for Teacher Preparation

Draft

Tad Watanabe, Towson State University

J. Randy McGinnis, University of Maryland at College Park

Mary Ann Huntley, University of Maryland at College Park

The preparation of this manuscript was supported in part by a grant from the National Science Foundation (Cooperative Agreement No. DUE 9255745). ... [T]he scientifically literate person is one who is aware that science, mathematics, and technology are interdependent human enterprises with strengths and limitations... . (Rutherford & Ahlgren, 1990, p. ix)

The above quotation from

As classroom teachers are increasingly expected to teach with an emphasis on connections between subject matters, in particular between mathematics and science, we must ask, "How does a teacher develop such an ability?" More specifically, for those of us who are involved in undergraduate teacher education, the question is "What kinds of undergraduate experiences will foster the ability to emphasize connections between mathematics and science?" It appears reasonable to conclude that preservice teachers' undergraduate programs should provide a wide range of opportunities to understand how mathematics and science relate to each other.

The Maryland Collaborative for Teacher Preparation (MCTP) is an NSF funded project attempting to create teacher education programs to prepare special mathematics and science teachers in the middle grades. The teachers who complete these programs will be able to teach mathematics and/or science at grades 4 through 8, emphasizing connections between these disciplines. One of the primary components of these special teacher education programs is undergraduate mathematics and science courses that will emphasize connections between mathematics and science. This paper will report findings from an investigation that is studying MCTP college instructors' perceptions about mathematics and science, how these disciplines relate to each other, and their efforts to teach mathematics or science with an emphasis on connections between these disciplines.

This paper consists of three sections. In the first section, we will briefly describe the MCTP project as a background for our research. In the second section, we will discuss an overview of MCTP research efforts. Finally, we will share our findings from the first two years of an on-going longitudinal study.

It has been said by many that teachers teach the way they were taught. For example, Lortie(1975) called the experience teachers had as a student an "apprenticeship of observation" (p.61). If that is the case, we must reformulate college level mathematics and science courses prospective teachers take as a part of their college programs. Those courses must also emphasize the connections between mathematics and science, and they must be taught in a manner that is appropriate for them to use in their own classrooms. These are two of the bases for the MCTP programs. In addition, constructivism and the notion of "less is more" are also central to the philosophy of the project. The project aims to design and implement such special teacher education programs at a number of state supported institutions of higher education in Maryland. The program is unique in that it involves not only mathematics and science educators but also a significant number of mathematicians and scientists who teach undergraduate mathematics and science courses. Specifically, the MCTP program consists of the following:

* specially designed courses in science and mathematics, taught by instructors committed to an inquiry based, interdisciplinary approach

* internship experiences with research opportunities in business, industrial and scientific settings, and with teaching activities in science centers, zoos, and other informal settings

* field experiences and student teaching situations with mentors devoted to the interdisciplinary approach to mathematics and science teaching

* modern technologies as standard tools for planning and assessment, classroom and laboratory work, problem-solving and research

* placement assistance and sustained support during the induction year in the teaching profession

The project began in the summer of 1993 with a summer meeting which brought together more than 60 university/college faculty members as well as 12 elementary and middle school teachers throughout Maryland. Many of the university/college content specialists were never involved in teacher preparation processes before - they might have had some prospective teachers in their classes, but preservice teacher education was not their primary concern. They stated that they taught mostly through lecturing, and few were aware of alternative pedagogical strategies or assessment techniques. Thus, for them, the first summer was an introduction to such ideas as cooperative learning, journaling, and alternative assessment. One of the final products of the first summer was a number of content modules crafted by the participants that attempted to emphasize mathematics-science connections. During the 1993-94 school year, many of the modules were field tested in university/college mathematics and science courses.

During the school year 1994-95, the first group of MCTP teacher candidates entered the programs, and approximately twenty MCTP courses were offered at six state supported colleges/universities. These were courses that were either developed or greatly modified by MCTP professors, and most of them were "content" courses. The instructors included experienced mathematics and science educators as well as university mathematicians and scientists. Classes varied in many ways. Some courses were offered under a specific discipline, like mathematics, biology, etc., while others were offered as "integrated science" courses. Some courses were team taught. Many classes were small in number - one class was team taught by two scientists with only four college students. The number of MCTP students was still very small; therefore, in many classes, the majority of students were not involved in the MCTP program. Some courses were designed only for MCTP students while others were open to non-MCTP students. A few courses satisfied the institutions' general education requirements, so, there were even several non-education students in those courses.

The MCTP programs continue to grow in Maryland. The number of MCTP teacher candidates has increased significantly in the 1995-96 school year to 175. The number of MCTP course offerings has also increased to more than 40, and several courses are being offered at a few community colleges, which reflects the changing demography of the four-year institutions involved. In many institutions, several MCTP courses are being offered. Most of the courses offered are introductory level mathematics/science content courses. More intermediate and advance mathematics/science courses will be offered as the programs continue to develop. The project is also working to develop a cadre of in-service teachers who can serve as mentors for the MCTP students once they reach their field-based experiences. In addition, the project aims to develop a support system for MCTP teachers during their induction years.

The MCTP is not simply a program development effort. The project has placed a significant emphasis on teacher education research. The research team includes two co-directors and several research assistants/associates, as well as institutional research representatives at all MCTP participating institutions of higher education.

In essence, the primary purpose of research in the MCTP is directed at knowledge growth in undergraduate mathematics and science teacher education. The unique elements of MCTP (particularly the instruction of mathematical and scientific concepts and reasoning methods in undergraduate content and methods courses that model the practice of active interdisciplinary teaching) are being longitudinally documented and interpreted from two foci: the faculty and the teacher candidate perspectives.

The following questions have served (and continue to serve) as

1 What is the nature of the faculty and teacher candidates' beliefs and attitudes concerning the nature of mathematics and science, the interdisciplinary teaching and learning of mathematics and science, the teaching of mathematics and science to diverse groups (both on the higher education and upper elementary and middle level), and the use of technology in teaching and learning of mathematics and science?

2 How do the faculty and teacher candidates perceive the instruction in the MCTP, especially with respect to being responsive to students' prior knowledge, addressing conceptual change, establishing connections among disciplines, incorporating technology, promoting reflection on changes in thinking, stressing logic and fundamental principles as opposed to memorization of unconnected facts, and modeling the kind of teaching/learning they would like to see on the upper elementary, middle level?

Answers to these questions will address the following global research questions driving teacher education research:

1 How do teacher candidates construct the various facets of their knowledge bases?

2 What nature of teacher knowledge is requisite for effective teaching in a variety of contexts?

3 What specific analogies, metaphors, pitfalls, examples, demonstrations, and anecdotes should be taught by content/methods professors so that teacher candidates will have some knowledge to associate with specific content topics?

Both numerical and qualitative data are being collected to address the MCTP research questions. Numerical data are derived from the administration of the "Attitudes and Beliefs About The Nature Of And The Teaching Of Mathematics And Science" survey, developed for the MCTP. Both participating faculty and students in MCTP classes contribute to this data (many of the MCTP classes surveyed also included a number of non-MCTP students as well as some non-education majors). Qualitative data are derived from semi-structured interviews with MCTP participants (both students and instructors), MCTP class observations, participant journals, and MCTP course materials. The data analysis process has been informed by standard qualitative data analysis techniques, such as constant comparison(Glaser & Strauss, 1979) and discourse analysis (Gee, 1990; Lemke, 1990).

The remainder of this manuscript will report the findings from the data collected during the first two years (1993 - 1995) of the project. The focus of this manuscript is how university/college instructors who were teaching MCTP mathematics and science courses perceived the nature of these disciplines as well as the connections between them. The two primary sources of the data for this analysis were the semi-structured interviews with individual instructors conducted during the 1994 - 95 school year and two content area debriefing meetings held during the summer of 1995.

Specifically, the following three questions will be addressed in this report:

* What are the perceptions of MCTP faculty about the "other" discipline?

* What are the perceptions of MCTP faculty about the connections between mathematics and science?

* What are some barriers in implementing mathematics and science courses that emphasize connections?

The instructors of MCTP courses were interviewed twice during the semester in which they were teaching MCTP courses. In addition, instructors who were not teaching an MCTP course during the second semester were interviewed once during that semester. The interviews were semi-structured in that there was a set of standard questions that were asked of all participants (see Appendix). However, additional questions were posed reflecting the responses of the participants. To answer the specific questions listed above, we have focused primarily on the participants' responses to the first question in both interview protocols. However, their responses to other questions, for example question 13 in interview 2, also related to the research questions, and participants' responses to other questions were also included as appropriate.

Altogether, forty interviews involving 16 mathematics and science instructors from four institutions were conducted (see table 1 for a summary). There were four mathematics instructors attending the mathematics debriefing meeting, while nine science instructors attended the science debriefing meeting. All interviews and group meetings were audio- and/or video-recorded and transcribed for subsequent analysis.

Mathematics Instructors Science Instructors Fall 1994: 5 10 Spring 1995 7 18

The findings are discussed in two sections. First, we will report findings on the MCTP instructors' perceptions of mathematics/science and their connections. In the second part, we will report some of the barriers in implementing mathematics and science courses that emphasized the connections, as identified by these instructors.

As we discuss the MCTP instructors' perceptions of the connections between math and science, we will begin with how the MCTP instructors looked at the "other" discipline. First, how do the MCTP science instructors perceive mathematics? One of the most common perceptions was the notion of "mathematics as a tool for science." As one science instructor put it:

Of course, that's from the point of view,

Accordingly, the examples several science instructors gave as the way they were integrating math in their classes involved calculations - probability in genetics, and pH in chemistry, for example. Other science instructors have also indicated the idea of mathematics as a language of science, the language that allowed more precise discussions of science.

Mathematicians and mathematics educators are often concerned about the "math-as-tool" perspective when working with scientists or science educators. This was also the case with the MCTP participants. For example, a mathematician said,

Every time we talk to the scientists, they say, "Oh, good, you're going to have more connections." And somehow the connections to the math kind of disappears because the essentials of organizing the mathematics in its own right is the part that they tend to want to leave out. (June, 1995)

A few MCTP science instructors, however, perceived mathematics to be more than just a tool for science. For example, a physical science instructor commented:

Before MCTP, I regarded mathematics as strictly a tool... After MCTP we have come, or at least I have come to appreciate math more in term of its intrinsic logic, its beauty, and a challenge to teach it, and I appreciate some interconnectedness that I did not really, you know, appreciated. (June, 1995)

Another science instructor stated:

I used never to worry about that. I used mathematics as a tool without worrying about

Thus, it appears that the "math-as-tool" perspective held by MCTP science instructors was complemented by their awareness of

How about the MCTP mathematics instructors? How do they perceive science? The prevailing perception can be described as "science as context for (mathematics) problem solving." This perception was realized in different ways. First, science is a source of problems to be investigated. A mathematics instructor commented that "science has provided a tremendous number of problems for us." Another instructor said that "things we model (in class) are very often from science."

Another important factor of science as a context to do problem solving was the issue of motivating students. One mathematics instructor stated,

So, science really plays a very, very important part with us as teachers because we need sometimes to attract or introduce the non-believer or the person who has a very poor image of (himself/herself)... But if you do it for something that's in the sciences, they might get interested. (June, 1995)

Thus, science can not only provide motivation for the development of mathematics as a discipline through problems they pose, but it can also motivate individual students in their learning of mathematics.

Other mathematics instructors felt that science helped students understand mathematics. One instructor noted that connections with science "help the visualization." Another instructor noted that science is helpful in giving meaning to mathematics:

... most of the research I did was in pure math that is probably never going to be useful... if mathematics keeps going as sort (of) an intellectual activity devoid of reality and devoid of science, it has no meaning ... (June, 1995)

Thus, science, as the MCTP mathematics instructors perceived, has provided problems to be solved, can be a useful tool to motivate students, and helps students understand mathematics more meaningfully.

How do the MCTP instructors perceive the connections between mathematics and science, and how did they attempt to emphasize those connections in their classrooms? The major themes that emerged in the ways MCTP instructors perceived the connections between mathematics and science were:

Some instructors noted that both mathematics and science used common tools. For example, a science instructor stated:

... we take the students to the computer lab one day a week, and we ask them to work up the data they've been collecting and prepare spread sheets, you know, use the same software, help one another learn how to use that, that kind of thing. (October, 1994)

Other common tools included other actual instruments such as scales, rulers, etc., as well as data sets to be analyzed. Another common "tool" that was mentioned was graphs. Some of the science courses utilized the Microcomputer Based Laboratory software, which produce graphs using data collected by the attached probes.

At a deeper level, MCTP instructors saw connections between mathematics and science in the process of problem solving. A science instructor noted that "we had problem solving in the physical sciences in the areas of physics and chemistry. ... that's where the math connection is in the physics and chemistry." A mathematics instructor said, "I've tried to have scientific questions be part of the ... environment in which we do the mathematics." Thus, many of the MCTP instructors emphasized the central role problem solving plays in both mathematics and science, which guided their efforts to highlight connections. This emphasis on problem solving in teaching of mathematics and science is consistent with recent recommendations in both mathematics and science education (e.g., Rutherford & Ahlgren, 1990; NCTM, 1989, 1991).

Another related theme that emerged was the idea of inquiry as a central activity of both mathematics and science. As a mathematics instructor put it,

I think science is an organized structure, it's the same as mathematics. It has its language. It has its syntax. It has its structure. People have an opportunity to explore new ideas and to kind of verify or refute or support conjectures and so forth. So, if you look at us in a parallel sense, I think we have many similarities with it. (June, 1995)

Another mathematics instructor phrased his ideas this way:

I think it's probably in the method, the Scientific Method, the method of inquiry, you'd probably get the same thing, I think of it as modeling approach, that we would apply and try to work from principles as well and to find out what the principles are. (June, 1995)

It appears that these MCTP instructors were saying that central activities in both mathematics and science are the activities of observing, conjecturing, and verifying or refuting conjectures. In other words, both mathematics and science are the activities of inquiry. MCTP instructors, both mathematicians and scientists, appear to perceive this notion of inquiry as another central commonality between their disciplines.

Another point often raised by the MCTP instructors was that both mathematics and science studied the same object, the "real world". For example, a mathematics instructor said,

But in a sense the mathematical system is the theoretical realization of it, and in the sense of a logical model you have an interpretation in the real world. And we hope that they're copies of the same thing, or shadows of the same reality. (June, 1995)

Another mathematics instructor added,

Science deals more with tangible phenomena that you can touch, and feel, and see, whereas mathematics deals with phenomena that are more abstract and mental. ... I described the mathematical model as a mental representation of some phenomenon. It stops becoming real in a certain sense. I mean, it's real to us, I guess, but in a sort of ... In the common usage of the word, it's more abstract. (June, 1995)

This perspective was also shared by a science instructor:

They (students) have to spend time describing in words the physical motions to complement what's visible in the graph and what is then associated with the mathematical description of the very same thing. So, it's a very rich array of physical behaviors represented by the transformation into the graph ... which is REAL STUFF! Real distances, real velocities, real accelerations that they can make reference back to their own personal decisions in the case of their personal motion, or describe the behavior of the fan cart. And so this is a very very close tie between as (an MCTP math instructor) would say describing things in words, seeing the transformation to the graph, finding that there is indeed a mathematical way of describing it. (September, 1994)

Thus, many of the MCTP instructors perceived that both mathematics and science study the real world, and the difference is how we approach those studies. This perception seems to be underlying the mathematics instructors' "science-as-context-of-problem-solving" perspective and the science instructors' "math-as-tool" perspective. Because both mathematicians and scientists are studying the real world, we can use mathematics in science and science in mathematics.

As the MCTP instructors tried to implement courses they designed or modified based on their participation in the project, they perceived a number of barriers which affected their effort to connect mathematics and science. In the following sections we will describe some of those barriers identified by the participants.

One barrier noted by some of the MCTP faculty was the issue of time versus content coverage. One mathematics instructor said,

... but as far as the content of the course, there is really not much room to give on that because to get all that content in just 6 semester hours, it's really a challenge. (September, 1994)

This same instructor noted that his mathematics content courses were already full of topics that were considered to be important by the NCTM

Another factor that negatively influenced the MCTP instructors' efforts to connect mathematics and science was, ironically, the students. Several instructors noted that their students did not want to see connections between mathematics and science. One science instructor reported a student "flat out said he doesn't like math, doesn't want to do it and wants to avoid it and please don't do any math in this course." Another science instructor said, "It was just a lack of willingness to try to integrate the math" that limited her attempt to emphasize connections between mathematics and science.

As the MCTP instructors tried to make connections between mathematics and science, a fundamental question emerged. A science instructor asked during the group discussion of the instructors of MCTP science classes, "Is it our (science instructors) job to teach mathematics in an integrated math/science course?" This question addressed two related yet distinct issues: what counts as legitimate content of an integrated math/science course; and who can teach those topics - the issue of competence. The issue of legitimacy led one science instructor to comment that "it's not our job to teach math as a discipline but rather an appreciation of math." This perspective was consistent with the comment made by another science instructor:

It doesn't mean that we take time out of the science to separately do the mathematics, but we do at least a metacognitive recognition when we've used it, when they've benefited from it, and it then becomes a natural part... because that's the way that these students benefit from their own growth in understanding of mathematics. (June, 1995)

Thus, for some MCTP instructors, the primary role of "the other" subject matter was to enrich learning opportunities, but not necessarily to learn topics from that particular subject matter.

The issue of competence also lead to another important point. A science instructor made the following statement during an interview:

I'm hitting against my, ... since I've never taught math, and it's been so many years since I've had that taught to me, and besides it probably wasn't taught to me correctly anyway, I just don't have a clue. (September, 1994)

The notion of "teachers teach the way they were taught" was a basis for the project to create and implement mathematics and science content courses for teacher candidates taught in a manner appropriate for middle grade learners, the target population for the MCTP teacher candidates. However, it appears that the MCTP instructors, especially the content area specialists, were caught in this dilemma of not knowing how to teach "correctly." They were not taught that way. A mathematics instructor commented during the summer instructor's meeting,

But I also grew up with the lecture method, and I don't think I've ever had a class in my life where I wasn't lectured to most of the time. So, it's hard to change your style. It really is. (June, 1996)

The same science instructor quoted at the beginning of this paragraph later stated,

... I feel that discipline faculty like myself will not be able to transform either themselves or their classes overnight. We tend to be attracted to our disciplines because we think the discipline itself (the "content") is inherently interesting and valuable. ... So many of the things we are being asked to do in this project do not come naturally to us. We need time to adjust. (December, 1994)

Thus, these MCTP instructors were trying to move away from the way they were taught. However, this issue of learning to teach remains one of the challenges many MCTP instructors face.

One way many MCTP instructors are dealing with these issues of legitimacy and competence is by talking with their colleagues from other departments. The science instructor quoted in the previous paragraph reported that he had been in frequent contact with a science educator at his institution. Another science instructor said,

... if we're teaching something we don't know, we ought to go ask somebody in another field and say, "I'm going in this direction. What can you do to give me some suggestions?" (June, 1996)

Another science instructor added,

I think the key is finding somebody else to bounce your course outline off of and say, "Is there some place particularly where I could explore developing ____?" I like the idea of trying to see if there are new mathematics concepts and skills that we can somehow ... find an appropriate place in our course to see that something good could come from this rather than just an application. (June, 1996)

Thus, the cross-discipline collaboration among MCTP instructors seems to have become an important part of their professional lives.

It appears that the MCTP university/college faculty members have developed a renewed sense of respect and appreciation for each other and each other's discipline. At the same time, they are still struggling with a number of issues. One such issue that is of particular interest to mathematics educators is the nature of mathematics in relationship to science. On the one hand, there is a tendency/desire on the part of mathematics instructors to treat mathematics as a distinct and independent discipline of its own right. This perspective reflected the concern on the part of mathematicians and mathematics educators that science instructors would simply treat mathematics as a tool and "the nature of what mathematics is is very often not explored in science" (mathematics instructor, June, 1995). On the other hand, there is also a perspective that mathematics

We've always said that mathematics was the queen of all sciences, and some of us even say that we want to talk about the mathematical sciences. So, I think we ourselves are part of science. (mathematics instructor, June, 1995)

In between these points of view was the concern that "math is more than just its connections to science" (math instructor, June, 1995). Thus, it appears that participation in the MCTP project has raised a fundamental question among mathematicians and mathematics educators concerning their own discipline, as well as the nature of the relationship between mathematics and science. Most, if not all, mathematics instructors agree that mathematics-science connections are important and useful; however, many appear to be grappling with the nature of these connections. Is there something special about the connections between mathematics and science that are not shared by connections between mathematics and, for example, economics? The quote at the beginning of this manuscript seems to imply that the answer to this question is yes. On the other hand, the recommendations of the NCTM

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Skemp, R. (1978). Relational understanding and instrumental understanding.

Faculty Interview Protocols

Interview 1

1 To what extent is the instruction in your class planned to highlight connections between mathematics and science?

2 To what extent will this class involve the application of technology, such as e-mail, CDs, computers, calculators, etc.?

3 To what extent will you make significant attempts to access your students' prior knowledge of a topic before instruction? What techniques will you use?

4 To what extent do the tests and exams stress reasoning, logic, and understanding over the memorization of facts and procedures?

5 In what ways do you think your teaching models the type of teaching that you believe should be done in grades four through nine?

6 To what extent will you explicitly encourage your students to reflect on changes in their ideas about topics in your class?

Reflecting over this semester's MCTP class, what new thoughts do you have on these areas (Question 1-6):

1 instruction planned to highlight connection among math and the science?

2 instruction involving the application of technologies

3 need to access students' prior knowledge of a topic before instruction

4 use of assessment techniques that stress reasoning, logic and understanding as opposed to memorization of facts and procedures.

5 modelling the type of teaching that you believe should be done in grades 4-9

6 need to explicitly encourage your students to reflect on changes in their ideas in the class

7 Reflecting back, have you seen what you have learned and experienced with MCTP courses and experiences come through in any other professional areas?

8 Reflecting over your course, what are the pieces unique to MCTP that stand out in your mind that worked well or that you might change?

9 Projecting into the future, do you have plans to teach another MCTP course?

10 How do you feel about teaching another MCTP course?

11 Has your involvement with MCTP enabled you to make connections with other MCTP faculty.

12 What kinds of things that have been part of the MCTP project have provided support to you or have contributed to your wanting to continue in the project?

13 What constraints?