Lee justified his use of round tables by stating the students were more comfortable talking with each other and asking questions during class. He believed this is significant to their learning process because they all learned when they assisted each other.

A Typical Day

    Lee typically began class by introducing an activity, collecting data, or presenting a problem. He allowed students to ask questions, answer questions, or state their own ideas. These discussions were spontaneous and loosely organized. Often questions or comments would lead them to investigate related topics in detail (Field Notes). In a follow up interview, Lee explained why he was unconcerned if they were seemingly off the subject.
    They have their own personal understanding of the statistical ideas they discover or that I present. It’s very unlikely that many of my students understand sampling distributions, the Central Limit Theorem, confidence intervals or hypothesis tests in the same way and certainly not how I understand it. I think when they’re allowed to discuss these ideas among themselves, they share their individual notions of what’s going on. Many of them do not think linearly (I certainly don’t), so other, seemingly unrelated ideas, are part of the discussion. And when numerous versions of reality come together, I think everyone participating in the discussion comes away with a better, or deeper, understanding of the concept. (personal correspondence, April 2, 1998)
    If students asked questions about specific homework problems, they would address those issues. However, he did not solicit questions directly related to homework assignments. "If homework questions get asked and answered, that’s ok," he said. "But it’s up to the kids to make me do that because ultimately we will talk about the homework content during class" (personal correspondence, March 23, 1998). Lee did cover the syllabus material thoroughly (Field Notes).
    Beginning in January, another type of typical day occurred weekly on "project day." Students met with their groups and discussed their projects. Lee was available for questions, but he did not circulate the room. If they needed information, permission to go to the library, or statistical advice, they approached him. He gave advice and made suggestions, but rarely gave them detailed instructions. I asked Lee why he gave them vague answers to their questions about their projects. "I think the project and my lack of guidance helped them to connect all the seemingly disparate pieces of the statistical puzzle," he said. "Before the project, they probably saw the ideas as being unconnected to much of reality" (personal correspondence, March 17, 1998). Findings from students’ interviews support Lee’s idea that the project did help students’ construct overall understandings of statistical concepts.

Lee as a Constructivist

    Confrey (1990) proposed a model that connects classroom instruction to the constructivist learning theory. This framework consisted of six components: the promotion of student autonomy, the development of the reflective process, the construction of case histories, the identification and negotiation of tentative solution paths, the retracing and group discussion of the paths, and the adherence to the intent of the materials (Confrey, 1990). Inherent in this framework was the teacher’s commitment to active learning. Confrey suggested that the responsibility for and control over learning must shift from the teacher to the student.
    The first component necessary to shift responsibility to students required that students make a commitment to their answers. To accomplish this instructionally, teachers can ask students if their answers are correct, engage students at least by requiring that they explain what they attempted, act primarily as a facilitator, and/or involve students in evaluating their own work. Lee accomplished each of these elements, although not with equal weight. The discussion began with a question, posed by Lee or by a student. When another student answered the question, sometimes he responded with a question, sometimes with silence allowing others to explain or refute, and other times he probed that individual students’ thinking and forced him or her to explain their idea (Field Notes). These interactions between Lee and his students were the norm. This mode of instruction guided most classroom sessions.
    Next, Confrey (1990) suggested that students must "modify and adapt their constructions" (p. 116). She suggested that they face situations that are problematic, take action to solve the problem, and examine their action to determine if the problem has been resolved. Lee’s teaching included these strategies. He frequently began class by distributing a handout or assigning a problem on the board. Students worked at the round tables but usually sought to answer the question individually. After an initial solo attempt, they consulted others by asking questions or checking their answers. Their discussions focused on trying to determine whose answers, and therefore procedures and processes, were accurate. Then, as a class, they discussed and answered the question. Lee consistently forced them to explain what the problem was. He then allowed students to describe their strategies and forced them to defend their choice of strategy.
    Confrey’s (1990) third component evolved from student teacher interactions. She referred to the teacher’s familiarity of a student’s knowledge and abilities as a "case history" (p. 118). She suggested that teachers learn their students’ strategies and use this understanding to develop appropriate actions toward their problem resolution. Lee was willing and interested in developing these case histories with every student. Since he had the desire, small class sizes (14 students in fourth period, 17 students in sixth period) made this easy. Some students, however, were more open than others and Lee’s knowledge of their learning processes was more detailed. His accomplishment of this became apparent in our informal conversations. When we discussed a specific project, for example, he frequently conveyed insights into their learning and processes that revealed how well he knew that student. He used this knowledge to direct that student toward appropriate actions that would address his or her concern.
    The fourth component of Confrey’s (1990) framework suggested that teachers use case history knowledge and flexibility to adapt the class objectives depending directly upon students’ questions and understandings. One of Lee’s strengths as a teacher was flexibility. In our interview, he revealed that whenever students digressed, he saw this as an opportunity to investigate their thinking. He was never concerned that "getting off the subject" was that. He believed that students’ questions and issues, which seemingly detracted from the day’s purpose, were important to investigate. Many days, he admitted, he came to class with little or no predetermined structure. He presented a problem and let the students’ actions and responses determine where the investigation led. He was never concerned that they would not eventually cover all the material. His saw his primary responsibility as leading the class to investigate and discuss whatever they believed was relevant to the problem and its solution.
    Confrey’s (1990) fifth component involved reviewing the students’ solutions. She suggested this provides a variety of positive opportunities for students: reflecting on their process, looking at the specific problem holistically, and contributing to a sense of accomplishment. It is unclear if this must be accomplished on an individual basis. Lee provided these opportunities, but most frequently within the group settings. After presenting problems or while reviewing corrected test papers, he encouraged questions and discussion while leading them to an overall sense of the problem and its solution.
    The last component of Confrey’s (1990) framework required a commitment from the teacher to the mathematics content. While the instructor might not have a specific agenda of how to cover the material, it is essential that he or she had a clear picture of the information that needed to be addressed. After observing Lee extensively, it is easy to conclude that Lee was aware of the material required to cover the syllabus and prepare his students for the AP Exam (Field Notes).
    Overall, Confrey’s (1990) framework provided suggestions for teachers willing to shift their pedagogy and emphasize student activity, construction, and reflection. She admitted these techniques may appear to impede progress as related to traditional teaching techniques to cover material. However, she believed the knowledge constructed through these types of activities and processes was more accurate, more meaningful, and more powerful.

Interviewed Participants

    As the researcher, I was interested in the effect of concept-oriented instruction as recommended by the AP Statistics Test Development Committee. Lee and I examined a variety of characteristics to choose the specific students. A total of 12 students were chosen to interview in depth. It was important to obtain a diverse group of students representing all levels of the academic talent spectrum. I was also interested in observing juniors and seniors, males and females. Another important criterion for each student was to be outgoing and receptive to my presence and research. For example, three boys sat at one table in fourth period who rarely spoke to the instructor, the class or to me. These three students were easily eliminated for their shyness. The instructor and I frequently engaged in conversations concerning students’ academic abilities and other factors we thought might influence students’ learning. For example, I was interested in the only sophomore student taking his first AP class. His parents, however, would not sign the consent form and he was eliminated.
    It was important to select informants from both observed classes. The atmosphere of these two classes was quite different. Fourth period students engaged in discussions contained to their tables and students at neighboring tables. A student would frequently converse with the instructor. Other students would join the conversation while some continued to stay within the confines of their table. Sixth period, on the other hand, was frequently loud and sometimes ignored the instructor altogether. They would be engaged in statistical problem solving, yet would carry on conversations with students across the room rather than enlisting the assistance of someone closer. The noise level of this class was consistently higher than fourth period. Frequently, this class was behind schedule with respect to the other two classes because they would slow down instruction of the material. Therefore, two different groups from each class were targeted to study their project progress in detail. I interviewed at least one member from each of the project groups that were chosen. Listed below are the reasons that each participant was chosen.

Adam (AD): As a member of the mathematics team he represented the upper academic spectrum for juniors. He enrolled in Calculus BC his senior year. He is of Indian decent.
Beverly (BH): She was a senior who was not recommended for the AP Statistics class. She approached the mathematics chair, Lee, and requested permission to take this class. She was the only student out of all 40 enrolled who had not taken Precalculus. She was interviewed in detail to represent the group project BDL (Beverly, David, Lisa).
Carter (CF): He attended Governor’s Honor Program (GHP) the previous summer in mathematics. His interest in mathematical theory and rigor were significant. He was an African American junior who was recommended to take Calculus BC his senior year.
David (DW): He is also an African American male. Compared to the other 30 students, his mathematical abilities were average. He enrolled in Calculus AB his senior year. His attitude was excellent.
Ethan (EB): As a member of the debate team, he would frequently converse with the instructor in a more detailed way than other students. He displayed stubbornness about giving up before he believed he understood the concept. He was a junior of average academic talent. He enrolled in Calculus AB for his senior year.
Fran (FE): She was selected because she had Calculus BC the prior year as a junior and made a 5, the highest possible score, on that test. She falls on the upper end of the academic scale and was a senior. She was the student selected to represent the group known as FMN (Fran, Mark, Nancy).
Gretchen (GH): Academically, she represented a below average junior girl. However, her interest in learning was strong. She asked the instructor many questions during class and worked well with the other members at her table. She sparked my interest because she sometimes had an idea of the concept, but would continue to question and pursue until satisfied. She also consistently did the assigned homework. She was placed into Calculus AB for her senior year. She was selected to represent the GRS (Gretchen, Rachel, and Sam) group project.
Hanson (IF): His family heritage at this school was prominent. Academically, he fell in the middle of the junior class and was recommended to take Calculus AB for the next year.
Ingrid (IS): She was chosen because she consistently indicated an interest in learning statistics and expressed curiosity for my research. She was recommended to enroll in Calculus BC this year. She was Hindu Indian. She was selected to describe the ITW (Ingrid, Terry and Wesley) group project.
Josh (JK): He was chosen to represent the lower academic end for juniors. He was recommended for Calculus AB his senior year.
Lisa (LB): She was chosen to represent the below average female junior. She enrolled in Calculus AB her senior year. Like the others chosen for this academic reason, she worked with others well, was not intimidated to ask questions of the instructor, and was receptive to my presence.
Sam (SD): He represented a second generation Hindu Indian. Academically, he fell in the mid range for junior males. He was recommended to take Calculus BC in his senior year.

These were all honors students and each junior was advised to take an AP Calculus class their senior year. Overall, of the 12 students chosen, 5 were recommended to take Calculus AB, 5 were recommended to take Calculus BC and 2 graduated. I interviewed 7 males and 5 females. Nine students interviewed attended sixth period and 4 attended fourth period. Other students are quoted as members of the four selected groups: Mark, Nancy, Rachel, Terry and Wesley.
    The interviews were conducted the first 2 weeks following the administration of the AP Statistics exam. I was concerned about their ability to recall details on a test that was completed and, in many ways, no longer relevant to them. Because seniors at this school completed their course requirements and graduated before other classes are completed, I interviewed seniors first. Others were interviewed as early as possible. Since I observed the classes for the entire academic year and knew the students, I opted to audio tape only. Students were interviewed individually and privately. For the interview, we simply found a quiet, secluded spot near the classroom. Student interviews were conducted during class time. Every student appeared comfortable during the interview. Several students who were not selected indicated interest in participating.
    I asked each participant interview questions designed to probe areas of interest related to the research questions. Each student addressed the same questions (see Appendix A), but not in the same order because "the sequence of questions varies with each respondent, depending on prior answers" (Romberg, 1992). Frequently a student would answer a question not yet asked, while others would stray off the topic altogether. If they digressed, we explored their emerging ideas before returning to the list of questions. Also, it was common for a student to respond, yet not directly answer the question. I allowed them freedom to speak about whatever they felt important. I realized some of this information was not relevant to my research questions, but it was critical that they felt completely at ease to say what they wanted. In any of these events, I would probe until I felt all questions had been answered or until I believed the student was unable to provide the requested information.

The Students’ Perspective of Statistics

    In each interview, I asked the students to tell me about their statistics class. All 12 replied that the course was different from other mathematics classes. Five students suggested the course material was more applied. "Math was really different this year," Ingrid said. "Last year we were just totally manipulating numbers. That’s all it was. If we were sitting there writing the equation for a semicircle, ok, big deal. The thing about stat, I can admit that I saw the point to it" (IS, Interview). Ethan expressed a similar opinion.
    It’s a lot clearer from the very beginning why, what the final goal is when learning all these different steps. Whereas opposed normally [sic], you learn all these steps and maybe eventually when you are done, you learn why you’re doing it. Here you know why you are doing what you are doing from the very beginning because you just have the whole concept of trying to compare numbers that have been [sic], everything seems like it’s just trying to make two numbers that are apples and oranges comparable, whether its different numbers of samples or different characteristics and you are trying to account for that. You know that’s the goal the whole time and you can kind of see how everything you’re doing works toward that goal. (EB, Interview)
    Gretchen agreed with Ethan and Ingrid. "We took problems of life, like trying to figure out averages and things like that, real situations," Gretchen said. "I liked it, because unlike precal I can see how it relates to life. It is usable" (GR, Interview). Sam connected the practicality to a specific field of study.

It was more practical than other math classes. In precal and calculus, I don’t see any practical use to it. What are you going to do with a parabola or a derivative? At least here you can actually tell where people get their numbers that they do and how they make confidence intervals. If you ever study medical papers, that’s all they have. My sister is studying to become a doctor and she had to take a stat class to figure out what all the intervals and stuff were. (SD, Interview)

Five out of 12 students interviewed described differences between statistics and other mathematics classes related to practical applications.
    Consistent with Lee’s teaching emphasis (Lee, Interview), seven students reported the difference between statistics and other mathematics classes as conceptual. Hanson stated this difference clearly. "It was just different from math because it is very conceptual instead of working a set pattern," he said. "It’s like an English type of math. You describe things instead of solving for them" (HF, Interview). Ethan described his idea of conceptual learning.
    In other math classes you are doing math, like number crunching. But here, a lot of the number crunching is a lot less important. You don’t get equation sheets in other classes because what you’re learning is the equations. Here, it’s not learning the equations, but what they mean. And, you have to know, it’s a lot less clear. In other math it’s written in stone which equation you use. But that’s what the goal is here. It’s not if you know these equations, but do you know when to use them and how they work. That’s the biggest difference--not knowing the equations but when to use them and how they work. (EB, Interview)
    Sam explained a similar impression of statistics. "Its not solving for this variable," he said. "It’s more like, it’s not exactly what I think of as math. You do have some numbers but it’s more important to understand the concepts. In math usually its just plug in this number and you solve for x. Statistics is more idea oriented" (SD, Interview). David reiterates this same idea. "He tried to make us shy way from just punching numbers in the calculator," he said, "and talked more about what it means, understanding what we were doing" (DW, Interview). Adam expressed the same idea. "You could spend more of your time doing the concepts instead of the nit picky details with it" (AD, Interview). Overall, half the students interviewed suggested that statistics was more conceptually oriented than other mathematics classes they had previously taken.
    Five students stated a somewhat similar opinion by expressing that statistical manipulations were unlike those in other mathematics classes. Adam stated his opinion. "I guess you had to do a lot of reading as opposed to deriving, less computation," (AD, Interview) he said. Josh had a different idea. "When you are solving something you are not going to get an exact answer," (JK, Interview) he said. Two students suggested that statistics is not mathematical. Fran was surprised.

KR: Tell me about the AP Statistics class
FE: I went into it and I thought it would be pretty math based. I thought it would be more like you have a situation and you have to create a way to model the situation and analyze it from the model. But then after the first couple of days I realized, I obviously had no idea about the different statistical processes, I thought that was pretty math based. But a lot of it, I was really surprised. It seemed at least to me 80% of the entire course was not the same kind of math logic as it is for like calculus or prealgebra. I’m better at really concrete math, like those other classes.

Carter took this opinion one step further.

KR: Was there anything in this class that was similar to other math classes?
CF: I really didn’t think so. This is the first math class that I have been able to go into with a calculator and no paper and function and sometimes not even the calculator. He would just give us a worksheet and all we would have to do, especially on the exam preparatory worksheets we got the last half of the year it was so, it just seemed kind of pointless to me that we could just for almost every answer on the page we could type something in the calculator or draw back on some further knowledge to come up with an obvious answer to the question without ever having to get out a pencil or conceptualize a problem in your head and then write it down and then solve it. It just seemed that so little of it actually had to do with math.

These two students who indicated that statistics was not mathematical had the best mathematics background out of all three classes. Carter attended Governors Honors Program the previous summer and Fran scored 5, the highest score, on the Calculus BC test the previous academic year.
    Attempting to get an idea of their overall attitude about mathematics and statistics, I asked them what was their favorite class and if they liked statistics. Nine students said they liked statistics and referred to the practical applications. The mathematically weakest student explained her overall impression of AP Statistics.

Beverly: I liked it because what I got out of the class I think mostly, was a good grasp of the general concepts. The people who had a stronger background, a more in depth background than I did, probably got more of the nit picky stuff. I got more of a general sense of what normality is, how to plan a statistical study, instead of the step by step formulas. I think for myself, since I don’t plan to become a mathematician, I plan to teach high school English, it really served me well because I have a general grasp of what’s going on in this general area. I think it was a good class for me. (BH, Interview)

Three students indicated they did not like statistics. These three were among the top four mathematical students. They said they missed the rigor of pure mathematics and proofs. These are the same two students who said statistics was not like other mathematics classes. No race or gender patterns emerged regarding opinion of statistics.

The Projects

    The third research question examined the effect that projects, that employ cooperative group problem solving and writing, had on students’ development of statistical concepts. The Test Development Committee recommended student projects as part of the AP Statistics curriculum. The AP Statistics Course Description stated reasons to include projects (College Entrance Examination Board, 1997).

I truly want them to find their own way in the project. I envision the
project (whether successfully or not) as a culmination and tying togetherof the AP Statistics experience. It is a very global understanding I am aiming for and for every detailed instruction I give them, the more localized the experience becomes. I wanted them to experience the frustration and/or exhilaration of focusing on an interesting question, designing a method and collecting the data to answer the question, analyzing the data, and finally trying to make sense of the data to see if it truly helped to deal with the original question. It's a monumental task for anyone, let alone 17-18 year olds, but it removes statistics from the classroom and brings the subject to life. (personal correspondence, April 14, 1998)