CHAPTER V Summary, Findings, Conclusions, and Reccomendations


CHAPTER V

 

SUMMARY, FINDINGS, CONCLUSIONS,

AND RECOMMENDATIONS

 

 

Introduction

Chapter V begins with the purpose of the study, research questions, hypotheses, and methodology.  Chapter V presents a thorough discussion of the findings and offers conclusions that connect these findings to the theoretical expectations presented in the introduction and in the literature review.  The implications of these findings for the assumptions stated in the literature are discussed with respect to each question and hypothesis.  Research often raises more questions than it does answers; recommendations for future research are made where relevant questions were not answered by the data analyzed.

Purpose of the Study

The purpose of this study was to describe AP calculus teachers’ and administrators’ perceptions of AP calculus program characteristics that have increased student enrollment in selected San Bernardino County public comprehensive high schools.

Research Questions

 

This study included the following seven research questions:

  1. What are the perceptions among AP calculus teachers and site administrators on how well students are prepared for AP calculus classes?
  2. What are the perceptions among AP calculus teachers and site administrators as to the amount of support AP calculus teachers receive from the school district?
  3. What are the perceptions among AP calculus teachers and site administrators as to the amount of support AP calculus teachers receive from site principals?
  4. What are the perceptions among AP calculus teachers and site administrators as to the amount of support AP calculus programs receive from parents?
  5. What strategies are AP calculus teachers and site administrators using to prepare students for success in the AP calculus courses?
  6. What are the perceptions among AP calculus teachers that site administrators underestimate the amount of support given to students?
  7. What perceptions exist among AP calculus teachers and site administrators on the criteria for students to qualify for the AP calculus programs?

Research Hypotheses

The research hypotheses included:

 

Hypothesis 1- There is no difference between administrators and teachers in their perception of teacher training.

Hypothesis 2.  There is no difference between administrators and teachers in their perception of resources and support available to the AP calculus program.

Hypothesis 3.  There is no difference between administrators and teachers in their perception of student commitment and access.

Methodology

Descriptive research methodology and a posttest-only, one-group design was used to identify the current characteristics of AP calculus programs in the county of San Bernardino.  The posttest-only design may be useful in exploring for researchable problems or developing ideas or devices, as in action research (Isaac and Michael 1997).  The posttest-only, one-group design was selected because it works best for collecting data from subjects who have unique experience when no control group is being used.  This simplified design can be fully effective only where there has been assignment of the subjects by randomization and there is no basis for believing that the two groups are not equivalent.  It avoids all the problems of the effects of the pretest on the posttest and the problems of how the pretest may alter the way the subjects react to the stimulus itself (Baker 1999).  In this study, teachers who had the unique experience of teaching AP calculus were asked to provide self-reported data using a questionnaire.  No data were collected before they had this experience, thus posttest only.  In addition, no control group was selected for the collection of comparison of data, thus, the one-group only.  The posttest-only, one-group design has threats to validity that are discussed in the limitations section.

Furthermore, the descriptive research methodology was selected because it is a means to describe systematically, factually, and accurately the characteristics of an existing phenomenon (Isaac and Michael 1997).  “A high percentage of reported research studies are descriptive, no doubt because it is useful for investigating a variety of educational problems and issues” (Gay and Airasian 2000, 275).  Gay and Airasian (2000) enunciated that “descriptive research sounds very simple, but there is considerably more to it than just asking questions and reporting answers” (276).  Once a descriptive problem has been defined, related literature reviewed, and, if appropriate, hypotheses or questions stated, the researcher must give careful thought to selection of the research participants and data collection procedures.  It is not always easy to identify the population that has the desired information; for example, there are many methods for collecting data.

The research strategy engaged a survey design as expressed by Isaac and Michael in 1997:

Survey design, as the term implies, is a systematic means of collecting descriptive information about the characteristics, practices, or attitudes of a defined population of participants.  Surveys, which are the most common tool used in applied research, are most often found in the fields of sociology, political science, business, and education to gather information necessary for policy decisions.  Although their strength is a relatively quick and cost-effective means of gathering data, their weakness lies in the standardizations of their questions, and in the risk of making complex issues oversimplified and superficial.  (219)

 

Design decisions depend on the purposes of the study, the nature of the problem, and the alternatives appropriate for its investigation (Isaac and Michael 1997).  The research design suited the research problem relating to the characteristics of AP calculus classes in the county of San Bernardino.  The research controlled for standardization of questions by including a wide variety of types of questions to acquire a nonstandardized sampling of information.  In addition, to exclude oversimplification of issues, the researcher utilized research-based sources to develop the survey questionnaire instrument.  Isaac and Michael (1997) state that surveys are the most widely used technique in education and the behavioral sciences for the collection of data.  They are a means of gathering information that describes the nature and extent of a specified set of data ranging from physical counts and frequencies to attitudes and opinions.  This information, in turn, can be used to answer questions that have been raised, to solve problems that have been posed or observed, to assess needs and set goals, to determine whether or not specific objectives have been met, to establish baselines against which future comparisons can be made, to analyze trends across time, and generally to describe what exists, in what amount, and in what context.

First, the researcher utilized a qualitative pilot survey questionnaire instrument that tested for reliability and validity by auditing all procedures, field-testing the instrument, conducting the surveys and questionnaire, completing procedures consistently, having an expert assist the researcher, and using triangulation.  The pilot survey questionnaire was mailed with a cover letter, instructions, and a self-addressed, stamped envelope to five AP calculus teachers not in the sample, in the county of Riverside.  The purpose of the pilot survey was to make sure respondents understood the instructions, wording, and sections.  The wording was clarified, and irrelevant questions were removed.  The questions were rearranged in response to the pilot recipient’s responses.

Next, the researcher surveyed a larger percentage, representative of the site administrators and teacher population of AP calculus classes in the county of San Bernardino.  In the database of the California Department of Education, there were twenty-nine identified classes of AP calculus offered in selected San Bernardino County public comprehensive high schools.  Only one high school offered two classes of AP calculus.  Out of the population of twenty-nine identified AP calculus classes, twenty-five teachers responded, or 86 percent.  In addition, out of the population of twenty-eight identified site administrators, twenty-seven, or 96 percent, responded and one abstained from completing the survey.  Follow-up letters were sent to the high schools that did not respond in a timely manner.  The researcher strictly adhered to a timeline.  The researcher organized the returned surveys by utilizing a matrix analysis.

The study involved the thorough examination and description of concerns from respondents.  The result will be the increased success rate of students in AP calculus and the general improvement of AP calculus programs.  The current study was a descriptive study of twenty-five classes of AP calculus in the county of San Bernardino and twenty-seven site administrators who completed the questionnaire sent out by the researcher.

Summary of Findings

Results of the analyzed data related to the seven research questions and three research hypotheses including description of sample and population were presented in chapter IV.  The key findings were as follows:

Key Findings for Research Question 1

 

What are the perceptions among AP calculus teachers and site administrators on how well students are prepared for AP calculus classes?

Research question 1 was answered by several questions in the questionnaire.  Question 23 of the questionnaire asked the participants to indicate whether students who wanted to take AP calculus classes were not admitted to the classes because they were not qualified.  The findings found no noteworthy differences between the administrators and teachers.  Administrators and teachers largely agreed that students were permitted to take AP calculus if they were not qualified to take AP calculus classes.

The findings for question 22a indicated that to some extent a larger percentage of teachers perceived that there were gaps to students’ preparation for AP calculus classes, as opposed to administrators.  Furthermore, the findings for question 22a indicated that a large number of administrators and teachers agreed that algebra skills contributed to gaps in students’ preparation for AP calculus classes.

Question 21 of the questionnaire is divided into four responses.  The findings for question 21a indicated a significant difference between administrators and teachers in their responses to what percentage of students were very well prepared to take AP calculus classes.  It was not possible to conduct a valid significance test on questions 21b through 21d because of the number of nonrespondents for these questions.

Question 21b of the research questionnaire asked the participants for a perception of the percentage of students who were somewhat prepared to take AP calculus classes.  The findings for question 21b were too varied between teachers and administrators, and it was not possible to conduct a valid significance test.

The findings for question 21c indicated that differences were found between administrators’ and teachers’ opinions that students were not prepared to take AP calculus classes.  The responses from administrators and teachers were too varied to conduct a valid significance test.  Nevertheless, the perceptions of teachers were more pessimistic toward students’ preparedness to take AP classes.

Question 21d of the research questionnaire asked the participants for a perception of the percentage of the students who were very unprepared to take AP calculus classes.  It was not possible to conduct a valid significance test on question 21d because of the number of nonrespondents for these questions.  However, 50 percent of the teachers who responded indicated that students were very unprepared to take AP calculus classes.

Findings Related to the Literature

The data and literature support the premise that academic preparation for advanced study should begin in middle school (National Academy of Sciences 2005).  Middle schools face a challenge to influence as many students as possible to have the desire and preparation necessary to aspire to advanced study.  Middle schools must ensure that students are prepared to take advanced courses in high school by offering them a curriculum starting in the middle school.  Middle school programs have the goal to helping students achieve a deeper conceptual understand of content and unifying concepts (Education Week 2002).

The literature indicated that states are moving in the direction of providing eighth-grade students with greater exposure to algebra topics, whether in full-fledged Algebra 1 or in prealgebra courses.  However, McKnight et al. (1987), Porter et al. (1993) and Shaughnessy (1998) discovered that the course titles provide only a rough indication of the content students actually receive.  An arbitrary sample of eighth-grade state curriculum guides in mathematics revealed that the curricula lacked focus, covered too many topics, were repetitious from grade to grade, and were implemented unpredictably across schools and classrooms, resulting in highly irregular disclosure to a variety of important curricular themes (Schmidt, Finch, and Faulkner 1992).  The lack of thoroughness and consistency could have lasting consequences for students, whether or not they decide to pursue advanced study later in high school (National Academy of Sciences 2005).

 

Key Findings for Research Question Two

 

What are the perceptions among AP calculus teachers and site administrators as to the amount of support AP calculus teachers receive from the school district?

Research question 2 was answered by question 27 of the questionnaire, which asked the respondents to respond to the question as to how much support they perceived they received from the district.  The findings indicated that there were significant differences between administrators and teachers.  A larger number of administrators perceived that there was a lot of support for AP calculus classes from the district.  On the other hand, the findings indicated that teachers were more pessimistic that AP classes received a lot of support from the district.

Findings Related to the Literature

The data and literature clearly support the premise that school districts need to create working environments in which teachers have time to meet and discuss, reflect on, and refine instructional practices.  All mathematics teachers need to understand calculus so that students can begin working on its underpinnings in algebra.  School districts need to create mechanisms for teachers to participate in more structured professional development during contract hours (National Academy of Sciences 2005).

The National Academy of Sciences (2005) stated that the availability of high-quality professional development activities and the establishment of support networks for AP mathematics teachers are crucial to promoting and maintaining excellence in these programs.

 

Key Findings for Research Question Three

 

What are the perceptions among AP calculus teachers and site administrators as to the amount of support AP calculus teachers receive from site principals?

Research question 3 was answered by question 28 of the questionnaire, which asked the respondents to respond to the question as to how much perceived support they received from site principals fro AP calculus classes.  The findings indicated that although a significant difference was found between administrators and teachers on this question, there might not be a meaningful difference.  Overall, administrators and teachers generally agreed that they received a lot of support for AP calculus classes from site principals.  They disagreed on the two categories of “some support” and “no support.”

Findings Related to the Literature

The literature supports the premise that an effective principal must inspire and lead by example; he or she should be a defender of academic integrity (NASSP 1999).  Teachers need to engage in activities associated with their growth, especially in an era in which knowledge expands exponentially.

The principal of a high school, as a model for the staff will pursue his or her own ongoing professional growth while helping to lead the professional development for the entire school.  The best principals are maestros whose sense of timing and ability to lead stamp an indelible mark on the ensemble that we call a faculty (NASSP 1999).

Key Findings for Research Question Four

 

What are the perceptions among AP calculus teachers and site administrators as to the amount of support AP calculus programs receive from parents?

The findings indicated that a significant difference was found between the responses of administrators and teachers as to the amount of support AP calculus classes receive from parents.  Teachers appear more pessimistic about parental support for AP calculus than administrators do.

Findings Related to the Literature

The data and literature support the premise that placement of students in algebra is more often based on the results of standardized tests, teachers recommendations, and parental requests.  It is not uncommon for parental requests to take precedence over test results.  The literature review supports the premise that high schools should engage students’ families as partners in students’ education.  High schools must tighten these bonds so that students know their parents know about their schooling and so that they can, in turn, benefit from the caring of parents who regard themselves as partners in the learning process (NASSP 1999).  In addition, despite changes in the American family, the family’s capacity for supporting scholastic achievement remains strong.  Indisputable evidence links the family to the scholastic achievement of students (Grissmer 1994).

Key Findings for Research Question 5

 

What strategies are AP calculus teachers and site administrators using to prepare students for success in the AP calculus courses?

Research question 5 was answered by several questions in the questionnaire.  Question 35 of the questionnaire asked the participants to indicate whether students were given practice AP calculus exams.  The findings indicated that there were no significant differences between administrators and teachers.  Administrators and teachers generally agreed that students were given practice AP calculus exams.

Question 36a of the research questionnaire asked the participants whether students were provided Saturday support sessions for AP calculus classes.  The findings found no significant differences between administrators and teachers.  Administrators and teachers generally agreed that students were not provided Saturday support sessions for AP calculus classes.

Questions 36b of the research questionnaire asked the participants whether students were provided after-school sessions for AP calculus classes.  The findings found no significant differences between administrators and teachers.  Administrators and teachers generally agreed that students were provided after-school sessions for AP calculus classes.

Question 36c of the research questionnaire asked the participants whether students were provided individual tutoring preparation sessions for AP calculus classes.  The findings found no significant differences between administrators and teachers.  Administrators and teachers generally agreed that students were provided individual tutoring preparation sessions for AP calculus classes.

Question 36d of the research questionnaire asked the participants whether students were provided guidance sessions for AP calculus classes.  The findings found no significant differences between administrators and teachers.  Administrators and teachers generally agreed that students were not provided guidance sessions for AP calculus classes.

Findings Related to the Literature

The literature supports the premise that to be successful, some students will require additional academic support, such as “double dosing” (taking the same course over more than one class period).  The National Academy of Sciences (2005) stated that after-school and Saturday academic classes, tutoring, and summer “bridge” classes can help them develop academic competencies or give them a head start on the curriculum they will be expected to learn should they pursue advanced coursework.  Successful strategies include classroom size reduction (Grissmer 1999) and eliminating low-level academic courses that do not prepare students academically (Frome 2001; NASSP 1999).  The National Academy of Sciences (2005) supports tactics that employ and sustain additional educational opportunities, and ever-increasing student access to skilled counselors and mentors who can help them plan and implement strategies for educational attainment and achievement.  As discussed previously, Dornbusch (1994) and Eccles and Harold (1996) pointed out useful strategies, which include hiring and retaining qualified teachers to teach to rural and inner-city schools, and providing information to parents about the long-term benefits of students’ participation in rigorous academic programs.

The National Academy of Sciences (2005) indicated that a curriculum for understanding is intentionally designed around the organizing principles and essential concepts of the domain and provides opportunities for in-depth exploration in a variety of contexts.  Such a curriculum emphasizes depth of understanding over breadth of coverage.  It is designed to provide genuine opportunities for high-quality instruction and multiple points of entry into mathematics and science (Au and Jordan 1981; Brown 1994; Heath 1983; Tharp and Gallimore 1988).

 

Key Findings for Research Question 6

 

What are the perceptions among AP calculus teachers that site administrators underestimate the amount of support given to students?

Research question 6 was answered by question 32 of the survey questionnaire, which asked respondents to respond as to whether teachers were lacking instructional materials for AP calculus classes.  The findings indicated that there were no significant differences between administrators and teachers.  Administrators and teachers generally agreed that they were not lacking instructional materials for AP calculus classes.


Findings Related to the Literature

As discussed previously, the literature supports the premise that secondary teachers are more likely than teachers at other levels to report feeling isolated and unsupported (NCES 2000).  Isolation is an especially frequent complaint of advanced study teachers.  The National Academy of Sciences (2005) stated that AP teachers should be offered workshops before beginning to teach AP courses, and these should be workshops that cover advanced topics for more experienced teachers.

Education Trust (2003) stated that new teachers and administrators have a special need for extra support.  Every school with an improvement plan should describe a mentoring program.

The literature strongly supports professional development for teachers of mathematics; as for all teachers, it must be a planned, collaborative, ongoing, and relevant process.  School districts need to create working environments in which teachers have time to meet and discuss, reflect on, and refine instructional practices.

 

Key Findings for Research Question 7

 

What perceptions exist among AP calculus teachers and site administrators on the criteria for students to qualify for the AP calculus programs?

Research question 7 was answered by question 26 of the survey questionnaire, which asked respondents whether passing grades in prerequisite courses were required for admission into AP calculus classes.  The findings found no significant differences between administrators and teachers.  Administrators and teachers generally agreed that passing grades in prerequisite courses were required for admission into AP calculus classes.  In addition, respondents were asked whether recommendations from other teachers were required for admission into AP calculus classes.  The findings found significant differences between administrators and teachers.  Administrators responded significantly higher that recommendations from other teachers were required for admission into AP calculus classes.

Furthermore, respondents were asked whether recommendations from specifically math teachers were required for admission into AP calculus classes.  The findings found no significant differences between administrators and teachers.  However, administrators responded more highly that recommendations from specifically math teachers were required for admission into AP calculus classes.

In addition, respondents were asked whether students were required to maintain a 3.0 GPA or higher in previous math courses for admission into AP calculus classes.  The findings found no significant differences between administrators and teachers.  Administrators and teachers generally agreed that students were not required to maintain a 3.0 GPA or higher in previous math classes for admission into AP calculus classes.

Lastly, the respondents were asked whether passing the writing proficiency exam was required for admission into AP calculus classes.  The findings found significant differences between administrators and teachers.  While no teachers responded “yes,” over half the administrators responded “yes” that passing the writing proficiency exam was required for admission into AP calculus classes.


Findings Related to the Literature

As discussed previously, the literature revealed that the process of determining which students will take advanced courses in high school begins with their placement in the first algebra course (Gamoran 1987; Horn, Nunez, and Bobbitt 2000).  The literature indicated that prerequisites are frequently used as gatekeepers to regulate enrollment in AP courses.  However, this practice is deemed necessary because there is limited space (National Academy of Sciences 2005).

As previously discussed in the literature, the placement of students in algebra is more often based on the results of standardized tests, teacher recommendations, and parental requests.  It is not uncommon for parental requests to take precedence over test results (National Academy of Sciences 2005).  Although counselors and administrators use test scores or current mathematics placement to bar low-income students from high-level courses, they permit middle-class students with similar qualifications to enroll when parents intervene on their children’s behalf (Orfield and Paul 1995; Paul 1995; Romo and Falbo 1996).

 

Key Findings for Hypothesis One

Hypothesis 1.  There is no difference between administrators and teachers in their perception of teacher training.

Null hypothesis 1 was rejected because teachers participate in and are connected to teacher training and staff development more than administrators think they are.


Findings Related to the Literature

As discussed previously, the literature suggested that teacher training is needed for staff teaching at all levels at a school.  At a minimum, staff development is needed for precalculus and calculus teachers.  Staff development should include all teachers from algebra to calculus (National Academy for Sciences 2005).  The fact that administrators have a lower assessment of teacher participation in training than the teachers themselves may provide teachers with more advantage to negotiate additional training and staff development.

 

Key Findings for Hypothesis Two

 

Hypothesis 2.  There is no difference between administrators and teachers in their perception of resources and support available to the AP calculus program.

The failure to reject null hypothesis 2 indicates a high level of agreement between administrators and teachers in their perception of resources available to the AP calculus program.

Findings Related to the Literature

As discussed previously, the literature supports the premise that AP teachers need the availability of high-quality professional development activities and the establishment of support networks for AP mathematics teachers as crucial to promoting and maintaining excellence in these programs (National Academy of Sciences 2005).  AP teachers need time to reflect on teaching and learning, both individually and with colleagues.


Key Findings for Hypothesis Three

Hypothesis 3.  There is no difference between administrators and teachers in their perception of student commitment and access.

The findings associated with hypothesis 3 suggest those administrators are more optimistic about the commitment and access of students to the AP calculus program than teachers are.  Here, the null hypothesis was rejected.

Findings Related to the Literature

As discussed previously, the literature supports the premise that states are moving in the direction of providing eighth-grade students with greater exposure to algebra topics (National Academy of Sciences 2005).  Schools must ensure that students are ready to take advanced courses in high school by offering students a curriculum starting in the middle school that prepares them.  In addition, programs must have the goal of helping students achieve a deep conceptual understanding of content and unifying concepts.  To attain that goal, a program’s curriculum should shy away from exhaustive coverage of content (Education Week 2002).

Conclusions

The key findings in terms of the research and hypotheses questions support the following conclusions:

  1. The findings of this study concluded that both administrators and teachers in selected high schools in the San Bernardino County perceived that there were gaps to students’ preparation for AP calculus classes.  Respondents agreed that algebra skills contributed to gaps in students’ preparation for AP calculus classes.  The literature supports the premise that strategies must be developed to ensure that students who enroll in calculus have an adequate background in algebra and trigonometry for subsequent work in mathematics (National Academy of Sciences 2005).
  2. The findings of this study concluded that teachers were more pessimistic than administrators, that teachers receive a lot of support for AP calculus classes from their school district.  As previously discussed, the literature recognizes the need for school district support for AP calculus teachers.  Schools and districts offering advanced study must provide frequent opportunities for continuing professional development so teachers can improve their knowledge of both content and pedagogy (National Academy of Sciences 2005).  Ongoing professional development opportunities should be improved, expanded, and made available to all AP teachers.  Furthermore, it was found that students with the best-prepared teachers made the greatest gains in standardized assessments (Haycock 1998).
  3. The findings of this study concluded that, generally, administrators and teachers in selected high schools in the San Bernardino County perceived that they received a lot of support from their site principal.  As previously discussed, the literature supports the premise that site principals play a significant role in engaging teachers in activities associated with their growth (NASSP 1999).  All teachers and administrators need to a have a special need for extra support (Education Trust 2003).  An effective school site principal must inspire and lead by example; he must be a defender of academic integrity (NASSP 1999).
  4. The findings of this study concluded that teachers appeared more pessimistic about the parental support for AP calculus classes than administrators.  The literature indicated that parental and family involvement must be encouraged at the high schools.  High schools should engage students’ families as partners in the students’ education (NASSP 1999).  Students benefit from the reinforcement of education in the home, and schools should do all they can to include families.  The literature concluded that students are more likely to take higher-level mathematics if their parents are highly educated and knowledgeable about the college admission process and help guide their children’s course selection (Ekstrom, Goertz, and Rock 1988; Horn, Nunez, and Bobbitt 2000; Lee and Ekstrom 1987; Useem 1992).
  5. The findings in the study concluded that, generally, administrators and teachers agreed that students were given practice AP calculus exams.  The literature revealed that all students enrolled in AP courses ordinarily should take the relevant external examinations as part of the course requirements.  If teachers expect that major ideas will be assessed rather than specific problem types, it is likely that instruction will encourage the development of students’ critical thinking and problem-solving abilities (National Academy of Sciences 2005).  Programs must employ regular assessment of student learning to guide instruction.  The end-of-course exams should be the final steps in the process of certifying a student’s mastery of the subject (Education Week 2002).

The findings of the study concluded that most administrators and teachers agreed that students were not provided Saturday support nor guidance sessions for AP calculus classes, but were provided after-school sessions and individual tutoring.  As previously discussed, the literature concluded that after-school and Saturday academic classes, tutoring, and summer classes can help students develop academic competencies or give them a head start on the curriculum students will be expected to learn, if they choose to pursue AP classes (National Academy of Sciences 2005).

  1. The findings of the study concluded that, generally, both administrators and teachers agreed that passing grades in prerequisite courses were required for admission into AP calculus classes and that students were not required to maintain a 3.0 GPA or higher in previous math classes for admission to AP calculus classes.  However, the findings of this study concluded that administrators perceived that recommendations from other teachers and, specifically, math teachers were required for admission into AP calculus classes.  The findings in this study found that over half of the administrators perceived that passing the writing proficiency exam was required for admission into AP calculus classes.  As previously discussed, the literature revealed that the process of determining which students will take advanced courses in high school begins with their placement in the first algebra course (Gamoran 1987; Horn, Nunez, and Bobbitt 2000).  In addition, the literature concluded that the placement of students in algebra is more often based on the results of standardized tests, teacher recommendations, and parental requests (National Academy of Sciences 2005).

Implications for Action

The implications for action can serve to inform AP calculus teachers and administrators of AP calculus programs as they generate strategies to increase student enrollment in AP calculus classes in selected San Bernardino County public comprehensive high schools.

  1. The data and literature are clear; all students who enroll in AP calculus should have had at least four years of college preparatory mathematics prior to AP calculus.  The primary goal of advanced study in any discipline should be for students to achieve a deep conceptual understanding of the discipline’s content and unifying concepts (National Academy of Sciences 2005).  Students need to understand the central ideas of the discipline in order to build a conceptual framework for further learning.
  2. Within the literature review, schools and school districts must find ways to integrate advanced study with the rest of their program by means of a coherent plan extending from middle school through high school.  Districts, with the support of advanced study programs, must provide substantial professional development opportunities for teachers, invest appropriately in laboratory facilities and materials, and develop academic support systems for those students who need them (National Academy of Sciences 2005).
  3. The literature review is comprehensible; program developers should ensure that the components of their curriculum, instruction, assessment, and professional development are consistent with what is known about how individuals learn, and work to foster students’ conceptual understanding.  Furthermore, program developers must include the role of students’ prior knowledge and misconceptions in building a conceptual structure, the importance of student motivation, and the substantial differences among learners (National Academy of Sciences 2005).
  4. Within this study, an effective curriculum development must be a collaborative effort conducted by teams of experienced teachers working with curriculum specialists and experts in the disciplines, in cognitive theory, and in pedagogy.  The curricula should be focused on a reasonable number of concepts that can be studied in depth during the time allotted (National Academy of Sciences 2005).
  5. As discussed previously, within the literature review students should be provided opportunities to experiment, analyze information critically, make conjectures and argue about their validity, and solve problems both individually and in groups.  Effective ways to use the Internet and other electronic resources should be encouraged and evaluated (National Academy of Sciences 2005).
  6. Within the literature review, end-of-course assessments must measure students’ depth of understanding and the students’ ability to transfer knowledge to unfamiliar situations.  Teachers and program developers should collaborate in developing assessments that measure student progress toward desired learning outcomes (National Academy of Sciences 2005).
  7. Schools and districts must provide frequent opportunities for continuing professional development so teachers can improve their knowledge of both content and pedagogy.  The data and literature review are clear; all professional development activities must be adequately funded and available to all teachers (National Academy of Sciences 2005).
  8. The data and literature review are apparent, college and university mathematicians should modify their introductory courses along lines similar to those for high school advanced study courses.  Departments should advise undergraduates about the benefits and costs of bypassing introductory courses.  Many students who participate in secondary advanced study later enroll in introductory college courses.  These courses need to evolve so that they will continue to be appropriate for audiences with diverse populations (National Academy of Sciences 2005).

Recommendations for Further Research

This study focused on the perceptions of AP calculus teachers and administrators on AP calculus program characteristics that have increased student enrollment in selected San Bernardino County public comprehensive high schools.  The implications of this study suggest further research possibilities.

1.   Replicate this study and follow up with a focus group discussion and interview of participants.

2.   Replicate this study by including additional variables: socioeconomics of students, parent educational level, and level of academic entrance skills of students into AP calculus classes.

3.   Conduct a similar study that compares students’, teachers’, administrators’ and parents’ perceptions of AP calculus classroom climate.

4.   Conduct a similar study that compares students’ perceptions of AP calculus classes.

5.   Replicate this study in other counties in the state of California.

6.   Conduct a similar study that compares the perceptions of students, parents, teachers, and administrators in other AP courses.

7.   Conduct a study on the strategies used by high schools to incorporate technology and the Internet into AP calculus classes.

8.   Conduct a study on the impact of how coordination between secondary schools and institutions of higher education can be enhanced to optimize student learning and continued interest in the mathematics discipline.

Concluding Remarks

This study described the AP calculus teachers’ and administrators’ perceptions of AP calculus program characteristics that have increased student enrollment in selected San Bernardino County public comprehensive high schools.  This study was based on research and strategies on student preparedness for AP calculus classes, student learning, placement, and the implications of that research for the design and integration of curriculum, instruction, assessment, and professional development.  The study provides further analysis of the perceptions among AP calculus teachers and site administrators as to the amount of support AP calculus teachers receive from site principals, parents, and school districts.

This study presents a set of recommended actions that could significantly improve existing programs for approaches to AP calculus courses, and serve to promote programs for advanced study in the mathematics discipline.  Indubitably, curriculum developers for AP calculus programs must also decide how to execute the recommendations within the unique structures of their respective organizations.  This study articulates recommendations and strategies for all individuals developing and teaching AP calculus classes.

Furthermore, this study has strong implications for both policy and practice.  Foremost, the thorough examination of the literature provides a resource for future investigations of AP courses.  Secondly, as it replicates a previously conducted, large-scale study of AP in California (Furry and Hecsh 2001) on a smaller, localized scale, and the findings are almost identical to the macro study, it may hold more meaning for local policymakers who sometimes perceive large study findings as too generalized.  In summary, taken collectively, this review of the literature, and the review of the previous study and its replication, provide a very clear policy recommendation for educators regarding the value and necessity of articulation and vertical teaming in subjects leading to AP courses.  Furthermore, in the perspectives of the major stakeholders in the Furry and Hecsh (2001) study and in this study, appropriate preparation for AP calculus classes, as considered within the literature of this study are critically lacking.  This study makes clear that while teachers feel supported by their administrators and feel they have adequate resources, what are missing are the resources, articulation, and time that it requires to provide pre-AP preparation, and to interface and communicate effectively with parents.

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