interactions focus on argumentation, group discussions, and co-learning:

  The standards of argumentation established in an inquiry classroom are
  such that the teacher and students typically challenge explanations
  that merely describe the manipulation of symbols. Further, acceptable
  explanations appear to carry the significance of acting on taken-as
  shared objects. Consequently, from the observer's perspective, the
  teacher and the students seem to be acting in a taken-as-shared
  mathematical reality, and to be elaborating that reality in the course
  of their ongoing negotiations of meaning (pg. 2-3)

Transforming the learning environment to resemble a community of validators is primarily dependent upon the teacher's initiative. The teacher must not only attach merit to this type of learning, but also have some vision of the type of curriculum and instruction that could foster it. Moreover, she must have substantive mathematical knowledge (Manouchehri & Goodman 2000), be willing to listen to children's voice and perspective (Confrey 1994), and be able to attach mathematical value to children's thinking (Voigt 1995). Designing educational experiences that facilitate teacher growth and development in these areas is of particular concern in mathematics teacher education.

In this article I report the outcomes of instruction in a graduate level course in mathematics education for high school teachers. My goal was to help teachers' mathematical and pedagogical--growth as they gained experience in acting as a community of validators. I hypothesized that such experience would provide the teachers with an imagery of learning and teaching compatible with those advocated by the reformers.

I used mathematical modeling as a vehicle to facilitate the constitution of a learning community among the teachers. This selection was intentional. The process of forming and using mathematical models is a qualitatively different type of mathematical activity than those used in applying specific algorithms for solving well-defined problems and questions. A major difference is that there are no precise rules in mathematical modeling and no correct answers (Cross & Moscardini 1985; Dym & Ivey 1980). The adequacy of a model is determined based on how accurately it describes and/or predicts the behavior of the real systems. Therefore, when exploring a similar situation multiple constructs may be offered. I had anticipated that in the process of tackling the problems I assigned teachers would come up with various models, some more sophisticated than others. I had also anticipated there would be disagreements and discussions about the various models they presented. The teachers then had to make a decision, as a community, about the most adequate model that represented the situation. These activities resonate well with what is expected to occur within a community of validators.

The research addressed three specific questions:

1. Does mathematical modeling provide a useful vehicle for creation of a community of validators?

2. What factors influence the functioning of the group as a community of validators?

3. What type of teacher learning and development occur as the result of exposure to the course activities?

The Mathematical Modeling, Course

The course commenced in July. The teachers in this class came together daily for approximately 4 hours each day for three weeks. The course was problem based. Each day I shared several real life problems with the participants. These problems came from various sources including business owners and other professionals. The problem presentation followed a sequence starting with simple modeling situations. The level of sophistication and difficulty of problems increased gradually. During the first week of class I began each instructional period by discussing a modeling problem in the large group. I introduced the participants to the techniques of mathematical modeling and elaborated on the mathematical structures I used to solve each problem. I routinely interacted with teachers during these presentations and encouraged them to agree or disagree with the methods I used to set up and/or solve each problem. I then presented them with a maximum of four new problems (or data sets) each day. The participants were assigned to work on problems of their choice in collaborative groups. I also requested that they work on some common problems. I believed these common problems would help create a shared-intellectual-space (Rochelle 1996) for the entire group. I envisioned that in the course of their presentations of solutions the participants would engage in mathematical arguments as they assessed their own methods in the presence of more or less sophisticated mathematical models presented by their colleagues. By immersing teachers in self-directed mathematics learning and by highlighting their intellectual achievements in the process I hoped they would recognize the power of collaborative verification processes for learning mathematics.

The participants discussed their models in the large group three days a week. In these occasions they were expected to exchange ideas and argue about the accuracy and efficiency of the models different groups or individuals presented. At the end of each day I also asked the teachers to reflect on their actions and to discuss with their peers implications of what they did and learned in our class for their own teaching.

The participants

The 15 teachers in this class came from various urban, suburban, and rural high school settings. Their teaching experience ranged from 7 to 25 years. Each of the teachers had completed graduate coursework in areas of: learning theory, curriculum and instruction, test and measurement (or assessment), reading and writing in the content area, and child psychology. The mathematical backgrounds of teachers varied (1). Among the 15 teachers 4 had Bachelor of Science degrees in Mathematics. These teachers had generally completed 18 hours of coursework in advanced mathematical studies that included courses in Advanced Calculus, Statistical analysis, Real analysis, and Abstract algebra.

Eight of the 15 teachers had Bachelor of Art degrees in Secondary Mathematics Education. They had completed a minimum of 12 hours of coursework in mathematical studies. This included courses in Calculus sequence, Probability and statistics, Numerical analysis, or Discrete Mathematics.

The remaining 3 teachers came from middle level teaching backgrounds. The mathematics preparation of this group consisted of a course in Pre-calculus, a course in Probability and Statistics, and two courses in Number Concepts and Geometry for teachers.

The diverse backgrounds among the group became an increasingly important point of analysis later in the study as I examined the nature of collaboration among the participants.

Data collection

Three distinct data collection procedures were employed. These included a pre-course survey, videotape of daily classroom sessions, and a final interview.

Pre-course survey: All participants completed a survey on the first day of class (See Figure 1). This survey collected base line data on their prior learning and teaching experiences. Participants were asked to identify aspects of innovative instructional practices they found easy or difficult. In addition, they were asked to rate their confidence about learning and teaching mathematics. Lastly, the survey obtained information about participants' knowledge of mathematical modeling and the frequency in which they used modeling situations in their instruction. The participants' responses to the initial survey allowed me to build a profile of the type of knowledge bases and teaching orientations they brought to the study. The data was used to monitor participants' actions in class and to trace their development as the course progressed.

[FIGURE 1 OMITTED]

Videotapes: All class sessions were videotaped. Two video cameras were planted in the classroom during the entire three weeks of instruction. In addition to videotaping all whole group discussion, the work of various small groups was recorded daily as well.

Final interview: An outside evaluator conducted a structured interview with each of the participants three months after the completion of the sum mer course. The purpose of this interview was twofold. First, it solicited qualitative data on the participants' thinking about their experiences during the summer. Second, it compiled evidence on ways in which teachers claimed their practices were influenced by the work they did in the summer. Teachers were also asked to share with interviewer examples of lessons they had implemented in their classes along with samples of students' work (2). With the exception of one, all interviews were tape-recorded and later transcribed.

Data Analysis

Videotapes of classroom discussions were coded using both qualitative and quantitative checklists in order to provide data relative to three questions:

1. How did teachers interact with each other?

2. What challenges did teachers face in the course of their mathematical work?

3. Were teachers successful in acting as a community of validators?

Qualitative checklist

The checklists documented the participants' collaboration along six specific dimensions: the nature of their conflict, their reactions to peer feedback and questions, ways in which they resolved differences and reached consensus on solutions, ways in which they reacted to public opposition, and ways in which they determined the adequacy of mathematical models.

(1) The nature of the participants' conflict: I coded participants' conflict according to two major categories: task oriented and social. A task specific conflict referred to disagreements among teachers over their interpretations of the problem, the information provided or asked, specific values of parameters, mathematical procedures used to solve a problem, and methods of assessing the adequacy of solutions. Social conflict pertained to those disagreements among the teachers that were not related to their mathematical work.

(2) Ways in which the participants reacted to peers' solution: I made distinctions among four types of reactions to peers' work: guiding, acceptance without testing, soliciting further explaining, and self-reflection. Guiding comments pertained to those statements that aimed to help their peers consider relevant concepts. Acceptance without testing pertained to occasions when the participants accepted solutions without examining their legitimacy. If participants asked questions to better understand the method their peers presented I marked them as soliciting further explanations. These comments required the participants to elaborate on their thinking or solution. If participants made comments that evidenced extended analysis of either self or peers' work I marked them as self reflection. "I see why mine would not fit this model," or, "Let me try and see if your method works the same way as mine," were examples of such reflective comments.

(3) Ways in which the participants responded to peers' questions: There was a need to also determine how the participants reacted to their peers' questions or feedback in class. To this end, I made distinctions among three types of responses: elaborate explanations, tutorial, and avoidance. Elaborate explanations were detailed explanations that participants used to justify their position or to share knowledge (i.e. At first I could not decide whether to use a quadratic or a cubic function, therefore I used the least squares method to find the best fit function for this set of data). Tutorial explanations included those comments that dictated convergence on particular actions among group. These responses did not mean to solicit feedback or to extend discussion of a solution. "Use quadratic formula," or "Set up a system of linear inequalities and solve it" were considered as tutorial comments made in class. If responses were not supported by data or a mathematical explanation and conveyed reluctance to consider peer input or question I coded them as avoidance. Statements such as "I just know," I don't believe it is true," or "If you say so" were coded as such.

(4) Ways in which the participants resolved differences or reached consensus on a solution: To determine whether the participants adopted to the norms of debate and argumentation as practiced within a community of validators, I marked their approaches to settling disputes along five criteria: looked for further evidence, abandoned the problem, validated various points of views without reaching consensus, changed their own answers, and asked for instructor intervention.

(5) Ways in which the participants reacted to public opposition: I made distinctions among four types of reactions: withdrawal from discussions, accepted feedback and looked for additional data to support a perspective, ignored the opposition, or intimidated the peers.

(6) Ways in which the participants determined the adequacy of mathematical models: I made distinctions between two methods: reliance on mathematical techniques, and acceptance without sufficient evidence. During the discussion of solutions, I paid particular attention to whether the participants used mathematical knowledge to assess the adequacy of solutions. I coded statements as knowledge whenever the participants referred to prior conceptual knowledge or made references to prior experiences and analogies they could use to verify answers or to solve a problem. If solutions were accepted without mathematical debate, they were marked as acceptance without sufficient evidence.

Quantitative checklist

In order to trace the development of the group over time, I counted the number of times the teachers confronted each other's solutions, the number of times they disagreed with one another's assumptions and models, the number of times they reached consensus on solutions based on data, and the number of times they asked for my intervention. The number of contributions of each teacher to small and large group discussions was also noted. Means and standard deviations were computed for each week. A comparison of descriptive statistics on each of the targeted measures occurred at the end of the third week to determine patterns of behavior that remained consistent and those that changed over time.

Two additional raters coded a random sample of videotapes from each of the class sessions. Each rater observed the video clips separately. In addition to completing the checklists, each rater wrote his own general impression of group's interactions. Inter-rated agreement among all three raters had to be met before making any conclusions. An inter-rater agreement of 86% was reached.

Interviews: Interview responses were used to assess the success of the course in helping teachers draw pedagogical implications for their practice. Each teacher's interview responses were coded individually first. Each interview was analyzed to record the teachers' claims to their own mathematical and pedagogical growth. Teachers were asked to identify: (1) aspects of the course they found difficult, (2) aspects of the course they found useful, (3) ways in which the course experiences impacted their thinking about mathematics and teaching, and (4) ways in which the course experiences impacted their practice and knowledge of mathematics.

Following each individual analysis, I cross-examined all interview responses to determine common themes among the participants' remarks. Conclusions concerning the pedagogical influence of the course on participants were drawn based on these common themes.

Results

The data indicates that mathematical modeling served as a useful vehicle for motivating the type of behaviors that characterize work within a community of validators. Although all teachers did engage in constructing mathematical models, there was some variability among them relative to the amount of mathematics they learned and the degree in which they contributed to the discourse of the class as a community of learners. Two critical variables included the prior mathematical knowledge base of the teachers and their confidence in their ability to do and learn mathematics. For five of the teachers whose knowledge of mathematics was limited, internalizing the process of constructing mathematical models was more challenging. It was also more difficult for these teachers to engage in productive mathematical argumentations. They rarely confronted their peers' perspectives. During the interview session these teachers argued that the content of the course was too complex for them to grasp. They suggested that the mathematical expectations of the course were too challenging for them and that they often found themselves confused by group discussions. Although these teachers recognized the value of group discussions in helping them construct new mathematical understandings, they were disturbed by my lack of direct intervention during these discussions and for not providing them with correct answers. They were also unsure of the efficiency of the instructional methods I used in class. They felt too much time was spent on solving a small set of problems.

All participants reported that their teaching was positively affected by the experiences provided for them during the summer course. In addition, they all articulated an appreciation for the epistemological status of mathematical knowledge as the result of their course experiences.

In reporting data relative to the usefulness of the mathematical modeling as a context for creating a community of learners I will draw evidence from the classroom discussions. I will also present episodes from the course sessions to illustrate the participants' struggles to establish discourse standards within their classroom community. In reporting the teachers' assessment of the influence of the course on their thinking and practice, I will draw evidence primarily from the interviews.

Initial data and teacher categories

The participants' responses to the initial survey questions provided data relative to their level of confidence about teaching mathematics and their ability to problem solve and to make sense of mathematical concepts. Combined with the data collected on how teachers exhibited their content knowledge in class during the first week of instruction, I classified the participants under two broad categories: moderately to highly, and least sophisticated. The sophistication referred to their competence in the use of mathematics and their confidence in their ability to do mathematics.

Moderately to highly sophisticated (n=10): The teachers in this category claimed having experience in the use of explorations and problem solving in teaching. They enjoyed being challenged, expressed an interest in solving mathematical problems, and felt confident in their ability to learn mathematics. These teachers felt it was necessary for students to learn to reason and to analyze data. These teachers used their textbooks only as a resource and deviated from it to include authentic tasks in their curriculum.

During the class discussions these teachers frequently made knowledge statements. They identified particular concepts or representations that could be used to solve problems. They provided elaborate explanations to questions their peers (or I) asked and made high-level interpretation statements when they examined problems. These teachers asked questions that required others to clarify their assumptions. They showed the tendency to test ideas. They were persistent in making sense of the problems they explored and exhibited greater tolerance for ambiguity. Lastly, these teachers drew connections between what they learned in the course and ways in which they could use that knowledge in their own classrooms. They often stated that they were curious to see how their own students would react to the problems they explored in the courses (3).

Less sophisticated (n=5): Teachers in this category were less confident in their ability to teach mathematics or to influence student learning. They expressed concerns about being in control of the classroom. These teachers claimed they followed their textbooks closely. They also claimed they never assigned questions to which they did not have an answer. They could often be heard in class stating that the problems I assigned would be confusing to their students.

These teachers had less mathematical tools. They relied mostly on symbolic manipulations and frequently asked question about the procedures and techniques they could use to solve problems. These teachers make low-level interpretations of problems and their solutions tended to be of primitive nature. They were unsure of how to proceed with problems and expressed difficulty articulating their thoughts mathematically. They also refused to tackle problems and often times expressed that problems did not make sense to them. They showed little or no tolerance for ambiguity and requested that I settle disputes in class by simply giving them the "correct" answer. They were reluctant to participate in the large group discussions and refused to ask questions, even when they claimed they did not understand their peers' explanations.

Although categorizing teachers according to the above criteria was not initially planned, it became evident that a coherent analysis of the course outcomes was futile in the absence of these considerations. I will elaborate on these issues in the following sections.

Learning mathematics and learning to build plausible mathematical models

Although on the initial survey a majority of the teachers (n=14) claimed to have regularly used mathematical modeling in their teaching only two of them provided examples that resembled modeling episodes. Seven teachers listed problems similar to standard textbook applications in which a specific algorithm is used to answer a question. These included determining the amount of interest paid on loans given specific rates, finding the probability of winning the Lottery, and finding the rate of bacteria growth or radioactive decay. Four teachers listed assignments that involved building physical models of various objects as modeling situations they had used in their classes. Two teachers did not provide examples. These responses allowed me to hypothesize that teachers had little knowledge about, and experience with mathematical modeling, either as learners or teachers. This assumption was substantiated during the first week of class. When placed in situations where neither the problems were well defined nor a direct path for arriving at solutions existed all teachers experienced difficulties. This new approach to doing mathematics was both challenging and frustrating to them.

To be successful in establishing mathematical models, the teachers had to resolve a problem that also confronts mathematicians in their working relationships. This problem concerns how one views and treats the parameters of the problem. The teachers had to decide what was central to the problem under investigation and what was peripheral. In turn, they had to decide which of the conditions that their peers proposed were worthy and which were not. All participants experienced difficulty when assessing the parameters of the data. A major challenge for them was simplifying the "real" system by making assumptions. The problematic nature of this phase was impetus to creation of a discourse community. During their discussions they argued about those elements they could ignore in order to adopt a mathematical structure that described the real system. Although the participants tended to reach consensus quickly when they worked in their small groups, their solutions were not finalized unless they were discussed both publicly and in the large group. These public discussions forced them to adhere to the standards of practice as exhibited by mathematics community as they tried to make sense of the opposing points of view and assessed their legitimacy.

Consider for instance the following episode from the second week of class during which the teachers were presented with the Rainfall Problem. The problem asked:

  Suppose you park your car in the parking lot and you have a mile to
  walk to your office. As soon as you leave your car it begins to rain.
  Though you do not have an umbrella you decide to take a chance and
  make the trip. How should you walk to minimize getting wet in the
  rain?

Prior to the whole group discussion the participants had worked in small groups to formulate a tentative approach for solving the problem. The groups joined later to examine these approaches. Three different ideas were put forth: 1) that the problem did not provide enough information, thus, it was impossible to solve, 2) the person had to run as fast as possible, and 3) certain conditions had to be set before a mathematical model could be built. After each group presented their arguments, a whole group discussion occurred.

  Arthur. I doubt there is one right answer to this problem. Maybe that
  is why we are having so much trouble with it. Maybe that is just the
  point of the problem, letting us see that there could be multiple
  answers.

  (All teachers look at me expecting a response. I ask if others agree
  with Arthur's statement).

  Gary: I still believe we can't solve this problem cause we don't have
  enough information here!

  Fox: That's what I said.

  Steve: I--we still believe the best strategy is to run as fast as
  possible. It is common sense, right?

  Arthur: I just don't see a problem here.

  Gary: Okay-Let's say the person has to run as fast as possible-If we
  say that then what is the point of the problem.

As Kuhn (1970) suggested, sometimes consensus is achieved only when the adherents of one point of view retire and fail to recruit new followers in the new generation. The group seemed to have reached a consensus, accepting "running as, fast as possible" as the final answer, when one of the participants objected to the approach. Emulating the work in the mathematics community, the group assumed that the problem was not solved unless all members agreed on the legitimacy of the answer. During the subsequent discussion several patterns of talk emerged, some of which were consistent with the conventions of argumentation within the mathematics community. In fact, in their attempts to reach consensus the participants began to rely more on mathematically persuasive strategies. This was most prevalent in the case of sophisticated and developing teacher categories.

  Fox: Does someone else have a different answer to this? I am curious
  to know if anyone else approached it differently.

  Jim: I don't like that answer--(pause) I just don't like that answer--
  See, let's assume a few things here. We can assume certain things
  here, can't we?

  I reassure them that they can.

  Mary: I disagree with Gary! Do you remember the problem we did
  yesterday? How we had to set up some assumptions in order to build a
  model for it? I think we should do the same here. I mean ... What is
  making this problem so difficult or different?

  Steve: There are just too many things we do not know-Like, we don't
  know if there is a shaded walkway-It depends on how fast the person
  can run-It depends on ... (Umm) it depends on whether it is pouring
  rain or just a drizzle ... It depends on ... I don't know, what else?

  Fox: I think it depends on whether the person is carrying a heavy bag
  or not.

  Rosa: Why would that be a factor?

  Arthur.--It slows down the person--It makes him walk or run slower.
  That means that he will be out in the rain longer.

  Rosa: We can go on forever and talk about all the factors that affect
  the situation. That's why I say--I mean in real life we just do what
  Arthur said-We just run as fast as we can and try to avoid getting
  wet. I know that if I give this same question to my students that is
  exactly what I will get from them.

  Mary: I can think of a million situations where we all instinctively
  do things in a certain way but it does not mean that what we do is
  necessarily right.

  Rosa (Cutting her off): I think you just like to argue for the sake of
  arguing. I don't see any point in getting into this with you now.

  Fox: There we go again, they are getting philosophical again (everyone
  laughs)-

  (Mary is quiet and seems withdrawn)

  I ask if anyone would like to challenge or support Rosa or Mary. No
  one comments. I ask Mary if she would like to continue with her
  argument. She declined.

As Latour and Woolgar (1979) suggested, the absence of a challenge may not signal consensus as much as the judgment of the class members that the social costs of a challenge exceeded the possible benefits. Both elements applied to the case at hand. Fearful of scrutiny, Mary was willing to give up her opposition and conform to the majority vote. At this point, I intervened to alleviate the social tension among the group members.

  I ask the group if they were in agreement that the factors identified
  by various people were important in the analysis of the problem. They
  all agree by shaking their heads. I ask if there are other factors
  that should be taken into consideration when reviewing the problem. I
  ask Monica, a very quiet member of the group, to state a few factors
  that she considered as important.

  Monica: This is just way too above my head ... I look at this problem
  and I know you probably want something like a formula but I can't even
  think of a formula for this. I am used to really straight forward
  questions, like word problems. I am real uncomfortable right now.

  The whole group is silent. I ask if others feel the same way. At least
  5 others voice the same concern as Monica.

  Scott: See, this is math like I have never seen before. I mean with
  what we did yesterday and what we are doing now--I just can't say who
  is right or what is right.

  At this point the group is ready to give up working on the problem. I
  ask Jim (the person who had initially insisted on finding a
  mathematical solution to the problem) if he would like to add anything
  else to the discussion.

  Jim: I am thinking here--I am thinking, (um) okay let's define a few
  things here. For example, I want to know what we mean by "wet, " like,
  okay, if it is you out in the rain, someone of your size, then it
  would take what 10 seconds and you ARE WET (everyone laughs), but if
  it is me in the rain, well, that is a different story-

  Scott: How do you mean?

  Jim: I mean I am bigger than Azita--so even if the same amount of rain
  is poured on each of us she will get wetter than me.

  Researcher. So, what is Jim saying here? How can we quantify this?

  Monica: I guess it means that the size of the individual is a factor.
  Like how big or how small a person is ...

  Researcher. Good, what else? How could the size be captured
  mathematically?

  Mary: Surface area? Of course! Surface area. Also, we need to find the
  volume of rain that is pouring. Am I on the right track here? (Looks
  at Jim for approval)--Jim here is my conscious. He is the King of
  calculus, so if he okays it, I know I am fine.

  Jim: I think we should try and standardize the conditions ... It makes
  it easier to handle--By standardizing it we won't need to worry about
  all these special cases.

  Researcher. What else should we standardize here? What else should we
  consider as constant so to avoid a lot of mess?

  Scott: How fast it is raining? I mean whether it is pouring cats and
  dogs, or is it just a drizzle.

  Janet: Right, it is like if you are out in a heavy rain for a minute
  you certainly get more wet than if you are out for the same amount of
  time but when the rain is lighter.

  Jim: Sounds reasonable to me. What do you think? Does it sound
  reasonable to you?

  Mark: If we stay with this line of reasoning then I want to know what
  we should consider as running speed I mean almost all of us said that
  we should run as fast as possible, but do we all run at the same
  speed? I guess I am going back to what was said earlier about whether
  the person is carrying a heavy load or not ... Perhaps it is best to
  just assume the person is running at a particular speed

  Rosa: If you two think it is reasonable then I guess I am for it too.

  Tom: I second that! (everyone laughs).

  Researcher: You need to convince each other, mathematically, that it
  does make a difference.

  Laura: I may be completely off here but while everyone else was
  talking I started doing some calculations ... I know that the distance
  is rate times time. So, if the person can run at say 15ft per second,
  then he is out in the rain for about 9 minutes, give or take a few
  seconds. Now to see how wet this person gets we need to know how fast
  it is raining and how hard. Otherwise we can't solve it.

  Mary: So, here we can make similar assumptions like you did with the
  running speed. We can assume that the rain is falling at a certain
  speed and it is coming at a particular intensity. Do we all agree on
  these?

  The majority of teachers nod their heads in agreement.

  Scott: I am not sure how you define "how hard" it is raining.

  Mary: We can define it in terms of how much water is accumulated in a
  certain amount of time. The same way they report it on T V.

  Scott: I think this is making more sense now. I mean I can actually
  see how it might be done.

  Jim: I am not sure we have accounted for all the variables here. We
  need to work on it a bit more. Did we agree on a particular size for
  the individual? (looks at the group) I think we should agree on
  certain things here.

  Joon: One more thing I meant to ask--It just came to me--I remembered
  those standard traveling upstream questions ... Are we saying that the
  wind is coming to play here? Or is it coming towards the individual or
  against it? It does affect the speed.

  Researcher. So, you need to consider, the individual's size, the
  direction of the wind and its speed. What else?

  Joon: I think we need "think" time in our group now. (Laughs)--This
  has been mind blowing and I need to think about it a little on my own.

  I ask the group whether they would like to discuss the problem further
  or to continue with the large group discussion. They request that they
  be allowed thinking time in their small groups.

Notice that although not explicitly stated by group members the participants abandoned their initial solutions and resorted to a system of reasoning that was more analytical. The emergence of mathematical rhetorical strategies during the class discussion was a promising development. The teachers adopted these standards and conventions because of their persuasive power rather than because I told them to do so.

To complete the mathematical modeling process the teachers had to test the accuracy of models in predicting the behavior of real systems. In mathematics community the negotiations about the veracity and accuracy of models and assumptions often play a central role (Kuhn 1970). During these discussions, the mathematicians rely on a number of standards and conventions. One of such standards is that the model has to account for more than special situations. Other standards include care in the use of mathematics, and attention to whether what is reported is consistent with patterns seen in the real condition as a whole. To function successfully as a community of validators the teachers had to use and respond to these mathematical standards.

Initially, when confronted with disagreements, teachers depended upon rhetoric strategy of shouting down the opposition. During the first week of class approximately 22 of such occasions occurred. The number was decreased to 14 during the second week. Only 6 occasions of similar nature were evident during the third week of class. The participants also commonly relied on the approval of those peers with greatest social status (those that taught advanced mathematics). This type of behavior occurred almost daily during the three weeks of data collection. Moreover, it was common for participants to insist that their own model was adequate even though they were unable to present arguments that persuaded their peers. In some cases, they rejected solutions without adequately examining their legitimacy. These rhetoric standards of argumentation seemed acceptable by all participants and unless I pressed them to establish rigorous conventions for their mathematical discourse they felt no need to do so. In essence, establishing consensus on problems without coercion posed a social and organizational challenge to the participants that they were initially incapable of meeting. They lacked shared standards and conventions for discourse and argumentations. They did not know how to compare their results or discuss their differences in ways that would ensure everyone a chance to be heard. They had difficulty deciding when and how differences in claims could be resolved.

Obstacles to Creation of a Community of Validators

Initially, in the presence of public disapproval many of the participants were reluctant to continue to defend their solutions. Even the most mathematically sophisticated teachers and those with a high degree of confidence in their solutions chose to withdraw from public discourse in fear of being ridiculed by the group, or for being perceived as "pretentious." Although a majority of the teachers continued to refine their own mathematical methods and models with a small core of colleagues, they refused to share their work with others in class. Those teachers with less sophisticated mathematics background, and those that possessed little confidence in their work frequently avoided public argumentations. These teachers rarely raised concerns about the mathematical models their peers offered and immediately agreed with those solutions that appeared more technical. The sense of community was most prevalent when all group members struggled with the same problem and when the need to provide affective support for the members of the group took priority over their need to resolve mathematical disputes. In fact, it was common for the teachers to compromise the need for mathematical rigor and accuracy in order to maintain the unity of the group. In addition, some participants, even with extensive social experiences at discourse, failed to meet the standards of practice commonplace within the mathematics community. The social status of particular individuals was most influential in how their peers received their solutions to mathematical problems, and whether their ideas were challenged at all. Although public testing of ideas became more commonplace among the participants, they occasionally accepted claims without challenging them. This was particularly evident when a peer with less mathematical confidence presented a solution to the large group. In these cases, the teachers were more concerned about building the confidence of the peer and refused to challenge the individual.

Connecting Learning and Teaching

The results of the interviews provided evidence that the course activities had seemingly influenced each of the participant's thinking, if not practice, in three distinct ways; 1) reconsidering the curriculum they taught, 2) reconsidering the teacher's role in the classroom, and 3) reconsidering views about worthwhile mathematical knowledge for students.

Reconsidering Curriculum

Ten participants provided a minimum of 6 lessons and sample assignments from courses they taught that indicated they had used resources from the summer course in their own teaching. All the teachers in this category were determined to continue using data driven tasks in their instruction. They attributed their enthusiasm for using authentic tasks to the high quality of their students' work on such assignments. They were surprised by their students' capacity to do "hard mathematics." These teachers reported an improvement in the students' reasoning and problem solving skills as the result of the use of modeling activities in their classes. The teachers also expressed concerns about the amount of time it took to implement such activities in class. Six teachers suggested that using modeling projects and problem solving contexts had become the "norm" in their instruction. These were the same teachers who showed greater skills in solving modeling problems, demonstrated greater confidence in their ability to reason from data, and frequently took on leadership roles among the group. These teachers claimed that they had begun departmental wide discussions regarding the adoption of new textbooks that supported the type of teaching and learning I promoted during the summer class.

Five teachers noted that due to time and curricular constraints they were unable to make any changes in their curriculum. The same teachers stated that their lack of confidence in their own ability to determine reasonable solutions to open ended tasks prevented them from using problem types similar to those they had studied. Interestingly, two of these teachers taught Algebra II and pre-calculus courses in their respective high schools.

Reconsidering Teaching

Although on the initial survey all 15 teachers had reported that they encouraged their students to share their ideas in class, during the final interview they admitted that they had developed a better understanding of what it meant to work towards establishing a community of learners. Twelve teachers claimed that they allowed students to explore problems independently. Fifteen teachers stated that they routinely asked "how" and "why" questions requiring students to express their thinking, encouraged students to determine the reasonableness of solutions as a group and by forming convincing arguments. Moreover, 14 teachers stated that they listened to students more carefully and tried to make sense of their ideas.

Nine teachers referred to the complexities and challenges associated with teaching in ways consistent with the model provided for them during the summer. These same teachers expressed that at times they felt inadequate about dealing with the various solutions students presented in class. They were unsure at times about how to help students adapt to the standards and conventions of discourse. They also articulated their lack of comfort with "letting go of being the authority" in class and found it difficult to say, "I don't know." In spite of the challenges they experienced, these teachers persisted that they would continue to learn more about the new teaching roles they had adopted.

Worthwhile Mathematics

The summer course activities appeared to have impacted the participants' perceptions of what constituted worthwhile mathematics. All teachers stated that they had never experienced doing mathematics in ways that I had expected them to do during the summer course. All of the teachers explicitly referred to the inadequacies they felt in applying their knowledge to "real" problems they explored during the summer course. All teachers expressed that they realized the value of inquiry and mathematical communication for their own students. Even the five teachers who seemed least affected by the content of the course articulated that they needed to devise plans to help their students learn to apply their knowledge rather than doing only exercises.

During the final interview all teachers also expressed an appreciation for collective inquiry, and the group's collective power in establishing "new mathematical knowledge." All fifteen teachers realized the value of the collective verification processes they had used in resolving mathematical dilemmas. According to all participants, this was the first time they had experienced "doing mathematics as a community activity." All fifteen teachers suggested that they were "amazed" by the amount of mathematics they learned from each other. The teachers also stated that their understanding of mathematics was increased as the result of their interactions with peers. Moreover, all teachers, even those that rarely participated in class discussions, expressed that they felt they had become better mathematicians as the result of their experiences in the course.

All teachers stated that they used small group activities more frequently, and encouraged collaborative decision making among their students. They admitted that they found it difficult to "stay quiet during students' explorations" without trying to directly influence their work.

Discussion

I am confident that a majority of participants in this study came to a richer understanding of the power of groups in problem solving, of understanding how models are grounded in observable phenomenon, and in developing social and intellectual tools that enabled them to participate in collective reasoning processes. Yet, I am still deeply concerned that these goals remained least fulfilled for those teachers with less mathematical confidence and skills. While the participants with a deeper mathematical understanding and greater confidence in their ability to do mathematics tended to appropriate the discourse and practices of the community as their own, this kind of appropriation was least visible in case of those with less mathematical confidence and sophistication. For five teachers the mathematics content of the course seemed too abstract to have practical merit. Although these same teachers claimed that their instruction was fundamentally affected by their experiences in the summer course, I am skeptical of their ability to establish the type of practice that would successfully advance their students' mathematical thinking without further mathematical training. In my class each of these teachers had minimal participation in mathematical discussions. They tended to quickly agree with others' arguments and seemed unable to assess the validity of the solutions their peers presented in class. These teachers were reluctant to engage in doing mathematics as learners. Mathematics explorations were filtered through their own classroom experiences. If they found an idea difficult, they assumed it would be confusing for children to do as well.

These results illustrate an issue that has yet to be fully resolved either in my own thinking or in the mathematics education community. This issue concerns limitations in our understanding about the nature of professional development activities designed for teachers of various mathematics backgrounds and the ability to establish a classroom culture that is conducive to collective inquiry in the presence of such diversity.

Professional Development Opportunities for Teachers of Mathematics

Creating a learning community in which pupils work collaboratively towards constructing mathematical knowledge is an enormously complex and difficult task. The magnitude of difficulty is increased when the members of the community share vastly diverse knowledge bases and backgrounds. To initiate and sustain the type of discourse that is displayed within the mathematics community, the group members need to have common knowledge of rules and standards of mathematics and of discourse. The kind of social and mathematical norms that are established and the ways in which activities are structured have much to do with the student engagement, and hence understanding. For those individuals whose understanding of how to settle disputes is based on their experience in traditional classrooms, establishing social norms for collective inquiry is a slow and difficult process.

Although the various levels of mathematical capabilities teachers brought to class contributed to the creation of a "community," it did hinder the establishment of a "community of validators" as envisioned by Cobb and Bausersfeld. On the one hand, the teachers shared similar frustrations when they faced challenging and unfamiliar problems. These common experiences as learners provided them with a shared language in which they articulated their feelings relative to course expectations. On the other hand, the diverse backgrounds of the teachers made it difficult for them to work as collaborators. Often times more successful teachers took on a leadership role in small and large group discussions. Their answers were almost immediately perceived as "correct." In addition, least successful teachers, though frequently had good suggestions seldom had any impact on what the group decided. These teachers either withdrew their ideas in fear of being challenged or abandoned them immediately for lack of mathematical tools and techniques to support them.

Despite the seemingly positive outcomes of the course as articulated by the teachers, I question the long-term influence of the experience on those participants who found the content of the course too advanced and my teaching inefficient. It appeared that those with a deeper knowledge of mathematics and greater facility with mathematical tools managed to benefit from the course content and drew implications for self-practice. This was evident in the quality of work they submitted during the course as well as in the sample lessons they provided during the final interview. Certainly further exposure to similar mathematical experiences will allow them to continually refine their curriculum and instruction. I am reluctant to draw the same conclusion for those with less sophisticated mathematical preparation and confidence. Studies of classroom discourse (Gee 1991; Michaels 1991) suggest that the relative lack of success by marginalized students is at least partially attributed to the instructional approach I used in class. I was attempting to help teachers develop the skills in working towards building sound mathematical arguments in collaboration with their colleagues. It was also my intention to enculturate them into the type of mathematical discourse and argumentation that is often practiced within the mathematics communities. In both of these respects, I was partly successful. Perhaps, a mathematics course designed to cater to specific level of knowledge of this group would have been more successful in making both the content and pedagogical points meaningful to them. Certainly, the possibilities of genuine education depend not so much on the already existing student's knowledge and experience as on the characteristics and the context of the learning environment that foster development and growth of individuals within that context (Vygotsky 1978). While the potential for knowing and development exists within the social context of a learning environment, a trajectory of this development cannot, and should not, be set in abstraction and independent of initial understandings that different individuals bring to the learning center. Teachers' prior knowledge should impact the content of the professional development activities they are offered. In light of this, I question the common wisdom of designing uniform learning opportunities for all teachers characterized as mathematics teachers of a certain school level. Clearly, the needs of teachers vary depending on their teaching orientation, background knowledge, current understanding of the curriculum they teach, and command of the representational tools effective for mathematics teaching. Planning common activities for individuals who posses vastly different levels of understanding potentially compromise the quality of experience for some of them. Either the needs of more advanced individuals are sacrificed at the cost of fostering the growth of less sophisticated individuals or the needs of less advanced group are neglected at the cost of accommodating those with more sophisticated knowledge base. Therefore, I argue here that considering differentiated learning experiences for teachers may be essential to their professional growth. This issue has not been considered in mathematics education community in the past.

In recent years, numerous studies have explored the outcomes of various innovative courses designed for teachers in terms of their impact on participants' beliefs and attitudes towards mathematics. A majority of this body of research tends to highlight the positive influence of such courses on teachers' outlook towards teaching mathematics. A substantial reference to mathematical development of participants enrolled in these courses is rarely made. There is a paucity of literature that investigate the work and growth of mathematical thinking of teachers in domains of mathematics that go beyond elementary concepts. The study I reported here partially addressed this gap. The results stress the need for a careful examination of research findings relative to teachers' mathematical growth as well as the impact of teachers' prior knowledge on what they actually gain from learning experiences that capitalize on collaborative work and discourse. A limitation associated with the design of the current study was its failure to utilize classroom observation as a component of assessing teacher development. Future studies may attempt to investigate the impact of similar initiatives on teachers' actual classroom practices by observing their instruction and documenting their work over time.

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Azita Manouchehri

Central Michigan University

(1) Classifying teachers according to the mathematics courses they had taken was virtually impossible since they had completed their degrees at various institutions across the country. These institutions vastly different degree requirements. Accordingly, in reporting teachers' content backgrounds I rely solely on the types of courses that seemed to have been of similar nature according to the teachers' description of their content.

(2) Participants were asked to provide videotapes of their classroom instruction as well. Although 7 teachers provided videotapes of their teaching, due to the poor quality of videotapes, it was impossible to either code them, or to make conclusions about the quality of classroom interactions. Therefore, these videotapes were not included in the data analysis process for the study.

(3) Comments included: "I cannot wait to use this in my class," or, "I cannot wait to give this to my students."