The Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) was issued to reflect what should be of value and to promote reform in mathematics education. The Professional Standards for Teaching Mathematics (NCTM, 1991) envisioned teachers as the primary agents for implementing the Curriculum Standards. In response, the Curriculum Standards have been incorporated into mathematics methods courses to prepare teachers to assume their roles as agents of change. However, success depends on the value that preservice teachers assign to the Standards and trends in these valuations remain unexamined.
Since its publication, the Curriculum Standards (hereafter referred to as the Standards), have had a highly visible impact on mathematics education. Besides being incorporated into mathematics methods courses, they have influenced K-12 curricula, and methods of assessment, as well as professional development programs (Ferrini-Mundy, 1996; Findell, 1996; Research Advisory Committee, 1998). In addition, publishers have aligned school mathematics textbooks with the Standards, although sometimes merely as addons (Battista, 1999; Chandler & Brosnan, 1994). The Standards have spawned numerous articles pointing to a myriad of effects they would have on schools and students (Crosswhite, Dossey & Frey, 1989; Lindquist, Dossey & Mullis, 1995; Loveless, 1997; Research Advisory Committee, 1990) and have been recently updated by the Principles and Standards for School Mathematics (NCTM, 2000).
Despite the visibility of reform in these contexts, other manifestations of reform are less obvious. It is difficult to determine to what degree teachers have implemented the NCTM's vision of how mathematics should be taught in the classroom. Widespread awareness or acceptance of reform was not the case in 1993, when a survey (Weiss, 1994) was conducted of 6000 teachers in grades 1-12. Only 56% of high school teachers and less than 28% of elementary teachers were "well aware" of the Standards. Also, when teachers from the elementary and high school levels were asked how important: instructional strategies suggested by the Standards were to effective instruction, they valued some strategies but rejected others.
The Standards called for a vision of mathematics teaching that encourages active student participation and problem solving. It fostered a vision in which students would be given opportunities to pose their own problems that involve everyday situations and have opportunity to read, write, and discuss meaningful mathematics. Students would be exposed to a variety of computation techniques, such as using paper and pencil, using calculators, and performing mental computation both exact and approximate. Ultimately, this style of teaching would encourage students to construct their own knowledge.
However, this vision is in conflict with both the way many preservice teachers learned mathematics and their conceptions of mathematics teaching (Frykholm, 1996; NCTM, 1991; Schram & Wilcox, 1988). All too often, they have been exposed to a style of mathematics teaching in which students are discouraged from being anything other than passive receptors of knowledge -- a style preoccupied with paper and pencil computation which emphasizes memorization of facts, rules, and formulas along with a diet of routine problems often lacking in meaning.
The Professional Standards (NCTM, 1991) acknowledged this conflict and recommends that preservice teachers be provided with opportunities to examine and revise their conceptions about mathematics teaching. Examination of their conceptions, before they take methods courses, might form a baseline for comparison and a compass for needed revisions. Preservice educators could benefit in knowing the initial value their students assign to the Standards, as well as trends in these evaluations. Brown and Baird (1993) state that in order for teachers to teach according to the vision presented in the Standards, they must believe in its value. The same is true for preservice teachers, since they are expected to eventually implement this vision to promote reform in mathematics education.
Teachers' conceptions of mathematics and its teaching, as Thompson (1992) suggests, will encompass beliefs, knowledge, preferences and, in this study, values as well. She explains that teachers' conceptions are often used as filters to evaluate their classroom practice. Consequently, identifying and addressing the conceptions held by preservice teachers is a necessary ingredient of reform. Other studies (Cooney, Shealy, & Arvold, 1998; Frykholm, 1996; Raymond, 1997a; Thompson, 1992; Wilson, Schram, Lappan, & Lanier; 1991) have pointed to the need for preservice teachers to acknowledge their own conceptions and to reflect on them to improve the quality of their mathematics teaching. Cooney et al. (1998) refers to the need for teachers to become "reflective connectionists" who integrate voices, analyze the merits of various positions, and come to terms with personal beliefs in a committed way. Such reflection by preservice teachers is warranted, owing to the potential conflict between their own conceptions and the vision of mathematics teaching advocated by the Standards.
Ernest (1989, 1991) defined three models that provide a basis for conceptions of mathematics teaching, as well as a framework to embed the vision endorsed by the Standards. In the Instrumentalist model, the teacher's role is that of instructor, and the intended outcome is skill mastery by the students with correct performance. Rote learning and memorization are emphasized in the mastery of skills, rules and procedures as separate entities. In this model, the textbook is followed strictly and the student's role is to master what the teacher is telling. In the Platonist model, the teacher's role is that of explainer, and the intended outcome is for students to have conceptual understanding with a unified knowledge. The teacher, who possesses all knowledge, transmits it to the students who are passive receptors. In this model, a textbook approach is used by the teacher to communicate the structure of mathematics. Lastly, in the Problem Solving model, the teacher's role is that of facilitator, and the intended ou tcome is for students to pose and solve problems including ones with personal relevance. Emphasis and value is placed on investigation and exploration with students constructing their own knowledge.
Both the Platonist and Instrumental models, which are characterized by a passive reception of knowledge, differ in construct from the Problem Solving model, which is characterized by an active construction of knowledge. Ernest's Problem Solving model of teaching resembles closely the vision advocated by the Standards. Moreover, what prevailed in most classrooms when the Standards were published was a combination of Platonist and Instrumentalist models of teaching (Mathematical Sciences Education Board and National Research Council [MSEB], 1989). Consequently, this study used the three teaching models to frame possible changes in conceptions held by preservice teachers.
In 1990, the Research Advisory Committee pointed to the need for monitoring the change that actually occurs in mathematics education owing to reform. The Standards have inspired many classroom reform efforts, and some of these may have directly or indirectly impacted preservice teachers' conceptions about mathematics teaching. Undergraduate preservice teachers have experienced a variety of mathematics teachers and courses in grades K-12. Clark & Peterson (1995) suggest that beliefs, a component of conceptions, are convictions or opinions that are formed either by experience or by the intervention of ideas through the learning process. Furthermore, findings (Ball, 1988; Thompson, 1992; Raymond, 1997b) indicate that preservice teachers' beliefs about mathematics teaching and learning are, for the most part, formed by their own K-12 experiences.
Even though the Standards were published in March of 1989, by the Fall of 1991, junior preservice students would have had little opportunity to experience the vision of mathematics instruction as presented in the Standards in their own K-12 classrooms. The youngest members of that class were, generally at the least, second semester high school seniors when it was published. Further, during their beginning college years and after, the reforms suggested by the Standards were not yet fully assimilated by most mathematics educators (Weiss, 1994).
The prospect of greater assimilation of the Standards and their potential impact on the conceptions of preservice teachers over time prompted this research, which was initiated the following spring and continued until, the fall of 1998. For the most part, by 1998, the youngest college junior would have been in sixth grade in 1989. They might have had the opportunity to experience instruction or textbooks impacted by the Standards over grades 6 through 12. Also, during their college years leading up to their junior year, it is possible that through self-study, by taking courses, by attending conferences, or by some other learning experience, they would have become familiar with the Standards. This in no way implies that if they had experienced them directly or indirectly, that the individual would value them; just that there was a longer period of opportunity to experience some curricular change or reform effort prompted by the Standards.
While previous research involving preservice teachers and the Standards has focused on time spans of two years or less, this study will explore trends over a seven-year period, which straddles the introduction of the Standards to, and their continued assimilation by, the mathematics education community. Since previous research has consisted of mostly case studies involving a small number of participants, it could be built upon with the study of a larger group of participants. Previous research has primarily focused on changes in the students' conceptions as they progress through programs encouraging reform. Most confirm the difficulty in changing the conceptions held by preservice teachers, and some offer direction for solutions and continued research.
Benken and Wilson (1996) studied a preservice secondary teacher who communicated a narror Instrumentalist (Ernest, 1989) view of mathematics teaching and learning. Her core conceptions remained impervious to change for the most part, despite her being inundated with reform themes. Similarly, Schram and Wilcox (1988) studied two elementary preservice teachers enrolled in two conceptually-based mathematics courses. The researchers reported how conceptions about mathematics and mathematics teaching limited their ability to teach mathematics emphasizing concept development and problem solving. One student, Denise, incorporated more talk about reasoning and exploring after completing the conceptually-based course. However, she used these terms in ways that showed little change in their thinking. For instance, she talked about exploring mathematics, but for her it meant persisting until she got the right answer.
A larger number of preservice secondary teachers (44) was studied by Frykholm (1996) over four semesters. He found that they had a high regard for the Standards and that they believed their instruction modeled them. However, the majority of lessons observed bore little or no resemblance to the Standards. This was attributed to several constraints, including little appreciation of reform efforts by cooperating teachers, lack of time to implement that type of instruction, and inaccurate student assumptions about what it means to implement the Standards. Again, Frykholm's study confirmed the difficulty in changing conceptions and that simply presenting the Standards to preservice teachers is inadequate.
Eggleton (1995) in his case study of a secondary preservice teacher, ruled out two other common strategies. Neither acknowledging choices among culturally established pedagogic perspectives nor experiencing alternative pedagogical practices is sufficient for providing a rich context that allows preservice teachers to examine their mathematics philosophies.
In contrast to these studies, Chauvot & Turner (1995) studied a secondary preservice teacher, Liz, who successfully changed her initial conceptions as she progressed through a reformed mathematics education program. At the start of the program, she completed a survey of her conceptions about mathematics and about the teaching and learning of mathematics. During her program she was given opportunities to examine, reflect, and revise her initial conceptions. This reflection allowed her to modify her views and to incorporate characteristics from the Problem Solving model into her student teaching experience. This evolution happened despite the fact that she had been exposed to a mostly teacher-centered classroom that was textbook-based (Instrumentalist and Platonist models). The researchers suggest that being aware of preservice teachers conceptions as they enter mathematics education programs may provide direction to help promote change endorsed by the Standards.
Further, Cooney et al. (1998) studied the beliefs and belief structures of four secondary preservice teachers to enhance the understanding of how preservice teachers construct meaning as they progress through teacher education programs. The inculcation of doubt and the posing of perplexing situations seemed vital to the process of change. Fuel for these "epistemological crises" might be most appropriately selected from the Standards themselves.
Research indicates that preservice teachers form their conceptions about how mathematics should be taught from their own experiences as students. Examining their conceptions before they take methods courses should, therefore, reveal much about the way they have been taught mathematics. By doing so over a protracted period, a picture could be developed of whether mathematics teaching is progressing toward the vision set by the Standards. This study, involving a large number of preservice teachers over 14 semesters, will look for trends and provide insight for preservice educators and especially those who teach methods courses. Specifically, this study will examine the value or importance assigned by preservice teachers to the vision of mathematics teaching endorsed by the Standards. And, pinpoint the Standards, if any, that are in conflict with the conceptions of beginning preservice teachers, so that reflection can be effectively directed to promote reform. Surveys and group interviews were conducted to answ er the following research questions:
1) Is there a trend over 14 semesters in the degree of importance assigned by preservice teachers to statements selected from the K-4 Standards?
2) Are there any trends over 14 semesters in the degree of importance assigned to statements selected from the individual Process Standards of Problem Solving, Communication, Reasoning, and Connection or to the individual Content Standards?
3) Which of the selected statements from the K-4 Standards, did preservice teacher's rate lowest in importance? What justification was given for such ratings?
A repeat cross-sectional design (Menard, 1991) was used to explore trends in the degree of importance assigned to the K-4 Standards. Surveys were administered to each successive class of preservice teachers over 14 semesters beginning in the Spring of 1992 through the Fall of 1998. During the last two semesters, demographic information was collected, and clinical group interviews were conducted, tape recorded, and transcribed to provide additional insight.
All enrolled elementary junior-level preservice teachers at an Eastern state public college participated in the study. There were a total of 1026 participants, with approximately 73 participating each semester. The academic profile of each class of preservice teachers remained steady over the years spanned by the research, but the GPA's of transfers showed an increase. The average verbal, mathematics, and combined Scholastic Aptitude Test (SAT) scores, as well as high school class percentile rank for all enrolled freshman education majors are presented in Table 1. Participants would have had the indicated entering profiles except for a small percentage who would have changed majors, into, or out of, education during their freshman or sophomore years. Furthermore for each class, there was a percentage of participants who transferred into the college with previous college credits. The percentage of each class who were transfers and their incoming average GPA is also listed in Table 1.
During the last two semesters, demographic information was collected from the enrolled preservice teachers (138). Their average age was 21.5 years, ranging from 20 to 40 years. Only 12(9%) participants were over the age of 25, and of the participants, 14(10%) were male. All of the participants were either elementary education majors (85%) or early childhood education majors (15%). The academic majors or areas of concentration outside of education was varied, with 14(10%) having mathematics as their concentration. Over the seven-year span of the study, the number of participants of white ethnicity was between 88% and 93% each semester, and over 95% were from the state of New Jersey. They represented a cross section of backgrounds from public, private, and parochial schools. Throughout the study, three instructors administered the surveys to their classes of preservice teachers. They reported similar student demographics from their observations of participants during prior semesters. Information was also colle cted concerning participants' mathematics background. The average number of yearlong high school mathematics courses taken was 4, with 110 (80%) taking 4 or 5 courses. The average number of college mathematics courses taken was 2.5, with the majority of participants, 83(60%), having taken exactly 2 courses. The ratings of their past experiences in mathematics were very positive, with 15 (11%) participants rating it excellent, 69(50%) very good, and 48 (35%) satisfactory, and only 6(4%) falling into the categories of less than satisfactory or poor.
A survey was developed to measure the degree of importance assigned by the preservice teachers to the Standards. To form the survey, statements were selected from the K-4 Standards, since the participants were either early childhood or elementary education majors.
The K-4 Standards consist of four Process Standards (Problem Solving, Communication, Reasoning, and Connections) and nine Content Standards (Estimation, Number Sense and Numeration, Concepts of Whole Number Operations, Whole Number Computation, Geometry and Spatial Sense, Measurement, Statistics and Probability, Fractions and Decimals, Patterns and Relationships).
Each of the 13 Standards is elaborated in a list of students' expectations, followed by a Focus section and a Discussion section with sample problems.
From the elaboration for each K-4 Standard, three statements were selected to represent the Standard on the survey. For example, from the Content Standard of Geometry and Spatial Sense, three statements were selected: To have students recognize and connect geometry to the world To have students draw and model different shapes, and To have students describe and classify different shapes. Combining these with three statements from each Standard, a random order 39-item paper and pencil survey was formed. The participants were instructed to score or rate each statement on a five-point scale: 1) of no importance, 2) of low importance, 3) of medium importance, 4) of high importance, 5) of extremely high importance.
Participants were surveyed either in the fall or spring at the beginning of their Junior Professional Experience (WE). The one-semester JPE consists of a block of courses including a practicum course which occurs at a local elementary school four mornings a week and four methods courses at the college: social studies and science, environmental awareness, health and physical education, and mathematics. The last of these, M342 (Teaching Mathematics in Elementary School) met for 2 hours weekly over 12 weeks and concludes with a two week full-time teaching experience at a local elementary school.
Surveys were completed at the beginning of the first meeting of the M342 class before the instructor made any introductory remarks. By administering the survey at this time, there was little opportunity for the instructor or class material to influence the results. Each of the three instructors who administered the survey informed the participants that this was a research activity to gather data to improve preservice education. No mention of, or connection to, the Standards were made.
For the final two semesters of the study, group interviews were conducted to assist in interpreting the results of the survey. Participants were asked to put their name on the back of the surveys so that they could be returned during a subsequent group interview. The students were assured that putting their name on the survey would in no way affect their grades. The clinical interviews took place during a regularly scheduled class time in each section of M342. They lasted for approximately 30 minutes each and occurred on four consecutive days approximately one month after they had taken the survey.
Survey Mean Scores
Using the 1026 surveys, a mean score for each semester over the 39 items was computed. A plot of the mean scores (1 being of no importance and 5 being of extremely high importance) is shown in Figure 1. In Figure 1, and For each subsequent figure, semester 1 will correspond to spring of 1992, semester 2 to fall of 1992, ..., semester 14 to fall of 1998.
Generally, the mean was greater than 4, indicating that students assigned high importance to the statements selected from the K-4 Standards. A Regression Analysis was performed to look for a trend in the line of best fit using the mean scores as the dependent variable. An alpha level of .05 was used for all statistical tests. The null hypothesis was that there was no significant trend in mean scores over the 14 semesters. The null hypothesis was accepted, since no significant trend was indicated f(l,1024) + 3.00, p<.11.
Process Standards Mean Scores (Using 3 Selected Statements
For each Process Standard, a mean score for each semester over the three selected statements was computed. A plot of the mean scores corresponding to each Process Standards over the 14 semesters is shown, along with the line of best fit, in Figure 2.
The results show that the students, in all semesters, assigned a score of 3.5 or more, to the statements selected from the Process Standards. In addition, a Regression Analysis was performed to look for trends in the line of best fit for each Process Standard. Significant increasing trends were indicated in the mean scores for the statements selected from Communication F(1,12) =42.24, p<.0l, Reasoning F(l,12) = 8.59, p<.0l and Connections F(l,12) 31.83, p<.01. However, the trend in the mean scores for the statements selected from Problem Solving was not significant F(1,12) = 0.43,p<.55. The 3 statements selected for each of Process Standards are shown in Table 2.
Process Standards Mean Scores (Using an Expanded Number of Statements)
While the K-4 Standards are divided into four Process Standards and nine Content Standards, the Process Standards form an umbrella for all of the K-4 Standards. Many of the statements selected from the Content Standard were also representative of one or more of the Process Standards. As an example, the statement: To have students recognize and connect geometry to the world is taken from the Content Standard of Geometry and Spatial Sense, bat also is representative of the Process Standard of Connections.
To enhance the measurement and improve the representation of the Process Standards, four experienced mathematics educators were asked to determine which of the statements selected from the Content Standards would also be representative of one or more of the Process Standards. A Content statement was then also assigned to represent a Process Standard, if it was chosen by 3 or 4 of the evaluators as representing a Process Standard. This resulted in the 4 Process Standards of Problem Solving, Communication, Reasoning, and Connections being represented by an expanded number of statements (16,12,13 and 14 statements, respectively). With the expanded number of statements assigned to each Process Standard, the internal consistency measured by Cronbach Alpha for Problem Solving was [alpha] = .86, for Communication [alpha] = .83, for Reasoning [alpha] = .82, and for Connections [alpha] = .82.
For each Process Standard, a mean score was again computed for each semester using the expanded number of statements. A plot of the mean scores corresponding to each Process Standard over the 14 semesters is shown, along with the line of best fit, in Figure 3. A Regression Analysis was performed to look for trends in the line of best fit for each Process Standard. In this analysis, scores given to the statements assigned to Connections F(1,12) = 2.92, p<.ll showed no significant trend. But as before, the scores given to the statements associated with Communication F(1,12) = 4.88,p<.04, and Reasoning F(1,12) = 10.45,p<.01, showed significant increases, and those associated with Problem Solving F(1,12) =0.11, p<.74 again showed no significant trend.
Content Standards Mean Scores
For each of the nine Content Standards, a mean score for each semester over the three selected statements was computed. A Regression Analysis was performed to look for trends in the line of best fit for each. Two of the trends associated with the Content Standards were significant. Both Estimation F(1,12) = 31.83, p<.01 and Geometry and Spatial Sense F(1,12) = 14.15, p<.01 exhibited strong positive trends.
Mean Scores of Individual Statements
The mean score for each individual statement (S1-S39) were computed and sorted from low to high. The statements with the overall lowest mean scores are listed in Table 3 with their means and standard deviations.
Two of the statements selected from Communication appeared on the list: S5 To have students write or talk about mathematics, and S6 To read and discuss literature with students concerning mathematics. In addition, three statements selected from Estimation appeared in the list: S13 To have students use estimation to check computation, S14 To have students use terms such as about, near, closer to, and between, and S15 To have students determine the reasonableness of results by estimation. Representing the Content Standard of Whole Number Computation were the statements S22 To show that the purpose of computation is to solve problems and S23 To use calculators in appropriate computational situations. Rounding out the statements with the lowest means was the statement: S33 To have students explore concepts of chance selected from the Standard of Statistics and Probability.
Survey mean scores over the 14 semesters were relatively high (about 3.9), which confirms Frykholm's (1996) findings that preservice teachers generally report a high regard for the Standards. Of note, the mean score from semester 10 (fall 96) spiked above those of the other semesters. During the previous semester, and only that semester, an instructor who was a strong advocate of the Standards and whose instruction modeled them, taught three sections out of eight offered in Foundations of Mathematics; a feeder course for the semester 10 participants. It is conjectured that a number of participants might have had a heightened awareness of the views expressed by the Standards due to that experience. Consequently, they may have assigned higher levels of importance to the statements selected from the K-4 Standards. Although, taken as a whole, survey scores over the 14 semesters did not show a significant trend.
The mean scores assigned to the Standard of Communication exhibited significant increasing trend, whether represented by three selected statements or an expanded set of statements. Further, the mean scores assigned to the Standards of Reasoning and Connections exhibited significant increasing trend when represented by three statements. When an expanded number of statements were considered, scores for both Standards exhibited increasing trends, however only those representing Reasoning were significant. Overall, the participants assigned increasingly higher scores to the selected statements for Communication, Reasoning, and Connections over the 14 semesters. Despite these results, the mean scores assigned to Problem Solving remained unchanged whether represented by three or an expanded number of statements.
This difference merited a closer examination of the statements representative of the individual Process Standards from the perspective offered by Ernest's models of teaching. The statements S5 To have students write or talk about mathematics, and S6 To read and discuss literature with students concerning mathematics associated with Communication, would likely be important components of the Problem Solving model where the teacher is a facilitator. However, these statements could be accommodated to a degree by the Instrumentalist or Platonist models. For example, the instructor or explainer might provide for some minimal student feedback, yet still dominate most communication. Also from Communication, S4 To relate physical materials, pictures, and diagrams to mathematical ideas, differs from S5 and S6 in that the word student is not used. In the Problem Solving model, students would be the manipulators of materials to actively construct their own knowledge. Since the manipulator is not indicated in S4, the exp lainer in the Platonist model could use physical materials to demonstrate concepts as the students passively watched. While the statements associated with Communication are important components of the Problem Solving model, they are not exclusive of the other two models and might only represent superficial changes to them.
As with Communication, the statements from Reasoning and Connections are not excluded as components of the Instrumentalist and Platonist models of teaching. For example, the statement S7 To have students justify and explain their reasoning might just mean showing or talking about each step in a procedure for an Instrumentalist. And in the Platonist model, statement S10 To have students link conceptual and procedural knowledge would be an expected outcome of an effective explanation to passive listeners. Hence, the statements representing the Standards of Reasoning and Connections were important components of the Problem Solving model, but they would not be excluded from being components, even if only superficially, of the other models.
In contrast to the other Process Standards, many of the statements from Problem Solving are critical to the Problem Solving teaching model, and at the same time antithetical to the Instrumentalist or Platonist models. Statement S1 To have students create their own problems from real world activities, and from the expanded set of statements, S18 To have students construct meanings through real world experiences are fundamental to the Problem Solving model where students pose problems and actively constructing their own knowledge. Yet, these would not be components of the other models where the instructor or explainer transmits all knowledge to passive receptors. Additionally, statements 52 To use problems that involve everyday situations, S3 To show how to solve problems in more than one way, and from the expanded set S22 To show that the purpose of computation is to solve problems convey a sense of exploration and investigation involving problems with personal relevance so important to the Problem Solving mo del. To use applications, or to show its purpose of computation would be uncommon components of an Instrumentalist model of teaching, which is dominated by rote learning of skills, facts, and procedures as separate entities. Showing how to solve problems in more than one way would be out of place in the Platonist model, since the explainer would probably have decided ahead of time the single best way to solve or explain the problem to the students. Nevertheless, these statements are more out of step with the Instrumentalist or Platonist models, and might have been foreign to their prior mathematics experience.
The assumption that the surveys provided insight into the participants' prior mathematics experience was supported during group interview with comments like:
"I believe that the reason why certain numbers on the survey are rated low or high is because we based our answers on our past elementary experiences."
This confirms previous research (Ball, 1988; Thompson, 1992; Raymond, 1997b) which indicates that preservice teachers' beliefs, which are a component of conceptions, about teaching mathematics are shaped by their own experiences as students of mathematics. As with the three Process Standards, the mean scores assigned to the Content Standards of Estimation, and Geometry and Spatial Sense exhibited significant increasing trends. Despite these results, group interviews revealed the restrictive and superficial conceptions held by some of the participants. This was especially true of the statements selected from Estimation, which were among the lowest rated statements on the entire survey: S13, S14, and S15. In justifying the low ratings given these statements, participant responses included:
"I just as soon add it together faster than they (his teachers) could explain estimation to me."
"It is only right on that day that you teach it. If the child is to give it on another day that you teach another concept and has to add and he estimates. It is close, but it is wrong! Est imation is wrong on any other day than the one you teach it on."
Students also indicated that the topic of estimation was seldom taught or assessed as in the following example.
"I rated it low because we never did estimation in my curriculum, so I rated it low. You must have an exact answer. You're looking for the exact answer in a problem but it is frustrating since estimation is not right or wrong."
The Standards call for an increased attention to estimation including mental computation, reasonableness of answers, estimation of quantities, estimation of measurements, and a decrease in the use of rounding to estimate. Yet, rounding was the only type of estimation mentioned by the participants.
Also, during group interviews students were asked to comment on the lowest rated statements from Communication which exhibited the widest deviations and some of the lowest ratings on the survey. Statement S5 was the lowest rated on the survey and 56 was among the lowest rated. It seems for statement 55 the word "write" in the same sentence with the word "mathematics" remained incongruent for a number of the participants each semester. Some participant responses involving statement S5 included: "I never experienced math as something to write or talk about" and "Math was always my best subject, but don't ask me to talk about it. I can do it, but it doesn't matter." These responses attest to the lack of student communication in their prior mathematics experience.
In addition to the lowest rated statements from Estimation and Communication, two others appear that have not been mentioned: S23 To use calculators in appmpriate computational situations, and S33 To have students explore concepts of chance. Calculator use remains controversial and participants repeatedly referred to the fear of encouraging dependency. Concern was also voiced for exploration of concepts of chance, since it might lead to gambling and its associated excesses. These responses are illustrative of the restricted conceptions held by some of the participants about statements contained in the survey, and consequently of the Standards themselves.
This study examined the conceptions held by elementary preservice teachers over a seven-year period which straddled the introduction and assimilation of the Standards by the mathematics education community. Preservice teachers, for the most part, assigned a relatively high degree of importance or value to the NCTM's Standards (1989). Also, significant increasing trends over the 14 semesters were found in the rating for the Process Standards of Communication, Reasoning, and Connections. A closer examination suggested that the increase in ratings were not necessarily representative of increasing value of the substantial changes in mathematics teaching called for by the Standards.
Examination of the statements assigned to Communication, Reasoning, and Connections suggested that they might be fitted into, or accommodated by the Instrumentalist and Platonist models of teaching in very superficial ways. As reported by Stigler & Heibert (1998), teaching is such a complex system and changes are often modified to fit in the pre-existing system instead of changing the system itself. Consequently, statements selected from these Standards posted little conflict for the preservice teachers' conceptions of mathematics teaching.
In contrast, the ratings of the statements assigned to Problem Solving remained unchanged over the 14 semesters. These statements conveyed an active, rather than passive construction of knowledge that is present in the Instrumentalist and Platonist models of teaching. The difference in ratings suggest that Problem Solving, in particular, posed conflict to the conceptions held by preservice teachers which supports previous work (Frykholm, 1996; NCTM, 1991; Schram & Wilcox, 1988). Since this study also confirmed that preservice teachers' conceptions about mathematics teaching have been strongly influenced by their prior mathematics experiences, it is likely that their prior mathematics experience did not conform to the Problem Solving model of teaching. But rather, as reported in 1989, what prevails ] most classrooms is a combination of Platonist and Instrumentalist teaching models. Models of teaching that are ineffective for long-term learning, higher order thinking, and for versatile problem solving. (MSEB, 1989). Problem Solving is antithetical to both of these teaching models and a fundamental construct of the Standards. Hence, questions about Problem Solving are central in revealing the degree to which schools are or are not progressing in their implementation of the vision of mathematics teaching advocated by the Standards.
This study also suggests that Problem Solving might be the best choice to provide the "epistemological crises" needed for the development of elementary preservice teachers as Reflective Connectionists, Cooney et at, (1998). suggests that opportunities for preservice teachers to experience conflict are crucial. Because statements associated with Problem Solving provided more conflict than those from the other Process Standards, they might be the best choice in bringing such conflicts to light. Also, it was this standard which played a crucial role in the successful evolution of conceptions for a preservice teacher studied by Chauvot & Turner (1995). When preservice teachers begin methods courses in mathematics education, their conceptions need i:o be recorded and reflected upon so that robust conceptions of the Standards can be developed, and this is especially true when it comes to Problem Solving.
Although preservice teachers assigned a relatively high degree of importance to most of the K-4 Standards, and most of the trends in these ratings showed increase, conceptions revealed during group interviews during the last two semesters were restrictive compared to the robust ones envisioned by the Standards. Their conceptions of the statements associated with Estimation and Communication were particularly revealing. For some, Estimation had a singular and restricted meaning, and Communication in mathematics did not involve writing about mathematics. Statements from other Standards involving the use of calculators and the exploration of chance also exposed other restricted conceptions that they held. These were similar to the type of conception exhibited by Denise, reported by Schram and Wilcox (1988), who talked of exploring mathematics, but meant much less. It is impossible to determine how many of the preservice teachers in this study held such restricted conceptions. Yet, despite the overall quality of their academic profiles, some did.
Additional research is warranted, given some of the restricted and superficial conceptions of the Standards held by elementary preservice teachers in this study. Even with the many manifestations of reform spawned by the Standards, preservice teachers must continue to be provided with opportunities to examine and revise their conceptions. In particular, Problem Solving is a fundamental construct of the Standards and preservice teachers must come to a greater understanding of, and belief in, its value, since they are expected to eventually implement mathematics teaching as envisioned by the Standards in their own classrooms. Also, preservice educators must recognize the needs of their students and provide direction for required revisions of their conceptions, if the vision of mathematics teaching championed by the Standards is to proceed.
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Academic profiles of part
Year 1989 1990 1991 1992 1993 1994 1995 1996
SAT Verbal ---- 579 585 577 571 576 575 576
SAT Math ---- 569 573 576 567 576 568 583
SAT Combined ---- 1148 1158 1153 1138 1152 1143 1159
HS Rank ---- 86 85 85 85 85 84 86
% of Transfers ---- ---- ---- 44 45 42 41 41
GPA of Transfers ---- ---- ---- 3.17 3.23 3.35 3.36 3.39
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Process Standards an and Scheduled Statements
S1 To have students create their own problems from
real world activities
S2 To use problems that involve everyday situations
S3 To show how to solve problems in more than one way
S4 To create physical matrials, pictures, and
diagrams to mathematical ideas
S5 To have students write or talk about mathematics
S6 To read and discuss literature with students
concerning mathematical ideas
S7 To have students justify and explain their
S8 To relate to students that the method of solution
of a problem is as critical as the answer
S9 To use patterns and relationships to analyze
S10 To have students link conceptual and procedural
S11 To relate or connect two different topics in
S12 To use mathematics in other curriculum areas
Statements with overall lowest means scores (standard deviation)
Statement Mean (s.d.)
S5 To have students write or 3.49 (.88)
talk about mathematics
S13 To have students use 3.67 (.82)
estimation to check
S33 To have students explore 3.72 (.79)
concepts of chance
S14 To have students use 3.76 (.86)
terms such as about,
near, closer to, and
S23 To use calculators in 3.81 (.81)
S22 To show that the purpose 3.82 (.76)
of computation is to
S15 To have students 3.87 (.75)
determine the reason-
ableness of results by
S6 To read and discuss 3.91 (1.02)
literature with students
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