Beginning mathematics teachers' beliefs of subject matter and instructional actions documented over time.

In this study we report the results of survey research that collected responses of an identical sample (31 beginning mathematics and science teachers, elementary and middle school level) that graduated from a reform-based mathematics and science teacher preparation program, the Maryland Collaborative for Teacher Preparation (MCTP). Our aim was to compare responses of the same beginning teachers over the two administrations of the survey. We administered the identical survey instrument in two separate batches spreading over nearly a four-year period (1st batch fall 1999 through fall 2001; 2nd batch summer 2002). The first administration (pre-test) was conducted soon after the beginning teachers graduated from the teacher preparation program and had not started full teaching. The second administration (post-test) was conducted after the new teachers had taught full time for a minimum of a full year, with the majority having taught for two years. The instrument was crafted to measure the constructs of interest, MCTP Teacher's Beliefs and Actions of Mathematics and Science. Results for teachers who taught for at least two years indicated that in all areas the MCTP teachers maintained their reform-based beliefs and actions after their induction years. These findings provide evidence for the sustainability of positive impact in the workplace resulting from a reform-based undergraduate teacher preparation program.

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Researchers in mathematics teacher education continually question and study the impact of teacher preparation on the future teachers who graduate from their programs (e.g., Ensor, 2001; Steele, 2001). Although many studies have been designed to create changes in the conceptions of prospective and practicing teachers (e.g., Cooney, Shealy, & Arvold, 1998; Raymond & Santos, 1995; Steele & Widman, 1997) very few studies have examined whether and how these changes in conceptions have been sustained through the beginning years of teaching. Motivated by a critical shortage of mathematics teachers and the need to have long-term access to changing and building future teachers' beliefs in support of reform efforts, the Maryland Collaborative Teacher Preparation (MCTP) program, was designed to generate new understandings in reform-based undergraduate mathematics and science teacher preparation. In this study, our aim was to compare responses of the MCTP teachers soon after they graduated from the reform teacher preparation program (pre-test) and after they had taught full time for a minimum of a full year, with the majority having taught for two years (post-test).

When teacher interns enter their initial practicum experience they are confronted with differing teaching philosophies, their own, their university professors, and their school mentors (Sullivan, Mousley, & Gervasoni, 2000). In this situation, teacher interns struggle to find their own niche in teaching and have to learn to reflect being both a learner and a teacher (Kelly, 2000). Too often, many of the pedagogical strategies emphasized in their college methods classrooms are lost when they enter their field-based sites. This is due to the barriers they face, such as differing teaching styles between mentors and interns, time factors, particular classroom situations, and lack of support from mentors and supervisors (John, 2001). Ziechner and Tabachnick (1981) described the attitude shift that student teachers make when they moved from university course work to school teaching. They highlighted the regression in students' attitudes toward more traditional viewpoints as they become immersed in the profession. There were only few studies (e.g., Huffman, Thomas, & Lawrenz, 2008; Loughran, 1993; Steele, 2001; Wingfield, Freeman, & Ramsey, 2000) that explored whether perceived changes in conceptions of teacher interns were sustained over time, that is, through the first years of teaching. Such longitudinal studies could help teacher educators and researchers deepen their understanding of how or whether teachers reconceptualize their teaching practice to include their new conceptions.

Huffman et al. (2008) examined the science and mathematics instruction of teachers who were initially prepared by the Collaboratives for Excellence in Teacher Preparation program (CETP) and compared them to a sample of teachers who were not prepared by CETP. The focus of their study was on examining the extent to which science and mathematics teachers (6th-12th grade teachers with 1-5 years of experience) used more reform-oriented instructional practices in their classes when they entered the teaching profession. This study included non-CETP and CETP science and mathematics teachers as well as their students. The authors found,

   An examination of the raw teacher and student survey
   results indicates that both science and mathematics
   teachers actually had relatively low levels
   of use of reform strategies. The majority of instructional
   strategies on the survey were only used in
   the 'seldom' to 'occasionally' range. In other
   words, both science and mathematics teachers used
   reform techniques less than envisioned in the standards
   (p. 144).

Steele (2001) contrasted four elementary teachers who were graduates of a teacher education program that incorporated a reform-based mathematics methods course. Her report provided results from a longitudinal study that extended from the time that the participants were teacher interns until the end of their second year of full-time teaching. The case studies indicate that two of the four teachers sustained their cognitively based conceptions about mathematics teaching and learning, and implemented these conceptions into practice. The other two did not. The analysis suggests that there were several factors that influenced the teachers' conceptions and the choices they made in their teaching: personal commitment, professional strength, curriculum planning, assessment, beliefs, knowledge, and support from the school administration.

In our study we focused primarily on the MCTP teachers' beliefs and attitudes towards mathematics and mathematics teaching. A variety of terms are used to define teacher beliefs. These include preconceptions, implicit theories, and orientations. While definitions differ, it is generally agreed that teachers' beliefs about what mathematics is and what it means to learn mathematics directly impact how they teach mathematics (Ernest, 1989; Szydlik, Szydlik, & Benson, 2003). For example, Thompson (1984) found that a teacher who viewed mathematics as a collection of facts and rules to be memorized and applied was more likely to teach in a prescriptive manner, emphasizing rules and procedures conveyed by the teacher. On the other hand, a teacher who held a problem-solving view of mathematics was more likely to employ activities that allow students to construct mathematical ideas for themselves.

When practicing teachers were young learners they often learned in teacher-centered classrooms by listening to lectures, memorizing information, and practicing rote computations. Ernest (1989) suggested that teachers who learned mathematics in these ways most likely to conceive of mathematics in the instrumentalist view, a conception that mathematics is a collection of unrelated facts, rules, and skills. Swats, Hart, Smith, Smith, and Tolar (2007) recently argued that "in mathematics education it is not uncommon for beginning pre-service teachers to come to their teacher preparation programs with a traditional view of what it means to know and do mathematics: a view of mathematics as a fixed body of knowledge to be delivered to children, usually through clear, organized presentations and lectures" (p. 325). In contrast, mathematics teachers, teacher educators, and researchers involved in the current reform movement in mathematics education recommend that students need to be actively involved in constructing their own knowledge and developing mathematical concepts that require them to explore, explain, and justify solution strategies to mathematical tasks (National Council of Teachers of Mathematics [NCTM], 2000). Teaching mathematics in this way coincides with the problem-solving view of mathematics, a conception that mathematics is expanding continually through human inquiry. For many teachers, this reform-based approach to teaching mathematics requires a change in beliefs about mathematics and about what it means to learn and teach mathematics (Ernest, 1989).

A teacher's past events create "guiding images" that act as a filter for new information. A belief structure created from an earlier experience may also be resilient enough to become the standard to which newer information is compared. For example, if a teacher changes conceptions of what quality teaching is, from a traditional whole group approach to a cooperative learning orientation, all new information about practice will be filtered through the cooperative learning belief structure (Blake, 2002).

Leinhardt (1990), as well as Bryan and Atwater (2002), demonstrated in their research on teacher thinking that teachers' beliefs about the teaching-learning process played a significant role in determining the nature of teachers' purposes in the classroom and directly affected many aspects of their professional work, including lesson planning, assessment, evaluation and teachers' decision-making during classroom interactions with students.

Studies from different perspectives were designed to document and interpret the effectiveness of the MCTP program. McGinnis (2002, 2003) measured MCTP and non-MCTP teacher interns' attitudes and beliefs about mathematics and science teaching and found the MCTP teacher interns' attitudes and beliefs to be more aligned with the overall program goals than the non-MCTP controls. Moreover, he found that over 2.5 years the MCTP teacher interns' attitudes and beliefs continued to move in the desired direction. Roth-McDuffie, McGinnis, and Graeber (2000) in a case study of MCTP mathematics content course found that the interns benefited from the transformative vision of mathematics portrayed by the instructor. In previous studies (Marbach-Ad & McGinnis 2008; McGinnis & Marbach-Ad, 2007), we measured the new MCTP teachers' beliefs, attitudes and practices and compared them with a national sample of teachers. The MCTP teachers' responses were more in alignment with desired responses. McGinnis, Parker, and Graeber (2004) also conducted case studies of the MCTP interns and MCTP new teachers and reported how they perceived their MCTP instructors as modeling reform-based teaching and how they later implemented that vision of teaching in their own teaching practices in precollegiate level classrooms.

In this study we used the same reliable and validated "MCTP Teacher's Actions and Beliefs of Mathematics and Science" survey to compare between teachers' attitudes and beliefs towards mathematics and mathematics teaching with which they entered workplace and after two or three years of full time teaching. We wanted to answer the question: To what extent do MCTP teachers maintain their beliefs and intended instructional actions in mathematics as they are inducted in schools? To test this question, we focused our study on these highly significant research questions:

1. Do MCTP teachers maintain their beliefs about the nature and teaching of mathematics over extended practice (induction years and beyond)?

2. Do MCTP teachers maintain their perceptions about students' skills required for success in mathematics over extended practice?

3. Do MCTP teachers maintain their familiarity with standard documents for mathematics over extended practice?

4. Do MCTP teachers maintain their intentions about implementing reform activities in mathematics classes aligned more with the reform-based recommendations over extended practice?

Research Design and Methodology

The Context of the Study--The MCTP Program

The MCTP was a statewide, standards-based program for undergraduate students who planned to become specialist mathematics and science upper elementary or middle level teachers. The goal of the MCTP was to promote the development of teachers who were confident teaching mathematics and science, who could make connections between and among the disciplines, and who could provide an exciting and challenging learning environment for students of diverse backgrounds (University of Maryland System, 1993). This goal was in accord with educational practice reforms advocated by the major professional mathematics and science education communities in the past decade and present day (American Association for the Advancement of Science [AAAS], 1993; National Academy of Sciences [NAS], 2006; National Council of Teachers of Mathematics [NCTM], 1989, 1991, 2000; National Research Council [NRC], 1996;). As such, the innovations of the MCTP included: (a) introduce future teachers to standards-based models of mathematics and science instruction; and, (b) provide courses and field experiences that integrate mathematics and science (see Appendix A for MCTP goals overview).

In practice, the MCTP undergraduate courses were taught by faculty in mathematics, science, and education who made efforts to focus on "developing understanding of a few central concepts and to make connections between the sciences and between mathematics and science" (MCTP, 1996, p. 2). Faculty also sought to infuse technology into their teaching practices, and to use instructional and assessment strategies recommended by the literature to be compatible with the constructivist perspective (i.e., address conceptual change, promote reflection on changes in thinking, and stress logic and fundamental principles as opposed to memorization of unrelated facts) (Cobb, 1988; Driver, 1987; Tobin, Tippins, & Gallard, 1994; von Glasersfeld, 1989).

Salient features of all the MCTP reform-based courses were that faculty lecture was diminished and student-based problem solving was emphasized in cross-disciplinary mathematical and scientific applications. Cooperative learning strategies were used extensively. Statewide, the MCTP offered nearly 90 reformed-based courses in mathematics, science, and methods.

Teacher interns selected to participate in the MCTP program were in many ways representative of typical teacher candidates in elementary teacher preparation programs. Where they differed prominently was in their willingness to specialize in the teaching of mathematics and/or science by making connections between the two disciplines. Once in the MCTP program, the MCTP teacher candidates were distinguished from all other candidates by taking 36 credits of mathematics and science courses (18 hours of each discipline). This was contrasted starkly with elementary teacher candidates who chose not to emphasize either mathematics or science as field of concentration who were required to earn a total of 11 credits in mathematics and 8 in science as well as 18 in their chosen field. Non-MCTP elementary teacher candidates who chose to emphasize mathematics (only) had to earn 18 credits in mathematics and 8 in science. Candidates with a science (only) emphasis had to earn 18 credits in science and 11 in mathematics. The MCTP content courses (open to all teacher interns) were transformed to conform to the MCTP program goal. Mathematics courses were distributed across four areas: algebra/number, probability and statistics, geometry, and calculus.

The "MCTP Teacher's Beliefs and Actions of Mathematics and Science" Survey. In this study we used a reliable and valid tool that was crafted for our previous studies and was used to compare between our group of graduate students and a national sample of teachers (McGinnis, 2002). The tool that we used, "MCTP Teacher's Beliefs and Actions of Mathematics and Science" aimed to measure the constructs of interest of the program's graduates. Appendix B contains a complete copy of the MCTP survey. The constructs we measured were "Teachers' beliefs about mathematics (9 items)," "Teachers' beliefs about science (9 items)," "Teachers' use or intended use of instructional practices in mathematics (7 items)," "Teachers use or intended use of instructional practices in science (7 items)," "Teachers' perceptions about student success in mathematics (6 items)," "Teachers' perceptions about student success in science (6 items)," "Teachers intentions about implementing reform activities in mathematics classes (6 items)," "Teachers' intentions about implementing reform activities in science classes (6 items)," "Teachers' knowledge of the mathematics and science standards" (3 items), and 4 items that asked background information. In this study we report on students' beliefs and attitudes towards mathematics and mathematics teaching only and not towards science and science teaching (Marbach-Ad & McGinnis, 2008).

To establish face validity of our instrument (i.e., that there existed a connection between the surface features of the instrument's content and the theoretical construct (Smith & Glass, 1987), we provided for inspection the draft instrument to a sample of mathematics content experts and a sample of mathematics pedagogy experts (we reported on its reliability in McGinnis & Parker, 2001). Our theoretical constructs consisted of beliefs about the nature and teaching of mathematics. Namely, we sought to determine if the MCTP graduates held beliefs about mathematics content that aligned with a traditional view of mathematics as a static and codified body of knowledge or a view of the discipline as a dynamic way of knowing driven by inquiry. Regarding the teaching of mathematics, our aim was to measure if our graduates held beliefs about the teaching of mathematics that were teacher-centered or learner-centered, as characterized by passive learning (lecture) or active learning (problem solving), respectively. The surface content of the instrument consisted of the items selected from a limited number of existing instruments as well as three new items that measured beliefs about subject matter integration and knowledge about the major standards documents. The content specialists included a mathematics professor, two mathematics methods professors, and three mathematics education doctoral students. The result of the inspection by our sample strongly supported the face validity of our instrument.

Sample and Data Analysis

For our study we used a pre-post analysis. We administered the identical survey instrument by mail to the MCTP program's graduates in two separate batches spreading over nearly a four-year period (1st batch fall 1999 through fall 2001, n = 113; 2nd batch summer 2002, n = 68). The pre-test was conducted soon after the beginning teachers graduated from the teacher preparation program and had not started full teaching. The response rate for the pre-test was 60% (n = 68). The post-test (only to those who responded to the first one) was conducted after the new teachers had taught full time for at least one full year, with the majority having taught for two years. The response rate for the post-test was 60% (n = 42), a relatively good response rate for survey research that targets new teachers. Responses came from graduates of all seven of the MCTP participating institutions with baccalaureate programs. We attribute the high level of response partially to the strategies for increasing return rates to mail-in surveys suggested by Dillman (1978) (i.e., sending a token honorarium such as a $2 bill or a $1 coin in the first mailing and a $20 honorarium in our final mailing, sending a later reminder letter with another copy of the instrument, and using email and telephone reminders). To enhance the validity of our analysis, we conducted a nonresponse bias check in both administrations by randomly selecting a sample of 8 non-responding MCTP graduates. Upon contact, we encouraged them to complete the survey. We then compared their responses to those responses we had earlier collected. No differences were detected. Therefore, we believe our sample represented the entire population of MCTP teachers who had responded earlier to the survey when they graduated from the program.

For comparison purposes we only report results from the surveys of the sampled MCTP graduates that at the time of the second survey had completed 2+ years of practice (31 graduates): Approximately 50% were middle school teachers and 47% were elementary teachers. We analyzed the survey responses in different ways using t-test and chi square analyses, and using analysis for the whole survey and for separate group of questions. We decided that due to the small sample and the large variability between the different items in the survey that we should simply compare percentages for each of the items and use chi square tests to compare between responses to each item on the pre- and post-test. Another reason for using the chi square analysis was that since we were using a survey for which we had reliability and validity documentation, we couldn't change the items or the structure of the survey without recalibrating the survey's reliability and validity. However, please note that in this application of the survey in one subsection we used a different Likert point scale. For the chi square analysis we condensed the scale always to two categories (see results section). The crosstab analysis for each item also enabled us to examine the differences between specific students and learn if they changed their perspectives towards each item between the pre- and post-test.

Data sources also included individual participant semi-structured interviews of all the MCTP beginning teachers (each 1 hour in duration, four times per year, recorded and transcribed); focus group participant interviews (each 2 hours in duration, twice per year, recorded and transcribed). For this study, we read carefully through the extensive interview transcriptions of the MCTP new beginning teachers (n =6) who graduated in the first cohort from one of the participating institutions.

Teacher 1: Female, teaching mathematics and science in fourth grade.

Teacher 2: Female, teaching mathematics and science in third grade.

Teacher 3: Female, teaching mathematics and science in seventh grade.

Teacher 4: Male, teaching mathematics and science in eighth grade.

Teacher 5: Female, teaching mathematics and science in seventh and eighths grades.

Teacher 6: Female, teaching mathematics and science in eighth grade.

We report illustrative comments by MCTP new teachers to enhance the MCTP teachers' written survey responses. While we make no claims that they are necessarily representative of all MCTP new teachers' viewpoints, we offer them as a way to interpret more fully the survey responses. The comments were culled from interview questions that were recorded and transcribed for analysis. Examples of the interview questions include: "To what extent, and in specific ways, do you think what you learned in your MCTP teacher preparation program has impacted how you are teaching mathematics and science?.... Do you see yourself making connections between mathematics and science?" "How about technology?" "How about alternative assessment for the MCTP program? In comparing your teaching of mathematics and science with other teachers at your school, both veteran and new, in what ways do you think your teaching is the same or different? How about school personnel?

In this paper, we used the selective MCTP new teachers' interview responses to support and illuminate how teachers interpreted related questions, and not to draw any collective conclusions. Therefore, we did not find that it is necessary to elaborate here on the background of these teachers and their schools. These interviews are fully documented in a previous study that takes an in-depth case study approach for research (McGinnis et al., 2004),

It is recognized that research based on statewide mathematics teacher preparation reform efforts are very rare (see Wingfield et al., 2000). Athough our sample size is on the small side for quantitative research (N = 31) we believe that this study adds significant information to the emerging literature on beginning mathematics teachers (a sample known as difficult to recruit in educational studies).

Results

We report our findings according to the four categories in the MCTP survey. First we examined to what extent the MCTP teacher responses to the pre-test (in their first year in school) aligned with the philosophy of the MCTP program. Next, we compared the responses of the first survey to the responses of the posttest (after two or three years in school) as a way to determine if the teachers maintained their beliefs after two or three years in their workplace.

1. Teachers' beliefs about the nature and teaching of mathematics

In this section teachers were asked to rate on a scale from 1 to 4 (1 = strongly disagree, 4=strongly agree) nine statements concerning their beliefs about the nature and teaching of mathematics (see Appendix B, 19). Table 1 shows the MCTP teachers responses to the first and the second surveys; the percentages in this table reflect the combined proportion of teachers who either agree or strongly agree with the statements. We report the number of respondents before each percentage, since the sample size was relatively moderate (N=31).

Table 1 shows that there were significant differences between students' responses in the first and the second survey regarding to the first item Math is primarily an abstract subject. Inspection of the data shoes that only two MCTP teachers came to their workplace (first survey) with the belief that "Math is primarily an abstract subject." However, in the second survey four more teachers agreed with this statement. We believe this finding makes sense, since in the two years of teaching the teachers could have experienced for the first time through their students eyes (and not through their own eyes) how elementary and middle school students perceive mathematics.

Most of the MCTP graduates (more than 77%, see Table 1) came to school with the beliefs that "Math is primarily a formal way of representing the real world" and "Math is primarily a practical and structured guide for addressing real situations." Based on earlier documentation of the MCTP mathematics and science content courses, we know that the MCTP graduates were prepared to teach mathematics problems using real situations. As result, their support of statements such as "Math is primarily a formal way or structured guide to address real situations" was expected, at least initially, when they began teaching full time. However, we wondered if the MCTP graduates would maintain these beliefs after an extended time in the workplace. We found that there were no differences between the percentages of teachers who agreed with these statements in the first and second survey. Therefore, after two years of teaching, the MCTP graduates maintained their beliefs that real world situations should be used in mathematics teaching. In an interview one MCTP teacher said:

   ... They [the students] did a thing where they had to
   do a graph. I had this data on different roller coasters
   --how long they are, how fast they go, how
   much time is on it. So they did a graph. You know,
   if it's this long it goes this fast ... (Teacher 2).

Regarding the teaching of mathematics, most of the MCTP teachers (more than 90%) agreed in the first, and even more in the second survey (Table 1), with the following statements: "A liking for and understanding of students are essential for teaching math" and "More than one representation should be used in teaching a math concept". These statements are aligned with the MCTP philosophy, which stresses that teachers should understand the variability in students' nature and learning styles and thus use multiple representations that are understandable for different types of students. As one MCTP teacher said in the interview:

   We have a real push in the school for differentiating
   math, because we don't switch for math, and so
   I have real, real low kids and real, real high kids--ability
   level, so we are pretty much forced to have
   two or three kinds of activities going on at once
   within an hour period.... I know that it is a real
   frustration for a lot of teachers ... but I don't have
   a problem with having two or three things going
   on at once ... (Teacher 3).

We also believe that the variation in students' ability level that this teacher characterized is likely one of the reasons that more than 70% of the teachers in both surveys agreed with the statement "Some students have a natural talent for math and others do not."

Most of the MCTP graduates did not agree (about 70%) in the pre-test, and even more in the second survey, with the following statements: "Math should be learned as sets of algorithms or rules that cover all possibilities" (87%), "If students are having difficulty, an effective approach is to give them more practice by themselves during the class" (87%), and "Basic computational skills on the part of the teacher are sufficient for teaching elementary school math" (75%). These findings are aligned with the MCTP philosophy, since the MCTP program was designed around the recommendations to teach math through inquiry-based learning and not through a memorization of a set of algorithms and rules, and to involve students with discussion and group study instead of more self-practice. In addition, in the schools the MCTP teachers took positions there was a major recommendation to use telecommunication and teaching based technology, even in the elementary schools.

It is noteworthy that teachers complained that when teaching math (more so than for those concerning science) it was really difficult for them to use inquiry-based teaching and innovative approaches. They voiced that their major constraint was the overloaded curriculum and material coverage expectations that their district's central offices enforced. One teacher said:

   I feel like I'm sort of handcuffed in a way [because
   of the county's assessment, monitoring, and reporting
   system] in the county, I'm limited and probably
   shouldn't cover subjects to the depth that I do, because
   I should moving on to the next objective"
   (Teacher 4).

Another teacher said:

   My main problem was that the objectives of the
   county really kind of hurry you through it.... you're
   supposed to cover eight objectives per quarter ... in
   the beginning I was really kind of worried about
   getting into trouble...but now I'm kind of thinking
   more about what's better for the kids and okay if
   we just don't get to that one at the end of the year
   (Teacher 3).

2. Teachers 'perceptions about students' skills required for success in mathematics

In this section teachers were asked to rate on a scale from 1 to 3 (1 = not important, 3 = very important) the importance of particular kinds of skills for success in the discipline. These skills have elements ranging from remembering through understanding to thinking creatively (Appendix B, 19-24). Figure 1 shows the MCTP graduates responses to the first and second surveys concerning students' skills required for success in mathematics. The percentages in this figure reflect the percentage of teachers who choose the category "Very important."

Figure 1 shows that the MCTP graduates were consistent in their responses to the first and second surveys concerning the aptitudes and skills students need to succeed in learning mathematics. In both administrations around 90% of the teachers consider it very important for students to understand concepts, to understand how the subjects are used in the real world, and to be able to support their results and conclusions. Around 60% of the teachers considered it very important to think creatively and less than 45% of the teachers considered it very important to think in a sequential manner and remember formulas and procedures.

The fact that only around 30% of the MCTP graduates marked "remember formulas and procedures" as very important is consistent with the overall goals of the MCTP. The major reform-based recommendations are to put the emphasis on meaningful learning (characterized by a focus on understanding) instead of rote learning (characterized by memorization of facts). The MCTP philosophy, while not rejecting the importance of "remembering," emphasizes that learning in school should expand to include a fuller range of cognitive processes, such as the ability to use what was learned to solve new problems, answer new questions, or facilitate learning new subject matter.

One MCTP teacher in an interview complained about the fact that her workplace made it difficult to teach for understanding:

   I'm not always able to use everything that I taught
   in MCTP, because time constraints, and because of
   curriculum requirements of the county, and because
   of the level of reasoning of some of the students....
   For students applying knowledge--that's
   one of the toughest things is when we will give
   them quiz or something that asks them to apply
   something that they have learned to a new situation
   ... they are better at just remembering or memorizing
   sorts of questions.

However, she specifically referred to the way the MCTP program impacted her teaching:

   ... A lot of the stuff that we learned about in MCTP
   was higher order thinking and questioning.... I do
   try and use hands-on sorts of things and that's
   something I know I learned at MCTP ... and projects
   in math, math projects sorts of things instead
   of giving big tests (Teacher 5).

3. Teachers 'familiarity with standards documents In this section teachers were asked to rate their familiarity with standards documents on a scale from 1 to 5 (1 = not at all familiar, 5 = familiar to great extent). In both surveys, only around half of the teachers marked that they are "fairly familiar" or "very familiar" with the NCTM (1989) standards.

4. Teachers' intentions about implementing reform activities in mathematics classes

In this section of the survey teachers were asked to report on the kind of reform activities they intended to or were able to implement in their classrooms (items 34-40). Table 2 summarizes the MCTP graduates responses. The percentages reflect the percentage of teachers who choose to answer "yes."

Overall, a large majority or all of the MCTP teachers reported that they used each of the instructional practices that were included in this section in their mathematics instruction. The exception being "using telecommunication supported instruction" which was lower (approximately 60%). These results are encouraging since it indicates that the MCTP teachers maintained over time there beliefs about the importance of using key reform-based practices, including "using authentic assessments (100%)," "using standards aligned curricula (100%)," and "making connections with science" (97%). In the following we elaborate on some practical aspects that teachers mentioned in their interviews.

A. Assisting all students to achieve high standards and providing examples of high-standard work

Almost all of the teachers in the first and the second survey (above 90%) answered that they assisted all students to achieve high standards and provided examples of high-standard work. MCTP teachers in interviews said that the background knowledge they acquired in the MCTP program enabled them to help students to better achieve:

   ... I do have a lot more background knowledge. I
   even noticed that I have more so than the veteran
   teachers that are here [her teammates].... It was
   part of the MCTP requirements... I'm teaching my
   kids [fourth graders] things that they learn in middle
   school. And the kids are getting it because I'm
   doing it hands-on type way and I'm not just throwing
   it at them ... I'm doing a lot extension stuff ...
   (Teacher 1).

B. Use authentic assessments

Almost all of the teachers in the pre-test (93.5%) and the post-test (100%) answered that they used authentic assessments, even though they also had to use their public school district's formal assessments:

   At the beginning of the year I was giving a lot math
   quizzes, every Friday, and tests ... Now, I've been
   giving the ISM's [ISM's were Instructional System
   in Mathematics countywide content exams for the
   students linked to the county's curricula], because
   we have to, but I've also been doing a lot of learning
   station things and walking around and assessing
   how they react to different problems ... and I
   keep math journal, a science journal.... We've
   done a lot hands-on things and discussions, and
   just seeing their responses in their journals, and
   that's how I've grading them--group participation
   in lab and things like that ... I definitely stepped
   away from the formal assessment (Teacher 1).

The MCTP program recommends strongly using authentic assessments. The call for "Science and Mathematics for all" (Fey, 2002, p. 119) meant that teachers would be required to assist all students to achieve high standards, and to reach all the students. There was a need to use different assessment strategies, since for many students the conventional testing paradigms do not give accurate readings of their knowledge (Fey).

C. Use standards-aligned curriculum, textbooks and materials

Most of the teachers in the pre-test (87.1%) and the post-test (100%) answered that they use standards-aligned curriculum, textbooks and materials in teaching mathematics. However, in the interviews, the MCTP teachers raised the dilemma about how to use textbooks. One MCTP teacher complained about her more veteran teammates who pushed her to use the textbook in a traditional way:

   ... They [other two third grade teachers] keep on
   pushing these textbooks in my face and they keep
   on saying, "you have to use this. They [the students]
   have to answer questions from it."' We do
   answer questions, I mean, we do write in a daily
   log and we do, um, discuss, but we don't read the
   whole section in book--a chapter and than answer
   corresponding questions to it.... They [the teacher's
   teammates] think they cover more things.... I say,
   "Well you may cover more things and the students
   may remember it short-term ..." (Forum group:
   Teacher 1).

D. Making connections between science and mathematics

Almost all of the teachers in the first and the second survey teachers (above 90%) answered that they were making connections between mathematics and science in their practices. One of the fourth grade MCTP teachers, in an interview, was asked, "How about making connections between math and science? Do you see yourself as the same or different as the teachers in your school?" She responded in this way:

   Well it's kind of hard for my other teammate to
   make connections because she doesn't teach the
   science... definitely because I'm so math and science,
   really concentrated on both subjects and
   teaching both of them, It's definitely not to the
   same level as I am. (Teacher 1).

Another teacher of 7th and 8th grades stressed the point that because she was teaching both mathematics and science she was better able to make connections between them in comparison to other teachers: "When I'm in math I know what they're studying in science, and so sometimes, I might pull something in from them" (Teacher 5). Another teacher stated:

   The county curriculum has it [mathematics and
   science] separated, but within, like it is easier for
   me to draw math into science ... Right now we're
   starting a unit on plants and so they'll be growing
   and we'll be graphing and, you know, taking down
   data and making generalizations ... (Teacher 6).

Teachers also talked about how students react to their effort to connect mathematics and science,

   ... The other day in math we were talking about
   distance and scientific notation and I was talking
   about distance to planets and looking at the angles
   of planets, and how people used to measure how
   far away stars were and all that stuff. And they'd
   say [the students], "How is it that you know this?
   You're a math teacher. And, oh, that's right. You're
   a science teacher too." So I can bring things from
   one area to the other and use them in class (Teacher
   5).

E. Using telecommunication-supported instruction

Only about 60% of the MCTP teachers reported in both surveys using telecommunication-supported instruction in mathematics instruction. These percentages are low in comparison to the other practices that MCTP teachers indicated that they used. In the interviews the MCTP teachers were asked to refer to their use of technology in general in their classrooms and the teachers' responses appropriately explain the low usage of technology and telecommunication:

   I'm not as pleased with that [the use of technology]
   as everything else I've been doing. I have a lot of
   complaints just about the way our technology is
   distributed ... I tried to use digital camera for something
   and nobody can find the chargeable batteries
   ... and they send you back ... and apparently
   there are all these color printers in our buildings,
   apparently there are 13 but nobody knows where a
   single one is". (TEACHER?)

Although there were a lot of frustrations about using technology in schools, some teachers also reported good experiences:

Teachers 'reports about their classroom experience in their second year of teaching

Based on the pre- and post-test of the MCTP surveys, we conclude that the MCTP reform-based philosophy was not "washed out" after two or three years of teaching. In fact, in some cases, the MCTP teachers felt more comfortable using the MCTP philosophy in the second year of teaching, as they were more experienced. One teacher said:

   The first year, I was trying to do all these cool lessons.
   Now I realize the lessons I did last year are
   not as strong as the ones I'm doing this year because
   I have so much more experience with it. Like
   I was doing little bits and pieces of everything. And
   the kids were like, "Wow, wow, wow!" But now
   I'm kind of able to like put together an entire thing.
   Like all-encompassing. And it's just complete....
   It's been really good because I'm seeing the lessons
   are so much more, they were structured, but
   now they're more creative (Teacher 2).

Another teacher said:

   What I do notice is that more and more experience
   I get, how much easier it is for me. And I think because
   I have that solid foundation already, it's really
   helped me teaching.... Now it just comes
   much easier to me because I have been taught in
   the constructivist way. You know, this whole idea
   of teaching. And it's just something I can't forget,
   I don't forget anything that I've learned in the
   courses that I took because it's been very helpful to
   me (Teacher 1).

Discussion and Implications

The goal of the MCTP was to graduate teachers who were confident teaching mathematics (and science) using technology, who could make connections between and among the disciplines, and who could provide an exciting and challenging learning environment for students of diverse backgrounds. Consistent with other research (i.e., Lawrenz, Huffman, & Graves, 2007; Swars et al., 2007), earlier MCTP studies presented evidence (using both quantitative and qualitative methodologies) that the new MCTP teachers had experienced transformative mathematic content and methods courses and began teaching with the skills, dispositions, and knowledge targeted by the MCTP program (McGinnis, 2002, 2003; McGinnis & Parker, 2001; Roth-McDuffie et al., 2000).

The current study, based on repeat survey methodology and informed by teachers' interviews, shows that the MCTP teachers report that they maintain over time their initial, reform-based orientation and mathematics practices with which they began their teaching careers. Along all measures, the present analysis indicates that the graduates of the MCTP program conveyed the targeted reform-based beliefs and perspectives to the workplace, and, most importantly, that they maintained them during their induction years and beyond. It is encouraging that the full time teaching experience in the schools did not undermine the teacher change in beliefs as was earlier reported by Ziechner and Tabachnick (1981). The stability of these beliefs during their two years of teaching suggests that the distinctive features of the teacher preparation program helped in developing well-established beliefs.

The MCTP teachers reported beliefs defined in the literature as a good indicator for their actions and teaching performances (Pajares, 1992; Richardson, 1996; Wilkins & Brand, 2004). However, we are aware that studying beliefs is somewhat problematic because they cannot be directly observed and must be ascertained by what people say and do. Sometimes, teachers do not possess the language with which to express their beliefs; other times, they may be reluctant to express unpopular beliefs (Leinhardt, 1990). Nevertheless, by analyzing our surveys we can conclude that the MCTP teachers are knowledgeable about the types of viewpoints they are expected to hold as transformative, reform oriented teachers. And through our analysis of MCTP teacher interviews we have additional support that the MCTP teachers carried out the philosophy of the MCTP program into their beginning years of teaching.

A clear implication for policy includes the critical need to continue to support innovative, transformative mathematics teacher preparation programs. In particular, the major implementation features in the MCTP that warrant support as a set of innovations in mathematics teacher preparation included: reformed based content and pedagogy courses taught by faculty who have engaged in sustained professional development; summer internships in mathematics rich environments; and well-prepared student teacher mentors in the schools.

Appendix A

The MCTP Philosophy

[ILLUSTRATION OMITTED]

Appendix B

MCTP Teacher's Actions and Beliefs of Mathematics and Science

Directions: Please select the letter response that best represents your actions and beliefs.

SECTION I.

To what extent do you agree or disagree with each of the following statements?

Choices:

(A) Strongly disagree

(B) Disagree

(C) Agree

(D) Strongly agree

Mathematics

1. is primarily an abstract subject.

2. is primarily a formal way of representing the real world.

3. is primarily a practical and structured guide for addressing real situations.

4. should be learned as sets of algorithms or rules that cover all possibilities.

5. A liking for and understanding of students are essential for teaching math.

6. If students are having difficulty, an effective approach is to give them more practice by themselves during the class.

7. More than one representation should be used in teaching a math concept.

8. Some students have a natural talent for math and others do not.

9. Basic computational skills on the part of the teacher are sufficient for teaching elementary school math.

Science

10. is primarily an abstract subject.

11. is primarily a formal way of representing the real world.

12. is primarily a practical and structured guide for addressing real situations.

13. Some students have a natural talent for science and others do not.

14. A liking for and understanding of students are essential for teaching science.

15. It is important for teachers to give students prescriptive and sequential directions for science experiments.

16. Focusing on rules is a bad idea. It gives students the impression that the sciences are a set of procedures to be memorized.

17. If students get into debates in class about ideas or procedures covering the sciences, it can harm their learning.

18. Students see a science task as the same task when it is represented in two different ways.

SECTION II.

To be good at mathematics [science] at school, how important do you think it is for students to [fill in the blank with each of the items below]?

(A) Not important

(B) Somewhat important

(C) Very Important

In Mathematics

19. remember formulas and procedures?

20. think in sequential manner?

21. understand concepts?

22. think creatively?

23. understand math use in real world?

24. support solutions?

In Science

25. remember formulas and procedures?

26. think in sequential manner?

27. understand concepts?

28. think creatively?

29. understand science use in real world'?

30. support solutions?

SECTION III.

What is your familiarity with the reform documents?

(A) Not at all

(B) Small extent

(C) Fairly

(D) Moderate extent

(E) Great extent

31. Mathematics standards document (Curriculum and Evaluation Standards for School Mathematics).

32. Science standards document Benchmarks for Science Literacy.

33. Science standards document National Science Education Standards.

SECTION IV.

Please indicate if you use (or would use if you taught mathematics and science) the instructional strategies listed below.

(A) No

(B) Yes

In Mathematics

34. Assisting all students to achieve high standards.

35. Providing examples of high-standard work.

36. Using authentic assessments.

37. Using standards aligned curricula.

38. Using standards-aligned textbooks and materials.

39. Using telecommunication-supported instruction.

40. Making connections with science.

In Science

41. Assisting all students to achieve high standards.

42. Providing examples of high-standard work.

43. Using authentic assessments.

44. Using standards aligned curricula.

45. Using standards-aligned textbooks and materials.

46. Using telecommunication-supported instruction.

47. Making connections with mathematics.

SECTION V

48. If you have taught since graduation, for what duration'?

a. in beginning year

b. 1 to 2 years

c. 3 to 4 years

d. > 4 years

49. If applicable, what grade level are you teaching this year?

a. 1 or 2

b. 3 or 4

c. 5 or 6

d. 7 or 8

e. other

50. If applicable, are you a specialized teacher (by content)?

a. yes

b. no

51. If you are a specialized teacher, what is your content area?

a. mathematics

b. science

c. both mathematics and science d. other

Author's Note

The National Science Foundation's Collaboratives for Excellence in Teacher Preparation Program (DUE 9255745 and DUE 9814650) and the Teachers Professional Continuum Program (Cooperative Agreement No ESI 0455752) supported this research. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We gratefully acknowledge the MCTP graduates who participated in this study. References

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Gili Marbach-Ad

J. Randy McGinnis

University of Maryland

Table 1
Comparison Between the Responses of the First and Second
Survey of MCTP Teachers' Beliefs About the Nature and
Teaching of Mathematics by Number (and Percentage)
Responding "Agree" or "Strongly Agree"

Item
                          First Survey   Second Survey

1. Math is primarily       2 (6.5%)       6 (19.4%) **
an abstract subject.

2. Math is primarily      42 (77.4%)     24 (77.4%)
a formal way of
representing the
real world.

3. Math is primarily      29 (93.5%)     29 (93.5%)
a practical and
structured guide for
addressing real
situations.

4. Math should be          7 (23.3%)      4 (13.3%)
learned as sets of
algorithms or rules
that cover all
possibilities.

5. A liking for and       28 (90.3%)     29 (93.5%)
understanding of
students are
essential for
teaching  math.

6. If students are        4 (12.9%)      4 (12.9%)
having difficulty,
an effective
approach is to give
them more
practice by
themselves during
class.

7. More than one          29 (93.5%)     31 (100%)
representation
should be used in
teaching a math
concept.

8. Some students          23 (74.2%)     25 (80.6%)
have a natural
talent for math and
others do not.

9. Basic                   9 (2.9%)       8 (25.8%) *
computational skills
on the part of the
teacher are
sufficient for
teaching elementary
school math. School
Science and
Mathematics

Note. * p < 0.05 ** p < 0.01.

Table 2
Comparison Between the Responses of the First and Second Survey
of MCTP Teachers' Use of Instructional Practices in
Mathematics by Percentage Responding "Yes"

                                                         Mathematics
Item
                                                         First   Second

34. Assisting all students to achieve high standards.     90%     100%
35. Providing examples of high-standard work.            93.5%    100%
36. Using authentic assessments.                         93.5%    100%
37. Using standards aligned curricula.                   93.5%    100%
38. Using standards-aligned textbooks and materials.     87.1%    100%
39. Using telecommunication-supported instruction.       56.7%    60%
40. Making connections with science.                     90.3%   96.8%

Figure 1. Comparison between the responses of the first and
second survey of MCTP teachers' perceptions of student skills for]
success in mathematics by percentage responding "very important".

Selected student skills

                     1st survey   2nd survey

Remember
formulas and
procedures             32.30%       35.50%

Think in
sequential
manner                 38.70%       45.20%

Understand
concepts               96.80%       93.50%

Think creatively       56.70%       53.30%

Understand math
use in the real
world                  90.80%       87.10%

Support solutions      96.80%       93.50%

Note: Table made from bar graph.